|UFDC Home||myUFDC Home | Help|
This item has the following downloads:
1 CHARACTERIZATION OF SURFACE ROUGHNESS OF BARE AGRICULTURAL SOILS USING LIDAR By JUAN CARLOS FERNANDEZ DIAZ A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 2010
2 2010 Juan Carlos Fernandez Diaz
3 To my family and to all of you who inspired me to be curious and work hard
4 ACKNOWLEDGMENTS First and foremost, I want to acknowledge the great support and guidance from the members of my supervisory committee: Ramesh Shrestha, Jasmeet Judge, William Carter, David Bloomquist and K Clint Slatton I thank them for sharing with me their knowledge, exper ience and spirit ; and for guiding m e to the completion of this endeavor I m also grateful to the persons and institutions that make the University of Florida Alumni Fellowship a reality and those who support the department of civil engineering. Through them I had the unique opportunity t o reach the summit of higher education in one of the top public universities in the United States of America. Next, I would like to thank my friends, classmates and colleagues from the Geosensing Systems Engineering program ; the N ational S cience F oundation (NSF) National Ce nter for Airborne Laser Mapping (NCALM) ; and the Center for Remote Sensing at the Department of Agricultural and Biological Engineering; who provided me with invaluable support to complete this work. Finally I would like to express my deep gratitude to my friends, parents and siblings. Their support and encouragement was crucial t o my success in the UF Geosensing Graduate Program.
5 TABLE OF CONTENTS page ACKNOWLEDGMENTS .................................................................................................. 4 LIST OF TABLES ............................................................................................................ 8 LIST OF FIGURES ........................................................................................................ 10 LIST OF ABBREVIATIONS ........................................................................................... 13 ABSTRACT ................................................................................................................... 14 CHAPTER 1 INT RODUCTION AND LITERATURE REVIEW ..................................................... 18 Introduction ............................................................................................................. 18 Literature Review .................................................................................................... 20 Characterization of Surface Roughness ........................................................... 20 Surface Digitization Techniques ....................................................................... 23 Multi Scale Nature of Agricultural Surfaces and Detrending Methods .............. 26 Current Limitations of Surface Roughness Characterization ............................ 28 Scanning light detection and ranging (LIDAR) technologies ............................. 29 2 CHARACTERIZATION OF GROUNDBASED LIDAR ERRORS AND ACCURACY ASSESSMENT OF DERIVED ROUGHNESS PARAMETERS FROM TWO-DIMENSIONAL (2D) PROFILES ....................................................... 36 Characterization of Groundbased LiDAR Errors .................................................... 39 Accuracy Assessment Based on Synthetic Generated Profiles .............................. 40 Accuracy of Derived Autocorrelation Function (ACF) ....................................... 41 Accuracy of Derived of Random Height Root Mean Square (RMSh) ............... 42 Accuracy of Derived Correlation Lengths ......................................................... 43 Accuracy Assessment Based on Roughness References ...................................... 43 Digitization Fidelity of the Instruments .............................................................. 45 Accuracy of D erived RMSh .............................................................................. 46 Validation with Agricultural Soil Measurements ...................................................... 47 Comparison of RMSh Values ........................................................................... 47 Comparison of Correlation Lengths .................................................................. 47 Autocorrelation Functions of Agricultural Soils ................................................. 48 Chapter Conclusions .............................................................................................. 49 3 TESTING ROUGHNESS CHARACTERIZATION ASSUMPTIONS USING A DATABASE OF 2D PROFILES OBTAINED WITH GROUNDBASED LIDAR ....... 68 Instrumentation and Datasets ................................................................................. 68
6 Mobile Terrestrial Laser Scanner (M TLS) ....................................................... 68 Data Collection and Preprocessing .................................................................. 69 2D Profile Database ......................................................................................... 71 Testing Roughness Characterization Assumptions ................................................ 72 Value Ranges for Roughness Parameters ....................................................... 72 Sensitivity of Roughness Parameters to the Detrending Procedure ................. 73 Correlation between RMSh and Correlation Lengths ....................................... 74 Exponential or Gaussian ACF Models .............................................................. 75 Chapter Conclusions .............................................................................................. 76 4 DERIVING THREE DIMENSIONAL (3D) ROUGHNESS METRICS FROM GROUNDBASED LIDAR DIGITAL ELEVATION MODELS (DEMS) ..................... 82 Datasets and Preprocessing ................................................................................... 83 Averaging Parameter Values Obtained From 2D Profiles ....................................... 85 Extending 2D Formulations to 3D ........................................................................... 86 Comparison between 2D and 3D Roughness Parameter Values ........................... 89 Impact of Roughness Underestimation on Microwave Observables ....................... 92 Advanced Detrending and Scale Separation Methods for 3D DEMs ...................... 93 Chapter Conclusions .............................................................................................. 95 5 THE APPLICATION OF AIRBORNE LIDAR TO MAP SURFACE ROUGHNESS OF LARGE AREAS .............................................................................................. 111 Airborne LIDAR Instrumentation and Datasets ..................................................... 112 Profiles from a Single Scan Line ........................................................................... 115 Footprint Scale Roughness from Return Waveform Analysis ............................... 116 Point Cloud Binning .............................................................................................. 120 Validation of Airborne Roughness Maps ............................................................... 120 Chapter Conclusions ............................................................................................ 122 6 CONCLUSIONS AND RECOMMENDATIONS ..................................................... 131 Conclusions .......................................................................................................... 131 Recommendations ................................................................................................ 133 Main Contributions ................................................................................................ 134 APPENDIX A MIC ROWAVE SCATTERING AND EMISSION MODELS APPLIED TO SOIL MOISTURE MAPPING ......................................................................................... 135 B IMPROVEMENT TO THE MESHBOARD PROFILE DIGITIZATION METHOD .... 139 C SELECTED MATLAB SCRIPTS ........................................................................... 147 Fit a Plane and Level a Point Cloud ...................................................................... 147 2D Detrending Function ........................................................................................ 147
7 2D Roughness Parameters Determination Function ............................................. 148 2D Accuracy Assessment with Roughness References ....................................... 150 3D Co rrelation Length from DEMs ........................................................................ 155 RMSh from Single Scan Line from Airborne LiDAR Data ..................................... 157 Meshboard Digitizing and Correction .................................................................... 158 LIST OF REFERENCES ............................................................................................. 165 BIOGRAPHICAL SKETCH .......................................................................................... 171
8 LIST OF TABLES Table page 2 1 Mean difference in the exponent of generalized power law for profiles with exponential and Gaussian ACF. ......................................................................... 59 2 2 RMSE (cm) in the estimation of RMSh for profiles with exponential ACF. .......... 60 2 3 RMSE (cm) in the estimation of correlation length for profiles with exponential and Gaussian ACF. ............................................................................................ 61 2 4 Mean difference (cm) of correlation lengths for profiles with exponential and Gaussian ACF. ................................................................................................... 62 2 5 Raw results obtained from the accuracy assessment experiments. ................... 63 2 6 Summary of results obtained from the accuracy assessment experiments. ....... 65 2 7 Roughness parameter values obtained from agricultural soils using meshboard and LiDAR. ...................................................................................... 66 2 8 Comparison of roughness parameter values of agricultural soils obtained from LiDAR and meshboard data. ...................................................................... 67 3 1 3D digital elevation models derived from LiDAR for the study. ........................... 81 3 2 Roughness parameter values in cm, derived from 20,072 profiles of agricultural soi ls. ................................................................................................. 81 3 3 Mean parameter and fit metric values for different ACF models. ........................ 81 4 1 Correlation coefficient and RMSE in meters bet ween correlation length values derived by averaging profile values and averaging the ACFs. .............. 104 4 2 Averaged oriented roughness parameter values obtained from 2D profiles extracted from the DEM s. ................................................................................. 104 4 3 3D roughness parameters values obtained from the full DEM and averaging values obtained from 2D profiles. ..................................................................... 107 4 4 Com parison of 3D roughness parameter values obtained from different approaches. ...................................................................................................... 109 4 5 Difference in microwave observables due to errors in roughness parameterization. .............................................................................................. 109 4 6 Roughness Parameters derived from 3D models of quasiperiodic surfaces. .. 110
9 5 1 Optech GEMINI Airborne Laser Terrain Mapper (ALTM) specifications ........... 130 5 2 Optech waveform digitizer specifications .......................................................... 130 5 3 Airborne LiDAR datasets .................................................................................. 130
10 LIST OF FIGURES Figure page 1 1 Two configurations of reference surface and height random components. ........ 32 1 2 Synthetic generat ed profiles with exponential ACF, an RMSh of 1cm, and a correlation length of 5 cm. .................................................................................. 32 1 3 Synthetic generated profile with Gaussian ACF, an RMSh of 1cm, and a correlation length of 5 cm. .................................................................................. 32 1 4 Common autocorrelation functions shapes. ....................................................... 33 1 5 A meshboard used for surface roughness studies. ............................................. 33 1 6 A pin profiler deployed over vegetated terrain. ................................................... 34 1 7 A groundbased LiDAR scanner mapping an agriculture soil. ............................ 34 1 8 Detrending effects on roughness parameter values. .......................................... 35 2 1 Test to characterize of LiDAR random noise.. .................................................... 51 2 2 DEM derived from a LiDAR scan of a smooth flat target. ................................... 51 2 3 Elevation deviations from the smooth surface, extracting and concatenating profiles along the row direction of the DEM. ....................................................... 52 2 4 Characterization of groundbased LiDAR random errors. ................................... 52 2 5 Random seed sequence from which correlated sequences ar e generated. ....... 52 2 6 Gaussian and exponential weighting functions used in the moving average method. .............................................................................................................. 53 2 7 Generated profile wit h a Gaussian ACF. ............................................................ 53 2 8 Generated profile with an exponential ACF. ....................................................... 53 2 9 Section of a pseudorandom roughness reference with an RMSh of 2 cm. ......... 54 2 10 Section of a correlated roughness reference with an RMSh of 2.7 cm. .............. 54 2 11 Scanning geometry of roughnes s references using groundbased LiDAR. ........ 54 2 12 Inputs into the Matlab script used to perform the accuracy assessment based on roughness references. ................................................................................... 55
11 2 13 Random component extracted from the reference, the meshboard, and LiDARderived profiles. ....................................................................................... 56 2 14 Results of the RMSh accuracy assessment. ...................................................... 56 2 15 Simultaneous profile digitizing with A) meshboard and B) LiDAR. ...................... 57 2 16 Comparison of RMSh values of bare agricultural soils obtained from meshboard and LiDAR. ...................................................................................... 57 2 17 Difference between meshboard and LiDAR derived correlation lengths of bare agricultural soils. ......................................................................................... 58 3 1 The M TLS system performing agricultural surface scanning operations. .......... 77 3 2 Point cloud renderings of a groundbased LiDAR dataset at different processing steps.. ............................................................................................... 77 3 3 Samples of studied soil surface and their groundbased LiDAR derived 3D DEMs. ................................................................................................................. 78 3 4 Effects of detrending on the derived roughness parameters.. ............................ 78 3 5 Dispersion plots of the random height component derived correlation lengths and RMSh for the different detrending methods. ................................................ 79 3 6 Histograms of the exponent values for the Power Law Spectrum ACF model obtained from the soil profiles. ............................................................................ 79 3 7 Histograms of the R2 values of the fit of the observed ACF model with respect t o the theoretical models. ................................................................................... 80 4 1 Detrending a 3D DEM with linear, quadratic models and a FFT low pass filter. .................................................................................................................... 97 4 2 Directional roughness. ........................................................................................ 98 4 3 Deriving 3D correlation length. ........................................................................... 98 4 4 3D correlation length as a function of the horizontal angle. ................................ 99 4 5 3D Correlation length of a DEM with linear features. ........................................ 100 4 6 3D Correlation length of a DEM after removing the linear features. ................. 101 4 7 Dispersion plots of the 3D RMSh and the averaged values of RMSh derived from profiles ...................................................................................................... 102
12 4 8 Dispersion plots of the aver aged 3D correlation lengths and the averaged from profiles ...................................................................................................... 102 4 9 Two stage wavelet bank used to decompose a 3D DEM. ................................ 103 4 10 D etrending of a quasiperiodic surface (#20). ................................................... 103 5 1 Geometry of airborne scanning LiDAR data collection. .................................... 124 5 2 The National C enter for Airborne Laser Mapping (NCALM) Gemini ALTM and waveform digitizer. ............................................................................................ 125 5 3 Airborne LiDAR roughness maps. .................................................................... 126 5 4 Surf ace roughness map of a bare soil area. ..................................................... 127 5 5 Validation areas for the airborne derived roughness maps marked as squares 1 and 2 in 53 A. ................................................................................. 128 5 6 Validation of the airborne derived RMSh from the three proposed methods compared to RMSh values derived from groundbased LiDAR data through similar processing. ............................................................................................ 129 A2 1 Me shboard and the marked control points. ...................................................... 145 A2 2 Flat smooth profile at different processing steps. ............................................. 145 A2 3 Coordinate different ials and distortion correction factors. ................................. 146
13 LIST OF ABBREVIATIONS ACF Autocorrelation Function ALTM Airborne Laser Terrain Mapper CCD Charged Coupled Device CL Correlation Length COTS Commercial off theshelf DEM Digital Elevation Model EDM Electronic Distance Measurement ESA European Space Agency FFT Fast Fourier Transform FWHM Full Width at Half Max IEM Integral Equation Model IFAS Institute of Food and Agricultural Sciences LiDAR Light Detection and Ranging. MicroWEX Mic rowave Water and Energy Balance Experiment M TLS Mobile Terrestrial Laser Scanner system NCALM National Center for Airborne Laser Mapping PSREU Plant Science Research and Education Unit RMSE Root Mean Square Error RMSh Root Mean Square of random height v ar iations SAR Synthetic Aperture Radar SMAP Soil Moisture Active & Passive SMOS Soil Moisture & Ocean Salinity UF University of Florida UF GEM University of Florida, Geosensing Engineering and Mapping center.
14 Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy CHARACTERIZATION OF SURFACE ROUGHNESS OF BARE AGRICULTURAL SOILS USING LIDAR By Juan Carlos Fernandez Diaz December 2010 Chair: Ramesh Shrestha Major: Civil Engineering The characterization of surface roughness is an active area of research with relevance and applications to many fields of sciences and engineering. For example, in geology it is used to infer what type of process gave origin to particular surface morphologies. In hydrology it is used to model surface water runoff. Here the goal is to facilitate the use of active and passive microwave sensors to map soil moisture. For this purpose, surface roughness is co nsidered the random height variations of the soil surface relative to a reference surface Roughness is thus considered a stationary, singlescale random process characterized by three parameters: the root mean square (RMSh) of the height variations, the autocorrelation function (ACF) and its c orrelation length ( CL) To obtain these parameters the general practice is to record the terrain height variation along two dimensional ( 2D ) transects using mechanical or non contact profilers. The data from these tra nsect s are first detrended to separate the reference surface and the random component and then the random component is used to compute the roughness parameters. The current definition and parameterization of surface roughness have the advantages of being simple, but it is generally accepted that the way soil roughness
15 needs to be measured and described for the modeling of microwave phenomena is not yet fully understood. Some practical challenges c urrent ly faced include: obtaining precise and accurate roughness parameter values from field instruments and techniques; obtaining measurements that do not contradict the currents assumptions of single scale, isotropy and simple Gaussian or exponential ACFs ; obtaining good agreement between soil moisture measurements and the values derived from the microwave models using scattering and or emission observations and in situ roughness measurements The research hypotheses tested in this work, that explain these limitations are: the existence of systematic and random errors that are not properly accounted for in the measurements; the inadequate practice of using 2D profiles to derive 3D characteristics of the surface; the scaling up of in situ roughness measurements under the assumptions of homogeneity and isotropy. To test the se hypotheses, Scanning light detection and ranging (LiDAR) technology was employed. LiDAR enables the digitization of surface height variations in threedimensions ( 3D ) and thus allows for an improved characterization of surface roughness. To addr ess the above mentioned challenges in the characterization of surface roughness with LiDAR technology a multi step approach was followed. The first step was to investigate the factors that affect the precision and accuracy of roughness parameterization fr om 2D measurements. The next step was to analyze and test the current assumptions of roughness characterization following the traditional 2D formulation. This was followed by developing methodologies to characterize roughness from 3D information; that is t ruly representative of the entire surface. Finally, the issue of scaling was address ed by developing methodologies to use airborne LiDAR to derive millimeter level roughness
16 maps of large areas and to prove the heterogeneity and anisotropy of roughness characteristics over large areas. The issue of the accuracy and precision of roughness parameters was studied by performing two accuracy assessments and a direct comparison between a meshboard and the groundbased LiDAR. The first accuracy assessment was base d on computer generation of random rough surfaces and the second based on measurements employing roughness references. Results indicate that to obtain accurate and repeatable parameter values it is necessary to properly characterize the instrument random errors. The height variation measurements obtained with any instrument are the result of the addition of two random processes: the surface roughness and the instrument random error. If the instrument error is not properly characterized and considered the c omputed roughness parameter values will be corrupted and thus inaccurate. Of the parameters the RMSh is the least sensitive to instrument noise and it was determined that this parameter can be derived from groundbased LiDAR with an accuracy of better than 1mm. The achievable accuracy in retrieving the ACF and associated CL depends on the relative magnitudes of the surfaces roughness and the instrument error. C orrelation lengths can be accurately determined to better than a cm if the surface RMSh is larger than 1 cm. With regards to the assumptions used to characterize roughness using the traditional 2D formulation, it was found that agricultural surface s exhibit multiscale roughness characteristics. This contradicts the singlescale assumption. However, it is possible to obtain roughness at a particular scale if the proper detrending techniques are applied. It was also determined that the exponential and Gaussian ACF models are
17 just two limiting cases, and that the majority of surfaces have characteristic ACFs intermediate between these two models. In contras t to what has been commonly reported, no correlation was found between the RMSh and CL However, it was found that at small scales there is a possible negative correlation between RMSh and the maximum observable CL 3D characterization of surface s of agricultural fields reveals that they are generally even more multi scale in terms of their roughness than is evident from the 2D formulation. Roughness parameters obtained from the 2D formulation underesti mate the characteristics of the surface; by 25% in terms of the RMSh and 30% in terms of the CL This is because profiles generally do not record the extremes of the surface in a single transect and do not necessary follow the trend of the entire surface. The assumptions of homogeneity and isotropy were proved to not be valid even for small areas. 3D digital elevation models (D EMs ) derived from groundbased LiDAR allow for the characterization of roughness with advance tools in the spatial temporal domain. When the characterization of millimeter level surface roughness of large areas is required, data from high resolution airborne LiDAR can be used. RMSh derived from airborne data was within 1 mm of the RMSh derived from groundbased LiDAR data. It is expect ed that the results from this work will motivate a paradigm shift in the way surface roughness data is derived, from a limited sample of insitu 2D transects to remotely determini ng 3D roughness of large areas.
18 CHAPTER 1 INTRODUCTION AND LIT ERATURE REVIEW Introduction C haracterization of surface roughness is an active research topic w ith potential applications in diverse fields of the basic and applied sciences including physics, geology, geomorphology, hydrology, ecology, agriculture, civil engineering, manufacturing, and remote detection and sensing  [ 5 ]. One use of surface roughness characterization is in the remote sensing and mapping of soil moisture using active and passive microwave sensors  T he microwave scattering or emission from a soi l surface depends not only on the wavelength, polarization, and angle of incidence of the radiation, but also on the soils moisture and its surface roughness [ 2 ]. For the purpose of modeling microwave emission and scattering f or the retrieval of soil mois ture, roughness is considered to be the surface height s random variations with respect to a reference Contact and noncontact instruments are used to digitize the surface height variations along two dimensional (2D) transects. The digitized transect is t hen detrended to separate the random height variations from topographic trends. The height variations are then described by three parameters : the root mean square of height (RMSh), the autocorrelation function, and the correlation length ( CL) Despite the relative simplicity of the definition of surface roughness and the traditional surface profiling methods several limitations and challenges have been encountered by researchers over the years. Based on these challenges a recent literature review by Verh o est et al.  concluded that the method to describe and measure soil surface roughness f or the modeling of microwave backscattering is not fully understood.
19 This dissertation describes the results of experiments designed to characteriz e and parameteriz e t he roughness of bare agricultural fields using scanning light detection and ranging (LiDAR) technologies. These experiments were performed using a multi step approach aimed at overcom ing some of these documented challenges and limitations T his chapter presents a literature review The review covers the aspects of characterization of surface roughness ; traditional methods and instruments used to digitize surface heights ; the current limitations and challenges to overcome to better characteriz e roughness for its use in microwave m odels ; and finally there is a short overview of LiDAR technologies. C hapter 2 describes the experiments that constit ute the foundation of this study. These experiments were conducted to determine what factors affect the precision and accuracy of the parameters used to characterize roughness derived from surface measurements. Once the ability of the groundbased LiDAR to derive accurate values of roughness parameters was established, the next step consisted of collecting a large database of 2D surface roughness measurements. These measurements were used to test some of the current assumptions used in the characterization of roughness. Chapter 3 presents t he collection and processing of the roughness database, a description of the perfo rmed tests and their results These tests are also a validation of the different computational tools created to derive 2D roughness parameters values a s the results obtained from them reproduce the current state of knowledge and the known limitations. Ha ving validated the instruments and tools used to perform the roughness characterization in the traditional 2D formulation, the next step was an extension to characterize roughness in a 3D formulation utilizing the 3D information obtained using
20 the groundb ased scanning LIDAR. This characterization of surface roughness in 3D is covered in C hapter 4. The ground based LiDAR proved to be an accurate and convenient instrument for the characterization of millimeter level roughness in both 2D and 3D but l imited in terms of the area it can characterize. For the characterization of surface roughness of much larger areas, the use of airborne LiDAR was explored. This is the topic covered in Chapter 5. In addition to the specific conclusions presented in each chapter, general conclusions of the overall work are presented in Chapter 6. Chapter 6 also includes a list of the major contributions of this work toward the general advancement of knowledge and recommendations for further work. A lso included as appendices to thi s dissertation are a description of the most common used microwave scattering and emission models an improved method to digitize surface transects using the meshboard and some of the M atlab scripts developed to perform this research. Literature Review Ch aracterization of S urface R oughness There are two main approaches to quantify surface roughness T he first and most common is to assume a mathematically tractable form of a statistical nature T he second approach consist s of measuring and cataloging topographic expressions of unrelated of surfaces without a priori assumptions about their final form [1 ]. Most current microwave scattering or emission models take the first approach and assume that soil surface roughness can be described as a single scale, st ationary Gaussian, random process. This means that roughness can be fully described by the mean, variance and autocorrelation function of the surface height variations and that these parameters will have the same value independent of the size of the surf ace under study [ 8 ] According
21 to Ulaby et al.  for microwave emission and scattering models, soil roughness can be considered as a stochastic varying height of the soil surface with respect to a reference surface. The reference surface can be a) the m ean surface if there are only random variations to it, or b) the unperturbed surface of a periodic pattern ( Figure 1 1 ) From the random component the roughness parameters are computed from the following formulas given in . F r om a representative sample of a continuous surface height random component (z(x,y)) with dimensions Lx and Ly centered at the origin, the first (mean) and second moments of the surface random component are given by: 2 2 2 2, 1x x y yL L L L y xdxdy y x z L L z (1 1) 2 2 2 2 2 2, 1x x y yL L L L y xdxdy y x z L L z (1 2) The h) or height RMS (RMSh) is: 2 2z z RMShh (1 3) If the random component surface z(x,y) is isotropic then the previous formulas can be reduced to two dimensions z and x or y. Also, t he general pract ice is to record the surface height variation along 2D profiles Zs(xiAft er detrending the surface profile Zs(xi) to obtain the random height component z(xi) the roughness parameters are then computed as described in  and as follows. Random component height mean (should be zero):
22 N i iz N z11 (1 4) The standard deviation of the height variations (h) or height RMS (RMSh): 2 1 21 1 z N z N RMShN i i h (1 5) The normalized autocorrelation function for lags h [ 2 ]: N i i j N i j i iz z z x j h1 2 1 (1 6) The correlation length ( l ) is defined as the lag for which [ 2 ] and : e l 1 (1 7) The normalized exponential autocorrelation function [ 2 ]: l he h (1 8) The normalized Gaussian autocorrelation function [ 2 ]: 2 2l he h (1 9) As shown in Figure 1 2, a surface with an exponential ACF is characterized by short range (high frequency) sm all amplitude height variations, whereas a surface with a Gaussian ACF ( Figure 13 ) appear s smoother at these higher frequenc ies or small er scales [1 0 ] However, the measured ACF of real soils are more complicated than these theoretical formulations. Some studies have observed that the ACF of natural surfaces fal ls somewhere between the Gaussian and the exponential ACF [ 11], and a generalized power law spectrum has been proposed to characterize their behavior [ 2 ]:
23 n l he h (1 10) With n = 1t his ACF becomes the exponential, with n = 2 this ACF becom es the Gaussian, and explain s the intermediate ACFs for 1< n <2 [ 11]. Figure 14 shows the shape of the normalized Gaussian, exponential and generalized power law autocorrelation functions for the same correlation length. To better describe surface roughness, some researchers suggest an additional parameter the RMSh slope, which is the second derivative of the autocorrelation functi on evaluated at the origin or [ 8]: 0 2 2 2 2 h rmsh dh d m (1 11) The RMS slope is infinite for surfaces with exponential and power law ACF as can be inferred from Figure 14 For surfaces with different ACFs the RMS slope can be approximated by [ 1 ]: l mh rms (1 12) Surface D igitization T echniques There are two types of instruments used to obtain surface profil es to compute the roughness parameters for microwave modeling: a) contact instruments such as the pin profiler and the meshboard, and b) noncontact instruments such as LASER profilers, stereo imagers, and acoustic backscatter systems [ 1 ], , [ 12] [1 4] The meshboard is usually a metallic plate, generally 2 to 4 meters in length, on which a regular grid has been drawn. The meshboard is placed over the soil surface and a photograph is taken from a perpendicular position. The photograph is then
24 processe d using special image processing software from which the digitized surface profile is obtained. Figure 15 show s a meshboard used for a surface roughness data collection over a plowed field. Among the advantages of the meshboard are its simplicity, ease of transportation and low cost. Sources of error present in mes h board derived profiles include: parallax and geometric distortion, which originate when the photograph is taken; and interpretation, digitalization, and interpolation errors which arise in the image processing steps [ 1 3] The pin profiler is in essence a meshboard with a frame that keeps the board suspended off the ground. A beam is welded at the lower edge of the board, on which uniformly spaced holes are drilled and through which metallic rods or pins can slide up or down. During a profile measurement the board is leveled and the pins are freed to make contact with the ground. Looking down on the meshboard plate, the heads of the pins delineate the soil surface profile Then the pins heads a gainst the meshboard are photographed. To obtain the digitized profile a similar procedure as that described for the meshboard method is applied , [ 1 ], [ 13] [ 15 ] Figure 16 shows a pin profiler used for surface roughness characterization of vegetated terrain. The main disadvantage of these mechanical methods is that they tend to disturb or destroy the sur face that is under study. Meshboards have to be hammered into the soil, while n eedlelike p rofilers tend to penetrate into the surface yielding noisy height measurements [ 13 ] An additional disadvantage to these methods is that data collection and processing are very time consuming. Modern surface digitizing methods are based on noncontact instruments, which have the advantage of not disturbing the st udy surface. Examples of these instruments
25 include such systems as laser profiler, optical imaging and acoustic backscatter ing A description of an experimental acoustic backscatter system can be found in [ 14 ], and descriptions of stereoscopic optical imagers can be found in [ 1 ], [1 2 ] [ 16] Of relevance to this dissertation are the laser based instruments a mong which, it is possible to make a distinction between prototype oneof a kind systems, designed specifically by research institutions to obtain sur face roughness profiles and commercial off the shelf (COTS) LiDAR scanners designed mainly for surveying applications to map surfaces in 3D Examples of the oneof a kind systems are the ESA CESBIO LASER profiler described in [13 ], the Wageningen University Micro LASER relief meter described in [ 17] and the instantaneous profile laser scanner covered in [ 18]. Increased availability of COTS LiDAR scanning systems over the past years has made possible their employment for surface roughness studies. However as of this date very few studies on the characterization of soil surface roughness employing scanning LiDAR have been published. Some examples include work by Davenport et al ., i n which airborne mapping LiDAR has been used to automatically classify areas with different surface roughness characteristics [2 0 ] and on the characterization of the instrument errors applied to surface roughness studies [2 1 ]. Perez Gutierrez et al. used a Trimble groundbased terrestrial LiDAR to obtain four 20 m x 20 m fieldsur face samples in the Duero Basin, in the northcentral region of Spain. Their work confirmed the validity of the use of groundbased LiDAR to digitize agricultural soil surfaces H owever, they did not publish the values f or the traditional roughness paramet ers (RMS h CL, ACL ) obtained from the LiDAR data [ 22]. Bryant et al. employed an Optech ILRIS 3D LiDAR scanner to digitize test areas at the Walnut Gulch experimental
26 watershed in southeastern Arizona. 2D profiles were extracted f rom the 3D surface data and compared to pin profiler measurements. They also attempted to characterize the LiDAR and pin profiler RMSh error by digitizing flat surfaces with zero RMSh. The obtained RMSh was assumed to be a positive bias present in all measurem ents [ 1 5 ]. Figure 1 7 shows an Il RIS 3D LiDAR scanner the same instrument employed by Bryant et al. for their study  These studies with COTS LiDARS are limited pilot experiments and the potential of LiDAR to better characterize surface roughness and to overcome many of the limitations of the traditional methods has not been fully exploited. Also, t o date there has been very little research on the characterization of 3D soil roughness using these digitization instruments. Most of them limit the 3D data obtained to extract 2D profiles and perform analyses based on the traditional transect methodology. Little is known about how the results obtained with the traditional 2D techniques compare to the ones obtained from 3D characterizations. COTS groundbased LiDARs have the potential to digitize height variations of surfaces with areas of tens of square meters at resoluti ons in millimeters, a significant advantage over the single short profiles obtained using traditional methods. Even with coarse r resolution, Airborne LiDAR has the potential to map and characterize surface roughness at watershed and larger scales Datasets of this extent so far have not been available, but would be extremely use ful for the retrieval of soil moisture data from largescale spaceborne radar and radiom eter observations [ 6 ] MultiS cale N ature of A gricultural S urfaces and D etrending M ethods Romkens and Wang [23 ] recognized that agricultural soils exhi bit roughness characteristics on four main scales: microrelief variations, which are due to individual
27 so il grains and aggregates; random roughness, which are surface height variations due to individual soil clods; oriented roughness, or surface height variations due to agricultural structur es such as rows and furrow s; and higher order roughness, which is th e result of height variations on topographic scales. The effects of these roughness scales on microwave signatures depends on wavelength and angle of incidence and it is generally accepted that the ones that need to be considered are the scales higher than 1/10 of the observational wavelength [ 24]. In principle, for use in microwave emission and scattering models random roughness is the one that needs to be parameterized; the oriented and topographic roughness should be directly accounted for in the models [ 25]. The practical challenge becomes how to properly separate the different comp onents of height variations at the different scales from field measured profiles In other words how to properly detrend the profile? The traditional approach for short prof iles is to remove the first order or linear trend. F or longer profiles four diff erent methods are recommended in [ 2]. These methods are the application of 1) higher order polynomials, 2) 1 meter linear piecewise detrending, 3) moving average filter s, and 4 ) a F ast F ourier T ransform (FFT) based filter that remove s higher wavelength components. In Figure 18 three detrending methods are applied to a single profile 5 meters l ong extracted from a groundbased LiDARderived surface grid. The first two of these detrending methods are based on linear and quadratic models w hile the third method is an FFTbased filter The FFT filter defines the trend from the frequency components with a spatial wavelength larger than a meter. Part A) of F igure 1 8 d epicts the original surface profile and the trends that correspond to the first order second order and FFT
28 trends. It also shows the coefficient of determination (R) of the trend with respect to the observed profile, defined as: 2 1 2 1 2 1 2 21 1o n i i i n i i n i i in m o o o m o R (1 13) w here, io is the ith height observation point of the profile and im is the respective modeled height according to the trend. The 2 R is a measure of the closeness of the modeled trend to the observe d data. P art B) shows the extracted height random component from the different detrending methods and their respective RMShs. In part C ) of the figure, the autocorrelation functions of the random height components for the different detrending methods are plotted. It can be seen that as the modeled trend fits better the observed data, both the RMSh and correlation length values are reduced. Optimal detrending will extract random height components at a single scale, which should yield a roughness parameter t hat reach an asymptotic value indifferent t o the profile length [ 2 ]. A study by Bryant et al. demonstrated the sensitivity of soil moisture retrieval to the detrending technique used, report ing a variation in retrieval of more that 9% per volume between tw o different methods [ 15]. I n a study based on synthetic surface profiles Lievens et al. determined that for soil moisture retrieval the difference between whether or not the proper detrending was applied could be as high as 25% per volume  Current L imitations of S urface R oughness C haracterization Despite the simplicity of the above definitions and equations, a recent literature review [ 7 ] concluded that soil surface roughness description and measurement for the
29 modeling of backscattering is not full y understood. Current challenges in the characterization of surface roughness can be summarized as: Obtaining precise (repeatable) and accurate values of roughness parameters even for the same surface using different techniques and methods [ 2 ],  [1 5 ]. This limitation has been attributed to reasons that include systematic and random instrument errors; nonstandardized field data collection practices; sensitivity of the roughness parameters to profile length, horizontal digitizing resolution, height vari ation accuracy and detrending techniques. Obtaining comprehensive results that support the assumptions that roughness is a stationary singlescale process and that random height variations are isotropic, homogenous and their autocorrelation functions can be explained either by the Gaussian or exponential ACF models [ 8 ] [1 2 ]. Obtaining good agreement between soil moisture measurements and results obtained from the inversion of physical analytical models that use spaceborne microwave sensor observations and in situ roughness measurements [ 2 6 ]  [ 28]. The hypotheses presented in this dissertation argue that these challenges arise because there are instrument errors that are not properly accounted for; the approach of deducing 3D roughness characteristic s from a limited number of 2D profiles is not appropriate ; that the scale at which roughness needs to be characterized has not been properly determined; and that roughness measurements from small sample profiles do not scale up to represent the roughness of large areas Scanning LiDAR mounted in both terrestrial and airborne platforms, is explored as a technological alternative to overcome these limitations. Scanning light detection and ranging (LIDAR) technologies Light detection and r anging (LiDAR) is an active remote sensing technique that uses electromagnetic radiation in the visible spectrum to measure the range to a target. Also properties from the target may be deduced based on the reflection, scattering, absorption, fluorescence or any other phenomena result ing from the interaction
30 between the radiation and the target. The origins of LiDAR can be traced to the late 1930s for two different applications : the study of atmospheric components in 1937 [ 29] and as electronic distance measuring (EDM) devic es in 1938 [ 19]. However, t he acronym LiDAR was introduced only in 1953 . In the early days focu sed light beams were used as energy source. Substantial advancement in LiDAR technology came w ith the invention of the laser in the late 1950s  T he la ser allowed for a coherent, single frequency and highly collimated beam of optical energy to be used as the energy source. With the implementation of the laser as the light source the acronym LaDAR for laser detection and ranging came to use, and this is the nomenclature m ainly used in military circles today In the late 1960s and early 1970 s, the laser based EDMs were capable of measuring distances of up to 60 km and soon were integrated into airborne platforms as the early LASER altimeters. In the late 1970s optical scanning mechanism s were added to the airborne laser altimeters, allowing the laser beam to be steered in a direction perpendicular to the line of flight . This was the origin of the current airborne scanning LiDAR that has allowed researchers and engineers to map the topography of large areas at fine resolutions with high precision and accuracy  . The technology continued to evolve and advances resu lted in the miniaturization of laser s, optomechanics, detectors and electronics S mall self contained scanning LiDARs for terrestrial use became available at the end of the 1990s . The principles of operation of these terrestrial or groundbased LiDARs are extensively discussed in .
