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The modeling of daily precipitation in Costa Rica

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The modeling of daily precipitation in Costa Rica
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Harrison, John Michael, 1960-
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vii, 240 leaves : ill. ; 29 cm.

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Rain ( jstor )
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Thesis (Ph. D.)--University of Florida, 1998.
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Includes bibliographical references (leaves 228-239).
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Typescript.
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Vita.
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by John Michael Harrison.

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THE MODELING OF DAILY PRECIPITATION IN COSTA RICA








BY

J014N MICHAEL HARRISON














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 1998






















For my father, the late Marshall Gene Harrison, who pointed me on the path I could never have taken otherwise; and my mother, Mary Ann Harrison, who made sure that I never strayed from it. My thanks and love, always.















ACKNOWLEDGEMENTS



A dissertation can never be the product of a single individual, and I regret that it is not possible to list as co-authors all the people that have helped make this work possible. I am grateful, on both a personal and professional level, for the support and friendship offered by Mark McLean, Chris and Kathleen Meindl, Matt and Mary Zom, Sharon Cobb, Ryan Jensen, and Lisa Walsh. These fine individuals have made life as a graduate student not simply bearable, but enjoyable, and I am proud to call them friends.

I have been blessed with an outstanding advisory committee, and without their advice and support I could not have survived the arduous process of completing my degree. I am extraordinarily grateful for the support lent by Dr. Joann Mossa, Dr. Michael Binford, and Dr. Jim Jones for their expert guidance throughout my career as a doctoral student. I am especially grateful to Dr. Cesdr Caviedes, who not only illuminated the mysterious ways of El Niflo, but also allowed me the opportunity to teach within the Department.

Most of all, my thanks go out to Dr. Peter Waylen, who has proven to be the

finest advisor I have ever encountered. It is safe to say that without his help, insight, and encouragement, I could not have completed my degree. Throughout this journey, he has been there to help me, in good times and bad, and I can only hope that this work, and all those that follow, will in some small way repay the confidence he has shown in me.



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TABLE OF CONTENTS

p jge

ACKNOW LEDGEM ENTS ............................................................................................ iii

ABSTRACT .................................................................................................................... vi

CHAPTERS

I INTRODUCTION ......................................................................................... I

Rainfall M odeling .......................................................................................... 2
General Characteristics of Costa Rica ........................................................... 4
Structure of this Dissertation ......................................................................... 5

2 LITERATURE REVIEW .............................................................................. 8

Probability M odels ........................................................................................ 8
Forecasting and Prediction .......................................................................... 15
El Niho-Southem Oscillation ...................................................................... 17

3 STUDY AREA AND DATA DESCRIPTION ........................................... 24

Physiography ............................................................................................... 24
General Climatic Characteristics ................................................................. 25
Previously Noted Costa Rica-ENSO Connections ...................................... 26
Daily Precipitation Data and Station Location ............................................ 29
Atmospheric Index Data .............................................................................. 30

4 M ETHODOLOGY ...................................................................................... 47

Precipitation Statistics ................................................................................. 48
Relationship to ENSO Processes Segregated Analysis .......................... 56
Relationship to Ocean-Atmosphere Processes Lag-Correlation
Analysis ........................................................................................... 56
Creation of Statistical Forecast models of Rainfall Parameters .................. 57





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5 RESULTS .................................................................................................... 61

M odel Identification .................................................................................... 61
Tim e Series Correlation of Transition Probabilities .................................... 67
Tim e Series Correlation of L-M om ents ...................................................... 72
M odel Choice, Validation, and Rainfall Sim ulation ................................... 76
Results Overview ........................................................................................ 91

6 DISCUSSION ........................................................................................... 182

M odel Identification .................................................................................. 182
Time Series Correlation L-Moments and Transition Probabilities ...... 194
M odel Choice, Validation, and Rainfall Sim ulation ................................. 202
Param eter Forecasting ............................................................................... 205

7 CON CLU SION ......................................................................................... 212

Statistical M odeling ................................................................................... 213
Clim atology ............................................................................................... 215
Application to Other Areas ........................................................................ 220

APPEN DIX L-M OM EN TS ........................................................................... 222

REFEREN CES ....................................................................................................... 228

BIOGRAPHICAL SKETCH ........................................................................................ 240






















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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

THE MODELING OF DAILY PRECIPITATION IN COSTA RICA By

John Michael Harrison

August 1998


Chairman: Dr. Peter R. Waylen
Major Department: Geography

The understanding of precipitation and its underlying processes is important to many human activities. Agricultural planning, hydroelectric resource management, and industrial infrastructure development all rely heavily on being able to make reasonable predictions concerning rainfall. The lack of sufficient rainfall can have devastating social and economic consequences for developing nations that are reliant on subsistence agriculture and hydroelectric power. This study examines the means by which daily precipitation in Costa Rica can be modeled, and how the El Nifto-Southem Oscillation (ENSO) affects the precipitation generating mechanisms.

A selection of three meteorological stations are used to test how daily rainfidl can be characterized by the occurrence and intensity of the individual rainfall events. The occurrence is modeled using a two-state first-order Markov model, which provides insight into the relative length of wet and dry spells. The intensity model uses L-moments to determine the optimum statistical distribution. These statistical parameters are used to



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understand the inter-annual and inter-seasonal variations in the precipitation-generating mechanisms as they are modified by the ENSO phenomenon. The parameters are also combined to create monthly rainfall simulations based on the state of the ENSO, as well as test whether accurate forecasts can be created up to one year in advance.

It is found that the ENSO plays an important role in the daily rainfall process, by altering the behavior of the Inter-Tropical Convergence Zone (ITCZ), the Northeast Trade Winds, and the advance of cold air masses from North America during the winter. The eastern Caribbean slope of the country receives proportionally more rainfall during El Niho events, while the western Pacific slope receives less rainfall during the same period. Cold front (norte) intrusion is minimized by the El Nifio, resulting in less winter rainfall during El Nifto years. Simulations based on the state of the ENSO are shown to be effective in recreating monthly time series of rainfall accumulations based upon the calculated occurrence and intensity parameters. Long-range forecasts are less effective, which is indicative of the complexity of the tropical precipitation process.























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CHAPTER I
INTRODUCTION


One of the most important of all climatic variables is precipitation. It touches on nearly every aspect of human existence, both from a social and economic standpoint and as a field of scientific study. Human societies rely heavily on precipitation as a necessary input to agricultural systems and the resultant runoff has for centuries driven machines, and more recently generated electricity. At the same time, water has had destructive sideeffects, such as floods, soil erosion, mass movement, and the disruption of transportation and communication infrastructures. Because humans are so reliant on rainfall, the ability to predict accurately its occurrence and intensity has been pursued with exceptional energy. In many parts of the developing world, the understanding of rainfall occurrence and intensity goes well beyond mere convenience. Much of the world's population continues to survive on subsistence fanning systems, and many developing nations depend heavily on the export of cash crops and the generation of hydroelectric power. For these reasons, fluctuations in the amount of rainfall in these, such as those prompted by the El Niho-Southern Oscillation phenomenon, can have dramatic and devastating economic and social results.

The nation of Costa Rica is an excellent case-in-point. It is a small Central

American country attempting to deal with the often conflicting needs of a highly agrarian economy moving toward a more industrialized status with a burgeoning population that is becoming increasingly urban. Over 60% of national electrical power is derived from hydroelectric sources, and is therefore highly sensitive to changes in the precipitation climatology of the region. In addition, the agricultural systems that represent 60% of the

I





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GNP are heavily reliant on rainfall, with less than 25% of all cropland subject to irrigation. Unlike many countries in the region, however, Costa Rica has a stable economic and social system, and this has allowed it to develop a extensive network of meteorological stations, with relatively long-term records.

This research will use a set of three rainfall stations in the Central Valley of Costa Rica to test the effectiveness of daily rainfall modeling and prediction within a tropical context. Specifically, it will be shown how the El Niho-Southern Oscillation (ENSO) affects the intensity and frequency of daily rainfall over the study area, and how the daily rainfall characteristics can be modeled with a high level of confidence. The creation of such models will represent a substantial benefit to climate researchers, agricultural engineers, and planners.


Rainfall Modeling

A wide variety of approaches exists in the study of precipitation and precipitationgenerating processes. One, usually associated with meteorologists, examines causal physical processes, and attempts to extend the findings in such a way as to explain spatial and temporal variations in precipitation. While such an approach is useful in illuminating the physical processes involved, it can lead to conclusions that are spatially and temporally limited. Physical processes are usually complex, and the resulting models are either unacceptably complex (i.e., lacking in parsimony, with many variables involved, thus making it difficult to discern the most important processes involved), or overly restrictive, thus preventing generalization of their results. Intensive data demands also make them generally less applicable to most regions of the world.