31 The main advantage that Scanning LiDAR brings to the problem of roughness characterization is that it is capable of accurately reproducing the entire surface rather than under representing it by transects. Groundbased LiDAR derived digital elevation models can cover areas of tens to low hundreds of square meters, with sample spacing as low as a few millimeters. This allows for the characterization of surface features at high spatial bandwidths and fine resolutions This in turn enables t he accurate separation of the r eference surface and the random components and thus an accurate determination of the roughness parameters In addition, information from 3D digital elevation models ( DEMs ) can be used to describe the directional properties and heterogeneous characteristics of a surfaces roughness. Air borne LiDAR can improve the way surface roughness is currently characterized, going from sampling with a few in situ pr ofiles or relatively small surface models to remotely collecting 3D information of large areas. With this 3D information of large areas an accurate characterization of surface roughness and its spatial variations can be obtained.
32 A B Reference surface Random height component Figure 1 1. Two configurations of reference surface and height random components. a ) Reference is a periodic surface. b ) Reference is the mea n surface. 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 -0.04 -0.02 0 0.02 0.04 Horizontal displacement [m]Height [m] Figure 12. Synthetic generated profiles with exponential ACF, an RMSh of 1cm and a correlation length of 5 cm. 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 -0.04 -0.02 0 0.02 0.04 Horizontal displacement [m]Height [m] Figure 13. Synthetic generated profile with Gaussian ACF, an RMSh of 1cm and a correlation length of 5 cm.
33 -3 -2 -1 0 1 2 3 -0.2 0 0.2 0.4 0.6 0.8 1 X Lags [Meters]Normalized Autocorrelation Exponential Gaussian PLS Figure 1 4 Common autocorrelation functions shapes. Figure 15 A meshboard used for surface roughness studies.
34 Figure 16 A pin profiler deployed over vegetated terrain. Figure 17 A groundbased LiDAR scanner mapping an agriculture soil.
35 A B C 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 -0.04 -0.02 0 0.02 0.04 Profile length [m]Profile height (m) 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 -0.02 0 0.02 0.04 Profile length [m]Random component (m) 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 -0.5 0 0.5 1 Lags [m]Correlation Profile 1st Order, R2=0.39 2nd Order, R2=0.85 FFT fc=1/m, R2=0.95 1st Order, RMSh=1.1cm 2nd Order, RMSh=0.5cm FFT fc=1/m, RMSh=0.3cm 1st Order, cl=56cm 2nd Order, cl=38cm FFT fc=1/m, cl= 5cm Fig ure 18 Detrending effects on roughness parameter values. A ) T he original pr ofile and the modeled trends. B ) T h e extracted random component. C ) Derived autocorrelation functions from the random component extracted from the different detrending methods
36 CHAPTER 2 CHARACTERIZATION OF GROUNDBASED LIDAR ERRORS AND ACCURACY ASSESSMENT OF DERIVE D ROUGHNESS PARAMETERS FR OM TWO DIMENSIONAL ( 2D ) PROFILES There are many aspects to the problem of obtaining repeatable (precise) and accurate roughness parameters from soil surfaces. Some issues include instrumental errors that affect the digitized profile; the capacity of the instruments to capture the minute height variations at fine resolutions ; and nonstandardized field data collection techniques in terms of digitized profile length and horizontal sample spacing. Of the roughness parameters the root mean square of height variations (RMSh) is the least problematic as it is the least sensitive to instrumental errors and multiscale effects. The dif ficulty in obtaining consistent correlation lengths is due to the fact that the retrieval of the autocorrelation function is highly sensitive to noise induced by the instruments, detrending techniques, profile length, and sampling spacing. In a simulation study Oglivy and Foster determined that to properly distinguish between exponential and Gaussian ACFs it i s necessary to digitize the profile at a sampling space of at most 0.1 of the correlation length, with a profile length of at least 60 times the co rrelation length [1 0 ]. In a similar, but more recent study Oh and Kay reported that to precisely determine the RMSh of a profile, its length had to be at least 40 times the correlation length; and for the precise determination of the correlation length, t he profile should be at least 200 times the correlation length. They also determined that the sampl e spacing should be no longer that 0.2 times the correlation length. . These r ules have the disadvantage that an a priori knowledge of the correlation length of the surface to be digitize d is required A more practical rule from Ulaby et al. states that the sampling spacing should be smaller than 0.1 of the microwave observation wavelength . In
37 addition, the studies presented in  and  do not consider instrumental errors that will corrupt the studied surface and will degrade the accuracy of the retrieve d ACF and its respective correlation length. In terms of adequate profile lengths Callens et al. used roughness parameters obtained for 25meterl ong profiles from natural soils at different tillage states as reference to compare roughness parameters derived from shorter profiles. They found that for smooth surfaces the profile needed to be at least 10 meters in length to obtain comparable RMSh values and profiles of at least 5 meters we re required for rougher surfaces. T hey also determined that long profiles (~ 25 m) are required to accurately estimate correlation lengths [ 2 ] Baghdadi et al. suggested that if a surface has a correlation length bet ween 2 and 20 cm, averaging parameter values obtained from ten 2 m profiles provides a precision of 5% for RMSh and between 5 to 15% for correlation length. If ten 1m profiles are used, precision will be reduced to 10% for RMSh and 20% for correl ation length. [ 24] It has also been found that a large range of roughness parameter values can be obtained from the same surface using different techniques. There have been several attempts to compare the roughness parameters derived from different methodologies with results that have been contradictory or inconclusive. One of the first was by Gatti et al. who compared mechanical and laser profilers They found that after averaging metrics derived from five to eight profiles the results between mechanical and LASER profilers agreed to within 20% [3 3 ] Mattia et al. compared roughness metrics obtained from profiles derived from a meshboard, a pin profiler and a LASER profiler. They found relatively good agreement between the correlation lengths obtained fr om the
38 LASER profiler and pin profiler data. The RMS h obtained from the pin profiler is generally overestimated in comparison with the ones obtained from the LASER profiler. Finally they found that values for the correlation length and RMS obtained from t he meshboard data contained significant systematic error s with respect to the ones obtained from the laser profiler data T hey attribute this to errors that occur during the digitization process [13 ]. Bryant et al. performed a study in which they compared pin profiler and groundbased LiDAR scanner RMSh results. By digitizing what they assumed to be a smooth surface with RMSh = 0 they determined that each method presents a characteristic positive bias of 1.5 mm and 3 mm respectively. When these biases wer e removed, the RMSh derived from the two methods had an R of 0.6  While these studies compared roughness metrics obtained from different methods, an accuracy assessment with respect to a known reference surface for each method remains to be done  This chapter presents several analyses that were performed to assess the accuracy and precision of roughness parameters derived from groundbased LiDAR measurements. The first analysis consisted on characterizing the instrument random noise present in 2D elevation profiles extracted from flat surfaces. Based on this characterized instrument noise the second analysis was performed. This consisted of generating 2D profiles with different roughness characteristics and corrupting them with characteristic ins trument noise T he effect that the random noise had on the accuracy of the derived roughness parameters was then quantified. The third analysis consisted of performing an accuracy assessment of the derived RMSh from real measurements using roughness references. Finally, as a validation of the accuracy assessments a
39 comparison was conducted between the roughness parameters obtained from meshboard and LiDAR for a sample of 21 profiles of bare agricultural surfaces. Characterization of Groundbased LiDAR Errors The ground basedLiDAR employed in this study is an Optech I LRIS3D scanner (shown in Figure 17). The ILRIS is a n instrument composed by a timeof flight LiDAR and a twoaxis scanner. A full description of the instrument design, operation, and performa nce can be found in . To characterize the instrument random error seve n smooth flat surfaces were scanned. Figure 21 A) shows a picture of one such reference surface, a smooth metal plate resting on a flat floor. Figure 21 B) shows a rendering from the LiDAR scanned point cloud. The points from the area corresponding to the metal plate were then cropped and gridded at 5 mm spacing, creating a 3D digital elevation model (DEM) shown in Figure 22 From the DEM 2D transects were extracted along the row direction. F or each point of the transect, the height deviation from the flat surface w as determined. Figure 23 shows a plot of the height deviations resulting from concatenating all the profiles. Figure 24 A) sh ows a histogram of the deviations, while Figure 24 B) shows the log arithm of the power spectral density resultant from applying an FFT to the height deviation sequence of Figure 23. From Figure 2 4 it can be observed that the magnitude of LiDAR random error follows a Gaussian distribution and its spectrum is almost uniform for all frequency components. This same process was applied to the other six reference flat surfaces. A total of 848 profiles were extracted, containing 2,012,304 elevation measurements From these it was found that when extr acting 2D profiles the LiDAR random error can be modeled as Gaussian white noise with a standard deviation of 2.8 mm This is consistent with the modeling accuracy quoted by the manufacturer.
40 Accuracy Assessment Based on Synthetic Generated Profiles To perform the first accuracy assessment of the derived roughness parameter values, a computer simulation approach was followed. First a moving average method was used to generate 2D profiles with Gaussian and exponential ACFs for five different values of RMSh ( 5 mm, 1, 1.5, 2 and 2.5 cm ) and correlation length ( 2, 8, 14, 20, and 26 cm ). A total of 50 unique profiles 50 meters in length with a sampling spacing of 1 mm were generated. These 50 profiles were corrupted by adding random characteristic noise from the LiDAR (white noise with a standard deviation of 2.8 mm) The original and corrupted profiles were then cut into ten 5 meter segments T he corrupted profiles were also resampled at 5 and 10 mm. From the original and the corrupted profiles the RMSh, cor relation length, and the fit metrics for the different theoretical ACF were computed. The accuracy assessment was performed by co mparing the roughness parameter values of the original and corrupted profiles. T o generate the random height component sequence s with a given RMSh, correlation length, and ACF the moving average method described in [10 ], [3 2 ] [ 34] was followed The purpose is to generate a sequence representing random component height variations of the following form: x i h x h zi i (2 1) w here i is the sample index and x is the sample separation or discretization interval. The first step is to generate a sequence Vi of 2N+1 uncorrelated random numbers from a standard normal distribution with a zero mean and a standard deviation of 1 (Figure 25 ). It is also necessary to generate a sequence of weights Wi each with a length of 2M+1 (Figure 26) These weights are defined depending on the type of
41 autocorrelation function, the required RMSh (h) and correlation length ( l ) For the Gaussian ACF the weight sequence is given by: 222 l x i h Ge l x i W ( 2 2) The weight sequence for the exponential ACF is given by: l x i K l x i Wh e 02 ( 2 3) w here 0K is the modified Bessel function of the second kind. The moving average method is then applied to obtain the correlated sequence as: M M j i j j iV W z. ( 2 4) The resulting correlated sequence s are of length 2(N M)+1 as shown in Figures 2 7 and 28 Accuracy of D erived A utocorrelation F unction (ACF) To characterize an autocorrelation function two parameters are required: the exponent of the generalized power law (Equation 110) and its correlation length. To obtain the value of the exponent from a measured ACF, an iterative linearized least squares method must be employed. The linearized least squares model for Equation 110 is: m l h m l h n m n m l h n n l hv e h v e h n l h l h e l h l h en n n m n 0 1 0 1 0 0 0 0 0 0 11 1 1 1ln ln (2 5)
42 w here l is the determined correlation length, 0n is the exponent value estimate for the current iteration, n is the correction to the estimated value of the exponent that needs to be determined, mh is the mth observed lag, mh is the value of the measured autocorrelation function for the mh lag, and mv are the residuals to be minimized. Table 21 presents the change in the exponent of the generalized power law (Equation 1 10) of the ori ginal ACF due to instrument induced noise. The results indicate that the effect of the instrument noise is a reduction in the exponent value. This means that a profile with a Gaussian ACF will look more like one with an exponential ACF. This effect is part icularly noticeable for smooth profiles characterized by small RMSh and large correlation lengths. For instance, original profiles with RMSh in the order of 5 mm, the exponent of the generalized power law ACF from the corrupted profiles will be less than 1. Original profiles with Gaussian ACFs will be more affected that those that follow exponential ACFs. Accuracy of D erived of R andom H eight R oot M ean S quare ( RMSh ) Table 22 summarizes the root mean square error (RMSE) in the estimation of the RMSh for orig inal profiles wit h exponential and Gaussian ACFs. The following is the interpretation of those results : For profiles with a true RMSh much higher than the noise level of the instrument (RMSh >1 cm), the estimated RMSh is accurate to approximately 1 mm For profiles with a true RMSh about three times the noise level of the instrument (RMSh ~1 cm), the estimated RMSh is accurate to approximately 1 mm for Gaussian ACF and 2mm for exponential ACF. For profiles with a true RMSh about double the noise level of th e instrument (RMSh <5 mm), the estimated RMSh is accurate to approximately 23 mm for Gaussian ACF and 3 mm for exponential ACF.
43 For profiles with ACF resembling the Gaussian model, the estimation of RMSh is less affected by the noise corruption than their exponential counterparts. Accuracy of D erived C orrelation L engths Table 23 summarize s the root mean square error (RMSE) in the estimation of the correlation lengths for original profiles with exponential and Gaussian ACFs. T he net effect of instrument random noise is a reduction in the measured correlation length values with respect to the true value. Table 2 4 give s the average reduction of the measured correlation lengths for the different combinations of RMSh, correlation lengths and ACFs. The larger differences occur for those surfaces with low RMSh and long correlation lengths ; exponential surfaces are more affected than those with Gaussian ACFs. The accuracy of the derived correlation lengths of exponential surfaces is degraded on average by a fact or of two compared to the surfaces characterized by Gaussian ACFs. The least accurate results of derived correlation lengths are obtained f r o m those surfaces with low RMSh and long correlation lengths that have an RMSE of 1 5 to 25 cm. For surfaces with Gaussian ACF, RMSh higher than 1 cm and correlation lengths lower than 20 cm the accuracy of instrument derived correlation lengths is within 2 cm. The accuracy of derived correlation lengths for surfaces with exponential and similar characteristics is withi n 4 cm. Accuracy Assessment Based on Roughness References The second accuracy assessment was based on real measurements conducted u n der laboratory conditions. To perform this evaluation, a set of references with known roughness characteristics w as construc ted from square wooden pegs of 0.96 cm with variable heights. The pegs were mounted on two 1.22 m aluminum U rails, yielding a single 2.44m long reference with 254 height values. These reference s provide a
44 spectral sampling of 0.04 wavelengths for L band t o 0.17 for C band. By using different height pegs in a pseudorandom sequence uncorrelated height profiles were generated with RMSh values of 0.9, 2, 3, and 3.8 cm. Figure 2 9 shows a section of one of such u ncorrelated roughness reference Correlated sequences with a generalized power law spectrum ACF (n~1.5) were also produced with RMSh values of 0, 0.9, 1.6, 2.7, and 3.7 cm. See Figure 210 for a roughness reference with a correlated height sequence. A total of 28 reference profiles (2.44 m in length), 16 pseudorandom and 12 correlated, were digitized using the meshboard and the groundbased LiDAR. Meshboard data were collected and processed using the procedure described in Appendix B. Using the LiDAR the references were scanned from an oblique view at a range of 8 9 m and with a sample spacing of 3 4 mm (Figure 211) The result ing point cloud was rectified and transformed into a regularly spaced 9.6 mm grid to match the spacing of the reference pegs. The final meshboard digitized profile and the LiDARd erived grids were analyzed using a script that extracts the study profile, detrends them to obtain the random height component and computes the roughness parameters. The correlation coefficient and coefficient of determination (R) between the observed he ight profiles with respect to refe rence profile were computed as measures of how well the instrument reproduced the reference target (i.e. digitization fidelity) Figure 2 12 is a graphic representation of the three different data sources that are loaded onto the Matlab script that performs t he accuracy assessment. Figure 213 shows a comparison of the height random component from the reference and those extracted from meshboard and LiDAR data, and from which the roughness parameters are computed. The individual results for each of the 28 tests performed for the accuracy
45 ass essment are presented in Table 25, while the overall results are summarized in Table 26 Digitization F idelity of the I nstruments Two metrics were employed to quantify how well the ins truments record the height variations of the study surface, the correlation coefficient and the coefficient of determination (R). These are defined as r o n i i i r o y xr r o o r o Cov 1 ,, (2 6 ) 2 1 2 1 2 1 2 21 1o n i i i n i i n i i in r o o o r o R (2 7 ) where io is the ith h eight observation point from the meshboard or LiDAR data and ir is the respective height according to the reference. The data show that the meshboard records the surface height variations with great fidelity for both uncorrelated and c orrelated profiles with correlation coefficients exceeding 0.96 and R exceeding 0.92. The LiDAR scanner has less fidelity than the meshboard, especially with uncorrelated height sequences for which it shows poor performance with R between 0.440.77. This is due mainly to the higher noise level injected by the LiDAR and because a rough uncorrelated surface will have rapid height variations T hese variations cause effects such as multipathing shadowing, and ringing that affect the LiDAR performance Howeve r, for correlated surfaces, which is the case of most natural soils, the scanner performs well with correlation coefficients greater than 0.94 and R greater than 0.87.
46 Accuracy of D erived RMSh Results of RMSh accuracy assessment are illustrated in Figure 2 14. It was found that the RMSh obtained from the meshboard match the reference RMSh to within 1 mm, independent of the surface RMSh for both uncorrelated and correlated s urfaces. The RMSE for meshboard derived RMSh of correlated surfaces (n = 12) was 1.1 mm and 0.9 mm overal l (n = 28). In previous studies by [ 1 3 ] and [33 ] data from a laser profiler were used as a benchmark to compare meshboardderived roughness parameters and not even a close agreement was achieved. This improvement can be attributed to t he proper removal of the systematic errors caused by image distortion as described in Appendix B The results of the accuracy of the RMSh values obtained from the LiDAR are different for uncorrelated and correlated surfaces. For uncorrelated surfaces, the RMSh is underestimated with respect to the reference and as the roughness of the surface increases the underestimation also increases. This is due to the multipath, shadowing, and ringing effects that tend to smooth the digitized surface. With respect to correlated surfaces, the experiments show that for a uniform flat surface, the LiDAR overestimates the RMSh by 4 mm, which is consistent with the results obtained in [ 15] However in their work they assumed that this was a positive bias present in all LiDA R derived RMSh values. In contrast, results from this experiment show that the RMSh values derived from LiDAR are positively biased only in the range between 0 to 9 mm, unbiased for surfaces with RMSh between 1 to 2 cm and for surfaces with RMSh between 2 and 4 cm are underestimat ed by only 1 mm. RMSE for LiDAR derived RMSh of correlated surfaces was found to be 2.5 mm.
47 Validation with Agricultural Soil Measurements As a final validation of the accuracy assessments, a comparison of roughness parameter values was performed. For this, 21 profiles of agricultural soils with different roughness characteristics were obtained each with a length of 2.13 m. Data collection w as performed simultaneously with the LiDAR scanner and meshboard as shown in Figure 215 In addition to the comparison, it was determined how well the measured soil surface ACF matched the Gaussian or exponential models. For this purpose the exponent of the power spectrum law was determined for each experimentally determined autocorrelation function. Also the coefficient of determination (R) was computed between the measured ACF and each of the theoretical models for two different intervals, one considering the entire lag range from 0 to the max imum lag and another just considering the lag range from 0 to the correlation length. The individual results for the 21 comparison of roughness parameters values are presented in Table 2 7 and summarized in Table 28 Comparison of RMSh V alues The comparison of the derived values for RMSh is illustrated i n Figure 216. A very good agreement was found between the values estimated by the two methods with an R of 0.947, which was much better that the 0.6 obtained in [ 15] for the Arizona set and the no agreement reported in [ 13] and [3 3 ] This is also in agreement with the results of the accuracy assessments performed with synthetic profiles and roughness references. Comparison of C orrelation L engths The agreement between the derived correlation lengths was weak, with a correlation coefficient of only 0.50 and a n R of 0.25. Figure 217 shows the difference in the derived correlation lengths. Where the s olid dots represent the LiDAR derived
48 RMSh and correlation length, the tip of the horizontal error bar corresponds to the correlation length obtained from the meshboard. As can be seen, t he largest differences occur for RMSh values below 1.5 cm. This was predicted from the accuracy assessment based on synthetic ly generated profiles w here it was demonstrated that is extremely difficult to obtain accurate values of correlation lengths from smooth surfaces due to the corruption of the autocorrelati on function from the instrument generated noise. In addition, these differences can be attributed to the modification of the surface profile by the meshboard and the inherently large statistical variation of correlation length values. Despite the low correlation, it is worth notic ing that 81% of the absolute differences in t he derived correlation lengths were below 5 cm, and 52% below 2 cm. Also, good agreements of the der ived correlation length h ave not been reported before. A utocorrelation F unctions of A gricultural S oils With regard to the ACF model that best describes natural soils, it was found that t he averaged exponent of a power law spectrum was 1.47 for the m eshboar d and 1.23 for the LiDAR. This result is also expected from the previous results of the accuracy assessment based on the synthetic profiles. Because the LiDAR has a higher noise level than the meshboard, the derived ACF from LiDAR will be closer to the exp onential ACF model with an exponent value of 1. In terms of the R comparing the measured ACF with the theoretical models; if only the lag range from zero to the correlation length is considered, the exponential model is marginally better than the Gauss ian However, if the entire lag range is considered, the Gaussian model is a marginally better fit than the exponential.
49 Chapter Conclusions From the analyses presented in this chapter it can be concluded that the main issue affecting the precision and accur acy of the derived roughness parameters is not properly considering the effects of the instrumental systematic and random errors. Surface heights and roughness are random processes Wh en digitizing the surface with any instrument, the instrument random err ors are mixed with the random surface heights. Thus, what is obtained from the instrument measurements is the addition of two random processes. To properly retrieve the roughness parameters, the instruments systematic errors must be removed and the random errors properly characterized. When the random errors are properly characterized, their effects on the surface measurements can be quantified and accurate estimat es of the parameters can be obtained. Of the roughness parameters, the RMSh is the least sens itive to instrument noise. The measured RMSh value will asymptotically approximate the true value as the instrument characteristic noise is much smaller than the surface RMSh. For surfaces with RMSh comparable to the instrument noise, the measured RMSh can be corrected by: 2 2 inst meas estRMSh RMSh (2 8 ) w here RMShest is the estimated value, RMShmeas 2 inst is the instrument characterized noise variance. To accurately determine the correlation length it is necessary to properly retrieve the autocorrelation function. The measured autocorrelation function is the ACF result ing f rom the addition of two random processes. In theory, i nstrument random noise is usually Gaussian white noise. This means that the instrument random errors have no
50 correlation length. In practice the correlation length is nonzero but relatively small. The effect of the instrument n oise on the retrieved ACF is a reduction of the value of the exponent of the generalized power law (Equation 110) and a reduction of the correlation length. The magnitude of the reductions is proportional to the relative size of the instruments noise standard deviation and correlation length with respect to the random heights variation RMSh and correlation length. The closer the instrument noise level is to the surface RMSh, the gr ater the reduction of exponent and correlation length values. It was also determined that due to its lower noise level, under laboratory conditions the meshboard has a higher digitization fidelity than the LiDAR. However, under field conditions, this performance is not achievable because the meshboard unavoidably disturbs the surface under study. Beside its destructive nature, an additional disadvantage of the meshboard method is that is very time consuming. When the errors from the groundbased LiDAR are properly accounted for, it is possible to derive RMSh values accurate to 1mm and correlation length values accurate to less than 1 cm from its data.
51 A) B) Figure 21. Test to characterize of LiDAR random noise. A) Photo of a smooth flat target. B) Rendering from the LiDAR scanned point cloud. g metersmeters 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 -0.02 -0.015 -0.01 -0.005 0 0.005 0.01 0.015 0.02 0.025 Figure 22. DEM derived from a LiDAR scan of a smooth flat target.
52 0 1 2 3 4 5 6 7 x 104 -0.03 -0.02 -0.01 0 0.01 0.02 0.03 SampleElevation deviation [m] Figure 23. Elevation dev iations from the smooth surface, extracting and concatenating profiles along the row direction of the DEM. A) B) -0.02 -0.015 -0.01 -0.005 0 0.005 0.01 0.015 0.02 0 2000 4000 6000 8000 10000 12000 14000 Error magnitude [m]Counts 0 1 2 3 4 5 6 7 x 104 -35 -30 -25 -20 -15 -10 -5 0 5 Spatial frequency [1/m]Log PSD [dB] Figure 24. Characterizat ion of groundbased LiDAR random errors A) Distribution of magnitudes. B) Power spectral density. 0 1 2 3 4 5 6 -5 -4 -3 -2 -1 0 1 2 3 4 5 Horizontal displacement [m]Amplitude [m] Figure 25. Random seed sequence from which correlated sequences are generated.
53 -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 0 1 2 3 4 5 6 x 10-3 Horizontal Displacement [m]Amplitude [m] Gaussian Exponential Figure 26. Gaussian and e xponential weighting functions used in the moving average method. 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 -0.1 -0.05 0 0.05 0.1 Horizontal displacement [m]Height [m] Figure 27. Generated profile with a Gaussian ACF. 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 -0.1 -0.05 0 0.05 0.1 Horizontal displacement [m]Height [m] Figure 28. Generated profile with an exponential ACF.
54 Figure 2 9 Section of a pseudorandom roughness reference with an RMSh of 2 cm. Figure 2 10 Section of a c orrelated roughness reference with an RMSh of 2.7 cm. Figure 2 11 S canning geometry of roughness references using groundbased LiDAR
55 A 0 0.5 1 1.5 2 2.5 0 0.05 0.1 0.15 Length [m]Height [m] 0 0.5 1 1.5 2 2.5 0 0.05 0.1 0.15 Length [m]Height [m] Length [m]Depth [m] 0 0.5 1 1.5 2 2.5 0 0.2 0.4 0 0.5 1 1.5 2 2.5 0.05 0.1 0.15 Length [m]Height [m] B C D Figure 2 12 Inputs in to the Matlab script used to perform the accuracy asses sment based on roughness references A ) The roughness reference profile. B ) Measured profile using the meshboard. C ) 3D grid derive d from the groundbased LiDAR. D ) Extracted profile from the 3D grid.
56 0 0.5 1 1.5 2 2.5 -0.08 -0.06 -0.04 -0.02 0 0.02 0.04 0.06 0.08 0.1 p p Length [m]Height [m] Reference Meshboard LiDAR Figure 213. Random component extracted f rom the reference the meshboard and LiDARderived profiles. 0.000 0.005 0.010 0.015 0.020 0.025 0.030 0.035 0.040 0.000 0.010 0.020 0.030 0.040 Measured RMS [m] Reference RMS [m] Meshboard LiDAR Meshboard LiDAR Figure 214. Results of the RMSh accuracy assessment.
57 A) B) Figure 215. Simultane ous profile digitizing with A) m eshboard and B) LiDAR 0.0 1.0 2.0 3.0 4.0 5.0 0.0 1.0 2.0 3.0 4.0 5.0 LiDAR derived RMSh (cm) Meshboard derived RMSh (cm) Post Affine tx Corrected R =0.95 R =0.92 Figure 2 16 Comparison of RMSh values of bare agricultural soils obtained from m eshboard and LiDAR
58 0.0 5.0 10.0 15.0 20.0 25.0 30.0 0 1 2 3 4 LiDAR CL [cm] LiDAR RMSh [cm] Figure 2 17 Difference between meshboard and LiDAR derived correlation lengths of bare agricultural soils. Solid dots represent the LiDAR derived RMSh and correlation length, the tip of t he error bar corresponds to the correlation length obtained from the meshboard.
59 Table 2 1 Mean difference in the exponent of generalized power law for profiles with exponential and Gaussian ACF. Exponential ACF Gaussian ACF Correlation Referenc e RMS h(cm) Reference RMS h(cm) length (cm) 0.5 1 1.5 2.0 2 5 0.5 1 1.5 2.0 2 5 Sampled every 1 mm 2 0.71 0.25 0.10 0.06 0.04 1.06 0.35 0.13 0.08 0.06 8 0.58 0.20 0.09 0.05 0.03 1.05 0.32 0.14 0.10 0.05 14 0.60 0.19 0.11 0 .06 0.04 1.01 0.38 0.20 0.13 0.06 20 0.65 0.26 0.10 0.06 0.03 1.10 0.40 0.17 0.07 0.05 26 0.68 0.22 0.05 0.06 0.05 1.11 0.39 0.15 0.07 0.07 Sampled every 5 mm 2 0.55 0.23 0.09 0.07 0.07 0.93 0.30 0.14 0.07 0.0 4 8 0.55 0.18 0.07 0.05 0.03 1.04 0.31 0.14 0.10 0.05 14 0.58 0.18 0.11 0.06 0.04 1.00 0.38 0.20 0.12 0.06 20 0.62 0.27 0.11 0.06 0.04 1.10 0.41 0.18 0.07 0.05 26 0.67 0.23 0.06 0.05 0.05 1.11 0.38 0.15 0.07 0.07 Sampled every 10 mm 2 0.41 0.14 0.13 0.08 0.07 0.69 0.19 0.17 0.02 0.01 8 0.50 0.17 0.08 0.04 0.02 1.03 0.29 0.13 0.09 0.05 14 0.55 0.16 0.10 0.07 0.03 0.99 0.38 0.20 0.13 0.05 20 0.60 0.26 0.11 0.06 0.04 1.0 7 0.39 0.18 0.07 0.05 26 0.65 0.23 0.05 0.05 0.05 1.12 0.39 0.15 0.08 0.07
60 Table 2 2 RMSE (cm) in the estimation of RMSh for profiles with exponential ACF. Exponential ACF Gaussian ACF Correlation Reference RMS h(cm) Reference RMS h(c m) length (cm) 0.5 1 1.5 2.0 2 5 0.5 1 1.5 2.0 2 5 Sampled every 1 mm 2 0.3 0.2 0.1 0.1 0.1 0.2 0.1 0.1 0.1 0.0 8 0.3 0.1 0.1 0.1 0.1 0.2 0.1 0.1 0.1 0.1 14 0.3 0.1 0.1 0.1 0.1 0.2 0.1 0.1 0.1 0.1 20 0.3 0.2 0.1 0.1 0.1 0.3 0.1 0.1 0.1 0.1 26 0.3 0.2 0.1 0.1 0.1 0.3 0.1 0.1 0.1 0.1 Sampled every 5 mm 2 0.3 0.2 0.1 0.1 0.0 0.2 0.1 0.1 0.1 0.0 8 0.3 0.1 0.1 0.1 0.1 0.2 0.1 0.1 0.1 0.1 14 0.3 0.2 0.1 0.1 0.1 0.2 0.1 0.1 0.1 0.0 20 0.3 0.1 0.1 0.1 0.1 0.3 0.1 0.1 0.1 0.1 26 0.3 0.1 0. 1 0.1 0.1 0.2 0.1 0.1 0.1 0.0 Sampled every 10 mm 2 0.3 0.2 0.1 0.1 0.0 0.2 0.1 0.1 0.1 0.0 8 0.3 0.1 0.1 0.1 0.1 0.2 0.1 0.1 0.1 0.1 14 0.3 0.1 0.1 0.1 0.1 0.2 0.1 0.1 0.1 0.0 20 0.3 0.1 0.1 0.1 0.1 0.3 0.1 0.1 0.1 0.1 26 0.3 0.2 0.1 0.1 0.1 0.2 0.1 0.1 0.1 0.0
61 Table 2 3 RMSE (cm) in the estimation of co rrelation length for profiles with exponential and Gaussian ACF. Exponential ACF Gaussian ACF Correlation Reference RMS h(cm) Reference RMS h(cm) length (cm) 0.5 1 1.5 2.0 2 5 0.5 1 1.5 2.0 2 5 Sampled every 1 mm 2 2.3 0.7 0.4 0.2 0.1 1.1 0.3 0.2 0.1 0.0 8 9.2 2.5 2.7 1.7 0.6 5.7 1.2 0.9 0.4 0.2 14 13.4 6.2 1.7 1.1 0.7 7.5 2.0 0.9 0.7 0.5 20 18.5 4.1 2.2 1.2 0.8 16.8 3.0 1.4 1.1 0.4 26 22.2 7.8 15.3 1.4 0.6 15.9 4.8 1.9 0 .8 0.3 Sampled every 5 mm 2 2.1 0.6 0.4 0.1 0.1 1.1 0.2 0.1 0.1 0.0 8 8.8 2.8 2.7 1.7 0.9 5.7 1.4 0.9 0.4 0.4 14 14.6 7.3 1.5 0.7 0.8 8.2 2.3 0.8 0.6 0.6 20 23.3 3.4 2.1 0.8 0.6 16.4 2.7 1.1 1.4 0.5 26 23.7 6.3 15.5 1.4 1.0 16.4 5.2 1.8 1.0 0. 3 Sampled every 10 mm 2 2.0 0.4 0.5 0.1 0.1 1.1 0.2 0.1 0.0 0.0 8 9.7 2.2 2.1 2.0 0.8 6.0 1.3 0.7 0.4 0.4 14 15.0 6.9 1.2 0.5 1.7 8.8 2.1 0.7 0.6 0.7 20 23.5 2.8 2.8 1.0 0.4 16.5 2.1 1.8 0.9 0.5 26 24.6 4.9 14.3 1.1 0.6 15.0 4.4 1.1 0.7 0.1
62 Table 2 4 Mean difference (cm) of correlation lengths for profiles with exponential and Gaussian ACF. Exponential ACF Gaussian ACF Correlation Reference RMS h(cm) Reference RMS h(cm) length (cm) 0.5 1 1.5 2.0 2 5 0.5 1 1.5 2.0 2 5 Sampled ever y 1 mm 2 0.7 0.2 0.1 0.1 0.0 0.4 0.1 0.0 0.0 0.0 8 2.9 0.8 0.8 0.5 0.2 1.8 0.4 0.3 0.1 0.1 14 4.2 2.0 0.5 0.4 0.2 2.4 0.6 0.3 0.2 0.2 20 5.8 1.3 0.7 0.4 0.2 5.3 1.0 0.4 0.3 0.1 26 7.0 2.5 4.8 0.4 0.2 5.0 1.5 0.6 0.2 0.1 Sampled every 5 mm 2 0.7 0.2 0.1 0.0 0.0 0.4 0.1 0.0 0.0 0.0 8 2.8 0.9 0.9 0.5 0.3 1.8 0.4 0.3 0.1 0.1 14 4.6 2.3 0.5 0.2 0.2 2.6 0.7 0.3 0.2 0.2 20 7.4 1.1 0.7 0.3 0.2 5.2 0.8 0.3 0.4 0.2 26 7.5 2.0 4.9 0.4 0.3 5.2 1.6 0.6 0.3 0.1 Sampled every 10 mm 2 0.6 0.1 0.1 0.0 0.0 0.3 0.1 0.0 0.0 0.0 8 3.1 0.7 0.7 0.6 0.3 1.9 0.4 0.2 0.1 0.1 14 4.7 2.2 0.4 0.2 0.5 2.8 0.7 0.2 0.2 0.2 20 7.4 0.9 0.9 0.3 0.1 5.2 0.7 0.6 0.3 0.2 26 7.8 1.5 4.5 0.3 0.2 4.7 1.4 0.3 0.2 0.0
63 Table 2 5 Raw results obtained from the accuracy assessment experiments. Ref erence M eshboard Raw M eshboard corrected LiDAR Dataset RMSh CL RMSh CL R2 CC RMSh CL R2 CC std_l CL R2_l CC 9mm_1cm_ABCD 0.0094 0.0059 0.0109 0.0061 0.8250 0.9097 0.0107 0.0061 0.8458 0.9206 0.0072 0.0086 0.4587 0.8261 9mm_1cm_DCBA 0.0094 0.0059 0.0099 0.0062 0.8810 0.9388 0.0095 0.0060 0.9075 0.9526 0.0076 0.0078 0.6614 0.8844 10mm _1cm_ABCD 0.0095 0.0062 0.0102 0.0070 0.9151 0.9568 0.0096 0.0063 0.9628 0.9813 0.0076 0.0090 0.5312 0.8416 10mm_1cm_DCBA 0.0095 0.0062 0.0100 0.0072 0.9149 0.9565 0.0095 0.0065 0.9676 0.9839 0.0083 0.0089 0.6044 0.8389 Average 0.0095 0.0060 0.0103 0.006 6 0.8840 0.9404 0.0098 0.0062 0.9209 0.9596 0.0077 0.0086 0.5639 0.8477 0.0001 0.0002 0.0005 0.0006 0.0425 0.0222 0.0006 0.0002 0.0570 0.0296 0.0004 0.0006 0.0881 0.0254 20mm_1cm_ABCD 0.0204 0.0065 0.0203 0.0067 0.9698 0.9848 0.0202 0.0066 0.9803 0.9902 0.0165 0.0081 0.8030 0.9422 20mm_1cm_BACD 0.0204 0.006 5 0.0204 0.0065 0.9685 0.9842 0.0203 0.0065 0.9804 0.9903 0.0162 0.0082 0.7714 0.9358 20mm_1cm_DCBA 0.0204 0.0065 0.0210 0.0068 0.9718 0.9858 0.0205 0.0067 0.9824 0.9912 0.0165 0.0081 0.7272 0.9112 20mm_1cm_CDAB 0.0204 0.0065 0.0206 0.0066 0.9546 0.9771 0.0206 0.0067 0.9638 0.9818 0.0164 0.0079 0.7664 0.9300 Average 0.0204 0.0065 0.0205 0.0066 0.9662 0.9830 0.0204 0.0066 0.9767 0.9884 0.0164 0.0081 0.7670 0.9298 0.0000 0.0000 0.0003 0.0001 0.0078 0.0040 0.0002 0.0001 0.0087 0.0044 0.0002 0.0002 0.0311 0.0133 30mm_1cm_ABCD 0.0297 0.0061 0.0298 0.0064 0.9508 0.9752 0.0297 0.0066 0.9601 0.9800 0.0233 0.0097 0.3174 0.7615 30mm_1cm_BACD 0.0297 0.0061 0.0298 0.0069 0.9542 0.9768 0.0300 0.0062 0.9599 0.9798 0.0226 0.0098 0.2581 0.7543 30mm_1cm_DCBA 0.0297 0.0061 0.0300 0.0064 0.9491 0.9743 0.0301 0.0069 0.9605 0.9802 0.0248 0.0084 0.6197 0.8576 30mm_1cm_CDAB 0.0297 0.0061 0.0302 0.0071 0.9542 0.9769 0.0302 0.0064 0.9591 0.9794 0.0262 0.0082 0.5625 0.8151 Average 0.0297 0.0061 0.0299 0.0067 0.9521 0.9758 0.0300 0.0065 0.9599 0.9798 0.0242 0.0090 0.4394 0.7971 0.0000 0.0000 0.0002 0.0004 0.0025 0.0012 0.0002 0.0003 0.0006 0.0003 0.0016 0.0008 0.1783 0.0486 38mm_1cm_ABCD 0.0378 0.0060 0.0392 0.0060 0.9906 0.9958 0.0387 0.0061 0.9913 0.9962 0.0284 0.0078 0.6353 0.9041 38mm_1cm_DCBA 0.0378 0.006 0 0.0392 0.0060 0.9866 0.9940 0.0395 0.0063 0.9874 0.9945 0.0294 0.0084 0.4580 0.8213 38mm_1cm_ABDC 0.0377 0.0062 0.0377 0.0068 0.9700 0.9849 0.0384 0.0064 0.9704 0.9851 0.0308 0.0072 0.6628 0.8833 38mm_1cm_CDBA 0.0377 0.0062 0.0375 0.0064 0.9917 0.9958 0.0359 0.0065 0.9929 0.9965 0.0312 0.0068 0.8146 0.9412 Average 0.0378 0.0061 0.0384 0.0063 0.9847 0.9926 0.0381 0.0063 0.9855 0.9930 0.0299 0.0075 0.6427 0.8875 0.0000 0.0001 0.0010 0.0004 0.0101 0.0052 0.0016 0.0002 0.0103 0.0054 0.0013 0.0007 0.1462 0.0502
64 Table 2 5 Continued. Raw results obtained from the accuracy assessment experiments Ref erence M eshboard Raw M eshboard corrected LiD AR Dataset RMSh CL RMSh CL R2 CC RMSh CL R2 CC std_l CL R2_l CC 0mm_inf_A 0.0000 NaN 0.0022 0.2718 0.0000 NaN 0.0012 0.2583 0.0000 NaN 0.0049 0.0070 0.0000 NaN 0mm_inf_B 0.0000 NaN 0.0026 0.2779 0.0000 NaN 0.0015 0.2680 0.0000 NaN 0.0040 0.0068 0.0000 NaN 0mm_inf_C 0.0000 NaN 0.0025 0.2543 0.0000 NaN 0.0015 0.2348 0.0000 NaN 0.0035 0.0065 0.0000 NaN 0mm_inf_D 0.0000 NaN 0.0027 0.2730 0.0000 NaN 0.0017 0.2645 0.0000 NaN 0.0047 0.0079 0.0000 NaN Average 0.0000 NaN 0.0025 0.2692 0.0000 NaN 0.0015 0.2564 0.0000 NaN 0.0043 0.0070 0.0000 NaN 0.0000 NaN 0.0002 0.0103 0.0000 NaN 0.0002 0.0149 0.0000 NaN 0.0007 0.0006 0.0000 NaN 9mm_18cm_ABCD 0.0092 0.1768 0.0099 0.1992 0.9700 0.9869 0.0094 0.1851 0.9855 0.9928 0.0086 0.1682 0.8914 0.9510 9mm_18cm_DCBA 0.0092 0.1768 0.0102 0.1996 0.9641 0.9860 0.0100 0.1980 0.9754 0.9900 0.0087 0.1641 0.8635 0.9365 Average 0.0092 0.1768 0.0101 0.1994 0.9670 0.9865 0.0097 0.1915 0.9804 0.9914 0.0087 0.1661 0.8775 0.9437 0.0000 0.0000 0.0002 0. 0003 0.0042 0.0006 0.0004 0.0091 0.0072 0.0020 0.0001 0.0029 0.0197 0.0103 16mm_18cm_ABCD 0.0160 0.1795 0.0154 0.1786 0.9775 0.9897 0.0162 0.1816 0.9955 0.9978 0.0162 0.1820 0.9704 0.9851 16mm_18cm_DCBA 0.0160 0.1795 0.0157 0.1776 0.9788 0.9899 0.0165 0.1775 0.9951 0.9978 0.0162 0.1807 0.9728 0.9863 Average 0.0160 0.1795 0.0155 0.1781 0.9782 0.9898 0.0163 0.1796 0.9953 0.9978 0.0162 0.1813 0.9716 0.9857 0.0000 0.0000 0.0002 0.0007 0.0009 0.0002 0.0002 0.0029 0.0003 0.0000 0.0000 0.0009 0.0017 0.0009 37mm_33cm_ABCD 0.0369 0.3271 0.0350 0.3236 0.9943 0.9991 0.0364 0.3284 0.9989 0.9996 0.0358 0.3304 0.9943 0.9977 37mm_33cm_DCBA 0.0369 0.3 271 0.0359 0.3105 0.9959 0.9990 0.0373 0.3158 0.9993 0.9997 0.0367 0.3358 0.9913 0.9957 Average 0.0369 0.3271 0.0355 0.3170 0.9951 0.9991 0.0369 0.3221 0.9991 0.9996 0.0362 0.3331 0.9928 0.9967 0.0000 0.0000 0.0006 0.0093 0.0011 0.0001 0.0006 0.0089 0. 0003 0.0000 0.0007 0.0039 0.0021 0.0014 37mm_33cm_BACD 0.0369 0.3062 0.0393 0.3165 0.9956 0.9995 0.0377 0.3105 0.9987 0.9994 0.0381 0.2747 0.8683 0.9325 27mm_32cm_ABCD 0.0267 0.3174 0.0258 0.3222 0.9951 0.9983 0.0268 0.321 7 0.9972 0.9986 0.0260 0.3258 0.9816 0.9913 27mm_32cm_DCBA 0.0267 0.3174 0.0252 0.3176 0.9922 0.9980 0.0258 0.3147 0.9960 0.9986 0.0262 0.3213 0.9877 0.9941 Average 0.0267 0.3174 0.0255 0.3199 0.9937 0.9981 0.0263 0.3182 0.9966 0.9986 0.0261 0.3236 0.984 7 0.9927 0.0000 0.0000 0.0004 0.0033 0.0021 0.0002 0.0007 0.0050 0.0009 0.0000 0.0001 0.0032 0.0042 0.0020
65 Table 2 6 Summary of results obtained from the accuracy assessment experiments. Values from roughness references Height RMS ( s ) [m] 0.009 0.020 0.030 0.038 0.000 0.009 0.016 0.027 0.037 Correlation l ength ( l ) [m] 0.006 0.006 0.006 0.006 0.177 0.180 0.317 0.327 # Samples 4 4 4 4 4 2 2 2 2 Values from meshboard derived data Profile c orrelation c oefficient 0.960 0.988 0.980 0.993 0.991 0.998 0.999 1.000 Profile R 0.921 0.977 0.960 0.986 0.980 0.995 0.997 0.999 Height RMS s [m] 0.010 0.020 0.030 0.039 0.001 0.010 0.016 0.026 0.037 s [m] 0.001 0.000 0.000 0.001 0.000 0.000 0.000 0.001 0.000 Correlation l ength l [m] 0.006 0.007 0.007 0.006 0.256 0.192 0.180 0.318 0.322 l ength l [m] 0.000 0.000 0.000 0.000 0.015 0.009 0. 003 0.005 0.009 Values from ground based scanning LiDAR data Profile c orrelation c oefficient 0.848 0.930 0.797 0.887 0.944 0.986 0.993 0.997 Profile R 0.564 0.767 0.439 0.643 0.877 0.972 0.985 0.993 Height RMS s [m] 0.008 0 .016 0.024 0.030 0.004 0.009 0.016 0.026 0.036 s [m] 0.000 0.000 0.002 0.001 0.001 0.000 0.000 0.000 0.001 Correlation l ength l [m] 0.009 0.008 0.009 0.008 0.007 0.166 0.181 0.324 0.333 l ength l [m] 0.0 01 0.000 0.001 0.001 0.001 0.003 0.001 0.003 0.004
66 Table 2 7 Roughness parameter values obtained from agricultural soils using meshboard and LiDAR. M eshboard L iDAR PSL R2 PSL R2 Dataset RMSh CL n E G P E cl G cl P cl RMSh CL n E G P E cl G cl P cl 20060308 0.008 0.155 1.43 0.501 0.598 0.585 0.870 0.884 0.981 0.007 0.113 1.52 0.532 0.627 0.616 0.843 0.929 0.988 20060310A 0.007 0.081 1.17 0.739 0.763 0.763 0.982 0.827 0.994 0.007 0.091 1.25 0.517 0.601 0.577 0.965 0.858 0.990 20060310B 0.004 0.053 0.94 0.555 0.517 0.555 0.985 0.695 0.984 0.004 0.030 1.13 0.480 0.453 0.484 0.991 0.865 0.998 2009_MB&Ltest01 0.012 0.136 0.99 0.502 0.465 0.503 0.997 0.597 0.996 0.012 0.190 0.77 0.454 0.466 0.357 0.934 0.212 0.990 2009_MB&Ltest02 0.00 8 0.034 1.47 0.523 0.535 0.542 0.951 0.966 1.000 0.009 0.055 1.30 0.465 0.442 0.462 0.966 0.905 1.000 2009_MB&Ltest03 0.023 0.156 1.17 0.497 0.580 0.542 0.980 0.767 0.998 0.018 0.157 1.01 0.459 0.533 0.462 0.998 0.601 0.998 2009_MB&Ltest04 0.023 0.115 1.57 0.388 0.494 0.481 0.857 0.953 0.991 0.025 0.117 1.51 0.358 0.481 0.458 0.875 0.934 0.992 2009_MB&Ltest05 0.001 0.190 1.14 0.349 0.480 0.399 0.864 0.564 0.870 0.003 0.238 0.43 0.367 0.402 0.21 0.003 2.24 0.885 2009_MB&Ltest06 0.002 0.170 1.01 0.451 0 .515 0.455 0.992 0.628 0.991 0.003 0.119 0.75 0.478 0.392 0.450 0.917 0.214 0.980 2009_MB&Ltest07 0.005 0.067 1.81 0.559 0.665 0.662 0.810 0.992 0.999 0.008 0.090 1.52 0.497 0.563 0.556 0.908 0.954 0.996 2009_MB&Ltest08 0.004 0.073 1.62 0.658 0.708 0.708 0.884 0.975 0.999 0.005 0.067 1.57 0.397 0.481 0.473 0.872 0.949 0.981 2009_MB&Ltest09 0.006 0.078 2.16 0.132 0.217 0.222 0.666 0.995 0.996 0.008 0.076 0.86 0.225 0.290 0.202 0.942 0.466 0.948 2009_MB&Ltest10 0.013 0.187 1.35 0.515 0.561 0.564 0.931 0.8 84 0.999 0.013 0.099 1.53 0.617 0.653 0.655 0.871 0.950 0.998 2009_MB&Ltest11 0.015 0.130 1.72 0.588 0.683 0.678 0.824 0.986 0.999 0.014 0.141 1.52 0.567 0.662 0.647 0.886 0.947 0.997 2009_MB&Ltest12 0.016 0.092 1.60 0.601 0.622 0.625 0.879 0.972 0.999 0.015 0.086 1.72 0.577 0.641 0.638 0.827 0.981 0.996 2009_MB&Ltest13 0.007 0.034 1.87 0.541 0.560 0.561 0.891 0.995 0.998 0.007 0.035 1.38 0.489 0.490 0.499 0.958 0.951 0.999 2009_MB&Ltest16 0.015 0.041 2.36 0.090 0.132 0.136 0.850 0.996 0.991 0.011 0.043 1.72 0.123 0.162 0.161 0.925 0.970 0.989 2009_MB&Ltest17 0.003 0.121 1.38 0.436 0.541 0.510 0.923 0.900 0.988 0.005 0.082 0.75 0.381 0.423 0.293 0.910 0.310 0.963 2009_MB&Ltest18 0.007 0.283 1.06 0.401 0.524 0.427 0.992 0.629 0.995 0.005 0.016 0.57 0.541 0.459 0.619 0.997 0.892 0.980 2009_MB&Ltest19 0.017 0.078 1.62 0.542 0.660 0.649 0.858 0.971 0.999 0.018 0.095 1.34 0.427 0.547 0.514 0.947 0.897 1.000 2009_MB&Ltest20 0.039 0.252 1.48 0.423 0.534 0.517 0.885 0.933 1.000 0.044 0.244 1.62 0.445 0.545 0. 538 0.825 0.967 1.000
67 Table 2 8 Comparison of roughness parameter values of agricultural soils obtained from LiDAR and meshboard data. Meshboard LiDAR Correlation coefficient of s (R) 0.97, (0.95) Correlation coeffi cient l (R) 0.50, (0.25) R exponential and real ACF 0.480.15 0.450.12 R Gaussian and real ACF 0.540.15 0.490.12 R exponential and real ACF to l 0.900.08 0.870.21 R Gaussian and real ACF to l 0.860.15 0.780.27 Exponent power law spectrum 1.470.38 1.230.40
68 CHAPTER 3 TESTING ROUGHNESS CH ARACTERIZATION ASSUM PTIONS USING A DATABASE OF 2D PROFI LES OBTAINED WITH GR OUNDBASED LIDAR Once it was demonstrated that measurements from the groundbased LiDAR yield precise and accurate values of roughness parameters the next step was to test some of the current assumptions of surface roughness characterization in two dimensions (2D) These assumptions include: the singlescale nature of roughness, possible correlation between roughness parameter values, and if the Gaussian or exponential models adequately descri be the ACFs agricultural soils. This is an intermediate step toward the characterization of threedimens ional (3D) roughness of agricultural soils which is covered in the next chapter To validate the current assumptions of 2D roughness characterization, a large database of agricultural surface transect data was collected. This chapter describes how this da ta base was collected and how it was used to test traditional characterization assumptions. Instrumentation and Datasets Mobile T errestrial L aser S canner (M TLS) D ata was collect ed utilizing the University of Florida mobile t errestrial laser scanning (M TLS ) system which is shown in Figure 31 The M TLS is a unique tool that enables UF geosensing engineering and mapping ( GEM ) researchers to acquire highdensity LiDAR point clouds from an elevated viewing point The core of the M TLS is an Optech ILRIS 3D, commerci al 2 axis time of flight groundbased laser scanner. The ILRIS is integrated to a mobile telescoping, rotating, and tilting platform that provides up to 6 degrees of freedom for performing scanning operations. The platform which can be elevated to a height of 10 meters above the surface i s mounted on the bed of a heavy duty 4x4 truck From an elevated position and employing the rotating
69 base, the M TLS can quickly generate 3D maps of agricultural surfaces of areas of hundreds of square meters with m illim eter level resolution and precision. The specifications and description of operation of the M TLS are covered extensively in . Data C ollection and P reprocessing The M TLS was used to obtain 3D point clouds of the soil surface from a height of 5 to 7 meters with a sample spacing that ranged from 5 mm to 3 cm Most datasets were collected at the University of Florida/Institute of Food and Agricultural Sciences (UF/IFAS) Plant Science Research and Education Unit (PSREU), located 20 miles south of G ainesville, at 2556 West Highway 318, Citra, Fl orida. S ince 2003, PSREU has been the location of the seasonal Microwave Water and Energy Balance Experiment (MicroWEX) MicroWEX is carried out by the Center for Remote Sensing of the Agricultu ral and Biologi cal Engineering D epartment at UF. Soil roughness measurements were collected as part of MicroWEX 4 in the spring of 2005 using the meshboard [ 35] and  C oncurrent meshboard and groundbased LiDAR observations were collected for Micr oWEX5 in the spring of 2006 , MicroWEX7 in the spring of 2007, and MicroWEX 8 in the summer of 2009. In addition to the roughness measurements collected as part of the MicroWEX seasonal experiments several other datasets were acquired at PSREU. And additional bare surf ace datasets were collected at St. Augustine and Hastings, Florida; along the shore of the Great Salt Lake in Utah, and Houston, Texas. A total of 112 groundbased LiDAR scans of natural surfaces were c ollected over a four year period. Raw LiDAR data from the M TLS are irregular point clouds with an arbitrary reference frame determined by the orientation of the ILRIS instrument at the moment of data collection. When mapping from a height of 5 to 7 meters the ILRIS instrument is
70 tilted downward. A s a result the raw point cloud will appear to be tilted upward as shown in Figure 32A. These tilted point clouds require rectifi cation. B ecause surface roughness metrics are computed with relative high variations and not absolute values, it is not crucial that the point cloud be leveled according to the local gravity vector at the moment of collection. A decimat ed sample of the point cloud first fits to a plane using a least squares regression following the model: cbY aX Z 3 1 w here X, Y, Z are the observed 3D coordinates of each point of the plane, and a, b, c are the parameters defining the plane. Those parameters also define the normal vector to the plane as [a, b 1]. From this normal vector the rotation angles around the X (xR) and Y ( yR ) axes which are required to l evel the plane, are determined using: b Rx 1tan 3 2 a Ry 1tan 3 3 If the surface presented spatial structure, such as that arising from agricultural rows or linear features, the point cloud was also rotated about the Z axis such that these features were aligned to the Y axis. The rotations w ere performed using a TerraScan 3D rotate and t ranslate transformation module. From the leveled and oriented point cloud, square regions of interest with dimensions that ranged from 3 to 6.75 meters to a side (9 to 45.65 m) were cropped. Finally, the coordinates of the cropped point cloud where modified by a simple arithmetic transformation, such that the lower left coordinat es corresponded to the origin (x = 0, y = 0), and the average of the Z components was zero. Figure 32 s hows renderings of a raw point cloud, along with the processed ( rotated and cropped) point cloud.