An alternate approach is to examine the time series of the end-result (i.e., the precipitation itself), as a means to understanding its pertinent properties. The rainfall






3


record then becomes the integrated output of all the contributing physical processes. This is an especially useful approach when the exact physical mechanisms are not well understood, are complex, and/or pertinent data are difficult to obtain. The statistical approach may mitigate many of the problems associated with modeling causes of climatic variability such as the ENSO, while maintaining the goal of making accurate predictions. In general, most statistical analyses of daily rainfall seek to model the occurrence of precipitation (as a sequence of wet or dry days), and the probability distribution of the intensity of the resulting precipitation.

From the standpoint of statistical modeling, this study seeks to address three

broad issues. First, the nature of the seasonal and spatial variations of daily rainfall over the Central Valley of Costa Rica will be examined. Models of occurrence and intensity of precipitation are to be developed, using rainfall records available from the appropriate agencies in Costa Rica. Second, the effect of the ENSO phenomenon on the spatial and temporal variability of the region's rainfall will be examined through its influence upon model parameters. Previous studies (e.g., Estoque et al., 1985; Ferndndez and Ramirez, 1991; Waylen et al., 1994) have demonstrated a link between Costa Rican annual and monthly rainfall and the state of the ENSO as represented by the so-called Southern Oscillation Index (SOI). It is important to establish the spatial extent and nature of the associations at the daily scale. Third, the ability of these models to predict accurately future daily rainfall characteristics will be examined. Several modeling techniques will be employed to create "hindcast" predictions, and the results will be compared to the measured rainfall during these periods.






4


This study yields statistical models with broad applicability to a variety of regions in Costa Rica and Central America that will help to illuminate the nature of the physical processes which affect precipitation in the tropics, particularly the ENSO phenomenon. Ultimately, correlations between daily rainfall characteristics and the indices describing the ENSO, such as SOI, could lead to a method for predicting ftiture rainfall variation based upon the state of the ENSO. This will have broad applications for both short-term precipitation forecasts and long-range climate prediction.



General Characteristics of Costa Rica

The nation of Costa Rica occupies an area of 51110 kM2, supporting a population of 3,032,000 (Goode's World Atlas, 1992) (Figure 1-1). Its economy is strongly linked to agriculture, but the trend has been towards increased urbanization, with well over half the population now living in urban areas (West and Augelli, 1989). This has placed increasing demands on the extensive hydroelectric projects that are being developed (hydroelectric power accounts for about 60% of the total generated electricity in Costa Rica), as well as pressing for increased availability of water for drinking and industrial concerns (Gleick, 1993). In addition, despite a shrinking rural population, rural access to sanitation services has fallen from 93% in 1975, to 89% in 1985, reflecting the increased competition between rural and urban populations for available water resources. The Costa Rican agricultural system is strongly dependent on regular, predictable rainfall, with rained cropland accounting for over 77% of cultivated land (Gleick, 1993).

Costa Rica has had a relatively peaceful history and stable government, and this social environment has allowed for the creation of a reasonably diverse and wellsupported meteorological network. Reliable precipitation records in this region date back





5


more that 150 years in some cases. It has also been shown that, due to its geographic position in the tropics and its susceptibility to meteorological influences from the midlatitudes, Costa Rica experiences considerable spatial variability in precipitation in response to the state of the ENSO (Waylen et al., 1996).



Structure of this Dissertation

This research is organized along the following lines. First, previous research into the nature of the statistical modeling of rainfall magnitudes and intensities will be recounted. This will be followed by a brief history of forecast modeling, with particular concentration on the ARIMA and multiple regression techniques currently in widespread use. Next, the evolution of the current understanding of the El Nifto-Southern Oscillation (ENSO) will be presented, with particular emphasis on its impact upon Central American climatology. In addition, a discussion of the atmospheric indices to be used in this study will be presented.

The study area is delineated in considerable detail in Chapter 3, especially emphasizing the known relationships between the spatial variability of Costa Rican rainfall and the ENSO phenomenon. The ma or climatological elements are introduced and discussed, as well as the measurement indices by which they are represented. The individual meteorological stations used in the study are evaluated for applicability, both in terms of geographic location and record length. Specific stations are identified for detailed examination, based upon their suitability to represent broad regions within the country.






6


A research methodology is then presented, which will outline the creation of daily rainfall intensity and occurrence models to be used in this study. The basic occurrence parameters are calculated using a Markov model, while the intensity statistics are computed using linear moments, or "L-moments." The resulting model parameters are examined in terms of whether the statistic occurred during an ENSO event, with the intention of evaluating the broad-scale influence of ENSO upon Costa Rican daily rainfall. Next, lag-correlations between several atmospheric indices and the calculated statistics will be performed, so that more subtle patterns of influence can be detected. Finally, the parameters will be used as the basis for a series of statistical hindeast forecast models, which will be evaluated on the basis of how well they approximate known rainfall conditions for periods of up to one year into the future.

Finally, the results of the study will be evaluated in terms of the statistical

importance of the correlations and forecasts at the individual stations, as well as in the broader context of regional climate variability. The model results will be placed into a format that can be directly accessible by researchers in other areas, such as reservoir and crop modeling, and an attempt will be made to reconcile the statistical models for Costa Rica with the known factors affecting the climate of the Central American region. It is anticipated that this study will provide much needed information for researchers in many other fields, as well as shed light upon the underlying physical processes governing the precipitation process in Costa Rica.








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CHAPTER 2
LITERATURE REVIEW



Pertinent research relating to the understanding and modeling of daily

precipitation and ENSO-based climate variability are discussed. The nature of chaindependent precipitation models is first reviewed, followed by a discussion of continuous probability models which have been employed to model precipitation magnitudes. A brief history of the climatological impacts of ENSO is recounted, with special emphasis on its effects with Central American climate.



Probability Models

The modeling of precipitation has often been viewed as an almost intractable problem, as it is unknown whether a physically realizable upper limit to the amount of rainfall that can fall during a given event exists, for a given set of environmental parameters (Knisel et al., 1979; Willeke, 1980; Hirschboeck, 1988). In addition, it is impossible to fully quantify all the different possible physical states that can exist during a given precipitation event. For these reasons, deterministic models are generally regarded as inadequate when evaluating the precipitation climatology of a region (Dutton, 1986).

Modeling of daily precipitation climatology therefore reduces to deriving the

proper distributional models for the occurrence and intensity of the precipitation (Coe and Stem, 1982; Woolhiser, 1992; Cong and Li, 1993). The occurrence is frequently represented using Markov models (Chin, 1977; Katz, 1977, 198 1; Coe and Stem, 1982; Stem and Coe, 1983; Guzman and Torrez, 1985; Hughes and Guttorp, 1994; Lall et al.,


8






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1996), while the magnitude is most frequently denoted by a variety of continuous probability models, such as the gamma, generalized extreme value (GEV), and Pareto distributions (Markovic, 1965; Haan, 1977; Hosking, 1985; Guttman, 1993; Hosking and Wallace, 1993).



Probability Models -- Occurrence

Access to meteorological data which has been discreetized into regular intervals allows the variable to be examined in terms of its dependence upon its previous values. This implies that the data sequence is stochastic in nature so long as the series is not excessively burdened by long-term serial dependence (Wilks, 1995). Stated simply, each value of the data stream (e.g., daily precipitation totals) must exhibit reasonable independence from values observed long in the past. If this assumption is met, the data are said to have "Markovian properties," the process which generated the data is referred to as a Markov process, and the resulting data model being called a Markov chain model (Haan, 1977). An important characteristic of a Markov model is that the value of the current data point depends only upon a fixed number of preceding measurements, and is independent of any other data. This implies that, in order to construct a Markov model of rainfall, the data must be analyzed both in terms of its temporal characteristics (i.e., how often precipitation occurs), and in terms of the measurement of the quantity (e.g., rainfall amounts). The degree of temporal dependence is determined by the order of the model (a first order model contains dependence only upon the value immediately preceding it in time, and is assumed independent of all other values), while the number of possible categories or classes within the basic time unit is referred to as the state of the model. For example, it is common to categorize rainfall occurrence as either "rained" or "did not rain'; this would represent a two-state model of rainfall (Haan, 1977; Wilks, 1995).






10


The model is typically produced by calculating the transition probabilities from the data sequence, by calculating the number of wet and dry days and conditioning these results on the basis of the prior day's state. Thus, a two-state first-order Markov model would contain four parameters: poo (the probability of a dry day following a dry day), pol (the probability of a wet day following a dry day), plo (the probability of a dry day following a wet day), and pi 1 (the probability of a wet day following a wet day). It is important to note that poo and p01 will always add to one, as will plo and p1 1. The creation of a matrix of four transition probabilities therefore requires the computation of only two values, which improves the parsimony of the resulting model (Wilks, 1995).