71 To facilitate the mathematical manipulation of the dat a the irregular point clouds were transformed into digital elevation m odels (DEM s) also known as grids. A DEM is essential ly a matrix composed of equally spaced elements called cells, organized in r ows and columns. Each cell has three attributes, two corr espond to a unique horizontal position (i.e. easting, northing or latitude, l ongitude) and the third to an elevation value representative of the entire cell. To create a DEM the first step is to establish a regular mes h using the horizontal position (x, y ) information. Cell h eight is obtained by interpolating height information from the irregular point cloud to the grid nodes For this study DEMs with a cell spacing of 1 cm were created by applying a triangulation with linear interpolation method. A tota l of 21 agricultural surfaces with different roughness conditions ranging from smooth to very rough were selected for the analyses presented in this and the following chapter. The details pertaining to the collection dates and locations, the surface condit ion s, and the square lateral dimension s in meters for the se 21 surfaces are presented in Table 3 1. Figure 33 shows four samples of the studied agricultural surfaces in photographs and, to each samples right its groundbased LiDAR derived DEM. 2D Profile Database To test the assumptions used in the characterization of surface roughness a large database of 2D profiles was created. This was done by extracting all the profiles along the row and column directions of the 3D DEMs obtained for the 21 study surf aces. A total of 20,072 profiles with lengths that ranged from 3 to 6.75 meters were obtained, totaling 100.4 km of height variation data. These profiles were detrended using the linear, quadratic and FFT filter models. Several analyses were performed wit h this
72 database of surface roughness information. These analyses include d the sensitivity of the roughness parameters to the detrending method, the existence of a correlation between RMSh and correlation length, and how well the theoretical exponential and Gaussian ACF models described the ACF of agricultural and natural surfaces. The results of these analyses are described next Testing Roughness Characterization Assumptions Value R anges for R oughness P arameters Because the different microwave emission and scattering models are valid over different roughness domains significant effort has been put into determining value ranges for the roughness parameters. In [38 ] a plan was described to establish global joint statistics for both RMSh and correlation lengt h for bare agricultural soils over a variety of tillage conditions based on a database of roughly 1,500 1meter profiles obtained in 6 European countries. The res earchers found that the RMSh can vary between 6 mm for seedbed soils to 2.7 cm for plow ed surf aces, with a mean value of 1.6 1.1 cm [3 8 ]. A survey of similar studies revealed that the RMSh for agricultural soils varies in the range from 2.5 mm for sown fields to 4.1 cm for plowed fields  With respect to the correlation length, it has been ext remely difficult to obtain consistent range of correlation lengths. In [38 ] it was found that the typical value for correlation lengths was 4.8 1.8 cm. A compilation from recent studies yields values in a range between 2 and 20 cm . The minimum, maxi mum, mean and standard deviation of the roughness parameters obtained from the profile dat abase for the three different detrending methods are summarized in Table 3 2. It can be seen that the RMSh values obtained from the first order detrending are consis tent with the reported values. A significant
73 difference arises with respect to the correlation length, where the maximum correlation length obtained was 81 cm, roughly four times the maximum reported in previous studies This can be attributed to the fact that longer profiles are available in this database than thos e previously used. A s shown in [ 2 ] longer profiles yield larger and more accurate values of the correlation length. It can als o be demonstrated that when applying E quation 16 the maximum observ able correlation length is close to 0.62 of the measured profile length. Consequently, short profiles will limit the value that can be determined for the correlation length. Sensitivity of R oughness P arameters to the D etrending P rocedure To assess the sens itivity of roughness parameters to the detrending procedure, the coefficient of determination (R) for the fit of the profile data to the modeled trend was computed using Equation 113. All the profiles were detrended according to the 3 ear lier models desc ribed and the R of fit was determined for each then the difference between the R of the first and second order trends as well as the difference between the first and FFT filter trend, were computed. In the same fashion, the difference s between the values of the associated roughness parameters were determined. Figure 3 4 show s the results from this analysis I t was found that as the R of the trend increases the values of RMSh and correlation length decrease If the first and second order detrending met hods are considered, the correlation coefficient between the differences in R and the differences in RMSh was 0.809, and the correlation coefficient between the differences in R with respect to the differences in correlation length was 0.786. Consideri ng the linear and the FFT filter detrending methods the correlation coefficient between the differences in R and the differences in RMSh was 0.786, and the correlation coefficient between the differences in R with respect to the
74 differences in correlat ion length 0.57. These results demonstrate the high sensitivity of the roughness parameters values to the detrending method, the multi scale nature of agricultural and natural soils and the need t o establish the proper detrending technique to characteriz e roughness at a given scale. Correlation b etween RMSh and C orrelation L engths Several studies have attempted to derive models that relate the correlation length to the RMS  or to perform empirical or semi empirical calibrations of the IEM by adjusting measured correlation lengths [ 27] and [ 28]. This is done to reduce the parameterization of surface roughness for its use in microwave emission or scattering models. It is also done to circumvent the problem of obtaining precise and repeatable values of c orrelation lengths. A dispersion plot shown in Figure 35 was generated from the extracted profiles database and show s the correlation between the RMSh and correlation lengths obtained f rom the different detrending methods. In contrast to the previously cited publications w h ere good correlation between RMSh and cor relation lengths were found, this result indicated a large spread between values of both roughness parameters However, it can also be seen that the spread is reduced when second order and FFT filter detrending is applied. The correlation coefficient between RMSh and correlation lengths was found to be 0.1 for first order detrending, 0.132 for second order detrending, and 0.355 for FFT filtering detrending. An additional observation only considering the spread for the FFT filter detrended data i s that there is a probable correlation between the RMSh and the maximum attainable correlation length. This is a negative correlation, i.e. as the RMSh increases the maximum observable correlation length decreases.
75 Exponential or Gaussian ACF M odel s Because t he results of soil moisture inversion using the Integral Equation Model vary strongly between the Gaussian and Exponential ACF models , the shape of the autocorrelation function is an essential desc riptor of the soil surface roughness. To explore this issue, the least squares method described with Equation 25 was used to determine the exponent n of the generalized power law s pectrum that best fits the ACF obtained for the random height component The R was used as a metric to determine how well the observed ACF matched the modeled exponential, Gaussian, and generalized power law spectrum ACF The R was computed twice, once considering the entire lag range, then again only considering the range fr om the zero lag to the correlation length. The results from this analysis are summarized in Table 3 3 and in Figures 3 6 and 3 7. The first conclusion that can be drawn from these results is that more profiles follow exponential than Gaussian ACFs, adding validity to the assumption that soil surfaces can be better explained with exponential ACFs. However, the vast majority of profiles follow ACFs somewhere between the theoretical Gaussian (n = 2) and exponential (n = 1) models. Also, the detrending method affects the shape of the autocorrelation function, and it can be seen from the histograms in Figure 36 that for higher order detrending methods there is a relative ly higher number of profiles with ACFs with exponents closer to the Gaussian model (n = 2). Fig ure 37 shows histograms with the R values for the fit between the observed ACF and the theoretical exponential or Gaussian ACF forms considering the entire lag range. These histograms show that the theoretical ACF does not represent well the real ACF for the entire lag
76 range. This is also evidence of the multi scale nature of agricultural and natural surfaces. Chapter Conclusions The main conclusion that can be draw from the analyses of 2D roughness presented i n this chapter is that natural and agricul tural surfaces have roughness components at different scales. This is in contradiction to the main assumption of roughness characterization, w h ere roughness is considered a stationary singlescale process. However, with high resolution, large extent data obtained from groundbased LiDAR and proper detrending techniques making it possible to characterize roughness at different scales. The question that still remains to be answered by the microwave remote sensing community is: What are the relevant scales that need to be considered? The theoretical Gaussian and exponential ACF models are just two specific cases of an entire universe of ACF that can be characteristic of agricultural and natural surfaces. Also, t hey do not explain the structural correlation of random components observed at lags longer than the correlation length.
77 Figure 31. The M TLS system performing agricultural surface scanning operations. Figure 32. Point cloud renderings of a g roundbased LiDAR dataset at different processing steps A) raw and B) rectified.
78 Height Variation [m] Across Row [m]Along Row [m] 0 1 2 3 4 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 -0.06 -0.04 -0.02 0 0.02 0.04 0.06 Height Variation [m] Across Row [m]Along Row [m] 0 1 2 3 4 5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 -0.06 -0.04 -0.02 0 0.02 0.04 0.06 Height Variation [m] Across Row [m]Along Row [m] 0 1 2 3 4 5 0 1 2 3 4 5 -0.06 -0.04 -0.02 0 0.02 0.04 0.06 Height Variation [m] Across Row [m]Along Row [m] 0 1 2 3 4 5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 -0.06 -0.04 -0.02 0 0.02 0.04 0.06 Figure 3 3 Samples of studied soil surface and their groundbased LiDAR derived 3D DEMs A B 0 0.2 0.4 0.6 0.8 1 -0.035 -0.03 -0.025 -0.02 -0.015 -0.01 -0.005 0 delta R2delta RMSh (m) 0 0.2 0.4 0.6 0.8 1 -0.8 -0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0 delta R2delta cl (m) 2-1 F-1 2-1 F-1 Fig ure 3 4 Effects of detrending on the derived roughness parameters. A) RMSh B) correlation lengths.
79 0 0.01 0.02 0.03 0.04 0.05 0.06 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 S ad Coeato egt RMSh (m)Correlation Length (m) First Order Second Order FFT Figure 35 Dispersion plots of the random height component derived correlation lengths and RMSh for the different detrending methods. A B C 0.5 1 1.5 2 2.5 0 500 1000 1500 2000 2500 3000 1st Order dt PLS Exponent 0.5 1 1.5 2 2.5 0 500 1000 1500 2000 2500 3000 2nd Order dt PLS Exponent 0.5 1 1.5 2 2.5 0 500 1000 1500 2000 2500 3000 FFT fc1/m PLS Exponent Fig ure 36 Histograms of the exponent values for the Power Law Spectrum A CF model obtained from the soil profiles. A) First order detrending. B) Second order detrending. C) Third order detrending.
80 A B C 0 0.5 1 0 1000 2000 3000 4000 p, st Ode dt 0 0.5 1 0 1000 2000 3000 4000 p, d Ode dt 0 0.5 1 0 1000 2000 3000 4000 p, dt D E F 0 0.5 1 0 1000 2000 3000 4000 5000 0 0.5 1 0 1000 2000 3000 4000 0 0.5 1 0 1000 2000 3000 4000 5000 Fig ure 37 Histograms of the R2 values of the fit of the observed ACF model with respect to the theoretical models A) Exponential, first order detrending. B) Exponential, second order detrending. C) Exponential, FFT detrending. D) Gaussian, first order detrending. E) Gaussian, second order detrending. F) Gaussian, FFT detrending.
81 Table 3 1. 3D digital elevation m odels derived from LiDAR for the study. # Lateral d im ensions Date Location Surface condition 1 6.00 18 Mar 08 St. Augustine, FL Natural smooth 2 4.37 8 Mar 06 PSREU Citra, FL Harrowed smooth 3 4.50 21 Jun 10 Houston, TX Natural smooth 4 5.00 10 Mar 06 PSREU Citra, FL Rolled 5 5.00 10 Mar 06 PSREU Citra, FL Rolled 6 3.50 11 Nov 07 Salt Lake, UT Natural clay soil 7 4.25 18 Nov 09 PSREU Citra, FL Harrowed 8 3.02 18 Nov 09 PSREU Citra, FL Harrowed 9 5.38 18 Nov 09 PSREU Citra, FL Natural 10 4.50 3 Sep 10 PSREU Citra, FL Plo w ed 11 6.00 2 9 Apr 07 PSREU Citra, FL Plo w ed 12 3.23 18 Nov 09 PSREU Citra, FL Gravel bed 13 5.00 3 Sep 10 PSREU Citra, FL Harrowed 14 5.72 18 Nov 09 PSREU Citra, FL Harrowed 15 5.24 18 Nov 09 PSREU Citra, FL Harrowed 16 5.00 3 Sep 10 PSREU Citra, FL Plo w ed 17 6. 00 29 Apr 07 PSREU Citra, FL Plo w ed 18 3.10 24 Jun 09 PSREU Citra, FL Furrowed sandy soil 19 5.01 8 Dec 09 PSREU Citra, FL Furrowed sandy soil 20 6.75 8 Dec 09 PSREU Citra, FL Furrowed sandy soil 21 4.00 28 Oct 06 Hasting, FL Ridged sandy soil Table 3 2. Roughness parameter values in cm, derived from 20,072 profiles of agricultural soils. 1st order 2nd o rder FFT filter RMSh m in imum / m ax imum 0.3 4.3 0.2 3.6 0.1 2.6 RMSh m ean s t andar d d eviation 1.4 0.7 1.2 0.7 0.8 0.4 C orrelation l ength min/m ax 0.9 81.1 0.8 56.2 0.6 14.2 C orrelation length, m ean s t andar d s eviation 24.6 15.8 17.3 9.6 6.7 2.3 Table 33. Mean parameter and fit metric values for different ACF models. Mean 1st order 2nd o rder FFT filter PLS exponent 1. 2463 1.2913 1.4258 R 2 Exponential 0.4427 0.4076 0.2853 R 2 Gaussian 0.4709 0.4398 0.3318 R 2 PLS 0.4461 0.4175 0.3161 R 2 e xponential to the correlation length 0.8758 0.8813 0.8943 R 2 Gaussian to the correlation length 0.6117 0.6818 0.8356 R 2 PLS to the correlation length 0.9852 0.9865 0.9935
82 CHAPTER 4 DERIVING THREE DIMENSIONAL ( 3D ) ROUGHNESS METRICS FROM GROUNDBASED LIDAR DIGITAL ELEVATION MODELS (DEMS) As mentioned in C hapter 1 of this dissertation, some of the current challenges in the character ization of surface roughness arise from the approach of using two dimensional ( 2D ) profiles to try to deduce 3D characteristics This approach is not adequate because profiles do not contain enough information to properly separate the random component from the surface trend, and it is unlikely that a single profile will record the extremes of a surface These two aspects affect the accuracy of roughness parameters A s has been shown in the previous chapters, the values of the parameters vary significantly depending on how the random component was extracted, and not recording the extremes of the surface will cause an underestimation of the parameters [ 7 ]. Finally profiling presents very limited information on the directional characteristics of the surface roughness. The characterization of 3D roughness of agricultural and natural surfaces for application in microwave modeling is virtually unexplored. This is due in part because until recently t echnology that a llows for the recording of finescale 3 D height v ariation of large areas has not been readily available. Two alternatives are stereoscopic imagers as described in [12 ] and [16 ] or LiDAR scanners [15 ] and [ 22]. Stereoscopic imagers have the advantage of producing 3D digital elevation models with high res olution and high accuracy, but of a very small area (~1 m). LiDAR scanners have comparable resolution and are able to digitize larger areas (~10s m), but with somewhat lower accuracy. To date, there has been very little research on the characterization of 3D soil roughness using these digitization instruments. Most research , ,  limit s the 3D information obtained to extract 2D profiles and derive roughness parameters based
83 on the traditional transect methodology. Little has been determined about how the results obtained with the traditional 2D techniques compare to the ones obtained from 3D characterizations. In the previous chapter the assumptions used for the characterization of surface roughness from 2D profiles were tested. Also, v alue ranges for the parameters were obtained using the traditional 2D formulation. This chapter extends the traditional 2D characterization by explor ing alternatives to characterize 3D roughness from groundbased LiDAR derived DEMs. It begins with a description of the methodology used to detrended 3D DEMs from which roughness parameters will be computed. A comparison between the roughness parameter values obtained from the traditional 2D methodology versus the 3D formulation is also performed. This is followed by an analysis of the impact in the modeled microwave emission and scattering due to difference in the 2D versus 3D parameterization of roughness. Finally, the advantage that 3D brings to the characterization of roughness is illustrated by a case of quasiper iodic agricultural surfaces. Datasets and Preprocessing The 3D DEMs from the 21 surfaces that were used in the previous chapter to build the database of 2D profiles are used in this chapter to perform the 3D roughness characterization. To derive roughness parameters from the 3D DEMs two ma in approaches were followed. T he first approach consists of extracting all the profiles along the rows and columns of the DEM, detrending them individually, computing the roughness parameters then averaging all the obtai ned values. The second approach consists of extending the 2D profile formulation to a 3D surface formulation and applying these to a detrended 3D DEM. Here is where the first difference between 2D
84 and 3D roughness characterization arise s, because the trend obtained from individual transects m ay not represent the overall trend of the surface Optimal detrending of a 3D DEM can be done by FFT filtering, high order polynomials and moving average filters as described in [ 2 ] For the computation of 3D roughnes s parameters the 21 DEMs were detrended using 3D linear and quadratic models and by applying a 3D FFTbased filter that removes the spectral components with spatial wavelengths longer than a meter. The first and second order 3D models applied for the pur pose of detrending the DEMs are: c bY aX Z (4 1) f eY dX cXY bY aX Z 2 2, (4 2) w here X, Y and Z are the coordinates of the observed points and a, b, c, d, e, and f are the coefficients of the models These coefficients are determined via a linear least squares process. The 2D Fast Fourier Transform applied to a matrix of M rows and N columns is: M x N y N vy M ux je y x f v u F1 1 2, (4 3) The result ing FFT is also a matrix of M rows and N columns but with complex elements. The detrending is achieved by setting to zero the elements of the FFT that correspond to a spatial wavelength larger than 1 meter. The modified FFT (F(u,v)) is then transformed back to the space domain using the inverse FFT: M x N y N vy M ux je v u F y x f1 1 2, (4 4) w here f(x,y) is the detrended DEM, a matrix of the same dimensions M x N with real value elements. Fig ure 41 shows an original DEM, its trend acco rding to the first and
85 second order models the FFTbased trend, and their respective detrended random component DEM s. Averaging Paramet er V alues Obtained From 2D Profiles Averaging RMSh values can be done by considering all the extracted profiles from the grids rows and columns or separating the profiles extracted from rows from those extracted columns and calculating the average for each group. This latter app roach has the advantage that it yields information on the directional properties of the roughness, i.e. the isotropy of the surface. In [ 39] a ratio between the averaged RMSh along the columns and the rows is proposed as a measure of the directionally of the surface roughness is as follows : row col RMShRMSh mean RMSh mean I (4 -5 ) I f RMShI is equal to 1, then the surface can be considered an absolute isotropic scatterer or emitter. With regard to obtaining average values of correlation lengths there are two ways described in literature. The first consists of averaging values of the correlation lengths for individual profiles; and the second consists of averaging the measured autocorrelation functions from which the correlation length is then determined. F or the 21 surfaces in this study both ways for determining correlation lengths were tested. The correlation coefficient and the RMSE between the obtained values of correlation lengths are summarized in Table 4 1 Results in dicate that despite high correlation coefficients between values, the root mean square difference between the values is close to 3 cm for first order detrending, and between 1 and 2 cm for second order detrending. These are significant differences between the values obtained from the two methods. F or the
86 FFT filter based detrending, h owever the correlation length va lues agree to better than 5 mm This indicates that for determining singlescale correlation lengths either method is equally valid. For the comparison s presented in the next sec tion s, the method of averaging the correlation length values of individual profiles is used. Extending 2D Formulations to 3D Equation 15 used to com pute RMSh, is applicable to data extracted fr o m both 2D profiles and 3D surface models. I f this formulation is applied to the DEM it provides an RMSh value truly representative of the entire surface. However, there is the disadvantage that it does not provide information of the roughness directional properties. An alternat ive to derive directional roughness from a detrended DEM consist of extracting 2D radial profiles starting at the center and extending to the edge of the DEM at a given angular separation ( ang ) from each other. For a square DEM with M r ows, N = M columns and a cell spacing of c the horizontal grid coordinates for each point j in the radial profile r are given by: c j ang r Cos M c j ang r Sin M Col Row 2 2 # # (4 6) For r = 1, 2,60/ ang and j = 0, 1,M/2 1. The height h( r j ) for each point is then obtained by using nearest neighbor or bilinear interpolation from the DEM cell values. The RMSh for each radial profile is then computed using Equation 15 Fig ure 4 2 A shows a first order detrended DEM and Figure 4 2 B shows its directional RMSh as a function of the horizontal angle. This horizontal angle is measured with the vertex at th e center of the grid, and with respect to a horizontal axis. This methodology provides a description of the roughness anisotropy of the surface. A quantitative measurement of this anisotropy is the eccentricity of the RMSh given by:
87 2 2max min 1 RMSh RMSh eRMSh (4 7) An eccentricity value of zero indicates a perfect isotropic surface. To determine the correlation length from a full surface model with M rows and N columns the normalized autocorrelation function for displacements in two directions hx and hy is computed by: N a M b b a j N a k M b k b j a b a y xz z z y k x j h h1 1 2 1 1 ,, (4 -8) This is a 3D ACF with two dimensions being the lags in the x and y directions and the third dimension being the normalized autocorrelation. Figure 43 A shows the 3D autocorrelation function from the detrended DEM shown in Fig ure 4 2 A. In the profile formulation, the correlation length is uniquely defined as the lag for which the ACF value is 1/e. For the 3D formulation, the correlation length is not an unique value. Instead it is a set of values defined as the distance from the (0,0) lag point to each point (hx,hy) that defines the contour that has a normalized correlation value of 1/e. Figure 43 B shows the 1/e co ntour extracted from the 3D ACF in part A. From the 1/e contour, t he correlation length at a dir ection i is: 2 2i iy x ih h (4 9) where the angle i is given by: i ix y ih h1tan (4 10)
88 Figure 44 illustrates the correlation length derived from the 3D formulation as a function of the angl e measured from the x lags axis. An important lesson learned with respect to the computation of correlation lengths using the 3D formulation, has to do with those surfaces that presented linear features as arising from the agricult ural row structures. Figure 45 A shows the detrended DEM of a surface exhibiting such linear characteristics. In Figure 4 5 B a histogram with the distributions of the correlation length values obtained from the traditional 2D formulation is presented. The distribution contains correlation lengths within the values of 4 and 24 cm. Figure 4 5 C shows the 1/e contour extracted from the 3D autocorrelation function and Figure 4 5 D presents the 3D correlation length as a function of the horizontal angle. From Figure 4 5 C and Figure 4 5 D it can be seen that the normalized autocorrelation is extremely elongated in the Y direction, and the correlation length reaches a value of almost 1.8 meters, more than nine times the average value obtained from the 2D formul ation. This elongation of the correlation lengths is due to the row structure as the height variations will be highly correlated in the row direction compared to the across row direction. To eliminate this elongation effect and to obtain coherent values of correlation lengths it is necessary to apply an additional detrending procedure that removes the row structure. To do this a simple M atlab routine was coded. For each column on the DEM, the routine determines the difference between the average height value of the column and the average height value of the entire DEM. T he script then removes this vertical difference (bias) on a columnby column basis. Figure 46 A shows the resultant DEM after applying the derow procedure to the DEM of Figure 45 A. Figure 46 C and Figure 46 D show the 1/e contour for the 3D ACF of the new DEM
89 and the correlation length as a function of the horizontal angle. It can be seen that the values of correlation length derived from the 3D formulation are within the same order of magnitude as the ones derived from the 2D formulation. Comparison b etween 2D and 3D R oughness P arameter V alues Several analyses were performed to compare roughness parameter values of a surface considering profiles extracted from the DEMS and the DEMs as a whole. The procedure proposed in [15 ], which state s that to obtain a representative value of RMSh that accounts for the heterogeneity of the surface a minimum of 20 3m transects should be acquired, was used as the baseline of the conventional profiling approach. The following comparisons were performed: Oriented roughness obtained by averaging values computed from all profiles extracted along the row or column directions with the values obtained from 20 random extracted profiles 3 m in length for the r ow or column directions. Full surface roughness obtained by averaging values computed from all profiles extracted from the DEM with the values obtained from 20 random extracted profiles 3 m in length (10 extracted from the rows and 10 from the column dire ction). Full surface roughness obtained by applying the 3D formulation from the DEMs with values obtained by averaging values computed from all profiles extracted, and from 20 random profiles 3m in length (10 extracted from the rows and 10 from the colum ns direction). Oriented roughness parameters were obtained by averaging values computed from profiles extracted along the rows and columns and detrending them with first order second order and FFT methods These values are summarized in Table 4 2 The co rrelation coefficient and root mean square error (RMSE) comparing parameter values from a sample of 20 profiles with respect to all the profiles along the row or column directions are also presented at the bottom of Table 42 Results indicate a relative ly good agreement between RMSh values obtained from the sample and all extracted
90 profiles, with correlation coefficients higher than 0.89. The RMS E in the estimation of RMSh using this sampling method was determined to be 3 mm for first order detrending, 2 m m for the second order detrending and 1 mm for the FFTbased detrending. In terms of correlation lengths the correlation coefficients ranged from 0 to 0.89. The RMSE was 10 11.5 cm for first order detrending, 5.5 6.6 cm for seco nd order detrending, and 1 2.4 cm for the FFT based detrending. This fairly large spread is due to the high sensitivity of the correlation length to the detrending procedure and the inherent high statistical variation observed for correlation lengths. It can be concluded that the sa mpling method proposed in [15 ] works well to produce a representative estimate of RMSh values but not so well for correlation lengths. Values of 3D roughness parameters obtained by averaging results from 2D profiles and from the 3D formulation are summari zed in Table 43, and results from their comparisons are presented in Table 44. When comparing averaged roughness parameter values derived from all the profiles with thos e obtained from 20 random profiles, results are similar to th os e obtained from the or iented roughness comparison described before. The correlation coefficients of RMSh values were 0.94, 0.97, and 0.98 with a RMSE of 3, 2 and 1 mm for the first order, second order and FFT based detrending method, respectively The RMSE in the estimation of correlation lengths from the sample of 20 transects is 10.2, 6.0, and 1.4 cm for the different detrending methods. These results demonstrate the importance of applying the proper detrending method to obtain consistent values of correlation lengths. Sign ificant differences were found in the results of the third comparison, where parameter values derived from the 3D formulation and those obtained from averaging
91 values from 2D profiles were compared. Table 4 4 presents correlation coefficients, RMSE and per centage difference s of the roughness parameters obtained from 2D and 3D formulations. Figure 47 shows dispersion plots for the RMSh derived from the 3D model and that derived by averaging the values obtained from all the profiles and from a sample of 20 r andom profiles. Figure 48 shows dispersion plots of the derived correlation lengths from 3D and 2D formulations. It is evident that RMSh values derived from profiles are significantly lower than the ones obtained from the 3D formulation. It was determined that if all the profiles extracted from the DEM were considered the RMSh is underestimated on average by 27.5 % independent of the detrending method used. The correlation length is also underestimated by between 27 and 35%. The most important comparison i s the one that considers the difference between digitizing the full surface and taking a sample of a few transects as is the current practice for traditional field methods. From this comparison, it was found a considerable underestimation of the surface r oughness parameters Deriving RMSh from profiles underestimates the surface RMSh by : 15 to 63% with a mean of 37% for the first order detrending; 25 to 47% with a mean of 35% for second order detrending; and 10 to 36% with a mean of 25% for the FFTbased detrending. Differences were also found when comparing the mean correlation length obtained from the 3D formulation with respect to the averaged values obtained fr om profiles, especially for those obtained from the limited sample of 20 profiles. On average the correlation length can be underestimated by 64% if obtained from all the possible transects extracted from the 3D model and by 50% if obtained from a sample of 20 profiles detrended using first and second order detrending. If the sample profiles are detrended using the FFTbased filter the
92 underestimation of the correlation lengths is on average 33%. Part of this difference can be attributed to the fact that the first and second order detrending models due not completely remove the trend and thus higher values of correlation lengths are obtained. Although it is larger than the underestimation of RMSh, t he underestimation of the correlation lengths has less impact than the error related to RMSh. This is because sensitivit y studies of the integral equation model (IEM) have shown that soil moisture retrieval is more sensitive to error in the parameterization of RMSh than the correlation length by a factor of 10 [25 ]. The impact of the underestimation of the roughness parameters on microwave observations modeled with IEM is discussed in the next section. Impact of Roughness Underestimation on Microwave O bservables The Integral Equation Model (IEM) [ 40] and the Advanced Integral Equation Model (A IEM) [41 ] provide estimates of the backscattering coefficient and emissivity from a randomly rough dielectric surface. These models take as inputs the dielectric constant, RMSh, correlation length, and correlation function of the surface; and microwave sensor parameters such as frequency, polarization and angle of incidence. For the mapping of soil moisture (dielectric constant) from microwave instrument signatures these models are inverted using a va riety of methods which include look up t ables, neural networks, least squares and iteration [25 ]. Performing a full sens itivity study of the inversion of IEM models to errors in roughness parameterization is extremely complicated because of the great number of combinations of input and output parameters. While t his is beyond the scope of this research the impact of roughness underestimation on scattering coefficient and brightness temperatures was modeled for a few scenarios. These scenarios were the combinations of three roughness states (3D RMSh = 0.5, 1.5,
93 2.5 cm), two levels of volumetr ic soil moisture (7% and 20%), two frequencies (1.2 and 5.3 GHz), with four microwave observables (TbH, Tb0 0 HH) considering surfaces with exponential ACFs and an incidence angle of 40 degrees. Table 4 5 summarize s the difference s in the microwave observables values due to errors in the parameterization of roughness from 2D profiles with respect to 3D DEMs. The results indicate that the sensor configurations that are more sensitive to errors in roughness parameterization are the radiometers that observe the surface in horizontal polarization irrespective of their band of operation, and radars that operate in L band that transmit and receive in horizontal polarization. Differences in brightness temperature in the horizontal polarization of up to 4.2 k elvin can be observed due to roughness underestimation under conditions of low soil moisture (7% by volume) and of up to 7 kelvin for high soil moisture levels (20%). In terms of scattering coefficients, the largest difference was observed for the HH polarization in the L band of 4.77 dB for low soil moisture and 4.58 dB for high soil moisture. Differenc es in the modeled brightness temperature in the vertical polarization also becomes significant at high roughness and high soil moisture conditions. Advanced Detrending and Scale Separation Methods for 3D DEMs In [ 42], microwave scat tering and emission models for three types of surfaces are covered: purely random, purely periodic and random superimposed over a periodic pattern. However, it is possible to encounter surfaces that exhibit random components superimposed over a quasiperiodic or nonstationary s urface. The key problem is how to separate the random and reference components of such a surface When traditional profiling methods are used, the approach is to only take transects along the row direction and consider this the random roughness, and the ro ughness component
94 across the rows is referred as the oriented roughness  For deriving roughness parameters from 3D models of quasiperiodic surfaces, three approaches were tried: E xtracting all the alongrow profiles from the DEM, and processing each individually then averaging the results G enerating a new DEM by extracting every profile along the column direction of the DEM and removing its vertical bias with respect to the entire surface mean height Generating a new DEM using a five stage wavelet bank to decompose the DEM in vertical, horizontal and diagonal detail at different scales then synthesizing a new DEM by remov ing the vertical detail at scale s comparable to the row structure. Wavel ets are the foundation of multi resolution analysis which is an extension of Fourier analysis. Fourier analysis only provides spectral information from temporal or spatial information while wavelet analysis provides spectral and temporal or spectral and spati al information simultaneously [5 4 ]. Wavelets decompose a signal into approximation and detail components at multiple scales. Figure 49 illustrates a two stage 2D wavelet decomposition bank use d to decompose a 3D DEM A t the top of the figure is the original DEM. This original dataset is considered to be at scale 0. The data at s cale 0 is then decomposed into four components: the level 1 approximation and the horizontal, vertical and diagonal details. Each component i s the size of the original scale. To obtain the components at the second scale, the appro ximation at scale 1 is again decomposed into the four elements. This decomposition process can be repeated until the components of the last scale have only 1 pixel. After decomposition, a new DEM at the original scale can be synthesized by inverting the pr ocess starting from the large scales and modifying or removing particular components at specific scales. To extract the 3D random component of quasiperiodic surfaces, the random component
95 DEM was synthesized by removing the vertical detail at scales comparable to the row structure Figure 410 shows an example of detrending a quasiperiodic surface using the vertical bias removal and the wavelet method. Values of roughness parameters obtained from the random component that was extracted using the different quasiperiodic detrending met hods are presented in Table 46 It can be seen that the result ing RMSh are comparable to each other independent of the detrending method, not so for the correlation lengths where the traditional method and the vertical bias r emoval closely agree, but the wavelet detrending method produce s lower correlation values. Chapter Conclusions The most important conclusion derived from the analyses presented in this chapter is that characterizing roughness from 3D datasets provides a much better description that th os e obtained with the traditional 2D profiling methods. The multi scale nature of agricultur al soils and natural surfaces becomes much more evident from 3D data, thus it is important to apply advanced detrending methods to char acteriz e roughness at a particular scale. 3D models have the resolution and extent necessary to properly separate the height random component from the reference surface using such advance d detrending techniques as FFT filters or wavel et decomposition. 3D m odels also allow for the characterization of roughness heterogeneit y and directional properties. This study demonstrated that roughness parameters obtained from a limited number of transects underestimates the roughness of the full surface On a verage, RMSh is underestimated by 25% and correlation lengths by 30%. This is due mainly to
96 2 reasons: it is unlikely that a single profile will record the extremes of the surface and trends derived from a single profile might not be representative of the general su rface trend. Simulations using IEM indicate that these errors in parameterization can cause errors in modeled scattering coefficients of up to 4.77 dB for the HH polarization, and in the modeled brightness temperatures of up to 7 kelvin for the H polarizat ion.
97 A Across Row [m]Along Row [m] 0 1 2 3 4 0 1 2 3 4 B C D st Ode ed, 055 Across Row [m]Along Row [m] 0 1 2 3 4 0 1 2 3 4 d Ode ed, 0808 Across Row [m]Along Row [m] 0 1 2 3 4 0 1 2 3 4 E F G Across Row [m]Along Row [m] 0 1 2 3 4 0 1 2 3 4 Across Row [m]Along Row [m] 0 1 2 3 4 0 1 2 3 4 Fig ure 41 Detrending a 3D DEM with linear quadratic model s and a FFT low pass filter A) Original DEM. B) Linear trend, R2 = 0.2755. C) Quadratic trend, R2 = 0.8108. D) FFT trend, R2 = 0.8685. E) R andom height component after linear detrending, RMSh = 1.8 cm. F) R andom height component after quadrati c detrending, RMSh = 0.08 cm. G) Random h eight component after FFT based filtering, RMSh = 0.07 cm.
98 A ) B ) g  Across Row [m]Along Row [m] 0 0.5 1 1.5 2 2.5 3 0 0.5 1 1.5 2 2.5 3 -0.05 -0.04 -0.03 -0.02 -0.01 0 0.01 0.02 0.03 0.04 0.05 0.01 0.02 0.03 0.04 0.05 30 210 60 240 90 270 120 300 150 330 180 0 RMSh[m] Fig ure 42 Directional roughness. A) Fir st order detrended DEM. B) RMSh as a function of the angle from a horizontal axis with origin at the center of the grid. A) B) X Lags [m]Y Lags [m] -3 -2 -1 0 1 2 3 -3 -2 -1 0 1 2 3 -0.5 0 0.5 1 -3 -2 -1 0 1 2 3 -3 -2 -1 0 1 2 3 X Lags [m]Y Lags [m] Fig ure 43 Deriving 3D correlation length A) 3D ACF from DEM in Fig ure 42 A B) The extracted 1/e contour from the 3D autocorrelation function.
99 -200 -150 -100 -50 0 50 100 150 200 0.4 0.5 0.6 0.7 0.8 0.9 1 Angle from the X axis [deg]Correlation length [m] Figure 44. 3D correlation length as a function of the horizontal angle.
100 A) B) X [Meters]Y [Meters] 1 2 3 4 0.5 1 1.5 2 2.5 3 3.5 4 -0.03 -0.02 -0.01 0 0.01 0.02 0.03 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 0.22 0.24 0 10 20 30 40 50 60 70 Correlation length [m] C) D) -4 -3 -2 -1 0 1 2 3 4 -4 -3 -2 -1 0 1 2 3 4 X Lags [m]Y Lags [m] -200 -150 -100 -50 0 50 100 150 200 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 Angle from the X axis [deg]Correlation length [m] Fig ure 45 3D Correlation length of a DEM with linear features. A) Detrended DEM. B) Distribution of correlation lengths obtained from the 2D formulation. C) The extracted 1/e contour from the 3D ACF. D) 3D correlation length as a function of the horizontal angle.
101 A) B) g X [Meters]Y [Meters] 1 2 3 4 0.5 1 1.5 2 2.5 3 3.5 4 -0.03 -0.02 -0.01 0 0.01 0.02 0.03 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 0.22 0.24 0 10 20 30 40 50 60 70 Correlation length [m] C) D) -4 -3 -2 -1 0 1 2 3 4 -4 -3 -2 -1 0 1 2 3 4 X Lags [m]Y Lags [m] -200 -150 -100 -50 0 50 100 150 200 0.13 0.14 0.15 0.16 0.17 0.18 0.19 Angle from the X axis [deg]Correlation length [m] Fig ure 46 3D Correlation length of a DEM after removing the linear features. A) Detre nded and derowed DEM. B) Distribution of correlation lengths obtained from the 2D formulation. C) The extracted 1/e contour from the 3D ACF D) 3D correlation length as a function of the horizontal angle.
102 A) B) C) 0 0.01 0.02 0.03 0.04 0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 3D RMSh (m)Average of 2D RMSh (m)st ode dt All Random 0 0.01 0.02 0.03 0.04 0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 3D RMSh (m)Average of 2D RMSh (m)d ode dt All Random 0 0.005 0.01 0.015 0.02 0.025 0 0.005 0.01 0.015 0.02 0.025 3D RMSh (m)Average of 2D RMSh (m) te dt All Random Fig ure 47 Dispersion plot s of the 3D RMSh and the averaged values of RMSh derived from profiles A) F irst order detrending, B) second order detrending, and C) FFT filter detrending. A) B) C) 0 0.5 1 1.5 0 0.5 1 1.5 Mean 3D cl (m)Average of 2D cl values (m) All Random 0 0.2 0.4 0.6 0.8 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Mean 3D cl (m)Average of 2D cl values (m) All Random 0 0.05 0.1 0.15 0.2 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 Mean 3D cl (m)Average of 2D cl values (m) All Random Fig ure 48 Dispersion plots of the averaged 3D correlation lengths and the averaged from profiles A) F irst order detrending, B) second order detrending, and C) FFT filter detrending.
103 Original 200 400 100 200 300 400 500 -0.05 0 0.05 A1 50 100 150 200 250 50 100 150 200 250 H1 50 100 150 200 250 50 100 150 200 250 V1 50 100 150 200 250 50 100 150 200 250 cD1 50 100 150 200 250 50 100 150 200 250 A2 20 40 60 80 100 120 20 40 60 80 100 120 H2 20 40 60 80 100 120 20 40 60 80 100 120 V2 20 40 60 80 100 120 20 40 60 80 100 120 cD2 20 40 60 80 100 120 20 40 60 80 100 120 Figure 49 Two stage wavelet bank used to decompose a 3D DEM. A B C Across Row [m]Along Row [m] 0 2 4 6 0 1 2 3 4 5 6 Across Row [m]Along Row [m] 0 2 4 6 0 1 2 3 4 5 6 Across Row [m]Along Row [m] 0 2 4 6 0 1 2 3 4 5 6 -0.05 0 0.05 -0.02 -0.01 0 0.01 0.02 -0.02 -0.01 0 0.01 0.02 Fig ure 4 10 Detrending of a quasiperiodic surface (#20). A) Original DEM. B) Random component extracted using vertical bias removal. C) Random component extracted using wavelets.
104 Table 41. Correlation coefficient and RMSE in meters between correlation length values derived by averaging profile values and averaging the ACFs First order Second order FFT Filter C c RMSE cc RMSE cc RMSE Rows 0.976 0.029 0.992 0.011 0.995 0.002 Columns 0.990 0.028 0.985 0.021 0.973 0.005 Table 42 Averaged oriented roughness parameter values obtained from 2D profiles extracted from th e DEMs. First order detrending Rows Columns DEM RMSh Correlation lengt h RMSh Correlation lengt h IRMSh # All Rand All Rand All Rand All Rand 1 0.007 0.005 0.228 0.112 0.006 0.005 0.181 0.124 0.853 2 0.008 0.008 0.148 0.124 0.005 0.004 0.113 0 .071 0.578 3 0.007 0.003 0.516 0.275 0.005 0.004 0.543 0.323 0.818 4 0.008 0.007 0.173 0.093 0.008 0.005 0.316 0.161 0.958 5 0.012 0.007 0.365 0.148 0.010 0.005 0.491 0.178 0.861 6 0.015 0.015 0.282 0.299 0.008 0.006 0.327 0.291 0.511 7 0.014 0.014 0. 185 0.167 0.012 0.009 0.126 0.064 0.819 8 0.017 0.016 0.163 0.115 0.011 0.011 0.192 0.223 0.676 9 0.020 0.014 0.321 0.150 0.009 0.008 0.099 0.081 0.459 10 0.021 0.022 0.150 0.148 0.010 0.010 0.090 0.082 0.489 11 0.021 0.018 0.190 0.145 0.013 0.010 0.26 8 0.171 0.625 12 0.019 0.018 0.218 0.184 0.016 0.015 0.200 0.163 0.817 13 0.022 0.024 0.201 0.185 0.017 0.012 0.281 0.181 0.752 14 0.023 0.018 0.230 0.152 0.015 0.015 0.127 0.123 0.679 15 0.026 0.021 0.278 0.198 0.018 0.018 0.130 0.109 0.710 16 0.029 0.022 0.421 0.237 0.017 0.015 0.317 0.225 0.597 17 0.023 0.022 0.197 0.152 0.023 0.013 0.440 0.208 0.986 18 0.022 0.022 0.084 0.084 0.006 0.006 0.109 0.116 0.295 19 0.032 0.032 0.189 0.181 0.010 0.009 0.095 0.085 0.303 20 0.035 0.034 0.216 0.182 0.008 0.007 0.210 0.132 0.241 21 0.032 0.033 0.174 0.170 0.016 0.014 0.359 0.273 0.505 C c 0.962 0.677 0.891 0.816 RMSE 0.003 0.100 0.003 0.115 % 0.114 0.241 0.157 0.263
105 Table 42 Continued Second order detrending Rows Columns DEM RMSh Correlation length RMSh Correlation length IRMSh # All Rand All Rand All Rand All Rand 1 0.006 0.005 0.192 0.065 0.005 0.005 0.1 46 0.077 0.843 2 0.008 0.008 0.132 0.121 0.004 0.004 0.085 0.066 0.561 3 0.003 0.002 0.323 0.192 0.004 0.002 0.370 0.195 1.312 4 0.007 0.006 0.108 0.072 0.005 0.004 0.186 0.117 0.807 5 0.007 0.006 0.153 0.114 0.005 0.004 0.224 0.135 0.805 6 0.015 0.01 3 0.288 0.274 0.006 0.005 0.238 0.173 0.422 7 0.014 0.013 0.157 0.128 0.010 0.008 0.066 0.045 0.713 8 0.012 0.012 0.073 0.072 0.011 0.011 0.168 0.174 0.885 9 0.018 0.012 0.179 0.122 0.009 0.009 0.088 0.079 0.491 10 0.021 0.021 0.146 0.139 0.010 0.010 0 .082 0.067 0.485 11 0.020 0.019 0.183 0.175 0.012 0.009 0.216 0.133 0.580 12 0.015 0.015 0.141 0.153 0.012 0.011 0.111 0.109 0.771 13 0.022 0.023 0.191 0.176 0.015 0.011 0.212 0.132 0.687 14 0.021 0.016 0.184 0.125 0.015 0.013 0.113 0.099 0.699 15 0.0 25 0.019 0.249 0.147 0.018 0.018 0.119 0.098 0.706 16 0.020 0.017 0.232 0.161 0.015 0.011 0.232 0.153 0.732 17 0.022 0.022 0.179 0.156 0.013 0.010 0.212 0.122 0.593 18 0.022 0.021 0.083 0.083 0.006 0.005 0.074 0.089 0.274 19 0.031 0.032 0.183 0.174 0.0 09 0.009 0.062 0.060 0.281 20 0.035 0.034 0.211 0.175 0.008 0.006 0.160 0.111 0.225 21 0.031 0.031 0.168 0.165 0.013 0.011 0.254 0.175 0.401 C c 0.970 0.737 0.951 0.893 RMSE 0.002 0.054 0.002 0.066 % 0.084 0.182 0.151 0.241
106 Table 4 2 Continued FFT filter detrending Rows Columns DEM RMSh Correlation length RMSh Correlation length IRMSh # All Rand All Rand All Rand All Rand 1 0.004 0.004 0.041 0.040 0.004 0.004 0.045 0.027 0.919 2 0.006 0.005 0.071 0.062 0.004 0.00 3 0.055 0.055 0.661 3 0.002 0.002 0.085 0.073 0.001 0.002 0.070 0.054 0.690 4 0.005 0.006 0.049 0.053 0.004 0.004 0.074 0.048 0.693 5 0.006 0.005 0.104 0.070 0.004 0.004 0.076 0.079 0.599 6 0.005 0.005 0.080 0.069 0.004 0.004 0.068 0.091 0.748 7 0.010 0.009 0.065 0.065 0.009 0.008 0.045 0.067 0.879 8 0.011 0.011 0.062 0.063 0.007 0.008 0.055 0.058 0.682 9 0.011 0.009 0.083 0.075 0.008 0.009 0.067 0.059 0.698 10 0.015 0.015 0.088 0.083 0.009 0.009 0.055 0.056 0.571 11 0.012 0.012 0.073 0.065 0.008 0 .008 0.066 0.060 0.643 12 0.011 0.011 0.053 0.060 0.009 0.008 0.056 0.069 0.839 13 0.013 0.013 0.075 0.069 0.009 0.009 0.085 0.064 0.687 14 0.012 0.012 0.081 0.079 0.012 0.011 0.057 0.091 0.954 15 0.013 0.013 0.079 0.072 0.014 0.014 0.067 0.050 1.013 16 0.012 0.011 0.096 0.089 0.008 0.009 0.078 0.088 0.700 17 0.015 0.015 0.079 0.074 0.009 0.009 0.067 0.094 0.612 18 0.020 0.020 0.074 0.074 0.005 0.006 0.038 0.078 0.273 19 0.019 0.020 0.088 0.092 0.008 0.007 0.039 0.097 0.399 20 0.019 0.021 0.088 0.0 85 0.006 0.006 0.046 0.090 0.299 21 0.014 0.012 0.075 0.067 0.008 0.008 0.074 0.054 0.560 C c 0.986 0.836 0.983 0.087 RMSE 0.001 0.010 0.001 0.0 24 % 0.040 0.059 0.008
107 Table 43 3D roughness parameters values obtained from the full DEM and averaging values obtained from 2D profiles. First order detrending # RMSh [m] Eccen t Correlation length [m] 3D All Random m Dir 3D min 3Dmean All Random 1 0.007 0.006 0.005 0.006 0.940 0.294 0.223 0.205 0.087 2 0.008 0.006 0.006 0.008 0.944 0.158 0.242 0.131 0.103 3 0.009 0.006 0.003 0.007 0.949 0.678 1.339 0.530 0.298 4 0.010 0.008 0.006 0.007 0.938 0.289 0.450 0.245 0.147 5 0.016 0.011 0 .007 0.011 0.967 0.686 1.046 0.428 0.177 6 0.017 0.012 0.009 0.009 0.984 0.318 0.541 0.305 0.293 7 0.017 0.013 0.011 0.014 0.938 0.175 0.292 0.156 0.119 8 0.019 0.014 0.014 0.015 0.950 0.190 0.322 0.177 0.170 9 0.021 0.015 0.013 0.016 0.977 0.305 0.127 0.210 0.123 10 0.022 0.016 0.015 0.019 0.977 0.152 0.130 0.120 0.123 11 0.022 0.017 0.015 0.020 0.931 0.219 0.318 0.229 0.174 12 0.023 0.017 0.015 0.017 0.970 0.310 0.277 0.209 0.172 13 0.024 0.020 0.020 0.020 0.976 0.207 0.509 0.241 0.200 14 0.024 0 .019 0.016 0.022 0.968 0.235 0.149 0.178 0.114 15 0.027 0.022 0.019 0.022 0.938 0.288 0.158 0.204 0.135 16 0.030 0.023 0.017 0.025 0.971 0.473 0.691 0.369 0.206 17 0.031 0.023 0.016 0.024 0.912 0.391 0.993 0.319 0.178 18 0.022 0.014 0.014 0.022 0.988 0 .085 0.130 0.097 0.090 19 0.032 0.021 0.021 0.029 0.988 0.191 0.081 0.142 0.125 20 0.035 0.022 0.022 0.032 0.988 0.220 0.251 0.213 0.150 21 0.036 0.024 0.020 0.027 0.981 0.185 0.361 0.267 0.210 Second order detrending # RMSh [m] Eccen t Correlation le ngth [m] 3D All Random m Dir 3D min 3Dmean All Random 1 0.007 0.006 0.005 0.006 0.934 0.288 0.203 0.169 0.069 2 0.008 0.006 0.006 0.008 0.961 0.153 0.151 0.109 0.086 3 0.004 0.003 0.002 0.003 0.982 0.438 0.975 0.347 0.193 4 0.008 0.006 0.005 0.006 0.948 0.165 0.290 0.147 0.118 5 0.008 0.006 0.005 0.006 0.914 0.226 0.519 0.188 0.116 6 0.016 0.011 0.009 0.009 0.988 0.296 0.446 0.263 0.225 7 0.017 0.012 0.011 0.014 0.932 0.169 0.267 0.111 0.097 8 0.015 0.011 0.011 0.014 0.959 0.113 0.314 0.120 0.12 9 9 0.019 0.014 0.011 0.015 0.977 0.180 0.119 0.133 0.110 10 0.022 0.016 0.015 0.019 0.976 0.151 0.127 0.114 0.099 11 0.022 0.016 0.014 0.020 0.930 0.207 0.290 0.200 0.123 12 0.018 0.013 0.013 0.016 0.948 0.202 0.190 0.126 0.121 13 0.023 0.018 0.017 0 .020 0.974 0.196 0.515 0.202 0.145 14 0.023 0.018 0.017 0.020 0.961 0.197 0.140 0.148 0.123 15 0.027 0.021 0.016 0.022 0.939 0.271 0.154 0.184 0.119 16 0.023 0.018 0.016 0.021 0.955 0.259 0.674 0.232 0.162 17 0.025 0.017 0.016 0.023 0.914 0.222 0.328 0 .195 0.125 18 0.022 0.014 0.014 0.022 0.991 0.084 0.127 0.079 0.082 19 0.031 0.020 0.019 0.029 0.988 0.185 0.077 0.122 0.101 20 0.035 0.021 0.019 0.032 0.991 0.212 0.187 0.185 0.108 21 0.035 0.022 0.025 0.027 0.985 0.179 0.357 0.211 0.171
108 Table 4. 3 Co ntinued FFT based filter # RMSh [m] Eccen t Correlation length [m] 3D All Random m Dir 3D min 3D m ean All Random 1 0.005 0.004 0.004 0.007 0.891 0.060 0.057 0.043 0.031 2 0.006 0.005 0.005 0.008 0.938 0.081 0.063 0.063 0.063 3 0.002 0.002 0.002 0.00 3 0.944 0.131 0.134 0.077 0.095 4 0.006 0.005 0.004 0.008 0.884 0.074 0.090 0.061 0.040 5 0.007 0.005 0.005 0.008 0.923 0.123 0.114 0.090 0.112 6 0.006 0.004 0.004 0.007 0.948 0.099 0.112 0.074 0.080 7 0.012 0.009 0.009 0.015 0.889 0.082 0.098 0.055 0. 057 8 0.012 0.009 0.009 0.020 0.953 0.073 0.078 0.058 0.062 9 0.013 0.010 0.008 0.020 0.957 0.097 0.098 0.075 0.085 10 0.016 0.012 0.013 0.027 0.970 0.096 0.086 0.071 0.084 11 0.013 0.010 0.010 0.018 0.886 0.084 0.119 0.070 0.068 12 0.012 0.010 0.010 0.018 0.933 0.069 0.077 0.055 0.038 13 0.015 0.011 0.012 0.021 0.960 0.089 0.198 0.080 0.075 14 0.015 0.012 0.013 0.026 0.955 0.090 0.089 0.069 0.068 15 0.017 0.014 0.013 0.023 0.897 0.088 0.098 0.073 0.068 16 0.013 0.010 0.009 0.018 0.915 0.096 0.196 0.087 0.095 17 0.017 0.012 0.011 0.021 0.890 0.096 0.145 0.073 0.051 18 0.020 0.012 0.013 0.033 0.988 0.074 0.073 0.056 0.074 19 0.020 0.014 0.014 0.025 0.979 0.090 0.052 0.063 0.093 20 0.019 0.012 0.014 0.028 0.982 0.090 0.067 0.067 0.084 21 0.016 0. 011 0.011 0.019 0.933 0.088 0.184 0.075 0.074
109 Table 44 Comparison of 3D roughness parameter values obtained from different approaches. First order detrending Second order detrending FFT based detrending CC RMSE Mean % CC RMSE cc RMSE mean %D RMSh, all profiles 20 r nd 0.94 0.003 14.0 0.97 0.002 9.1 0.98 0.001 0.4 l, all profiles 20 random 0.78 0.102 27.5 0.80 0.060 23.6 0.76 0.014 3.2 RMSh, 3D all profiles 0.97 0.007 26.8 0.97 0.007 28.0 0.98 0.0 04 25.0 l, 3D all profiles 0.93 0.329 27 4 0.85 0.236 34 5 0.72 0.051 32 1 RMSh, 3D 20 random 0.93 0.009 37.0 0.97 0.008 34.8 0.97 0.004 24.8 l, 3D 20 random 0.70 0.428 49 4 0.69 0.288 50 9 0.48 0.053 33 3 RMSh, 3D mean directional 0. 97 0.004 18.2 0.98 0.003 14.8 0.98 0.001 10.0 RMSh, 3D max directional 0.95 0.008 30.3 0.96 0.008 36.0 0.96 0.007 42.1 Table 45 Difference in microwave observables due to errors in roughness parameterization. 3D 2D Low Soil Moisture: 7% by v olume High soil moisture: 20% by volume RMSh l RMSh l T b h [k] T b v [k] 0 hh [dB] 0 vv [dB] T b h [k] T b v [k] 0 hh [dB] 0 vv [dB] 0.5 10 0.4 7 0.02 0.18 0.49 0.95 0.06 0.25 1.55 0.95 L 1.5 10 1.1 7 2.92 0.07 4.77 1.60 4.77 0.99 4 .58 1.62 2.5 10 1.9 7 4.26 1.79 4.05 1.14 7.18 4.39 4.20 1.17 0.5 10 0.4 7 0.02 0.18 2.09 0.39 0.06 0.25 2.10 0.42 C 1.5 10 1.1 7 2.89 0.08 2.27 2.02 4.73 0.96 2.31 1.26 2.5 10 1.9 7 4.22 1.76 0.07 0.07 7.12 4.32 0.04 0.22
110 Table 46 Roughness Parameters derived from 3D models of quasiperiodic surfaces. RMSh (m) Correlation Length (m) # Cols V Bias Wavelet Cols V Bias Wavelet 21 0.016 0.024 0.006 0.359 0.361 0.060 18 0.006 0.008 0.005 0.109 0.130 0.028 19 0.00 9 0.011 0.007 0.095 0.081 0.037 20 0.008 0.01 0 0.006 0.21 0.251 0.035
111 CHAPTER 5 THE APPLICATION OF AIRBORNE LIDAR TO MAP SURFACE ROUGHNESS OF LARGE AREAS As demonstrated in the previous chapters, groundbased LiDAR is a convenient and accurate tool to characterize millimeter scale roughness. I t is limited however, to map areas that cover a few hundred square meters. Some applications require the characterization of surface roughness over much larger areas, such as the mapping of soil moisture at the watershed, regional, and global levels. To date, obtaining a good agreement between soil moisture measurements and results obtained from the i nversion of physical based models that use spaceborne microwave sensor observations and roughness measurements has been challenging [ 27] [ 28]. This may be due to the scaling of roughness measurements. These measurements are made in situ and cover a relative ly small area, whereas the microwave sensor footprint can be up to several square kilometers in area. The roughness measurements are scaled up under the assum ption that roughness is isotropic and homogenous over the area of study These sparse roughness dataset s, in both the spatial and temporal domains, limit the applicability of inversion algorithms based on physical models such as the IEM to retrieve soil moisture from microwave signatures using observations at satellite scales [ 26]. Airborne LiDAR observations have the potential to provide surface roughness characterization for large areas as described i n preliminary studies by Davenport et al. [2 0 ] [2 1 ] These early experiments on the application of airborne LiDAR for surface roughness characterization were limited because of the decimeter accuracy of airborne LiDAR data and the sparse sample densities obtainable by the technology of the late 1990s and early 2000s of one or two points per square meter. Since then, sensors and
112 algorithms have undergone exponential development and it is now possible to obtain sampling densities of tens of points per m2 wi th subdecimeter accuracy [ 30], [ 3 1 ]. C urrent capabilities of airborne LiDAR systems still do not allow for the fine sampling resolution required to determine cm level correlation lengths. However, it is possible to derive RMSh and studies such as [ 27] and [ 38] have been aimed at deriving relationships between RMSh and correlation lengths This chapter explores methods to derive RMSh maps of large areas using airborne LiDAR data. Airborne LIDAR Instrumentation and Datasets To properly describe some of the methods proposed in this chapter it is important to cover some basic aspects of the geometry of airborne LiDAR scanning. There are several airborne l aser scanning mechanisms the most common being an oscillating mirror. As illustrated in F igure 5 1, scanning from an airborne platform is performed in two dimensions T he first is accomplished by employ ing the forward motion of aircraft and the second is obtained using an oscillating mirror The mirror steers the laser beam in a direction perpendicular to the line of flight. This mechanism distributes the laser pulses over the ground in a saw tooth pattern. The scanning angle and platform flight height determines the swath width. T he scanning frequency in conjunction with PRF determines the across track resolut ion. The aircraft ground speed in and the scan frequency determine s the downtrack resolution. A survey is designed to meet specifications in terms of resolution, point density and accuracy This is achieved by modifying the PRF, scanning angle range and frequency, platform ground speed, and flying height. For a more detailed description of the basics of airborne LiDAR technology and terminology used in the following sections the reader is referred to the papers by
113 Baltsavias [ 4 5] or Wehr and Lohr . F or descriptions of current capabilities and future technological developments the reader may refer to [ 30] and [ 3 1 ]. Data for this work was collected on September 2nd, 2010, employing an Optech Gemini Airborne LiDAR Terrain Mapper (ALTM) equipped with a wa veform digitizer owned and operated by the National Center for Airborne Laser Mapping (NCALM). The Gemini system is capable of pulse repetition rates of up to 166 kHz and can operate in multiple pulse m ode which means that the system is able to fire a sec ond laser pulse while still waiting for the return of the first. The Gemini uses a constant fraction discriminator to obtain up to four discrete returns for each outgoing pulse. An additional capability of the Gemini system is that it is possible to select between two beam divergence modes, a narrow divergence mode with full angle aperture of 0.25 mrad and the wide divergence mode with an aperture of 0.80 mrad. An addon signal digitizer and computer allows for the sampling and recording of full waveform da ta. The ALTM detector output voltage is sampled at 1 GHz (every 1 nanosecond) and is quantized on an 8bit scale between 0 and 1 volts. Detailed specifications for the Gemini ALTM and the waveform digitizer ar e presented in Tables 51 and 52; photographs of the full airborne system are shown in Figure 52. The airborne LiDAR data collection was conducted over the grounds of the University of Florida Plant Science Rese arch and Education Unit (PSREU) located near Citra, Florida. The general area that was mapped is shown in F igure 5 3 A. T his area was selected because of its heterogeneous composition which includes a variety of bare surfaces, vegetation with different characteristics, buildings, roads and other manmade structures. Data collection was performed at a PRF of 70 kHz which is the
114 maximum rate at which the waveform digitizer can operate, recording waveform information for every fired pulse. A single flight line was collected three times from a flying height of 600 meters above ground level varying the scan angle and beam divergence to test the effects of thes e adjustable sensor parameters on the ability to map small scale roughness. A summary of the main configuration parameters use d for each line collection is presented in Table 53. One line was collected with a scan angle of 14 a scan frequency of 60 Hz and using narrow divergence mode. With these parameters the swath width was 298.75 m with a point density of 3.6 points/m the cross track resolution is 0.513 m, and the downtrack resoluti on is 0.542 m Two lines were collected with a scan angle of 5 and a scan frequency of 65 Hz, one with narrow divergence and the second one in the wide beam mode. For these lines the swath width was 104.56 m with a point density of 10.26 points/m the crosstrack resolution wa s 0.195 m and the downtrack resolution was 0.5 m. From a flying height of 600 m at nadir the laser footprint in the narrow divergence mode is 15 c m and using the wide divergence mode is 48 c m Figure 53 B shows a rendering from the swath collected from 600 meters AGL with 14 scan angle. The rendering is color coded by elevation, with co o l colors for low elevations and warm colors for high elevations T he luminance of the color is modulated by the intensity of the return signal Two data products were produced by NCALM from the airborne survey. The first processed data product from the ALTM is the point cloud that contains the UTM coordinates (easting nort hing and ellipsoid height), intensity, echo type, range scan angle, and GP S time for each collected return. The point clouds were delivered in two
115 formats, the binary ASPRS .LAS format and an ASCII multi column text format. The second data product was the recorded raw binary waveform in the Optech proprietary NDF format. This ra w waveform data consist s of 8 bit scaled signal voltage sampled every nanosecond for the outgoing pulse and the return waveform. Both of these data products were used to develop the roughness maps T o create roughness maps three different methods based on two approaches we re proposed. The first approach is thematic mapping, which consist s of attributing a roughness value (RMSh) to each element in the point cloud. This approach wa s used on the first two methods: profiling from a single scan line and the fo otprint scale roughness from return waveform analysis. The second approach consists of creating an array with pixel elements that cover a given area, then grouping the individual points to a corresponding pixel element and computing the RMSh for the group. Each method and its respective results is described next. Profiles from a Single Scan Line Davenport et al. in [ 2 0 ] and [ 21] demonstrated that despite the decimeter accuracy obtained from airborne LiDAR due to navigation uncertainties, the precision of se quential returns is significantly higher and can be used to automatically distinguish between different surface preparations. Also because roughness parameters are based on relative height variations rather than absolute height values, it is possible to compute RMSh for transects generated from sequential points along a single scan line. Thus this method produces directional roughness maps in an orientation nearly perpendicular to the flight line. To test this method, the swath collected with a scan angle of 5, scan frequency of 65 Hz and narrow divergence was employed. With these parameters and at the flying height of 600 meters the points in the line are
116 separated by 19.5 c m with a beam footprint of 15 c m This is almost a continuous sampling of elev ations along the scan line. Flying at a ground speed of 60 m/s, the points at nadir between two successive sca n lines are separated by 50 cm. This method was implemented f using t he ASCII multi column text format, which provides a time tag, easting, northing, height, intensity, range, and scan angle for each return point. The timing and/or scan angle information can be used to separate the individual scan lines. Then, for each point in the line, the neighboring points that are within a given distance on either side are identified. Profiles 3 m in length were considered, following the recommendation in [ 15]. From these points a profile is built by determining the horizontal distance between the points and the hei ght of each point. The resulting profile is then detrended and its RMSh is computed. Next t he RMSh is attached to each point as the roughness attribute. The roughness maps can then be created in the same way elevation maps are rendered from the raw point clouds but using the RMSh attribute instead of the height Fig ure 53 C shows the created RMSh map from the entire swath. Figure 5 4 A shows the RMSh map derived with this method from the section of bare soil delineated with a yellow dashed rectangle, as shown in Fig ure. 53 B. Th e RMSh map shown in Figur e 54A is rendered using a color scale between 0 and 3 cm which demonstrates that it is possible to derive fine scale roughness maps from airborne LiDAR data. It also demonstrates the heterogeneity of surface roughness of this 160 m x 40 m area. Footprint Scale Roughness from Return Waveform Analysis T he return from each laser shot is treated as the reflection from a point source even though the beam footpri nt on the ground has some extent The LiDAR footprint size depends on the flying height above the ground and the beam divergence. For a
117 fl ying height of 600 m above ground level, the beam footprint diameter is 15 cm for the narrow divergence mode and 48 cm for the wide divergence mode. Also LiDAR systems emit laser pulses that have an extent in the tim e domain. The pulse amplitudetime profile or waveform is usually Gaussian in shape, characterized by a pulse width (FWHM) or by its time variance or standard deviation. This outgoing pulse waveform is modified as the signal is reflected off the target. If the target has a smooth surface oriented normal to the laser beam the return waveform will be a perfect mirror image of the outgoing waveform. For imperfect reflectors the return waveform will be distorted through a convolution process making it possible to deduce geometric properties of the surface illuminated by analyzing the return waveform One such geometric propert y is the surface roughness. The advantage of this method is that roughness can be determined for each individual laser shot and, in pri nciple, is immune to the positioning and navigation errors that degrade LiDAR accuracy. The methodology developed in [ 47] to map surface roughness on Mars from satellite laser altimetry was adapted to map surface roughness using airborne LiDAR. This approach is based on the principle that the pulse width of a reflection from a rough surface will be longer than the original emitted pulse width, as described by : 2 2 2 2 ir l t r (5 -1) w here 2 r is the variance of the received wavef orm, 2 t is the terrain height variation for the footprint, 2 l is the emitted laser pulse variance and 2 ir is the variance due to the detector impulse response. Using the speed of light in the pr oper medium these time variances can be converted into spatial variances ; and by taking into account the beam