Due to the fact that rainfall is often caused by conditions which linger for multiple days (or longer), a certain amount of serial correlation will be apparent in a Markov precipitation chain. This is known as persistence, and can be measured using the lag-l autocorrelation parameter, also known as the persistence parameter r, which is the difference between the pi 1 and pot parameters, and reflects the degree of serial dependence of the model (Wilks, 1995). Perfect independence between days of rainfall would imply r, = 0, which generally does not occur in natural systems. This would seem to indicate that the necessary conditions for applying Markov models to rainfall occurrence have been violated, since having serial correlation from one day to the next would indicate a degree of correlation for days separated by k-days, according to the

k
formula rk =(rl) .As Wilks (1995) points out, this does not violate the basic Markovian assumptions, which state that successive conditional probabilities be independent, rather than the actual data points.






I11


Past research has indicated that, with few exceptions, Markov models of the first order (i.e., those, in which a day's precipitation occurrence is conditioned only upon the previous day's occurrence, and ignoring events prior to that time) are the most applicable models for rainfall modeling (Caskey, 1963; Katz, 1977, 1981; Coe and Stem, 1982); however, these studies involve the use of mid-latitude rainfall records, and little published research indicates whether this holds for tropical precipitation. Wilks (1995) demonstrates how maximum likelihood estimates can be created for the applicability of first order versus higher order Markov models to a given data sequence.

Markov models have previously been used extensively to perform realistic

simulations of precipitation occurrence which are subsequently input into deterministic models, for purposes such as climate change impact assessments (Pickering et al., 1994; Wilby, 1994), as well as to gain insight into the underlying physical processes which govern rainfall occurrence (Katz, 1977). Refinement of these procedures have shown how periodic and quasi-periodic physical processes responsible for seasonality in precipitation can be assessed (Woolhiser and Roldan, 1986; Woolhiser, 1992; Woolhiser et al., 1993; Katz and Parlange, 1993; Hughes and Guttorp, 1994). Probability Models -- Magnitudes

Deriving the underlying distributional form for the intensity (i.e., the daily totals) of precipitation has been more problematic, due to the open-ended nature of extreme precipitation events and the resulting highly skewed rainfall histograms (Woolhiser, 1992). A further complicating factor is the large number of candidate continuous distributions from which to choose. Many two- and three-parameter probability distributions are applicable to a variety of hydrologic and climatological problems;






12



among those that have found broad appeal are the exponential, Gumbel (Haan, 1977), log-normal (Cong et al., 1993; Stedinger, 1980), generalized extreme value (Haan, 1977; Hosking, 1985), gamma and Pearson type III (Cong et al., 1993; Haan 1977), Pareto (Cong et al., 1993; Hosking and Wallis, 1987), Weibull (Haan, 1977), mixed exponential (Woolhiser et al., 1993), and Wakeby distributions (Landwehr et al., 1979).

Most intensity models involve the manipulation of an exponential function; the

rationale for this is that the exponential function can easily be made to represent a rapidly decreasing probability density function, while still preserving many of the desirable mathematical properties that make the exponential function so attractive (Haan, 1977). Examples of this type of function are the exponential, gamma, and Pearson distributions.

One of the major difficulties in working with precipitation data is the presence of a lower boundary on the data values; there can never be a negative precipitation, and zero amounts are modeled as "occurrence/non-occurrence" within the context of the Markov model. For this reason, distributions that are continuous in both the negative and positive directions are generally not applicable. However, it is possible, in some cases, to mimic the behavior of a normally distributed variable by taking the logarithm of the amounts, and using the transformed values as a normally distributed variable; this can often give acceptable results, and the resulting distribution is called a "log normal" (or "generalized normal") distribution (Haan, 1977). This procedure has been used in modeling hydrologic data such as stream discharges (Stedinger, 1980), but has found limited utility when modeling daily rainfalls due to the availability of other functional forms.

The lack of an effective upper limit on precipitation values forces the

consideration of more complex distributions which can take into account highly skewed data. Most of these distributions are some variation on the exponential function. A pure exponential, however, is unusable because of its fixed skew (equal to 2/mean); for a function to be useful, its skew must be capable of fitting a broad spectrum of data. A






13


more generalized version of the exponential is the "gamma function," which has a variable skew; the basic gamma function has two parameters, which together define the variability and skew of the resulting probability distribution (Haan, 1977). Gamma distributions have been used extensively to represent rainfall amounts (Markovic, 1965; Mooley and Crutcher, 1968).

In hydrologic research in general, and more specifically, studies of precipitation, the extremes of a set of events is often of interest. The extreme value of a set of random variables is itself a random variable, with the resulting distribution depending on the sample size and the underlying distribution of the individual events; these are known as 66extreme value distributions" (Haan, 1977). The most common type is the generalized extreme value distribution, which can be shown to be the asymptotic limiting distribution for a variety of exponentially based distributions with one tail unbounded. The lognormal, exponential, and gamma distributions all share this trait (Haan, 1977). The generalized extreme value distribution has three-parameters, allowing the location, scale, and shape to vary (Jenkinson, 1954; Hosking, 1985).

A very useful class of distributions was developed by Pearson (Elderton, 1953)

which can be manipulated to form several well known variants. In particular, the Pearson Type III distribution represents a three-parameter generalization of the gamma distribution, and has found broad applicability to flood and precipitation peak research (Haan, 1977). By increasing the number of associated parameters, the distribution can better model the mode of a data set, allowing a better fit than if the mode were fixed, as in a two-parameter gamma.

Distributions can also be mixed, if there is reason to believe that the precipitation originates from more than one process. Often, these models can better represent the data, but they do so at the expense of parsimony; mixed models, those composed of two or more lower order distributions, have as many parameters as all of the underlying






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component distributions. Conceivably, any set of distributions can be mixed to assemble new distributions, but understanding the nature and statistical properties of the resulting model can be difficult at best, and will always result in a larger number of parameters, thus reducing the total degrees of freedom of the calculations. Examples of such models are the mixed exponential distribution (Woolhiser et al., 1993); and Wakeby distribution, which is simply a mixed Pareto model (Landwehr et al., 1979).

Parameter estimation. The traditional means of estimating the underlying

distribution parameters has been to employ the "method-of-moments" approach, in which the statistical moments (mean, standard deviation, etc.) are computed from the data record (Wallis et al., 1974; Stedinger, 1980). However, this method is handicapped by the fact that the extreme precipitation values tend to be poorly represented (Vogel and Fennessey, 1993; Harrison, 1995). Since these extremes are of critical importance in many practical applications, more efficient parameter estimation methods are needed.

Recently the use of "linear moments," or L-moments, originally developed to deal with the problem of highly skewed data sets (Greenwood et al., 1979), have shown some promise in the realm of climatological studies. They have the advantage of producing distribution parameters that are more stable than those derived with the traditional method-of-moments approach (Hosking, 1989, 1990; Hosking and Wallis, 1993; Harrison, 1995). The reason for this is that, as the name implies, they are linear combinations of probability-weighted moments, rather than power-driven combinations, as with traditional moments. In calculating the values of traditional moments such as variance, skew, and kurtosis, the variation from the mean is raised to a power, thereby causing outlying values to exert an inordinate amount of influence, leading to unstable results with highly skewed data such as daily precipitation. It was shown by Hosking and Wallis (1993) and Harrison (1995) that L-moments can give more stable results than traditional moments when rainfall data are used. Mason et al. (I1996a, 1996b) demonstrate





15


that L-moments highly effective when used to estimate the return periods for rainfall and changes in flood frequencies over South Africa. Guttman (1994) further illustrates that Lmoments can be used with confidence for the determination of skew and kurtosis if as few as 60-70 samples are available, making the method well suited for long-term precipitation analyses. Vogel and Fennessey (1993) have shown similar results and provide an analytically simple algorithm for calculating L-moments from a data set, and Hosking (1989) demonstrates how they can be used to calculate the parameters and quantiles of the underlying distributions.


Forecasting and Prediction



One important aim of modeling is to understand how current and past

observations will lead toward the future state of a system. It is important to note, however, that the means by which a forecaster achieves this aim using a statistical model will often differ significantly with those of research which seek an explanation of the underlying physical processes. In many cases, it is not required that a forecast variable have a causal linkage to the variable being forecast, only that some degree of statistical correlation between the predictor and the predictand be established. Such models are referred to prediction models.

A broad view of the literature casts most prediction techniques into two

categories. The first rests on an understanding of the statistical and harmonic character of the time series of the predictand itself, without using (in a direct sense) any other variables to aid in the prediction process. Examples of such techniques include harmonic and spectral analysis (including Fourier analysis and singular spectral analysis) (e.g.,






16


Mitra et al., 199 1), and modeling the time series as an autoregressive process (Wilks, 1995, page 306). These methods presume that all of the information needed to make viable predictions is contained within the predictand time series itself, since ultimately the predictand represents the aggregate of all processes which contribute to its formation, and therefore adequate predictions can be created without resorting to other information sequences.

The second category attempts to mitigate these problems by employing predictor variables, which presumably have some relationship with the predictand (although this relationship need not be causal). A simple linear regression is an example, with some variable (or set of variables) expressing a linear relationship with the variable being predicted. This system can be easily extended to include non-linear relationships between the predictor variables and the predictand.

A serious limitation of basic regression techniques is that the same set of

predictors is used for the entire time series of the predictand. Thus, if one set of predictors will only fit a subset of the predictand time series, the remainder of the series will continue to be affected by its presence in the regression model. An alternative to standard regression techniques is using polynomial-based splines (De Boor, 1978; Friedman and Silverman, 1989).

A spline is simply a function that is used to fit only a portion of the data series

being modeled. Thus, each set of splines being used to approximate the behavior of some sequence of data act as "basis functions" (functions which are used to form the basic equations for the fitted model) only for that portion of the data being modeled, and each set of basis functions are piecewise continuous with all other basis function sets which






17


span the predictand series (Friedman and Silverman, 1989). These individual basis functions can be of any type which fit the needs of the model, but the most common type are linear and cubic splines; cubic splines are especially popular due to their desirable quality that they are differentiable. The spline sets are joined together in a piecewisecontinuous fashion such that the entire predictand series is modeled; the joining locations are known as knots (Friedman and Silverman, 1989; Friedman, 1991).

Splines are problematic due to their piecewise and possible discontinuous nature. However, Friedman (199 1) has published an algorithm which outlines the procedure by which a spline set can be tested for optimal fit and parsimony, which he has referred to as Multivariate Adaptive Regression Splines, or MARS. This technique works through the successive testing and eliminating of the spline basis functions, and then refitting the knot locations by regressing the results at each stage of the process and comparing with the original predictand series. In this way, nearly every combination of predictors is tested, and only those that reduce the overall piecewise error are kept. The error is measured using the generalized cross validation (GCV) criterion introduced by Craven and Wahba (1979), which is output by the MARS procedure. This algorithm has been employed by Lall et al. (1996) and Sangoyomi et al. (1996) to examine the non-linear characteristics of the depth and volume variations of the Great Salt Lake, and by Lall and Mann (1995) to measure low-frequency climate variability in the Great Basin region of the U.S.


El Nubt Southern Oscillation

Few areas of climatic research have garnered as much attention in recent years as the El Nubt-Southern Oscillation (ENSO) phenomenon. For many researchers, ENSO represents the epitome of ocean-atmosphere couplings, and its effects reach far beyond





18


the tropics. With greater varieties of data sets and more advanced observational and computational tools at their disposal, climatologists are now better able to comprehend the underlying mechanisms surrounding the ENSO, and have a clearer understanding of the tropical/extratropical ocean-atmosphere connections that influence the earth's climates.

The periodic appearance of anomalous precipitation in northwestern South

America has been recognized for centuries, and first appears in the literature in the 1800s. It was noted that the precipitation arrival often coincided with Christmas, and the colonists christened the event "El Nifio," after the Christ child (Enfield, 1992); the term "El Niflo" would later be used to refer to the warrn ocean currents associated with the anomalous precipitation. While the increased rains were often welcomed in the otherwise and region, the rainfall sometimes reached such extreme levels that flooding occurred, causing farmland and structures to be destroyed. In addition, there were massive periodic fish and bird deaths that were often associated with these events (Sharp, 1992).

In the early twentieth century, another atmospheric phenomenon was discovered, which was to be closely related to the El Niflo. Sir Gilbert Walker, a British statistician stationed in India, noted an inverse correlation between the pressure fields in Indonesia and the central Pacific Ocean (Walker and Bliss, 1932). This pressure relationship became known as the "Southern Oscillation," noting its tendency to shift phases in a quasi-periodic manner. The high and low pressure regions would modify over a period of two to five years on average (Rasmussen and Carpenter, 1983), roughly the same frequency of change associated with the El Niflo, as pointed out by Berlage (195 7). Nevertheless, many atmospheric scientists were reluctant to couple the two phenomena without firm evidence that they were related, especially since not all oscillation swings were tied to El Nifto events (Deser and Wallace, 1987; Enfield, 1989).





19


While the physical processes may not have been understood, many of the

manifestations of these processes certainly were, and this led to the development of measures of the severity of the El Nifho and the Southern Oscillation. One measure of the pressure gradient between the central Pacific Ocean and the Indonesian archipelago was the Southern Oscillation Index (SOl), which represented the normalized sea level pressure differences between Tahiti and Darwin, Australia (Caviedes, 1975, 1984; Diaz and Pulwarty, 1992). Relatively strong negative values of this index were associated with low east-west pressure gradients in the equatorial Pacific, and weak easterly Trade winds, while strong positive values were associated with stronger Trade winds (Diaz and Kiladis, 1992). The term "El Nifio" has come to refer to the former condition, while the latter is variously referred to as "anti-El Niflo," "La Nina," and "El Viejo." (For purposes of brevity, the term "La Nifia" will be used in this document to identify the opposing conditions to El Niflo.)

The connection to wind fields was important, because it was well understood that upwelling was caused by wind shear across the ocean surface. Upwelling is an important aspect of ocean-atmosphere relations, both from a biological and physical standpoint. Upwelling helps to recycle deep nutrient-rich waters in several ecologically sensitive regions, notably off the coast of western South and North America (Ochoa and Gomez, 1987). Upwelling can also drive climatic conditions within a region, particularly when the upwelled waters are from below the primary thermocline; cold upwelling can create domes of cool air, which have a stabilizing effect on the local meteorology, and thus lead to regions of relative aridity (Steiner and Khalsa, 1987). This is especially apparent in the coastal regions of Peru, Ecuador, western Central America, and California (Umatani and Yamagata, 1991).

In the 1960s, Jacob Bjerknes began formulating ideas that attempted to link the El Nifio phenomenon with the Southern Oscillation using internal dynamic mechanisms,





20


rather than phenomenologically as with previous research. His arguments linked the local El Nifto phenomenon experienced in South America with the global relaxation of the prevailing Trade winds in the equatorial Pacific (Bjerknes, 1966, 1969). Bjerknes believed that the El Niho effects were the direct result of a similar local relaxation of the winds offshore of South America. While this was later shown to be incorrect, his ideas laid the foundation for most future research into ENSO (Enfield, 1989).

Bjerknes was no doubt hampered by the lack of high-resolution observational

data, which became available after 1975, due in part to increased research interest in the ENSO after the intense 1972 El Nifio episode. It was found that, contrary to common assumptions, the winds offshore Peru did not slacken along with the remainder of the equatorial Trades during low phase periods of the SOI (Enfield, 1989). This forced a reconsideration of Bjerknes' theories concerning the presence of warm waters offshore of South America during El Nifto periods. What has evolved instead is a much more complex model of ocean-atmosphere interaction.

The application of wave dynamics to the coupled ocean-atmosphere system has given rise to the most successful theories regarding the ENSO phenomena, and demonstrates how the atmospheric conditions of the western equatorial Pacific can directly impact on the oceanic conditions off the coast of South America. The Trade winds in the vicinity of the equator generally exercise enough shear on the surface waters to force them westward, thereby causing a rise in sea level in the west, and a commensurate depression of the primary thermocline (Philander, 1989). During periods when the Trades slacken, the warm surface waters in the west would begin to move eastward. Due to the Coriolis effect, water north and south of the equator is forced towards the equator, thus entraining the eastward propagating wave of water. This is known as a Kelvin wave (after Lord Kelvin, who is credited for having first described them mathematically) (Enfield, 1989).






21


When the wave encounters the east continental boundary, three things can happen to the energy being carried with it. First, a certain amount of the energy is reflected directly back towards the west; in this situation, the Coriolis effect works against the wave train, driving northern water more towards the north and southern waters more towards the south, thus quickly dispersing the wave (Cane and Gent, 1977; Horel and Wallace, 198 1; Knox and Halpern, 1982). Second, much of the energy is transferred into westward propagating counter-rotating eddies called Rossby waves (Horel and Wallace, 1981; Keen, 1982). Third, the sea levels are pushed upward (and the thermocline depressed) along the continental boundary; it is this effect which causes the temperature of the waters along northwest South America to rise during an El Niiio event, since the upwelling now comes from warmer waters having been advected from the west (Wyrtki, 1975, 1984; Harrison and Schopf, 1984).