118 divergence, surface slope, and incidence angle, the surface variance or RMSh for the footprint is given by: 2 2 2 2 2 2 2tan tan 4 c R RMSht s (5 -2) w here R is the oneway laser range, c is the speed of light in the proper medium, is the beam divergence and is the incidence angle with respect to the surface local nor mal. To implement this method, both discrete and digitized waveform data w ere used. The data collected with a scan angle of 5, scan frequency of 65 Hz and wide divergence. With these parameters and at a flying height of 600 m the points in the line sca n line are separated by 19.5 cm. Flying at a ground speed of 60 m/s, the points at nadir between two successive scan lines are separated by 50 cm. Operating the system in wide divergence mode from 600 m AGL yields a beam footprint of 48 cm. The combination of alongtrack and across track resolution, plus the footprint size, allows for an overlapping and complete illumination of the surface. The digitizer data provides 8 bit amplitudes of the outgoing and return waveform s sampled at 1 nanosecond intervals. To obtain the values of 2 r and 2 l an iterative least squares Gaussian fit is applied following the model: 2 2 mod2 expT t A t wf (5 3) w here t wfmod is the modeled Gaussian waveform as a function of time (t), with parameters amplitude ( A ), time for the maximum amplitude ( T ) and pulse time variance (2 0) The linearized iterative least square model is:
119 n T t n T t T t n T t n T t T t T t T tv e A wf v e A wf T A e T t A e T t A e e T t A e T t A en n n 2 0 2 0 2 0 2 0 1 2 0 2 0 1 2 0 2 0 2 0 2 0 2 0 2 0 1 2 0 2 0 1 2 0 2 0 12 0 1 2 0 1 2 3 0 2 0 0 2 2 0 0 0 2 2 3 0 2 0 1 0 2 2 0 0 1 0 2 (5 4) w here n is the number of waveform samples obtained, nt is the time for sample n nwf is the digitized amplitude value of sample n. 0A, 0T 0 are the approximation of the Gaussi an parameters for the current iteration, and 0 are the correction factors for the parameter approximations to be determined for every iteration. For each laser shot fired the time variance of the outgoing and return pulse wav eforms were determined. T his yield s two of the variable values used in Eq uation 5 1 To estimate the time variance due to the systems impulse response a smooth flat area was selected for which it is assumed that 2 t = 0 F rom 5032 obse rvations it was determined that 2 ir is 2.72 nanoseconds2. With this the 2 t for each emitted pulse was computed. T hen Eq uation 5 2 is applied, and the footprint scale RMSh can be obtained. The RMSh is then attached to each element of the LiDAR point cloud as an additional attribute. The roughness map can then be generated using a point cloud rendering software in the same fashion as the last method. Figure 53 D shows the RMSh map rendered from the entire swath and Fi g ure 5 4 B for the bare soil surface. By comparing Fig ures 53 C and 5 3 D it can be seen that this method provides finer spatial detail but it is also very sensitive to digitizer noise that can affect the correct determination of the fit parameters.
120 Point Cloud Binning A final method to create RMSh maps from high density point clouds consists of creating an array or grid of square surface elements (cells or bins) of a given area. Each bin has well defined boundaries. Then the individual points are assigned to a given bin based on its horizontal coordinates. Once all the points are segregated into their respective bins, the point s coordinates are used to determine the 3D trend. After separating the random height component the RMSh is computed and this value is assigned to the respective bin. Repeating the process for all the bins a raster map is generated. To test this method, the swath collected with a scan angle of 5, scan frequency of 65 Hz and narrow divergence was employed. With these parameters, a flying height of 600 m and a ground speed of 60 m/s the nominal point density is 10.26 points/m2. Fig ure 53 E shows an RMSh map derived from this data using a bin size of 1x1 m. Fig ure 5 4 C shows the RMSh map for the bare soil surface Validation of Airborne Roughness Maps To validate the RMSh maps obtained from the airborne data, a groundbased LiDAR was employed to digitize two samples of a bare soil surface These surface samples are delineated by black squares in Fig ure 54A Figure 55 shows photos of the validation areas and DEMs derived from the groundbased LiDAR. To validate the RMSh obtained from airborne data using the profiling method, transects were extracted from the groundbased DEM in the northto south direction LiDAR. These transects were then detrended and the RMSh computed for each one. Figure 56 shows a histogram of RMSh values obtained from the profiling method from airborne and groundbased data. Values derived from airborne data have a larger spread, but, their
121 means differed by only 1mm. The mean RMSh derived from groundbased measurements is 2.2 cm and from airborne is 2.1 cm. RMSh results obtained by applying the binning method to groundbased and airborne data are also shown in Figure 56 Similar to the results from the profiling method, the airborne derived RMSh values have a larger spread but their means also only differ by 1 mm. The mean RMSh from groundbased data was determined to be 1.8cm and from airborne data it was found to be 1.7 cm. To validate the RMSh results obtained through the return waveform analysis method, the groundbased data was analyzed using the binning method considering a bin size of 0.43 meters which comprise s an area equivalent to the footprint area with a diameter of 48 cm. The average RMSh obtained f rom the groundbased data following this approach was 1.4 cm. However, it was not possible to obtain a coherent set of RMSh values from the waveform analysis for the validation areas. In fact, to obtain RMSh values below 3 cm using this method requires the determination of the time variances with an accuracy better than 0.1 nanoseconds. In the current implementation of the waveform digitizer this level of accuracy can not be reached. However, technology will continue to improve and in the near future the ca pability t o determine finescale RMSh from waveform data will be reached. It is a lso worth not ing that this method is applicable to characterizing vegetation structure, especially low grasses and shrubs that are not fully characteriz ed using discrete LiDAR data. A cross validation between RMSh results obtained from the profiling and binning methods from airborne data was performed. To have a consistent comparison, RMSh obtained from the profiling method considering transects 3 m in length was compared to
122 t he values obtained for bins 3 m x 3 m in area. A histogram of the obtained RMSh from both methods is presented in Figure 56 The average RMSh of the entire bare soil surface obtained with the profiling methods is 2.1 cm, while the average value obtained b y applying the binning method is 2.7 cm. This result is consistent with the results obtained in Chapter 4, w h ere RMSh values obtained from 2D profiles is lower by 2 5% on average than the values derived from 3D data. Finally, it is worth mentioning that the airborne data was collected in a way t hat ensures consistency between the three proposed methods and to record the waveform of every fired pulse. The digitizer limited the sampling capabilities of the ALTM by half. Operating the ALTM at its maximum PRF (1 66 kHz) would have yielded sampling densities m ore than twice the density of those collected, improving the results obtained with the pr ofiling form a single scan line and the point cloud binning methods. T he technology that was employed to collect this airborne dataset is a generation old. The current generation of ALTM includes a system that integrates three scanning mechanism s in to a single unit and it is capable of PRFs of 400 kHz. Faster digitizers are also available as electronic modules that have a sampling speed of 2 GHz or twice the current digitizing capabilities thus improving the sensitivity of footprint scale roughness by a factor of 2. Chapter Conclusions Data from g roundbased LiDAR is useful to characterize m m level surface roughness of areas that can encompass a few hundred square meters. For the characterization of roughness of larger areas high resolution airborne LiDAR is a viable option. It was demonstrated that there are several ways to derive accurate mm level roughness maps for surfaces with RMSh higher than 1 cm. These include directional
123 roughness derived from transects extracted from a single scan line and full surface roughness derived from data collected over a given area. Unfortunately c urrent technology does not allow for t he determination of LiDAR footprint sca le roughness bel ow the 10 cm RMSh level.
124 Swath Width Flight Direction Scan Line Cross Track Resolution Down Track Resolution Discrete Return Discrete Return Beam Footprint Figure 5 1. Geometry of airborne scanning LiDAR data collection.
125 A B Figure 52. The National Center for Airborne Laser Mapping (NCALM) Gemini ALTM and waveform digitizer. A) Front side of the control and electronics r ack and digitizer computer. B) Rear side of the racks and later side of the sensor head.
126 Figure 53 Airborne LiDAR r oughness maps. A)Photo of the general area of PSREU that was surveyed with the airborne LiDAR, the orientation of the photo is east to the right, west to the left. B) Point cloud rendering from a swath 14 from 600m AGL. C) RMSh map using the profile along a single scan line method. D) Footprint scale RMSh map from analysis of the return waveforms. E) RMSh map from airborne LiDAR using point cloud binning method.
127 Fig ure 54 Surface roughness map of a bare soil area marked as a yellow dashed rectangle in Figure 5 3 A) From profiles from single scan line method. B) Footprint s cale roughness RMSh from return waveform analysis. C) From the point cloud binning method.
128 Figure 55 Validation areas for the airborne derived roughness maps marked as squares 1 and 2 in Figure 53 A. The yellow rectangles represent the areas from w hich the DEMs were created. A) Photo of validation area 1, looking east. B) Groundbased LiDAR derived DEM of validation area 1. C) Photo of validation area 2, looking south. D) Groundbased LiDAR derived DEM of validation area 2.
129 0 0.02 0.04 0.06 0 0.2 0.4 0.6 0.8 ProfilesGround-based 0 0.02 0.04 0.06 0 0.2 0.4 0.6 0.8 Footprint 0 0.02 0.04 0.06 0 0.1 0.2 0.3 0.4 Bins 0 0.02 0.04 0.06 0 0.1 0.2 0.3 0.4 Airborne 0 0.02 0.04 0.06 0 2 4 6 8 x 10-3 RMSh [cm] 0 0.02 0.04 0.06 0 0.1 0.2 0.3 0.4 Figure 5 6 Validat ion of the airborne derived RMSh from the three proposed methods compared to RMSh values derived from groundbased LiDAR data through similar processing.
130 Table 5 1. Optech GEMINI Airborne Laser Terrain Mapper (ALTM) specifications Parameter Specif ied Value Operating a ltitude 150 4000 m, n ominal Ho rizontal a ccuracy 1/5,500 x altitude (m AGL); 1 sigma Elevation a ccuracy 5 30 cm; 1 sigma Range c apture Up to 4 range measurements, including 1 st 2 nd 3 rd last returns Intensity c apture 12 bit d ynamic range for all recorded returns, including last returns Scan FOV 0 50 degrees; Programmable in increments of 1degree Scan f requency 0 70 Hz Scanner p roduct Up to Scan angle x Scan frequency = 1000 Roll c ompensation 5 degrees at full FOV m ore under reduced FOV Pulse rate f requency 33 167 kHz Position orientation s ystem Applanix POS/AV 510 OEM includes embedded BD950 12 channel 10Hz GPS receiver Laser wavelength/c lass 1047 nanometers / Class IV (FDA 21 CFR) Beam d ivergence nominal ( fu ll angle) Dual Divergence 0.25 mrad (1/e) or 0.80 mrad (1/e) Table 5 2. Optech waveform digitizer specifications Parameter Specified Value Amplitude resolution 8 bits (0 255) Sample interval 1 nanosecond Maximum acquisition and recording rate 70 kH z T0 record length 40 samples / 40 ns Return waveform record length 440 samples / 440 ns Full scale input range 0 1 V Table 5 3. Airborne LiDAR datasets Strip m AGL Scan a ngle Scan f requency Divergence Points S1C1_Field_245b_008 600 14 60 Hz Narr ow 2,411,841 S1C1_Field_245b_009 600 5 65 Hz Narrow 2,319,168 S1C1_Field_245b_011 600 5 65 Hz Wide 2,099,178
131 CHAPTER 6 CONCLUSIONS AND RECOMMENDATIONS Conclusions The work presented in this dissertation was performed as a multi step appr oach to address the current limitations and challenges of the characterization of surface roughness for its application to microwave emission and scattering models applied to the problem of soil moisture mapping. The first step in the process was to charac terize instrument errors and perform an accuracy assessment of the roughness parameters derived from measurements obtained from these instruments (meshboard and groundbased LiDAR). Once the accuracy of these instruments and the comparative baseline between the roughness parameters obtained from their measurements was establish g roundbased LiDAR was used to collect an extensive database of 3D bare soil surface measurements. This database was used to test the va lidity of current assumptions that surface roughness is a single scale random process that could be characterized from its standard deviation, correlation length, and the shape of its autocorrelation function. The 3D information was also used to determine the impact on roughness parameter values obta ined from 2D profiles with respect to the values obtained from full surface digitization. As a final step, the issue of assuming homogene ity of roughness characteristic s used when scaling a limited number of in situ roughness measurements from a large area was addressed with airborne LiDAR. As referred to in several chapters of this dissertation, there has been much discussion of the characterization of surface roughness as a singlescale process when results of many studies indicate multi scale characteris tics. The data collected and analyzed for this work indicate that in fact, natural and agricultural surfaces have height
132 variations at different scales. However, if 3D information with sufficient resolution and extent is collected it is possible to separate the height variations at any particular scale. The question that remains is at what scale or scales does roughness have to be characterized to be an accurate input to the current singlescale microwave emission or scattering models ? An alternative is to use multiplecontribution models that consider the effects of roughness in two or more scales. It was also determined from results of the experiments that roughness characteristics of agricultural soils are not isotropic and homogenous as has been assum ed for microwave modeling. In addition, it was observed that the autocorrelation functions of agricultural soils are not fully explained by the theoretical exponential or Gaussian ACFs. Microwave emission and scattering models need to be improved to account for generalized power law spectrum ACF and for the anisotropy and heterogeneity of a gricultural and natural surfaces. Roughness parameter values are very sensitive to instrument errors, especially the autocorrelation function and its correlation function. To accurately determine the parameter values it is necessary to remove all systematic errors from the height variation measurements. It is also necessary to properly characterize the instrument random errors. The raw height variation measurements obtained with any instrument is the result of the addition of two random processes: the surface roughness and the instrument random noise. Accurate estimation of the surface roughness parameters depends on compensating for the effects of the instrument noise. Du e to the anisotropic and heterogeneous characteristics of natural and agricultural surfaces sampling a surface in transects is not adequate. Parameters
133 obtained from 2D data considerably underestimate the surface roughness and do not describe its directional properties. 3D information derived from groundbased and airborne LiDAR is adequate f o r the proper characterization of surface roughness. Recommendations The work presented in this dissertation was described as a multi step approach to address the curr ent limitations and challenges of the characterization of surface roughness At this point there are several additional steps that might be taken in future work to continue building on the foundation presented here. Future work might include the extension of the accuracy assessments performed to 3D. Also analytical models to quantify and correct the effect of random instrument errors in the derived roughness parameters could be developed. A n additional lateral direction with high relevance that needs to be explored further has to do with determining what the adequate scale or scales are from which roughness parameters need to be determined. In other words w hat is the appropriate detrending method than needs to be applied in order to obtain the best agreement between the observed and modeled microwave signatures? In the dataset used in this study there are several roughness measurements that were performed concurrently with microwave observations and soil moisture measurements during the different MicroWEX experiments. Combining the roughness measurements with the MicroWEX experiments for this purpose would represent an additional advancement in the current state of knowledge. A forward direction t hat also needs to be explored involves expanding the experiments employing airborne LiDAR to characterize surface roughness of large areas, considering a variety of surface preparation conditions (rolled, plowed, furrowed) and combining airbornederived roughness maps with microwave spaceborne
134 observations and in sit u soil moisture measurements. An alternate, related approach is the assessment of ICESAT1 data to derive surface roughness maps at a footprint scale. Main Contributions Ma j or achievements that contribut e to the advance of knowledge presented in this work include: Performing the first assessment of the digitizing fidelity and accuracy of derived roughness parameters for the meshboard and groundbased scanning LiDAR Creation of a large database of 2D profiles (total length 100 km) that was used to: determi ne the roughness parameter value ranges at different scales; test the hypothesis of the existence of a correlation between the RMSh and correlation length; and test the applicability of Gaussian and exponential ACF to explain the observed ACF from natural and agricultural surfaces. Development of a methodology to derive the traditional surface roughness metrics from 3D DEMs obtained from g roundbased LiDAR Compare roughness parameter values obtained from single profiles to those obtained from the full surf ace using first and second order detrending. Quantified the underestimation of surface roughness parameters when determined from a limited number of surface profiles Developed a lternate methodologies to characterize 3D random roughness from DEMs of surfac e that exhibit quasiperiodic or nonstationary trends, such as those ar ising from row or furrow structures resulting from agricultural preparations Development of three al ternate methods to produce fine scale roughness maps of large areas employing airborne LiDAR. Demonstrating that it is possible to obtain mm accuracy for surfaces with RMSh higher than 1 cm and validating it against high resolution data obtained from groundbased LiDAR. Development of a method to model and remove systematic errors (geometric distortion and parallax) present in meshboard data, thus being able to obtain fidelity and accuracy in the study profiles and their derived roughness metrics higher than ever reported before in literature.
135 APPENDIX A MICROWAVE SCATTERING AND EMISSI ON MODELS APPLIED TO SOIL MOISTURE MAPPING The water cycle describes how this compound in its different phases (solid, liquid and vapor) moves through the atmosphere, the land, and bodies of water. Water plays an important role in the carbon and energy cyc les. It enables the growth of vegetation, which in turn, through the evaporation and transpiration processes moves vapor and energy from the land to the atmosphere, mo difying the weather and climate [ 48] Of the water in the planet, roughly 97.5% is salt or mineralized water. Of the remaining 2.52% uncombined fresh water 0.006% is present in rivers and streams, 0.26% in fresh water lakes, and 0.001 % in the atmosphere The rest of the fresh water component occurs as soil moisture, permanent snow cover, mars hes, and active groundwater [ 49] Even when the total amount of water on the planet is considered fixed, on a global scale, there are gaps in knowledge about how and at what pace thes e reserves are bei n g transformed [ 48] Soil moisture, which is defined as the water held between soil particles, albeit representing only 0.001% of the total water and 0.05% of the fresh water [ 50] is a critical factor to many hydrology, biology, biochemistry and ecology processes. It is crucial in regulati n g the water and ener gy exchanges between the land and lower atmosphere, and it is the dominant factor in vegetation growth and crop yield. Soil moisture also affects strongly the runoff into streams and rivers. Despite its importance, current dataset s are limited to few in si tu measurements. Global dataset s are urgently required to improve our understanding of the global water cycle and to improv e the cur rent weather and climate models [ 48] and [ 51] In order to obtain large scale datasets remote sensing techniques must be applied. Different remote sensing technologies have been
136 researched over the years to map soil moisture, i ncluding the analysis of reflectance on visible, near and shortwave infrared; thermal infrared emittance; brightness temperature and scattering coefficient in the microwave bands. To date the most promising results are achieved through the use of active and passive microwave sensors [ 6 ]. M icrowave remote sensing of soil moisture is based on the principl e that there is a large difference in the values of th e dielectric constant of dry soil (~3.5) and water (~80)  S oil scattering and emission properties in the microwave region are directly related to the surface roughness and dielectric constant; they are also dependent on sensor characteristics (wavelength and polarization) and observ ation geometry (incidence angle) [ 2 ]. Currently several models exist ranging from the empirical to purely physical analytical, that can be used to retrieve soil moisture from the backscatter coefficient obtained from active radar sensors or emissi vity obtained from passive radiometers  The physical analytical models provide estimates of the microwave scattering or emission as a function of the surfaces roughness and dielectric constant. In addition, many models provide estimates of the soil dielectric constant based on its composition and water content. If the roughness of a surface and its scattering or emitting properties are determined then is possible to invert the microwave models to obtain estimates of the dielectri c constant and applying the soil dielectric mixing models is possible to derive estimates of soil moisture. Among the main analytical models some worth mentioning are: T he Small Perturbation Model (SPM) published by S.O. Rice in 1951 [ 52] T he Kirchoff A pproximation (KA) developed by P. Beckman and A. Spizzichino, which was published
137 in 1963 [ 53 ]. More recently there is the Small Slope Approximation (SSA) published in 1985 by A.G. Voronovich [ 54] T he Integral Equation Model (IEM) developed by Fung, Li and Chen from the Wave Scattering Research Center, University of Texas at Arlington in 1991 to describe the backscattering from a randomly rough dielectric surface [ 4 0 ]. Finally, the Advanced Integral Equation Model (A IEM), which is a refined version of the IEM suitable to model emission of a rough surface, was developed by Chen, Wu, Tsa ng, Li, Shi and Fung in 2002 . These models are based on the assumptions of single scale isotropic surface roughness. They have t heir intrinsic limitations and regions of holding. For instance the SPM is valid for slightly rough surfaces, the KA is applicable for a rough surface with a large surface curvature, SSA and IEM bridge the gap between the SPM and KA Results from IEM are in good agreement with the ones obtained f rom SPM and KA in their respective region of holding  and [ 4 1] According to Thoma et al., IEM is currently the most widely used scattering model for bare soil studies [ 26] with a validity surface RMS h [ 4 1 ]. IEM has been successfully validated in laboratory settings but has yield ed contradictory results from real field experiments. Explanations for the discrepancy have been offered in the literature by accounting for several factors: the hig h heterogeneity of roughness and moisture conditions present in agricultural fields, physical approximations considered in the model which had not been checked a posteriori and finally that the current mathematical description of the surface roughness is not sufficient to capture its real complexity .
138 Several researchers have recognized the limitation in the description of surface roughness and have proposed alternate models to better explain the microwave backscattering. Zribi et al. has proposed a model for the backscattering on soil structure described by plane facets [12 ]. Mattia and Toan proposed an enhancement to IEM that described surface roughness as a multiscale fractal random process [ 55]. All the above models only consider the effect of random roughn ess The effects of artificial machined agricultural structures (oriented roughness) such as rows and furrows on microwave backscattering are considered by other analytical models. Ulaby et al. in  considers the case of a random roughness component super im posed over a stationary periodic surface along one dimension as a description of row structure typical of many agricultural fields and mathematically characterized by: y Cos A n y z y x z2 1 (A 1 1) w here A is the amplit ude of the periodic oscillations and is its spatial period . Kong et al. provides a solution for the emission of one such surface by applying modal theory , while Johnson et al. provides solutions for the scattering and emis sion of periodic surfaces in one and two dimensions . These physical analytical models provide estimates of the microwave emission and scattering based on the dielectric constant of the soil, the surface roughness and the sensor parameters (polarizat ion, wavelength, and incidence angle). To obtain the dielectric constant based on the soil composition and its water content several mixing models exist To learn more of these mixing models refer to [ 58] [ 61].