The shifting of the equatorial waters is responsible for many of the atmospheric phenomena that are associated with ENSO. During a "normal" phase of the ENSO (i.e., an SOI near the long term average), the warm waters held in check by the Trade winds in the western Pacific destabilize the atmosphere over the region, causing widespread convection due to warm air over the warm water. In the east, cold waters near South America stabilize the atmosphere, causing subsidence and precluding widespread precipitation. The subsiding air in the east is carried westward by the trade winds, lifted, and carried back towards the east, completing the so-called "Walker circulation" of air over the equatorial Pacific (Julian and Chervin, 1978; Rasmussen and Carpenter, 1983). (Similar atmospheric circulation features have been noted over the Indian and Atlantic Oceans.) Thus, the prevailing arid conditions over Peru and Ecuador, as well as the wet conditions over Indonesia, can in large measure be attributed to the coupling of the prevailing winds and the advection of warm equatorial waters of the Pacific Ocean (Gill and Rasmussen, 1983; Goldberg et al., 1987).





22


If the pressure gradient is reduced, causing a relaxation in the Trades and the triggering the onset of an El Nifio, the migration of warm waters causes the areas of maximum convection to be shifted towards the east (Philander, 198 1). This disrupts the Walker circulation, causing subsiding conditions to move over Indonesia, often resulting in drought conditions because of colder than normal ocean waters (Julian and Chervin, 1978). In the east, warm waters destabilize the air over Peru and Ecuador, sometimes causing intense rains and flooding.

The ENSO effects extend well beyond the tropics. Through the process of

teleconnection, semi-permanent pressure centers in the mid- and high-latitudes can be shifted, causing dramatic changes in the mid-latitude jet stream patterns, and resulting in sudden and significant changes to the temperature and precipitation patterns of regions well outside the tropics (Van Loon and Madden, 198 1). The primary reason is that the zonal shortening of the Walker circulation associated with ENSO forces a commensurate meridional increase in the Hadley circulation, which is responsible for variations in the Ferrel and Polar circulation cells, and therefore the polar and subtropical jet streams as well (Horel and Wallace, 198 1). These changes in the jets are responsible for variations in mid-latitude weather system movement. For example, an eastward shift in, or a deepening of, the Aleutian Low by just a few hundred miles can result in winter storms impacting the California coast far to the south of their expected strike zone (Emory and Hamilton, 1985); this occurred during the 1982-83 El Nifio episode (Baumgartner and Christensen, 1985; Simpson, 1984). A shift in the opposite direction can cause extreme drought in the Southwest United States. Changes in the mean storm track into the west coast of the U.S. is often reflected in the so-called Pacific-North American (PNA) index, which compares 500 millibar geopotential heights for locations in the north-central Pacific Ocean, the southeastern U.S., and the northern Great Plains of North America; a positive correlation between the PNA and the SOI has been noted during the boreal






23


winter months, which is indicative of anomalous high-amplitude shifts in the polar jet stream, which tends to direct more storms into southern California and the Southeast U.S. during ENSO events (Epstein, 1992; Halpert and Ropelewski, 1992). A similar linkage between the ENSO and precipitation in the Southwest United States was noted by Woolhiser et al. (1993). Katz and Parlange (1993) further demonstrate the ability to condition stochastic rainfall occurrence parameters in the United States on an atmospheric index. This invites investigation into whether the daily rainfall properties of tropical Central America can be similarly conditioned on an index such as the Southern Oscillation Index (SOI), which gives an indication of the state of the ENSO.

Several studies have examined the effects of ENSO on the Central American

isthmus. An examination on the relationship between precipitation and the state of ENSO by Ropelewski and Halpert (1987) and Halpert and Ropelewski (1992) demonstrated the prevalence of drought-like conditions over much of the region during El Niflo events. Estoque et al. (1985) shows that El Nifio periods are associated with decreased rainfall in most of Panama, while northwest Panama receives less rainfall during the same period; this supports the overall conclusions of Ropelewski and Halpert, while hinting at further complexity in the region. Further studies in Costa Rica by Fermindez and Ramirez (1991) reiterated these conclusions, and noting a connection between the Trade winds and the west-to-east contrast in the rainfall characteristics. Further research by Waylen et al. (1996) showed that the pattern of ENSO effects on rainfall in Costa Rica was more complex, with increased precipitation during El Nifio events occurring in the eastern part of the country, and thus opening the door to further speculation on the nature of ENSObased rainfall modification in the region.














CHAPTER 3
STUDY AREA AND DATA DESCRIPTION The primary physiographic features and climatic processes which influence the

formation of precipitation and facilitate the interpretation of daily precipitation models in Costa Rica are discussed. The study area is delineated, meteorological stations are identified, and a subset of three stations is selected for examination. The amount of daily rainfall data available from each station is determined. In addition, the climate indices used in the correlation analyses are identified.

Physiography

The topography of Costa Rica is a complex mixture of humid coastal lowlands to the east and semiarid plateaus to the west, separated by cordilleras which roughly bisect the country from northwest to southeast, with an embedded Central Valley (Figure 3-1). Four primary cordillera are evident. The northern part of the country is dominated by the Cordillera Guanacaste, Tilaran, and Central which are volcanic in nature; and in the south by the Cordillera de Talarnanca, which represents an exposed batholith. Elevations range from sea level to over 3000 meters in the cordilleras, with the greatest elevation found in the south. The cordillera are punctuated with narrow passes, which allow dominant winds from one side of the country to penetrate to the lee side; these passes are more numerous in the northern cordillera, which have lower elevations and less east-west extent than the southern cordillera.



24






25


General Climatic Characteristics

Costa Rica lies at the confluence of several weather-generating processes, which give rise to the seasonal precipitation patterns seen in Figures 3-2 and 3-3. The region is dominated by the northeast Trade winds, which keep the eastern lowland well-watered, but result in a rainshadow to the west. The Trade winds reach their greatest strength during the boreal summer months, when the pressure gradient between the Atlantic Subtropical High and the ITCZ is maximized. This low-level flow results in the advection of an large amount of precipitable water from the Caribbean, which is condensed when the moist air is forced over the cordillera. The ITCZ experiences a north-south migration in the eastern Equatorial Pacific, from approximately 3 degrees north in the boreal winter to 10 degrees north in the summer. The ITCZ promotes considerable instability in the region, bringing heavy rains to the western areas of Central America. In addition, crossequatorial westerlies arise south of the ITCZ, bringing frequent precipitation to the extreme south of the country as the ITCZ reaches its maximum northerly extent (Hastenrath, 1976). It is also during the northward transit of the ITCZ that a strengthening of the Trade wind flow occurs, resulting in the "Veranillos de San Juan" (or simply veranillos, literally, "little summer"), during which the rainfall abates somewhat; this generally occurs during July, and ends when the ITCZ comes to its most northerly position in late summer.

During the boreal summer, there has also been reported a connection between the occurrence of tropical storm activity in the Caribbean basin and rainfall on the Pacific coast of Costa Rica; it has been posited by Vargas and Trejos (1994) that this is due to a





26


reversal in the pressure gradient over this region during periods of tropical storm activity, altering the overall wind fields over the Central American isthmus.

A third seasonal influence is the incursion of polar air from the north during the winter months, following cold front penetration into the region; these are referred to as the notes. This results periods of prolonged precipitation in the northeastern part of the country, a consequence of stress differential induced convergence by the northerly winds (Bryson and Khun, 1961; Schultz et al., 1997, 1998). While less effective at creating precipitation over long periods than the summer climate components, the notes are nevertheless important rainfall components, since the rains they produce come at a time when the country is experiencing considerably drier conditions than in the summer. The norte-driven rainfalls are often locally quite intense, resulting in the unusual situation where the annual maximum rainfalls for the area will occur during the winter months.

Previously Noted Costa Rica-ENSO Connections

The convergence of these processes makes the climate of Costa Rica extremely responsive to changes in circulation in many parts of the global climate system; Waylen et al. (1996) has reported significant correlations between monthly rainfall and SOI measures with lags of up to +/- two years, demonstrating the long-term sensitivity of the region's climate to changes in global circulation.

Extremes in the state of the ENSO, as measured by the SOI, have been shown to affect the climatic conditions over the Central American isthmus. El Niho conditions tend to enhance the subtropical jet stream to the north during the boreal winter, while increasing the Trade winds in the south over Costa Rica and Panama; this is due to the increased pressure gradient that exists between the equatorward north Atlantic high






27


pressure system and the lower pressures associated with the warm waters off of Peru and Ecuador (Estoque et al., 1985; Ferndindez and Ramirez, 1991; Waylen et al., 1994, 1996). During low phase summers, northward movement of the Pacific ITCZ is restrained, causing significant precipitation deficits in northwest Costa Rica (Waylen et al., 1996). In the Atlantic during the boreal summer, increased Trade winds arise due to anomalous movement of the ITCZ, resulting in a decrease in the incidence of tropical storm activity (Vargas and Trej os, 1994).

During La Nifia conditions, the northward migration of the ITCZ during the

summer allows increased precipitation to the west coast of Central America, and reduced Trade wind activity over the Atlantic (Waylen et al., 1996); these periods are often associated with increased Atlantic tropical storm activity (Gray, 1 984a,b), and greater Westerly winds along the Pacific coast. During the winter of La Nifia, an increased meridional pressure gradient between North and Central America allows greater cold front intrusion into Nicaragua and Costa Rica, resulting in more intense nortes, and increased rainfall in the northeast regions (Schultz et al., 1997, 1998).

In terms of regional impacts, the progression of a typical ("canonical") El Nubt event is fairly well documented. During the year of the onset of ENSO ("Year 0"), the waters offshore of northwestern South America begin to warm beyond the seasonal averages, the result of the advected waters associated with the Kelvin wave moving eastward in the equatorial Pacific. The warm waters result in lower pressures in the eastern Pacific, which intensifies the pressure gradient between the Azores high pressure in the Atlantic and the eastern Pacific, bringing an increase in the Trade winds across the Central American isthmus. The Trade winds increase the rainfall on the eastern slope of






28


Costa Rica, and increase the upwelling associated with the Costa Rica Dome, an area of colder waters offshore of western Costa Rica. This causes a more stabilized atmosphere to develop over the western half of the region; the ultimate result is reduced summer precipitation in the west, and higher than normal summer rainfall in the east. Increased wind stress over the Caribbean (resulting from increased Trade winds) causes cooler water to surface, raising the atmospheric pressure in the region and reducing the overall pressure gradient between the North American continental high and the Caribbean low. The lower pressure gradient results in a reduced capacity for cold air masses to move southward, causing fewer norte events to occur during ENSO year 0. Figures 3-4 and 3-5 show how Year 0 seasonal rainfall differs from the long-term average.

In the year following an ENSO event ("Year I"), the climatological impacts in the region are reversed from the Year 0 conditions. This reflects the reestablishment and strengthening of the ITCZ, the increasing Trade winds over the central Pacific equatorial waters, and the resulting westward advection of warm waters (Figures 3-6 and 3-7). The resumption of cold conditions offshore of South America reduced the pressure gradient between the North Atlantic High and the western Pacific, resulting in an overall decrease in the Trade wind activity across Central America, and a commensurate reduction in summer rainfall in the eastern part of the country, as well as an increase in convective activity in the west. Additionally, norte activity increases during the winter months of Year 1, probably resulting from increase latent heat being drawn into the Caribbean basin as a consequence of a more northerly ITCZ during this time; this helps create an increased pressure gradient between the wintertime North American high pressure and the low pressure over the Caribbean, which allows greater cold front activity farther






29


south. The increase in norte activity bring increased Year I rains to the northeastern part of the country. Schultz et al. (1997, 1998) hypothesize a differentiation between the strength of the notes between Year 0 and Year 1, referring to them as "cool" and "cold" notes, respectively, based upon the strength of the cold air masses as they are modified by the El Niho.

Daily Precipitation Data and Station Location

The greatest difficulty faced in performing a detailed climatic analysis is the

acquisition of reliable, long-term data. This is especially true when the analysis entails the use of daily precipitation data. In many cases, such records were not kept in their raw state, but rather were aggregated into monthly and annual records, rendering them useless for this particular study. In Costa Rica, however, many of the original daily rainfall measurements exist in digital form, and they are considered a commercially valuable commodity by the agencies that control them.

For reasons of expense, it is important to narrow the focus for data acquisition to that which is absolutely necessary for the completion of the study. In this case, the focus is on describing a transect which encompasses the most populous (and thus most densely monitored) regions of the Central Valley, stretching eastward from the Caribbean coastline (enclosing an area approximated by the drainage basin of the Reventaz6n River), and westward to the Pacific Ocean (to include the basin of the Tarcol6s River). There are 41 stations available within this transect; however, only eight stations possess at least 30 years of records. It is from these eight stations, which places significant limitations on the number of stations that can be used for long-term analysis. Table 3-1 lists the these stations, and Table 3-2 shows their geographic disposition and data






30


availability. Figure 3-8 shows the locations of the stations, and the boundary of the study area.

Three stations were chosen from the eight long-term stations (those with records in excess of 30 years) to be used for performing detailed statistical analyses. The stations were chosen to be representative of the Caribbean, Central Valley, and Pacific coast regions of the study area, because of their geographic locations and relatively long data records. Sanatorio Duran (station ID number 073011) was selected to represent the Caribbean region, Villa Mills (073033) was chosen for the Central Valley/transition region, and Nagatac (080005) was chosen to represent the Pacific coast. These stations are highlighted on Figure 3-8. The monthly rainfall characteristics of these three stations in relation to ENSO is shown in Figures 3-9 to 3-11.

Atmospheric Index Data

Atmospheric index data, provided by NOAA via the World Wide Web (NOAA, 1996) was used to perform lag-correlations on the computed daily rainfall statistics. Four indices were selected to represent the likely areas of broad-scale climate impact upon the Costa Rican rainfall generation process. The indices selected were the Southern Oscillation Index (SOI), the North Atlantic Oscillation index (NAO), the central equatorial Pacific sea-surface temperature index (NIN034), and the Quasi-Biennial Oscillation index (QBO5O).

The SOI is calculated by subtracting the normalized monthly sea level pressure (SLP) at Darwin, Australia from the normalized SLP at Tahiti; the SOI provides a reasonable representation of the overall pressure gradient that exists between the central and western Pacific regions, and is the traditional means by which the strength of an El






31


Nif'io event is measured. The NAO is calculated in the same fashion as SO!, substituting the SLP values for the Azores and Iceland; the NAO measures the strength of the SLP dipole over the North Atlantic, and as such might give an indication into the role of the North Atlantic high pressure zone in the development of Trade winds across Costa Rica. In the case of both the SOI and the NAO. each monthly measurement is normalized by subtracting the average SLP for a month from the long-term average SLPs at that site, and dividing by the standard deviation of the long-term SLPs. The data for both indices have been recorded continuously since the 1 800s, and are readily available to any researcher.

The NIN034 index is important due to the strong association between abnormally warm ocean temperatures of the equatorial Pacific and the onset of an ENSO event. The NIN034 measurements represent a monthly averaging of the equatorial sea surface temperatures surrounding the 180 degree meridian, and are calculated by using a combination of historical shiptrack data augmented in recent years by remotely-sensed satellite data.

The QB050 index is an average of the zonal equatorial stratospheric winds at the 50 millibar level. These winds have been shown to possess quasi-periodic cycles of approximately 27-30 months, during which time they generally change direction twice; from this cycle is derived the designation "Quasi-Biennial Oscillation" (QBO). While the physical mechanisms that govern the interaction of tropospheric and stratospheric climate systems are poorly understood, a number of researchers have noted strong statistical associations between climate anomalies and the QBO (Kodera, 1991; Mason and Tyson, 1992). Gray (1 984a, 1 984b) has demonstrated a correlation between both the QBO and






32


SOI and the formation of tropical cyclones in the Atlantic basin. Vargas and Trejos (1995) noted a possible connection between Atlantic tropical cyclone activity and Central American rainfall, and for this reason the QB050 index was included in the analysis.

For many of the analyses it was necessary to segregate the data record into "El Niflo," "La Nifia," and "other" year-types. This segregation was based upon the classification scheme used by Diaz and Kiladis (1992) and supplemented with NOAA data to include the most recent events. The resulting year listing is shown in Table 3-3.