139 APPENDIX B IMPROVEMENT TO THE M ESHBOARD PROFILE DIGITIZATION METHOD The meshboard used to record surface profiles is a 2.14 m long by 0.9 m high and 0.003 m thick aluminum plate marked with a 2 cm x 2 cm grid as shown in Figure A2 1 To obtain a profile the meshboard is placed over the study surf ace, then a digital photo is taken as perpendicular to the meshboard as possible. The photograph is then loaded into imageprocessing software where the profile is manually digitized by an operator or by an automatic edge detection algorithm. The digitized profile then needs to be converted from pixel coordinates to meshboard or real space coordinates by a transformation model. For the first meshboard data collections, the meshboard photo was loaded into the image processing software SigmaScan Pro5, where the user selected control points on the meshboard grid and then digitized the soil profile. One of the digitizing methods employed consisted in picking a height measurement at every point where the slope of the surface is changing or at least every 1 cm. F or one 2.14 m long mesh board, each digitized picture yields 214 data points if there is no missed sampling point. The digitize profile was output into a MS Excel csv file with two columns for the x and y coordinate pairs. A more detailed description of the meshboard photo digitizing process using SigmaScan Pro can be found in [ 3 6] (Mi Young et al., 2005). The use of the proprietary software SigmaScan P ro was abandoned because early comparisons between profiles digitized using LiDAR and Meshboard showed sy stematic errors in the meshboard data causing the edges of the profiles to curve up or down. Figure A22 illustrates these distortions result ing from digitizing a flat smooth surface. To try to understand and correct these errors and to have full control over the entire
140 process Matlab scripts were developed to digitize and process the meshboard data which are described in the next paragraphs. When deriving measurements from a photograph, it is important to take into account that the image is subjected to s patial deformation due to imaging geometry and e ffects introduced by the lens T he t raditional photogrammetry approach used to obtain accurate measurements from photos consist of first characterizing the lens by determining its systematic distortion parameters, removing those effects present in the image and then determining the scale factors necessary to convert the pixel coordinates to real space coordinates through different transformation models  This approach is not applicable when using inexpensi ve digital cameras with an autofocus option because the distortion parameters vary for each photo taken. A novel two step approach was followed to minimize the effects of distortion and parallax. First 5 control points for which coordinates in both im age and meshboard coordinates w e re kn own w e re used to determine the parameters of an affine transformation. This transformation is applied to both the study profile and 107 points along one of the marked horizontal lines of the meshboard just on top of the soi l profile. The transformed profile and horizontal line contain the effects of geometric distortion but because derived and true real space coordinates are available for the horizontal line points it is possible to calculate coordinate differences between t hese coordinates as function of the pos ition over the meshboard. Thi s allows the researcher to characterize and remove the distortion present in the digitized profile. For this work a Canon PowerShot S2IS 5 megapixel camera was employed to obtain the meshboard digital images. To minimize the effect of lens distortion and
141 parallax, the photos were taken such that meshboard and the cameras focal plane were parallel the meshboard covered between 80 and 90% of the frame, and the study profile was as close as possible to the horizontal axis of the lens To process the photograph the Matlab script loads it to memory and in the first step displays the entire picture on the screen. At that zoom level it is hard to digitize control points and the profiles in an accurate fashion. So the user then uses the pointer to position and select 12 points that correspond to upper left and lower right corners of 6 areas to be zoomed into as marked on Figure A2 1. The first five of these six areas contain 5 control points mark ed on the meshboard. T he last area contains the profile to be digitized. After the twelve points have been selected five windows will automatically be displayed one at a time for which the user has to position and click over the displayed control point as accurately as possible. From these control points both image and meshboard coordinates are known, so it is possible to determine a single scale factor or the parameters of an affine transformation. The single scale factor has been used by some research ers to obtain a digitized profile in meshboard coordinates and it can be determined as: pixels j j i i m pixels length mb meters length mb ssfcp cp cp cp 2 1 2 2 1 21336 2 ( A 2 1) w here icp2 and icp1 are the pixel row coordinates of control points 2 and 1, corresponding to the meshboard top right and left corners. jcp2 and jcp1 are the pixel column coordinates of control points 2 and 1. The affine transformation model is:
142 F j E i D Y C j B i A X F j E i D Y C j B i A X F j E i D Y C j B i A X cp cp cp cp cp cp cp cp cp cp cp cp 5 5 5 5 5 5 2 2 2 2 2 2 1 1 1 1 1 1 ( A 2 2) o r in matrix form: F E D C B A j i j i j i j i Y X Y Xcpn n cp cpn n cp cp cp cp cp cpn cpn cp cp1 0 0 0 0 0 0 1 1 0 0 0 0 0 0 11 1 1 1 1 1 ( A 2 3) w here icp n and jcp n are the pixel row and column coordinates of control point n and Xcpn and Ycpn are the meshboard metric coordinates of control point n. A, B, C, D, E and F are the parameters of the affine transformation of which A, B, D and E are scale factors and C and F are translations [ 62]. These parameter s are determined through a least squares procedure. After the control points are selected and transformations are computed, six windows are sequentially displayed containing 6 segments of the profile at an appropriate zoom level to be digitized accurately. The user then digitizes the profile on the computer screen using the mouse cursor T he script records the digitized pixel coordinates After digitizing the study profile, the user has to digitize the 107 intersection points of the vertical lines along one of the marked horizontal lines of the meshboard just on top of the soil profile for is uses in the correction of the remaining distortions. For this purpose a new set of 6 windows are sequentially displayed containing 6 segments
143 of the meshboard at an appropriate zoom level. After the profile and horizontal line are digitized the Matlab script uses the single scale factor and the 6 parameters of the affine transformation to convert the digitized profiles and horizontal line in pixel coordinates to meshboar d coordinates by applying: n n n n n npp pp pp pp pp ppi i ssf Y j j ssf X max min ( A 2 4) and D j E i D Y C j B i A Xn n n n n npp pp pp pp pp pp ( A 2 5) Equation A24 is applicable for the s ingle scale transformation model and Equation A2 5 is applied for the affine transformation mode, w here ssf is the single sc ale factor determined accor ding to Equation A21, Xppn and Yppn are the metric meshboard coordinates for the nth profile point, and ipp n and jppn are the pixel row and column coordinates of the nth digitized profile point. A, B, C, D, E and F are the scale and transformation parameters. Fig ure A 3 2 provides an illustration of the distortions suffered by a smooth flat surface (RMSh = 0) when digitized with the meshboard not parallel to the focal plane of the camera for different transformation techniques Note that besides the curved distortion the meshboard length is distorted from 2.13 to 2.5 m. To remove the distortion effects the coordinates from the 107 interception points of the vertical lines along one of the marked horizontal lines of the meshboard are used. Because derived and true real metric meshboard coordinates are available for these points it is possible to calculate differences between these coordinates as function of the position over the meshboard of the form:
144 det detn n n nlp True lp lp True lpY Y y X X x ( A 2 6) w here Xlp True and Ylp True are the true x and y coordinates meshboard for the line interception points and Xlp det and Ylp det are the determined coordinates after the application of the affine transformation. These coordinate differ entials are plotted as dots in Figure A2 3. These differentials were then modeled as high order polynomial following the forms: x l x l x l x l x l x l x lG X F X E X D X C X B X A X x f det 2 det 3 det 4 det 5 det 6 det det ( A2 7) y l y l y l y lD X C X B X A X y f det 2 det 3 det det ( A 2 8) The coefficients Ax, Bx, Cx, Dx, Ex, Fx, Gx, Ay, By, Cy and Dy are determined via lea st squares. O nce the coefficients are known E quations A2 7 and A2 8 are applied to obtain the correction factors for each individual point of the digitized profile, the final corrected profile coordinate points are given by: det det l lp corr lpX x f X Xn n ( A 2 9) det det l lp corr lpX y f Y Yn n ( A 2 10) Figure A2 2 shows the final corrected profile. Once the profile is corrected the Matlab script writes a text file containing the x and y coordinates for the profile points.
145 Figure A21. Meshboard and the marked control points. 0.01 0.005 0 0.005 0.01 0 0.5 1 1.5 2 2.5 random component (m) displacement (m) 1 Scale Affine Corrected Figure A22. Flat smooth profile at different processing steps.
146 0 0.5 1 1.5 2 2.5 -0.06 -0.05 -0.04 -0.03 -0.02 -0.01 0 0.01 0.02 Distortion Corrections Meshboard x [m]Coordinate Differential [m] dx dy tdx tdy Figure A23. Coordinate differentials and distortion correction factors.
147 APPENDIX C SELECTED MATLAB SCRI PTS Fit a Plane and Level a Point Cloud planefit.m %fit plane clear all close all clc %Load XYZ Point Cloud File data=load('F:\ TEMP Juan Backup\ 68 Roughness Calibration\ CoastalLab_floor_FF\ xyz\ CoastalLab_floor_FF_CropB.xyz'); sized=size(data); npoints=(sized(1)); minx=min(data(:,1)) miny=min(data(:,2)) meanz=mean(data(:,3)) %Fit a plane to the Points Z=aX+bY+c X=data(:,1); Y=data(:,2); Z=data(:,3); obs=[X,Y,ones(npoints,1)]; par=obs\ Z %par=[a;b;c] %Residuals res=obs*parZ; res2=res.*res; rms=sqrt(sum(res2)/npoints) %Normal Vector NV=[par(1);par(2);1] %Rotations to level the plane Rx=atan(par(2)/1) Ry=1*atan(par(1)/1) Rxdeg=Rx*180/pi() Rydeg=Ry*180/pi() 2D Detrending Function detrend_fn_np.m function [R2,rc] = detrend_fn_np(prf)
148 % %Detrend %USAGE: [cl,R2exp,R2exp2l,R2gau,R2gau2l,n0,R2pls,R2pls2l] = detrend_fn_np(prf) %Input (prf) 2D vector %Output [npoints,ncols]=size(prf);%Number of points & Number of Columns sspace=prf(2,1)prf(1,1);% Sample or Grid Space %Detrend %First order par_1=[prf(:,1),ones(npoints,1)]\ prf(:,2); %Trend Parameters t_1=prf(:,1)*par_1(1)+par_1(2); % Z values Trend rc_1=prf(:,2)t_1;% Z Random Component R2_1=1(rc_1'*rc_1)/(var(prf(:,2))*npoints);%R2 of trend %Second order par_2=[prf(:,1).^2,prf(:,1),ones(npoints,1)]\ prf(:,2); t_2=prf(:,1).^2*par_2(1)+prf(:,1)*par_2(2)+par_2(3); rc_2=prf(:,2)t_2; R2_2=1(rc_2'*rc_2)/(var(prf(:,2))*npoints); %FFT filter nfsamples=round(sspace*(npoints1)); prf_f=fft(prf(:,2)); prf_fm=prf_f; prf_fm(1:nfsamples+1)=0; prf_fm(npointsnfsamples+1:npoints)=0; rc_f=real(ifft(prf_fm)); t_f=prf(:,2)rc_f; R2_f=1(rc_f'*rc_f)/(var(prf(:,2))*npoints); %Group Ouput R2=[R2_1,R2_2,R2_f]; rc=[rc_1,rc_2,rc_f]; 2D Roughness Parameters Determination Function acf_fn_np.m function [acf_res] = acf_fn_np(acf) % %ACF Analysis function %USAGE: [cl,R2exp,R2exp2l,R2gau,R2gau2l,n0,R2pls,R2pls2l] = acf_fn(acf) %Input (acf) normalized ACF in 2 cols, first lags %Output clear i0; clear icl; [npoints,ncols]=size(acf);
149 %Determine correlation length i0=ceil(npoints/2)+1; icl=ceil(npoints/2); while acf(icl,2)>=1/exp(1);%find the 1/e crossing point icl=icl+1; end cl=interp1(acf(icl1:icl,2),acf(icl1:icl,1),1/exp(1));%Correlation Length %Build ACF expacf=exp(1*abs(acf(:,1))/cl);%exponential autocorrelation function gauacf=exp(1*(acf(:,1).*acf(:,1))/cl^2);%Gaussian Autocorrelation function diffexp=expacfacf(:,2); R2exp=1(diffexp'*diffexp)/((std(acf(:,2)))^2*npoints); %R2exp2l=1(diffexp(i01:icl)'*diffexp(i01:icl))/(acf(i01:icl,2)'*acf(i01:icl,2)); R2exp2l=1(diffexp(2*i0icl2:icl)'*diffexp(2*i0icl2:icl))/((std(acf(2*i0icl2:icl,2)))^2*(2*(icli0)+3)); diffgau=gauacfacf(:,2); R2gau=1(diffgau'*diffgau)/((std(acf(:,2)))^2*npoints); %R2gau2l=1(diffgau(i01:icl)'*diffgau(i01:icl))/(acf(i01:icl,2)'*acf(i01:icl,2)); R2gau2l=1(diffgau(2*i0icl2:icl)'*diffgau(2*i0icl2:icl))/((std(acf(2*i0icl2:icl,2)))^2*(2*(icli0)+3)); %Fit the Power Law Spectrum n0=1.5; %Initial Guess dn=0.2; obs=(acf(:,2)); while abs(dn)>0.05; Un=(abs(acf(:,1))/cl).^n0; cmp=exp(1*Un); dfdn=1.*cmp.*Un.*log(abs(acf(:,1))/cl); dn=dfdn(i0:icl+1,1)\ (obs(i0:icl+1,1)cmp(i0:icl+1,1)); n0=n0+dn; end Un=(abs(acf(:,1))/cl).^n0; cmp=exp(1*Un); diffpls=cmpacf(:,2); R2pls=1(diffpls'*diffpls)/((std(acf(:,2)))^2*npoints); %R2pls2l=1(diffpls(i01:icl)'*diffpls(i01:icl))/(acf(i01:icl,2)'*acf(i01:icl,2)); R2pls2l=1(diffpls(2*i0icl2:icl)'*diffpls(2*i0icl2:icl))/((std(acf(2*i0icl2:icl,2)))^2*(2*(icli0)+3));
150 %Concatenate results acf_res=[cl,R2exp,R2exp2l,R2gau,R2gau2l,n0,R2pls,R2pls2l]; 2D Accuracy Assessment with Roughness References cal_v2_05_20mm_1cm_ABCD.m clear all; clc; close all; %Load Referece Profile ref=load('C:\ Dissertation\ Data\ calibration\ 20mmRMS_1cmCL_ABCD\ 20mmRMS_1cmCL_A BCD_ref.txt'); sizeref=size(ref); nrefpt=sizeref(1); refsp=(ref(2,1)ref(1,1)); %Load Meshboard data mbd=load('C:\ Dissertation\ Data\ calibration\ 20mmRMS_1cmCL_ABCD\ 20mmRMS_1cmCL_A BCD_mbcor.txt'); sized=size(mbd); npoints=(sized(1)); %Obtain extremes minx=min(mbd(:,1)); maxx=max(mbd(:,1)); miny=min(mbd(:,2)); maxy=max(mbd(:,2)); %Load LiDAR Derived Grid X=load('C:\ Dissertation\ Data\ calibration\ 20mmRMS_1cmCL_ABCD\ 9mmbins2\ X.grd'); %FOR BINS Y=load('C:\ Dissertation\ Data\ calibration\ 20mmRMS_1cmCL_ABCD\ 9mmbins2\ Y.grd'); Z=load('C:\ Dissertation\ Data\ calibration\ 20mmRMS_1cmCL_ABCD\ 9mmbins2\ Zmax.grd ); gsize=size(X); nrows=gsize(1); ncols=gsize(2); grdsp_l=X(1,2)X(1,1); %Extract Profiles frow=21;%%%%%UPDATE THIS startc=10;%%%AND THIS endc=265;%%%%AND THIS cbefore=startc1; cafter=ncolsendc; procl=[X(frow,1:ncolscbeforecafter);(Z(frow,startc:endc)+Z(frow+1,startc:endc))/2]';%Selected Profile %procl=[X(frow,1:ncolscbeforecafter);Z(frow,startc:endc)]';%Selected Profile
151 %procl=[X(frow,:);(Z(frow,:)+Z(frow+1,:))/2]';%Selected Profile %%%%Detrend Profiles %Reference param_ref=[ref(:,1),ones(nrefpt,1)]\ ref(:,2);%[m,b], [X 1]\ Y tr_ref=ref(:,1)*param_ref(1)+param_ref(2); randc_rf=ref(:,2)tr_ref; %Meshboard param_mb=[mbd(:,1),ones(npoints,1)]\ mbd(:,2);%[m,b], [X 1]\ Y tr_mb=mbd(:,1)*param_mb(1)+param_mb(2); randc_mb=mbd(:,2)tr_mb; %Interpolate Random Component Meshboard Data grdsp=(maxxminx)/nrefpt;%Gridspacing xvec=(minx:grdsp:maxx)'; %randc_mb_g=interp1(mbd(:,1),randc_mb,xvec);%Linea Interpolation randc_mb_g=interp1(mbd(:,1),randc_mb,xvec,'nearest'); ngrids=size(xvec); nsamples=ngrids(1,1); %LiDAR %param_l=[procl(:,1),ones(ncols,1)]\ procl(:,2);%[m,b], [X 1]\ Y param_l=[procl(:,1),ones(ncolscbeforecafter,1)]\ procl(:,2);%[m,b], [X 1]\ Y FOR BINS tr_l=procl(:,1)*param_l(1)+param_l(2); randc_l=procl(:,2)tr_l; %%Compute RMS of Height %Reference std_ref=std(randc_rf,1)% Standard Deviation %Meshboard std_mb=std(randc_mb,1) std_mb_g=std(randc_mb_g,1) %LiDAR std_l=std(randc_l,1) %%Compute R2 %Meshboard wrt to Reference diff_mb=randc_mbrandc_rf; R2_mb=1(diff_mb'*diff_mb)/(std_mb^2*npoints) %LiDAR wrt to Reference diff_l=randc_lrandc_rf; R2_l=1(diff_l'*diff_l)/(std_l^2*nrefpt)
152 %Compute Correlation Coefficient %Mesboard Reference ccm_mbrf=corrcoef(randc_rf,randc_mb); cc_mb=ccm_mbrf(1,2) %Lidar Reference ccm_lrf=corrcoef(randc_rf,randc_l); cc_l=ccm_lrf(1,2) %%Compute AutoCorrelation %Reference hcorr_ref=xcorr(randc_rf); maxhcorr_ref=max(hcorr_ref); nhcorr_ref=1/maxhcorr_ref*hcorr_ref; lagsv_ref=(((1*nrefpt+1):1:(nrefpt1))*refsp)'; i_ref=nrefpt; while nhcorr_ref(i_ref)>=1/exp(1);%find the 1/e crossing point i_ref=i_ref+1; end corl_ref=interp1(nhcorr_ref(i_ref1:i_ref,1),lagsv_ref(i_ref1:i_ref,1),1/exp(1))%Correlation Length % expaf_mb=exp(1*abs(lagsv_mb)/corl_mb);%exponential autocorrelation function % gauaf_mb=exp(1*(lagsv_mb.*lagsv_mb)/corl_mb^2);%Gaussian Autocorrelation function %Meshboard hcorr_mb=xcorr(randc_mb_g); maxhcorr_mb=max(hcorr_mb); nhcorr_mb=1/maxhcorr_mb*hcorr_mb; lagsv_mb=(((1*nsamples+1):1:(nsamples1))*(xvec(2,1)xvec(1,1)))'; i_mb=nsamples; while nhcorr_mb(i_mb)>=1/exp(1);%find the 1/e crossing point i_mb=i_mb+1; end corl_mb=interp1(nhcorr_mb(i_mb1:i_mb,1),lagsv_mb(i_mb1:i_mb,1),1/exp(1))%Correlation Length % expaf_mb=exp(1*abs(lagsv_mb)/corl_mb);%exponential autocorrelation function % gauaf_mb=exp(1*(lagsv_mb.*lagsv_mb)/corl_mb^2);%Gaussian Autocorrelation % function % % %%LiDAR hcorr_l=xcorr(randc_l); maxhcorr_l=max(hcorr_l); nhcorr_l=1/maxhcorr_l*hcorr_l; %lagsv_l=((1*(ncols)+1):1:(ncols1))*grdsp_l; %i_l=(ncols);
153 lagsv_l=((1*(ncolscbeforecafter)+1):1:(ncolscbeforecafter1))*grdsp_l;% FOR BINS i_l=(ncolscbeforecafter); while nhcorr_l(i_l)>=1/exp(1);%find the 1/e crossing point i_l=i_l+1; end corl_l=interp1(nhcorr_l(i_l1:i_l,1),lagsv_l(1,i_l1:i_l),1/exp(1))%Correlation Length % expaf_l=exp(1*abs(lagsv_l)/corl_l);%exponential autocorrelation function % gauaf_l=exp(1*(lagsv_l.*lagsv_l)/corl_l^2);%Gaussian Autocorrelation function %%%%%%%%%%% %Plot Data figure(1); set(gcf,'Color',[1,1,1]); subplot(4,1,1) plot(ref(:,1),ref(:,2)) title('Reference Profile'); xlabel('Length [m]'); ylabel('Height [m]'); axis ([0 2.5 0.9*min(ref(:,2)) 1.1*max(ref(:,2))]); grid; subplot(4,1,2) plot(mbd(:,1),mbd(:,2)) title('Profile from Meshboard'); xlabel('Length [m]'); ylabel('Height [m]'); axis ([0 2.5 0.9*min(ref(:,2)) 1.1*max(ref(:,2))]); grid; subplot(4,1,3) imagesc([X(1,1),X(1,ncols)],[Y(1,1),Y(nrows,1)],Z); title('Gridded Surface from LiDAR') %imagesc([minx,maxx],[miny,maxy],intth); xlabel('Length [m]'); ylabel('Depth [m]'); axis xy; axis image; %colorbar; subplot(4,1,4) plot(procl(:,1),procl(:,2)) title('Extracted Profile from LiDAR Grid'); xlabel('Length [m]'); ylabel('Height [m]'); axis ([0 2.5 min(min(Z)) max(max(Z))]); grid; figure(2) set(gcf,'Color',[1,1,1]);
154 plot(ref(:,1),randc_rf,xvec,randc_mb_g); title('Random Component extracted from profiles'); legend('Reference', 'Meshboard'); xlabel('Length [m]'); ylabel('Height [m]'); grid figure(3) set(gcf,' Color',[1,1,1]); plot(ref(:,1),randc_rf,ref(:,1),randc_mb); title('Random Component extracted from profiles Refrence X'); legend('Reference', 'Meshboard'); xlabel('Length [m]'); ylabel('Height [m]'); grid figure(4) set(gcf,'Color',[1,1,1]); plot(ref(:,1),randc_rf,procl(:,1),randc_l); title('Random Component extracted from profiles'); legend('Reference', 'LiDAR'); xlabel('Length [m]'); ylabel('Height [m]'); grid figure(5) set(gcf,'Color',[1,1,1]); plot(ref(:,1),randc_rf,ref(:,1),randc_l); title('Random Component extracted from profilesRefrence X'); legend('Reference', 'LiDAR'); xlabel('Length [m]'); ylabel('Height [m]'); grid figure(6);%AutoCorrelation Functions set(gcf,'Color',[1,1,1]); plot(lagsv_ref,nhcorr_ref,lagsv_mb,nhcorr_mb,lagsv_l,nhcorr_l) line([2.4 2.4],[1/exp(1) 1/exp(1)],'Color',[0 0 0])%1/e axis line grid; xlabel('X Lags [Meters]'); ylabel('Normalized AutoCorr'); title('Height AutoCorrelation'); axis ([2.4 2.4 0.2 1.2]); legend('Reference', 'Meshboard', 'LiDAR'); figure(7);% Difference set(gcf,'Color',[1,1,1]); plot(ref(:,1),diff_mb,ref(:,1),diff_l); title('Height Difference wrt Reference not considering x'); legend('Meshboard', 'LiDAR'); grid;
155 3D Correlation Length from DEMs plotterraincorr.m close all; clear all; clc; %Define Path path='C:\ Dissertation\ Data\ 20060308\ Sample4x4\ 10mmgrids\ ; %path='C:\ Dissertation\ Data\ 200711SL\ 10mmGrids\ '; %Load data files xgrid=load(strcat(path,'X.grd')); ygrid=load(strcat(path,'Y.grd')); zgrid=load(strcat(path,'Z2dtdr.grd')); %zgrid=load(strcat(path,'Z2dt.grd')); corrgrid=load(strcat(path,'xcorZ2dtdr.grd')); %corrgrid=load(strcat(path,'xcorZ2dt.grd')); sizearray=size(xgrid); ncol=sizearray(2); nrows=sizearray(1); %Derive Plot Limits minx=xgrid(1,1); maxx=xgrid(1,ncol); miny=ygrid(1,1); maxy=ygrid(nrows,1); maxcorr=corrgrid(nrows,ncol); gridspc=xgrid(1,2)xgrid(1,1); %Compute grid Height Mean and RMS meanz=mean(mean(zgrid)); meanzgrid=meanz*ones(nrows,ncol); hdiffgrid=zgridmeanzgrid; sdiffgrid=hdiffgrid.*hdiffgrid; rmshg=sqrt(mean(mean(sdiffgrid))) %Compute normalized Auto Correlation nrmcorr=1/maxcorr*corrgrid; %Create x & y grids for Coorelation Array [Xcor,Ycor] = meshgrid((1*(ncol1)):1:(ncol1),1*(nrows1):1:(nrows1)); %test Contour Def [ctest,htest]=contour(nrmcorr,[1/exp(1) 1/exp(1)]); sizectestv=size(ctest); npcnt=sizectestv(2); %Obtain contour in absolute coordinates orix=ncol*ones(1,npcnt);
156 oriy=nrows*ones(1,npcnt); fncntr=[(ctest(1,:)orix)',(ctest(2,:)oriy)']*gridspc;%Final Contour fncntr(1,:)=;%Remove First countour point % Compute Contour Distance vectors & Mean Distance %Distance to Each Point on the Coutour corlvec=sqrt((ctest(1,:)orix).^2+(ctest(2,:)oriy).^2);% grid Units clvt=corlvec'*gridspc;%in Meters ang=(rad2deg(atan2(ctest(2,:)oriy,ctest(1,:)orix)))'; %Remove Elements clvt(1,:)=; ang(1,:)=; %clvt(469,:)=; %ang(469,:)=; %Obtain Results meancorl=mean(corlvec); stdcorl=std(corlvec); % Results [min max mean std] RES=gridspc*[min(corlvec(1,2:npcnt)),max(corlvec(1,2:npcnt)),mean(corlvec(1,2 :npcnt)),std(corlvec(1,2:npcnt))] binmin=floor(meancorl2*stdcorl); binsize=round(((meancorl+2*stdcorl)(meancorl2*stdcorl))/40); binvec=binmin:binsize:binsize*42+binmin; %dlmwrite(strcat(path,'CorlV_ZFdtdr.txt'), [ang,clvt], 'delimiter', ,'newline', 'pc','precision', 6); %Plot Graphics figure(2); set(gcf,'Color',[1,1,1]); subplot(1,2,1) imagesc([minx,maxx],[miny,maxy],zgrid); title('Terrain Height'); axis xy; axis image; colorbar; xlabel('X [Meters]') ylabel('Y [Meters]') subplot(1,2,2) imagesc([1*ncol*gridspc,ncol*gridspc],[1*nrows*gridspc,nrows*gridspc],nrmcorr); title('Height Autocorrelation'); axis xy;
157 axis image; colorbar; xlabel('X Lags [Grid units]') ylabel('Y Lags [Grid units]') figure(7); set(gcf,'Color',[1,1,1]); subplot(1,2,1) plot(fncntr(:,1),fncntr(:,2)) axis([1*ncol*gridspc,ncol*gridspc,1*nrows*gridspc,nrows*gridspc]) xlabel('X Lags [m]'); ylabel('Y Lags [m]'); subplot(1,2,2) plot(ang,clvt, '.') xlabel('Angle from the X axis [deg]'); ylabel('Correlation length [m]'); RMSh from Single Scan Line from Airborne LiDAR Data AbPC_Rssl_v1.m clear all; close all; clc; %Define Desired Profile Length pl=3; %Load Pointcloud data pc=load('C:\ Dissertation\ Data\ 20100902_Airborne\ Strip009\ S1C1_ascii_245b_0091_c.txt'); % 1 Time, 2 Easting, 3 Northing, 4 Height, 5 Intensity, 6 Range, 7 ScanAngle [npoints,ncols]=size(pc); %Create Empty arrays RMSh=NaN(npoints,1); %Extract Arrays xy1=pc(1:npoints1,2:3); xy2=pc(2:npoints,2:3); %Compute Distance dist=sqrt((xy2(:,1)xy1(:,1)).^2+(xy2(:,2)xy1(:,2)).^2); mdist=mean(dist); %Compute scan angle difference between succesive points dsa=diff(pc(:,7)); %Determine number of points in profile npp=pl/mdist; npph=ceil(npp/21);
158 %Cycle through Point Cloud for i=npph+1:(npoints npph); %obtain vectors hv=pc(inpph:i+npph,4); % create displacement vector dv=zeros(npph*2+1,1); for j=2:npph*2+1 dv(j)=dv(j1)+dist(inpph+j2); end %linear detrend of profile par=[dv,ones(npph*2+1,1)]\ hv;%aD+b=Z tr=par(1)*dv+par(2)*ones(npph*2+1,1); %Obtain RMSh RMSh(i)=std(hvtr); end %Fill first and last values RMSh(1:npph)=RMSh(npph+1); RMSh(npointsnpph+1:npoints)=RMSh(npointsnpph); %Write XYZis PC %dlmwrite('C:\ Dissertation\ Data\ 20100902_Airborne\ Strip009\ 245b_0091_c_xyzis.xyz', [pc(:,2:5),RMSh],'delimiter',' ','newline', 'pc','precision', 10, 'append'); Meshboard Digitizing and Correction meshboard_C.m %Mesboard Digitization and Distortion Correction clear all; close all; clc; tic; %Set Path path='C:\ Dissertation\ Data\ 20100903\ 20100903_PD&MA\ ; outnm='PD&MA_A'; %Load Image