33


Table 3-1. Station locations*

Station ID Station ID Code Station Name Latitude Longitude Elevation**
Number
073011 SANDURAN Sanatorio Durain* 9.93 -83.88 2337
073016 JUANVINA Juan Vinas 9.83 -83.78 990
073026 TAPANTI Tapanti 9.77 -83.83 1203
073027 ELCANON El Cafi6n 9.68 -83.90 2460
073028 ELHUMO El Humo 9.80 -83.72 680
073029 ELLLANO El Llano 9.77 -83.87 1572
073030 CORDONCI Cordoncillal 9.75 -83.78 1240
073033 VILLAMIL Villa Mills* 9.57 -83.72 3000
073034 BELEN Bel6n 9.73 -83.88 2010
073036 TSEIS T-Seis 9.72 -83.77 2000
073037 ELDESTIE El Destierro 9.70 -83.75 2020
073038 OJOREVEN Ojo de Agua 9.62 -83.82 2960
073039 TRESDEJU Tres de Junio 9.67 -83.85 2200
073040 BERMA Berma 9.67 -83.82 2480
073044 LASUIZA La Suiza 9.85 -83.62 620
073045 TAUS Taus 9.77 -83.72 900
073046 CACHIPLA Cachi (plantel) 9.82 -83.80 1018
073047 TUCURRIQ Tucurrique 9.85 -83.75 770
073055 LAAMISTA La Amistad 9.98 -83.90 560
073074 ANTONTUR San Antonio de Turrialba 9.97 -83.72 1190
073080 LAMUERTE Cerro La Muerte (Repet.) 9.57 -83.77 3365
073096 ELSAUCE El Sauce 10.00 -83.65 870
080005 NAGATAC Nagatac* 10.07 -84.55 450
084005 AVANCEDE Avance de Tres Rios 9.98 -83.97 1870
084006 HACONCEP Hca Concepcion Tres Rios 9.92 -84.00 1320
084018 REDONDO Rancho Redondo 9.95 -83.95 1780
084019 LAGUNA Laguna 9.97 -83.87 3140
084024 OJOTARCO Hcda Ojo de Agua 9.93 -84.22 850
084027 SANRAMON Subestaci6n San Ram6n 10.08 -84.47 1061
084034 EMBALAGA Embalse La Garita 9.95 -84.35 460
084037 ANTONESC San Antonio de Escazu 9.90 -84.13 1380
084041 IPISDEGU Ipis de Guadalupe(Planta) 9.97 -84.03 1280
084043 SDROBLE Santa Domingo del Roble 10.07 -84.17 1320
084047 PRESALAG Presa La Garita 9.93 -84.30 471
084048 ANTECAMA AntecAimara La Garita 9.93 -84.35 466
084063 VOLCANPO Volcin Poas 10.18 -84.23 2564
084074 PAVAS Pavas 9.97 -84.13 997
084103 ELDESCAN El Descanso 10.02 -84.20 800
084104 LAMARAVI La Maravilla 9.90 -83.77 -9.9
084105 SUBHERED Subestacion Heredia 10.00 -84.13 -9.9
084110 LAGUACIM La Guacima 9.97 -84.25 -9.9
Bolded stations were used for in-depth focus in this study.
* A value of -9.9 indicates that the elevation has not been accurately surveyed.






34


Table 3-2. Station data lengths.

Station ID Station ID Station Name Start End Percent Available
Number Code Year Year Missing Years
Data of Record
073011 SANDURAN Sanatorio Durin 1943 1991 2.01 48.0
073016 JUANVINA Juan Vinas 1975 1991 0.52 16.9
073026 TAPANTI Tapanti 1939 1991 7.38 49.1
073027 ELCANON El Cafi6n 1954 1991 7.09 35.3
073028 ELHUMO El Humo 1954 1991 4.6 36.3
073029 ELLLANO El Llano 1957 1991 4.2 33.5
073030 CORDONCI Cordoncillal 1960 1991 9.15 29.1
073033 VILLAMIL Villa Mills 1942 1991 6.66 46.7
073034 BELEN Bel6n 1959 1991 22.03 25.7
073036 TSEIS T-Seis 1962 1991 13.99 25.8
073037 ELDESTIE El Destierro 1962 1991 23.11 23.1
073038 OJOREVEN Ojo de Agua 1959 1991 10.34 29.6
073039 TRESDEJU Tres de Junio 1962 1991 2.19 29.3
073040 BERMA Berma 1962 1991 19.76 24.1
073044 LASUIZA La Suiza 1962 1991 2.62 29.2
073045 TAUS Taus 1962 1991 26.21 22.1
073046 CACHIPLA Cachi (plantel) 1952 1991 0.16 39.9
073047 TUCURRIQ Tucurrique 1964 1991 0.21 27.9
073055 LAAMISTA La Amistad 1966 1991 13.03 22.6
073074 ANTONTUR San Antonio de Turrialba 1965 1991 2.93 26.2
073080 LAMUERTE Cerro La Muerte (Repet.) 1970 1991 3.19 21.3
073096 ELSAUCE El Sauce 1975 1991 3.51 16.4
080005 NAGATAC Nagatac 1959 1997 11.7 34.4
084005 AVANCEDE Avance de Tres Rios 1980 1984 3.29 4.8
084006 HACONCEP Hca Concepcion Tres Rios 1975 1984 5.81 9.4
084018 REDONDO Rancho Redondo 1975 1984 0 10.0
084019 LAGUNA Laguna 1975 1984 0 10.0
084024 OJOTARCO Hcda Ojo de Agua 1980 1984 6.63 4.7
084027 SANRAMON Subestaci6n San Ram6n 1978 1990 16.29 10.9
084034 EMBALAGA Embalse La Garita 1963 1990 7.24 26.0
084037 ANTONESC San Antonio de Escazu 1980 1983 0 4.0
084041 IPISDEGU Ipis de Guadalupe(Planta) 1969 1984 1.68 15.7
084043 SDROBLE Santa Domingo del Roble 1959 1990 4.77 30.5
084047 PRESALAG Presa La Garita 1971 1981 0.05 11.0
084048 ANTECAMA Antecamara La Garita 1971 1980 4.25 9.6
084063 VOLCANPO Volcin Poas 1971 1990 8.25 18.4
084074 PAVAS Pavas 1975 1979 10.3 4.5
084103 ELDESCAN El Descanso 1976 1990 3.8 14.4
084104 LAMARAVI La Maravilla 1976 1990 19.79 12.0
084105 SUBHERED Subestacion Heredia 1976 1982 20.12 5.6
084110 LAGUACIM La Guacima 1980 1990 5.95 10.3





35


Table 3-3. ENSO classification of years since 1900

El Nifio Years La Nifia Years "Other" Years
1902 1903 1900 1952
1905 1906 1901 1955
1911 1908 1904 1956
1914 1916 1907 1958
1918 1920 1909 1959
1923 1924 1910 1960
1925 1928 1912 1961
1930 1931 1913 1962
1932 1938 1915 1966
1939 1942 1917 1967
1940 1949 1919 1968
1941 1954 1921 1971
1946 1964 1922 1974
1951 1970 1926 1976
1953 1973 1927 1978
1957 1975 1929 1979
1963 1988 1933 1980
1965 1935 1981
1969 1936 1983
1972 1937 1984
1977 1943 1985
1982 1944 1987
1986 1945 1989
1991 1947 1990
1993 1948 1992
1997 1950 1994






36









11.00 0



10.50


Sanatorio Duran
10.00 4Xs (o(073011)




9.50
Nagatac (080005) Villa Mills (073033)
9.00



8.50




-85.50 -85.00 -84.50 -84.00 -83.50 -83.00
Longitude


0 375 750 1500 2250 3000 4000 elevation (m) Figure 3-1. Costa Rica topography.






37




11.50 .

DJF
11.00


10.5010.00,


9.50 2700
,. 2700


9.00- 2400

2100
8.50
1800
8.00 ...
-86.00 -85.50 -85.00 -84.50 -8400 -83.50 -83.00 -82.50 1500
11.50-1200 11.00- 900

V .i7 600
1050 .

-- &300
10.00


9.50 (millimeters)


9.00


8.50
MAM
8.00---
-86.00 -85.50 -85.00 -84.50 -84.00 -83.50 -83.00 -82.50

Figure 3-2. Costa Rica seasonal precipitation Top: Boreal winter; Bottom: Boreal spring.






38





11.50

JJA
11.00


10.5010.00- -9.50 2700


900 2400

2100
8.50

1800

8.00- --
-86,00 -85.50 -85.00 -84.50 -84.00 -83.50 -83.00 -82.50 1500

11.501200 11.00 900

--600


300
10.00, A
--0

9.50- (millimeters)



9.00


8.50"

SON
8.00 ..
-86.00 -85.50 -85.00 -84,50 -84.00 -83.50 -83.00 -82.50


Figure 3-3. Costa Rica seasonal precipitation Top: Boreal summer; Bottom: Boreal autumn





39



11.50DJF
11.00

10.50
650
o :: .ill,569
10.00-488
9.50- 406
i ".325
9.00 244

163
8.50 :-
-81
-0
8.00- ...
-86.00 -8550 -85.00 -84.50 -84.00 -83.50 -83.00 -82.50 -81
11,50 -. -163

-244
11.00.-325

..,-406
10.50
-488
10.00- j -569
-650
9.50.. (millimeters)

9.00- --....