159 MBimg=imread(strcat(path,'IMG_9733.JPG')); imsize=size(MBimg); imrows=imsize(1); imcols=imsize(2); %Display Image figure(1); set(gcf,'Color',[1,1,1]); imshow(MBimg); %Get Corners of ROI %Top Left and Bottom Right points around: %1,2. MB TL corner %3,4. MB TR corner %5,6. Mark @ 100,41 %7,8. Mark @ 10,51 %9,10. Mark @ 200,11 %11,12. Profile Extremes [xc, yc] = getpts; controlp=fix([xc,yc]); %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %Display Subimages to obtain control points %Control Point 1 close all; figure(1); set(gcf,'Color',[1,1,1]); imshow(MBimg(controlp(1,2):controlp(2,2),controlp(1,1):controlp(2,1),:)); [xc1, yc1] = getpts; %Control Point 2 close all; figure(1); set(gcf,'Color',[1,1,1]); imshow(MBimg(controlp(3,2):controlp(4,2),controlp(3,1):controlp(4,1),:)); [xc2, yc2] = getpts; %Control Point 3 close all; figure(1); set(gcf,'Color',[1,1,1]); imshow(MBimg(controlp(5,2):controlp(6,2),controlp(5,1):controlp(6,1),:)); [xc3, yc3] = getpts; %Control Point 4 close all; figure(1); set(gcf,'Color',[1,1,1]); imshow(MBimg(controlp(7,2):controlp(8,2),controlp(7,1):controlp(8,1),:)); [xc4, yc4] = getpts; %Control Point 5 close all; figure(1); set(gcf,'Color',[1,1,1]);
160 imshow(MBimg(controlp(9,2):controlp(10,2),controlp(9,1):controlp(10,1),:)); [xc5, yc5] = getpts; %Single scale scf=2.1336/sqrt((xc2+xc(3)xc1xc(1))^2+(yc2+yc(3)yc1yc(1))^2);%m/pixel %Affine Transformation Parameters %X=Ay+Bx+C %Y=Dy+Ex+F %X,Y Real Coordinates, x,y Pixel Coordinates CPC=[0;0.9144;2.1336;0.9144;1;.41;.1;.51;2;.11]; Amat=[yc1+yc(1),xc1+xc(1),1,0,0,0; 0,0,0,yc1+yc(1),xc1+xc(1),1; yc2+yc(3),xc2+xc(3),1,0,0,0; 0,0,0,yc2+yc(3),xc2+xc(3),1; yc3+yc(5),xc3+xc(5),1,0,0,0; 0,0,0,yc3+yc(5),xc3+xc(5),1; yc4+yc(7),xc4+xc(7),1,0,0,0; 0,0,0,yc4+yc(7),xc4+xc(7),1; yc5+yc(9),xc5+xc(9),1,0,0,0; 0,0,0,yc5+yc(9),xc5+xc(9),1]; param=Amat\ CPC; % [A B C D E F]' %# Pixels Meshboard Profile npixels=controlp(12,1)controlp(11,1); npixelinc=floor(npixels/6); %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %Display Subimages to obtain Profile points %Profile Fragment 1/6 close all; figure(1); set(gcf,'Color',[1,1,1]); imshow(MBimg(controlp(11,2):controlp(12,2),controlp(11,1):controlp(11,1)+npix elinc,:)); [xp1, yp1] = getpts; %Profile Fragment 2/6 close all; figure(2); set(gcf,'Color',[1,1,1]); imshow(MBimg(controlp(11,2):controlp(12,2),controlp(11,1)+npixelinc:controlp( 11,1)+npixelinc*2,:)); [xp2, yp2] = getpts; %Profile Fragment 3/6 close all; figure(3); set(gcf,'Color',[1,1,1]); imshow(MBimg(controlp(11,2):controlp(12,2),controlp(11,1)+npixelinc*2:control p(11,1)+npixelinc*3,:)); [xp3, yp3] = getpts;
161 %Profile Fragment 4/6 close all; figure(4); set(gcf,'Color',[1,1,1]); imshow(MBimg(controlp(11,2):controlp(12,2),controlp(11,1)+npixelinc*3:control p(11,1)+npixelinc*4,:)); [xp4, yp4] = getpts; %Profile Fragment 5/6 close all; figure(5); set(gcf,'Color',[1,1,1]); imshow(MBimg(controlp(11,2):controlp(12,2),controlp(11,1)+npixelinc*4:control p(11,1)+npixelinc*5,:)); [xp5, yp5] = getpts; %Profile Fragment 6/6 close all; figure(6); set(gcf,'Color',[1,1,1]); imshow(MBimg(controlp(11,2):controlp(12,2),controlp(11,1)+npixelinc*5:control p(11,1)+npixelinc*6,:)); [xp6, yp6] = getpts; %Concatenate Profile Points in Pixel Coordinates ppcoor=[xp1+xc(11),yp1+yc(11);xp2+xc(11)+npixelinc*1,yp2+yc(11);xp3+xc(11)+np ixelinc*2,yp3+yc(11);xp4+xc(11)+npixelinc*3,yp4+yc(11);xp5+xc(11)+npixelinc*4 ,yp5+yc(11);xp6+xc(11)+npixelinc*5,yp6+yc(11)]; %Final Profile Points in MB Coordinates fp=[ppcoor*[param(2);param(1)]+param(3),ppcoor*[param(5);param(4)]+param(6)]; %Single Scale ssp=scf*[ppcoor(:,1)min(ppcoor(:,1)),max(ppcoor(:,2))ppcoor(:,2)]; %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %Display Subimages to obtain correction line %Profile Fragment 1/6 close all; figure(1); set(gcf,'Color',[1,1,1]); imshow(MBimg(controlp(11,2):controlp(12,2),controlp(11,1):controlp(11,1)+npix elinc,:)); [xl1, yl1] = getpts; %Profile Fragment 2/6 close all; figure(2); set(gcf,'Color',[1,1,1]); imshow(MBimg(controlp(11,2):controlp(12,2),controlp(11,1)+npixelinc:controlp( 11,1)+npixelinc*2,:)); [xl2, yl2] = getpts;
162 %Profile Fragment 3/6 close all; figure(3); set(gcf,'Color',[1,1,1]); imshow(MBimg(controlp(11,2):controlp(12,2),controlp(11,1)+npixelinc*2:control p(11,1)+npixelinc*3,:)); [xl3, yl3] = getpts; %Profile Fragment 4/6 close all; figure(4); set(gcf,'Color',[1,1,1]); imshow(MBimg(controlp(11,2):controlp(12,2),controlp(11,1)+npixelinc*3:control p(11,1)+npixelinc*4,:)); [xl4, yl4] = getpts; %Profile Fragment 5/6 close all; figure(5); set(gcf,'Color',[1,1,1]); imshow(MBimg(controlp(11,2):controlp(12,2),controlp(11,1)+npixelinc*4:control p(11,1)+npixelinc*5,:)); [xl5, yl5] = getpts; %Profile Fragment 6/6 close all; figure(6); set(gcf,'Color',[1,1,1]); imshow(MBimg(controlp(11,2):controlp(12,2),controlp(11,1)+npixelinc*5:control p(11,1)+npixelinc*6,:)); [xl6, yl6] = getpts; %Concatenate Line Points in Pixel Coordinates lpcoor=[xl1+xc(11),yl1+yc(11);xl2+xc(11)+npixelinc*1,yl2+yc(11);xl3+xc(11)+np ixelinc*2,yl3+yc(11);xl4+xc(11)+npixelinc*3,yl4+yc(11);xl5+xc(11)+npixelinc*4 ,yl5+yc(11);xl6+xc(11)+npixelinc*5,yl6+yc(11)]; %Final Line Points in MB Coordinates fl=[lpcoor*[param(2);param(1)]+param(3),lpcoor*[param(5);param(4)]+param(6)]; disp('Time to process Mesboard Image') toc %%%%%%%%%%%%%%%%Correct Profile for Distortions ref=fl;% Correction Reference Line Coordinates prof=fp;% Digitized Profile coordinates [nrefpt,ncols]=size(ref); %Create reference Vectors Xref=(0:.02:(nrefpt1)*.02)'; Yref=0.11*ones(nrefpt,1);%%%%%%UPDATE THIS WITH PROPER REFERENCE LINE dx=Xrefref(:,1);%Xx=dx dy=Yrefref(:,2);%Yy=dy
163 %Obtain Correction Polynomials %dx 6th order polynomial Ax^6+Bx^5+Cx^4+Dx^3+Ex^2+Fx+G param_dx=[ref(:,1).^6,ref(:,1).^5,ref(:,1).^4,ref(:,1).^3,ref(:,1).^2,ref(:,1 ),ones(nrefpt,1)]\ dx;%[A,B,C,D,E,F,G] tr_dx=ref(:,1).^6*param_dx(1)+ref(:,1).^5*param_dx(2)+ref(:,1).^4*param_dx(3) +ref(:,1).^3*param_dx(4)+ref(:,1).^2*param_dx(5)+ref(:,1)*param_dx(6)+param_d x(7); %dy 3th order polynomial Ax^3+Bx^2+Cx+D param_dy=[ref(:,1).^3,ref(:,1).^2,ref(:,1),ones(nrefpt,1)]\ dy;%[A,B,C,D] tr_dy=ref(:,1).^3*param_dy(1)+ref(:,1).^2*param_dy(2)+ref(:,1)*param_dy(3)+pa ram_dy(4); %Correct Profile X=prof(:,1)+prof(:,1).^6*param_dx(1)+prof(:,1).^5*param_dx(2)+prof(:,1).^4*pa ram_dx(3)+prof(:,1).^3*param_dx(4)+prof(:,1).^2*param_dx(5)+prof(:,1)*param_d x(6)+param_dx(7); Y=prof(:,2)+prof(:,1).^3*param_dy(1)+prof(:,1).^2*param_dy(2)+prof(:,1)*param _dy(3)+param_dy(4); %%%% Plots %Final Profile Pixel Coordinates close all; figure(1); set(gcf,'Color',[1,1,1]); plot(ppcoor(:,1),ppcoor(:,2),lpcoor(:,1),lpcoor(:,2)); title('Pixel Coordinates') xlabel('MBx [pixels]'); ylabel('MBy [Pixels]'); legend('Affine Profile', 'Cor Line') figure(2); set(gcf,'Color',[1,1,1]); plot(ssp(:,1),ssp(:,2),fp(:,1),fp(:,2),fl(:,1),fl(:,2),X,Y); title('MB Coordinates') xlabel('MBx [m]'); ylabel('MBy [m]'); legend('Single Sc', 'Affine' 'Cor Line', 'Corrected') figure(3) set(gcf,'Color',[1,1,1]); plot(ref(:,1),dx,ref(:,1),dy,ref(:,1),tr_dx,ref(:,1),tr_dy); title('Distortion Corrections'); legend('dx', 'dy', 'tdx', 'tdy'); xlabel('x [m]'); ylabel('y [m]'); grid %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %Write Output Files %_mbor for original, %_mbdxdy for Correction %_mbcor for corrected dlmwrite(strcat(path,outnm,'_mbor.txt'), fp, 'delimiter', ', 'newline', 'pc', 'precision', 5);% Distorted Profile
164 dlmwrite(strcat(path,outnm,'_mbss.txt'), ssp, 'delimiter', ', 'newline', 'pc', 'precision', 5);% Distorted Profile dlmwrite(strcat(path,outnm,'_mbdxdy.txt'), fl, 'delimiter', ', 'newline', 'pc', 'precision', 5); %Reference Line dlmwrite(strcat(path,outnm,'_mbcor.txt'), [X,Y], delimiter', ', 'newline', 'pc', 'precision', 5); %Corrected Profile dlmwrite(strcat(path,outnm,'_rep.txt'), 'Affine Transformation Parameters', 'delimiter', ', 'newline', 'pc'); %Report dlmwrite(strcat(path,outnm,'_rep.txt'), 'Ay+Bx+C ; Y=Dy+Ex+D ; [A B C D E F]' 'delimiter', ', 'newline', 'pc', append'); %Report dlmwrite(strcat(path,outnm,'_rep.txt'), param 'delimiter', ', 'newline', 'pc', 'precision', 5, append'); %Report dlmwrite(strcat(path,outnm,'_rep.txt'), 'delimiter', ', 'newline', 'pc', append'); %Report dlmwrite(strcat(path,outnm,'_rep.txt'), 'Single Scale' 'delimiter', ', 'newline', 'pc', append'); %Report dlmwrite(strcat(path,outnm,'_rep.txt'), scf 'delimiter', ', 'newline', 'pc', 'precision', 5, append'); %Report dlmwrite(strcat(path,outnm,'_rep.txt'), 'delimiter', ', 'newline', 'pc', append'); %Report dlmwrite(strcat(path,outnm,'_rep.txt'), 'Correction Polynomials' 'delimiter', ', 'newline', 'pc', append'); %Report dlmwrite(strcat(path,outnm,' _rep.txt'), 'dx = Ax^6+Bx^5+Cx^4+Dx^3+Ex^2+Fx+G' 'delimiter', ', 'newline', 'pc', append'); %Report dlmwrite(strcat(path,outnm,'_rep.txt'), param_dx 'delimiter', ', 'newline', 'pc', 'precision', 5, append'); %Report dlmwrite(strcat(path,outnm,'_rep.txt'), 'dy = Ax^3+Bx^2+Cx+D' 'delimiter', ', 'newline', 'pc', append'); %Report dlmwrite(strcat(path,outnm,'_rep.txt'), param_dy 'delimiter', ', 'newline', 'pc', 'precision', 5, append'); %Report
165 LIST OF REFERENCES  M.K. Shepard, B.A. Ca mpbell, M.H. Bulmer, T.G. Farr, L.R. Gaddis, J.J. Plaut The Roughness of Natural Terrain: A Planetary and Remote Sensing Perspective, Journal of Geophysical Research Vol. 106, N o E12, pp 32,777 32,795, December 25, 2001  M. Callens, N.E.C. Verhoest, M.W .J. Davidson Parameterization of TillageInduced SingleScale Soil Roughness From 4m Profiles, IEEE Transactions on Geoscience and Remote Sensing, vol.44, no.4, pp. 878 888, April 2006.  C.J. Dodds and J.D. Robson, The Description of Road Surface Roughness, Journal of Sound and Vibration, 31 (2), pp. 175 183, 1973.  T.R. Thomas, Characterization of Surface Roughness, Precision Engineering, vol. 3, iss. 2, pp. 97 104, April 1981  J. M. Bennett, Recent Developments in Surface Roughness Characterization, Measurement Science and Technology, vol. 3, pg. 1119 1127, 1992.  M.S. Moran, C.D. Peters Lidard, J M. Watts, and S McElroy Estimating soil moisture at the watershed scale with satellitebased radar and land surface models, Can adian Journal of Remote Sensing, Vol. 30, No. 5, pp. 805 826, 2004  N.E. Verhoest, H., Lievens, ; W. Wagner, J. lvarez Mozos, M..S. Moran, F. Mattia. On the Soil Roughness Parameterization Problem in Soil Moisture Retrieval of Bare Surfaces from Synthetic Aperture Radar. Sensor s 2008, 8, pp. 4213 4248, 2008.  W. Dierking, Quantitative roughness characterization of geological surfaces and implications for radar signature analysis, IEEE Transactions on Geoscience and Remote Sensing, vol.37, no.5, pp. 2397 2412, September 1999.  F. T. Ulaby, R. K. Moore, A. K. Fung, Microwave Remote Sensing: Active and Passive ; Artech House: Boston, MA vol. II, ch 11 1982.  J.A. Ogivily and J.R. Foster Rough surfaces: Gaussian or exponential statistics?, Journal of Physics D: Applied Physics, vol. 22, pp. 1243 1251, 1989.  Q. Li, J. Shi, and K.S. Chen, A general power law spectrum and its application to the backscattering of soil surfaces based on the integral equation model, IEEE Transaction on Geoscience and Remote Sensing, vol. 40, no. 2. F ebruary 2002.  M. Zribi, V. Ciarletti, O. Taconet, P. Boissard, M. Chapron, and B. Rabin, Backscattering on soil structure described by plane facets International Journal on Remote Sensing, vol. 21, no. 1, pp. 137 153, 2000.
166  F. Mattia, M.W.J. Davidson, T. Le Toan, C.M.F. DHaese, N.E.C. Verhoest, A.M. Gatti,M. Borgeaud,"A comparison between soil roughness statistics used in surface scattering models derived from mechanical and laser profilers," IEEE Transactions on Geoscience and Remote Sensing, vol. 41, pp. 1659 1671, 2003.  M.L. Oelze, J.M. Sabatier, R. Raspet, Richard, Roughness Measurements of Soil Surfaces by Acoustic Backscatter Soil Science Society of America Journal Vol. 67, pp. 241 250, 2003  R. Bryant, M.S. Moran, D P Thoma, C D Holifield Collins, S Skirvin, M Rahman, K. Slocum, P Starks, D Bosch, M P. Gonzlez Dugo, Measuring surface roughness height to parameterize radar backscatter models for retrieval of surface soil moisture IEEE Geoscience and Remote Sensing Letters vol. 4, no. 1, 2007.  P. Marzahn and R. Ludwig, On the derivation of soil surface roughness from multi parametric PolSAR data and its potential for hydrological modeling, Hydrology and Earth System Sciences 13, pg. 381 394, 2009.  L. Zhixiong, C. Nan, U.D.Perdok, W.B. Hoogmoed, Characterization of Soil Profile Roughness, Biosystems Engineering, vol. 91, 369 377, 2005.  F. Darboux and C.H. Huang An instantaneous profile laser scanner to measure soil surface microtopography, Soil Science Society of America Journal vol 67, pp. 92 99, 2003  J. Fernandez Diaz, Scientific applications of the Mobile Terrestrial Laser Scanner (MTLS) system M.S. thesis, Department of Civil Engineering, University of Florida, Gainesville, Florida, 2007. Available: http://purl.fcla.edu/fcla/etd/UFE0021101  I.J. Davenport, J. Holden, R.J. Pentreath, "Derivation of soil surface properties from airborne laser altimetry," in Proceedings of the International Geoscience and Remote Sensing Sym posium (IGARSS '03) Toulouse, France, pp. 4389 4391, 2003.  I.J. Davenport, N. Holden, R.J., Gurney, "Characterizing errors in airborne laser altimetry data to extract soil roughness," IEEE Transaction on Geoscience and Remote Sensing, vol. 42, pp. 2130 2141, 2004.  C. Perez Gutierrez, J.M. Martinez Fernandez, N. Sanchez, J. lvarez Mozos, "Modeling of soil roughness using terrestrial laser scanner for soil moisture retrieval," in Proceedings of the International Geoscience and Remote Sensing Symposium (IGARSS 2007) Barcelona, Spain, pp. 1877 1880, 2007.  M.J.M. Romkens and J.Y. Wang, Effect of tillage on surface roughness, Transactions of the ASAE vol. 29 (2), March April, 1986.
167  N. Baghdadi, O. Cerdan, M. Zribi, V. Auzet, F. Darboux, M. El Hajj and R.B. K heir, Operational performance of current synthetic aperture radar sensors in mapping soil surface characteristics in agricultural environments; applications to hydrological and erosion modeling, Hydrological Processes 22, pp. 920, 2008.  H. Lievens, H. Vernieuwe, J.Alvarez Mozos, B. De Baets and N. Verhoest, Error in radar derived soil roughness parameterization: An analysis based on synthetical surface profiles, Sensors 9, pp. 1067 1093, 2009.  D.P. Thoma, M.S. Moran, R. Bryant, M. Rahman, C.D. Holifi eld Collins, S. Skirvin, E.E. Sano, and K. Slocum, Comparison of four models to determine surface soil moisture from C band radar imagery in a sparsely vegetated semiarid landscape, Water Resources Research vol. 42, (1), pp. 1 12, 2006.  N. Baghdadi, I. Gherboudj, M. Zribi, M. Sahebi, C. King and F. Bonn, Semi empirical calibration of the IEM backscattering model using radar images and moisture and roughness field measurements, International Journal of Remote Sensing vol 25, no. 18, pg. 3593 3623, Sept ember 2004.  N. Baghdadi, C. King and L. Bonnifait, An empirical calibration of the Integral Equation Model based on SAR data and soil parameter measurements, In Proceedings of the International Geoscience and Remote Sensing Symposium (IGARSS 2002), vol .5., pp. 2646 2650, 2002.  U. Wandinger, Introduction to Lidar, in Lidar: RangeResolved Optical Remote Sensing of the Atmosphere, C. Weitkamp, Ed. New York, Springer, pp. 1 18, 2005.  Carter, W. E., R. Shrestha and K.C. Slatton, 2007, Geodetic Laser Sca nning, Physics Today, Vol. 60, Number 12, pp 41 47, 2007.  K. C. Slatton, W. E. Carter, R. L. Shrestha and W. Dietrich, Airborne Laser Swath Mapping: Achieving the resolution and accuracy required for geosurficial research, Geophysical Research Letters, Vol. 34, 2007.  Y. Oh and Y.C. Kay, Condition for precise measurements of soil surface roughness, IEEE Transactions on Geoscience and Remote Sensing, vol 36, no. 2, pp. 691 695, March 1998.  A.M. Gatti, F. Mattia, C.M.F. DHaese, N.E.C. Verhoest, M.W.J. D avidson, T. Le Toan, M. Borgeaud, "A comparison between Laser needlelike and meshboard techniques for soil roughness measurements," in Proceedings of the 3rd International Symposium, Retrieval of Bioand Geophysical parameters from SAR Data for Land Applications Sheffield, UK, 11 14 September 2001, ESA SP475, January 2002.  J.A. Ogivily Computer simulation of acoustic wave scattering from rough surfaces Journal of Physics D: Applied Physics vol. 21, pp. 260 277, 1988.
168  J. Casanova, L. Tzu yun, L. Mi young, J. Kai Jen, C. Tien, J. Judge, O. Lanni, L. Miller "Field Observations During the Fourth Microwave Water and Energy Balance Experiment (MicroWEX 4): March 10 June 14, 2005,"IFAS report available: ht tp://edis.ifas.ufl.edu/AE362  J. Mi Young,C.T. Kai Jen, J. Casanova, J. Judge, "Measurements of Soil Surface Roughness during the Fourth Microwave Water and Energy Balance Experiment from April 18 through June 13, 2005," IFAS report available: http://edis.ifas.ufl.edu/AE363  J. Casanova, F. Yan, J. Mi young, J. Fernandez, J. Judge, C. Slatton, J. Kai Jen, C. Tien, L. Tzu yun, O. Lanni, L. Miller "Field Observations During the Fifth Microwave Water and Energy Balance Experiment: from March 9 through May 26, 2006,. IFAS report available: http://edis.ifas.ufl.edu/AE407  M.W.J. Davidson, F. Mattia, G. Satalino, N. E. C. Verhoest, T. Le Toan, M. Borgeaud, J. M. B. Louis, E. Attema, Joint Statistical Properties of RMS Height and Correlation Length Derived From Multisite 1m Roughness Measurements IEEE Transactions on Geoscience and Remote Sensing, vol 41, no. 7, pp. 1651 1658, July 2003.  O. Taconet, V. Ciarletti, Estimating soil roughness indices on a ridgeandfurrow surface using stereo photogrammetry, Soil & Tillage Research vol. 93, pg. 64 76, 2007.  K. Fung, Z. Li, and K. S. Chen, Backscattering from a randomly rough dielectric surface, IEEE Transactions on Geoscience and Remote Sensing, vol. 30, pp. 356 369, Mar. 1992.  K. S. Chen, S. Wu, L. Tsang, Q. Li, J. Shi, A. K. Fung, "Emission of Rough Surfaces Calculated by the Integral Equation Method With Comparison to ThreeDimensional Moment Method Simulations,", IEEE Transactions on Geoscience and Remote Sensing, vol. 41, pp. 90 101, Jan. 2003.  F. T. Ulaby, R. K. Moore, A. K. Fung, Microwave Remote Sensing: Active and Passive ; Artech House: Boston, MA 1982; vol. III, ch 19. 2001.  J.E. Gilley and E.R. Kottwitz, Random roughness assessment by the pin and chain methods, Applied Engineering in Agriculture, vol. 12, (1), pp. 39 43, 1995.  R.C. Gonzales and R.E. Woods, Digital Image Processing, Second Edition, Prentice Hall, Upper S addle River, New Jersey, c h. 7.  E.P. Baltsavias, Airborne Laser scanning: Basic relations and f ormulas ISPRS Journal of Photogrammetry & Remote Sensing, vol. 54, pp. 199 214, 1999.  A. Wehr and U. Lohr, Airborne laser scanni ng an introduction and overview, ISPRS Journal of Photogrammetry & Remote Sensing, vol. 54, pp. 68 82, 1999.
169  G.A. Neumann, J. B. Abshire, O. Aharonson, J. B. Garvin, X. Sun, and M. T. Zuber, Mars Orbiter laser Altimeter pulse width measurements and footprint scale roughness, Geophysical Research Letters vol. 30, no. 11, pp. 151 154, 2003.  National Research Council, Space Studies Board, Earth Science and Applications from Space: National Imperatives for the Next Decade and Beyond, National Academies Press, ch. 11, 2007.  M. Murakami, Managing Water for Peace in the Middle East: Alternative Strategies. Tokyo: United Nations University Press, 1995. Available: http://www.unu.edu/unupress/unupbooks/80858e/80858E00.htm#Contents  V.I. Korzun, World water balance and water resources of the Earth, Paris, UNESCO, pp. 1756, 1978.  European Space Agency, SMOS: ESA's Water Mission Paris, 2009. Available: http://esamultimedia.esa.int/docs/SMOS/BR 278_low res.pdf  S.O. Rice, Reflection of electromagnetic waves from slightly rough surfaces Communications in Pure and Applied Mathematics, 4, pp. 361 378, 1951.  P. Beckman, and A. Spizzichino, "The Scattering of Electromagnetic Waves from Rough Surfaces," London, Pergamon Press A., 1963.  A.G. Voronovich, SmallSlope Approximation in wave scattering from rough surfaces Soviet Physics Journal of Experimental and Theoretical Physics 62, pp. 6570, 1985.  F. Mattia, T. Le Toan, Backscattering properties of multi scale rough surfaces Journal of Electromagnetic Waves and Applications vol 13, pp. 491 526, 1999.  J.A. Kong, S.L. Lin and S.L. Chuang, Microwave Thermal Emission from Periodic Surfaces IEEE Transactions on Geoscience and Remote Sensing, vol GE 22, no. 4, pp. 337 382, July 1984.  J. T. Johnson, R. T. Shin and J. A. Kong, Scattering and thermal emission from a two dimensional periodic surface, Progr ess In Electromagnetics Research vol. 15, 303 333, 1997.  M. T. Hallikainen, F. T. Ulaby, M. C. Dobson, M. A. El Rayes, and L. K. Wu, Microwave dielectric behavior of wet soil Part I: Empirical model and experimental observations, IEEE Transactions on Geoscience and Remote Sensing vol. 23, pp. 25 34, Jan. 1985.  M. T. Hallikainen, F. T. Ulaby, M. C. Dobson, M. A. El Rayes, and L. K. Wu, Microwave dielectric behavior of wet soil Part II: Dielectric mixing models IEEE Transactions on Geoscience and Remote Sensing, vol. 23, pp. 25 34, Jan. 1985.
170  N. R. Peplinski, F. T. Ulaby, and M. C. Dobson, Dielectric properties of soils in the 0.3 1.3GHz range, IEEE Transactions on Geoscience and Remote Sensing, vol. 33, pp. 803 807, May 1995.  V. L. Mironov, M. C. D obson, V. H. Kaupp, S. A. Komarov, and V. N. Kleshchenko Generalized Refractive Mixing Dielectric Model for Moist Soils, IEEE Transactions on Geoscience and Remote Sensing, vol. 42, no. 4, pp. 773 785, April 2004.  P.R. Wolf, Elements of Photogrametry, With Air Photo Interpretation and Remote Sensing Second Edition, New York: McGraw Hill, 1983
171 BIOGRAPHICAL SKETCH Juan Carlos Fernandez Diaz was born in 1976 in Teguci galpa, Honduras, to Venancio Fernandez and Ana Maria Diaz. H e also has one sister M aria Esther and two brothers David and Jose Venancio. From a very young age, he developed a strong interest towards science and technology especially earth and space science, aviation, telecommunications and electronics. He was fortunate to attend an Am erican school (Elvel School) from kindergarten to 11th grade, where the teachers motivated his scientific curiosity. In 1993, he graduated from High School from a program that fulfills the requirements from both the American and Honduran academic curricul um. That same year he enrolled in the Electrical Engineering program of the Universidad Naciona l Autonoma de Honduras (UNAH). Not find ing college challenging enough, he decided to work full time while pursuing the bachelors degree. His first position was as an instructor of the universitys astronomical observatory where he acquired knowledge and expertise related to the design, use and maintenance of astronomical instrumentation as well as astronomical data processing and analysis. During this period, he also participated in a traineeship at the European Space Agency Satellite Tracking Station in Villafranca del Castillo, Spain and received a Summer Undergraduate Research Fellowship (SURF) from the California Institute of Technology (CALTECH) to perform s cientific research at the Jet Propulsion Laboratory. Soon after this experience he accepted a new position with the Honduras nat ional t elecommunications c ommission (CONATEL) as a spectrum planning and engineering technician. He obtained the BS degree in electrical engineering in June 2001,
172 complimenting the formal academic knowledge with solid experience in telecommunications, space science and technology. From 2002 to 2005, he continued his career in telecommunications holding positions at a Wireless se rvice provider were he performed functions such as network planning engineer and quality assurance chief. During that same time, he obtained a M aster of Business A dministration degree with a summa cum laude distinction from the Universidad Catolica de Honduras in 2005. During 2004, he applied for a Fulbright S cholarship to participate in a M asters program in the fields of Satellite Applications (Navigation, Communications and Remote Sensing). He was fortunate to receive the scholarship and to be accepted to the University of Florida, Geosensing Systems Engineering graduate program. He s tarted the program during the fall of 2005, and obtained the Master of S cience degree in the summer of 2007. After the MS degree, Juan received a University of Florida alumni fellowship to continue his graduate education in pursuit of a PhD degree. During his PhD studies he also participated in the summer program of the International Space University in 2007 in Beijing, China and 2008 in Barcelona, Spain. He served as a teachi ng associate for team design project s, which aimed to identify and propose space technologies to monitor and respon d to geophysical hazards. In the summer of 2009, Juan participated in an internship program sponsored by the University of Maryland Baltimore County. During this internship he worked at the Microwave Instruments and Technology Branch at the National Aeronautics and Space Administration ( NASA) Goddard Space Flight Center. Juan hopes to keep enhancing his multidisciplinary experience and continue to explore his interest in space science and technology.