8.50

MAM
8.00 .
-86.00 -85.50 -85.00 -84.50 -84.00 -83.50 -83.00 -82.50

Figure 3-4. Costa Rica seasonal precipitation changes, average El Nifho years, Year 0 Top: Boreal winter; Bottom: Boreal spring






40



11.50
JJA
1100


10.5010.00


9.50 650
569
9.00 488
406
850- 325
244
8.00- ... 163
-86.00 -85.50 -85.00 -84.50 -84.00 -83.50 -83.00 -82.50
11.50o ----..- 81
___0 11.oo- -81
)9 -163
10.50

-~ -325
10.0006-
-, .. -- --488
9.50- -569

1-650
9.00
(millimeters)

8.50

SON
8.00- -- _......- ... .
-86.00 -85.50 -85.00 -8450 -84.00 -83.50 -83.00 -82.50


Figure 3-5. Costa Rica seasonal precipitation changes, average El Nifio years, Year 0 Top: Boreal summer; Bottom: Boreal autumn






41




11 50
DJF
11 00


1050


10 0

650
9.50
569
488
9 .....406

325
8.50 -24
85o 244
-- 163
800
-8.00 -85.50 -85.00 -84.50 -84.00 -83.50 -83.00 -8250 81
11.50- --. 0
-81
11.00 -163
-244
10.50- ; -325

> ;-406
1000-. ;- -488
-569
50-650
9 50-. ,- // .. .

.(millimeters)
9.00 .... ...


8.50-

MAM
800 ..
-86.00 -85.50 -85.00 -84.50 -84.00 -83.50 -83.00 -82.50


Figure 3-6. Costa Rica seasonal precipitation changes, average El Nifio years, Year 1 Top: Boreal winter; Bottom: Boreal spring





42

11.50JA
JJA
11.00


10.5010.00- i '
I "'1 > R ) 650
569
9.50 -- 569
K / 488
: : "406
9.00 40
9.=- '325

8.50 244
--163
8.00 81
-8600 -85.50 -85.00 -84.50 -84.00 -83.50 -83.00 -82.50
11.50- .0
-81
11.00 -163


1050 -4
-325
1.-o- -406
-488
-569
9.50-.
-650

9.00 (millimeters)

8.50

SON
8.00 ---....
-86.00 -85.50 -85.00 -84.50 -84.00 -83.50 -83.00 -82.50

Figure 3-7. Costa Rica seasonal precipitation changes, average El Nifio years, Year 1
Top: Boreal summer; Bottom: Boreal autumn






43









11.0010.50


Nagatac (080005) + 10 Sanatorio Duran (073011)
10.00, + +
++. : .*+* o




9.50
Villa Mills (073033)

Years of Available Record
9.00- 4 to 10
+ 10 to 20 O 20 to 30
30 to 40
40 to 50
8.50,




-85.50 -85.00 -84.50 -84.00 -83.50 -83.00


Figure 3-8. Costa Rica station location.






44
















350








E
zJ0
C

L
*.

0
E
10D








JAN~. FEB W AMR IVY IN 1 AIJG UP OCT NOV EBZ find

-- Al Ym -a 8 N Yn -6- La 46 Yars~*-iw





Figure 3-9. Sanatorio Duran average monthly precipitation.







45




















500





400 300

CL

0c
E







100 JAN FEB MAR APR MAY JUN~ AL AUG SEP CT N'CW DEC nmnt ~AJ Y~s~B Nlo Yas -.-N&f~Y aYewsFigure 3-10. Villa Mills average monthly precipitation.






46


















70D








.00 100


0





AN FEB NAR APR W~Y ANI' 1 AUG SEP CT NOV DEC 11 -PJI Y s--B WbYm--- Lza e Ys --- dtu Y4 Figure 3-1 1. Nagatac average monthly precipitation.














CHAPTER 4
METHODOLOGY


Modeling of the occurrence and intensity of rainfall permits the creation of a comprehensive representation of daily precipitation across a variety of spatial and temporal scales. In addition, examination of the parameters describing each component facilitates the statistical correlation of model characteristics with indices describing the atmospheric processes related to the ENSO phenomenon. Within the context of the present research, the following questions will be addressed:

1. How can the characteristics of daily Costa Rican precipitation be modeled? In

particular, can the occurrence and intensity probabilities be used effectively to describe the changes in rainfall that arise as a result of ENSO-related weather

anomalies at individual locations? How well do each of the models compare to the actual precipitation measurements, both in terms of the goodness-of-fit

and parsimony?

2. To what extent do the resulting models indicate changes in weather conditions

as a result of ENSO?

3. Are the modeling results consistent with the current understanding of the

climate mechanisms governing the region? How well do these results

correspond to previous studies of rainfall in the region, especially those

conducted at more aggregated (i.e., annual) time scales?



47





48


4. Can the modeling results be used to create reasonable estimates of future

rainfall characteristics? That is, can forecasts of rainfall be created using

predictions of ocean and atmospheric conditions. How well do these forecast techniques compare to each other, and to simple climatology (i.e., long-term

averages)?



The general strategy for performing this research is as follows: (1) create statistics representing the occurrence and magnitude of daily rainfall events, using three longrecord stations representing geographical regions within the study area, which will be aggregated into a monthly time series for each station; (2) segregate the statistics by yeartype, using the augmented Diaz and Kiladis (1992) classification previously discussed, and analyze the resulting patterns; (3) perform lag-correlations on each statistic for the stations against the monthly atmospheric indices previously identified, for lags of +/- 36 months, to identify possible temporal relationships between the statistics and the antecedent or following atmospheric conditions; (4) use the parameters created in step (1) to perform statistical goodness-of-fit tests to determine the best-fit distributions used to describe precipitation magnitudes for each of the stations, utilizing both Chi-squared tests and bootstrap simulation experiments; and (5) use the preceding results to construct several types of forecast models, to test the feasibility of creating "real world" forecasts, for application to agricultural and hydrological models.

Precipitation Statistics

The primary difficulty in aggregating daily rainfall data into statistics which can be meaningfully compared to existing atmospheric indices is arriving at a time scale





49

which will minimize the amount of inter-seasonal climate change while allowing the maximum amount of information to be represented by the statistics. Generally, a monthly time scale is used; however, this allows a considerable amount of "noise" into the transition probability measurements. For this reason, monthly triads, represented by the middle month, are computed per year of data, with each triad overlapping the previous and following triad by one month. The triad statistics are then used as a monthly time series for each of the occurrence and magnitude statistics, and can be considered "smoothed" in the sense that inter-monthly variation has been distributed across monthly boundaries, thereby reducing the "noise" and minimizing the possibility that seasonal variations will be masked in the computation of the parameters. Precipitation Modeling -- Occurrence

Markov chain models are created by conditioning the probability of the

occurrence of rainfall on one day upon whether measurable rainfall was observed on the previous days. These are characterized by "transition probabilities":

Pr ( Xt, I I Xt, Xt- 1, Xt-2, ... X I I = Pr ( Xt+ I I Xt, Xt- 1, ... Xt-,) (4-1)

in which the "order" of the model is given by m, the number of previous days to be examined for rainfall.

Conceivably, the number of states to be used in a Markov rainfall model is unlimited, but more generally a two-state model is used: Xt = I wet day (4-2a)
Xt = 0 dry day (4-2b)


Using this notation, the transition probabilities for a two-state Markov precipitation model of order m is given by:





50


Ptjk. in Pr{IX+i m I rn -i1, Xt- = m-2, X-2 = m-3, ... ,Xt(m- 1) =j, Xtrim = i 1 (4-3)

where ij,k,... refer to the previous days' state (i.e., wet or dry).

The maximum likelihood estimators for the transition probabilities is given by:



nhA

for a multi-order, two-state Markov model, where n represents the number of transitions which have been counted for a given time period, and ij,k... represent the state of previous days upon which the transition probability is conditioned.

To decide what order to use in modeling the rainfall process, it is necessary to determine how much information is being included in a model of given order, and compare this to other model orders. Of concern is the actual amount of information about the rainfall occurrence process, as well as the number of parameters to be generated to account for this information. Ideally, we desired the maximum amount of information and the fewest parameters (i.e., greatest degree of parsimony); in other words, a balance must be struck between having too many, and too few, parameters generated. "Information criteria" are employed to make this determination. The two most common are the "Akaike Information Criteria" (AIC) (Akaike, 1974) and the "Bayesian Information Criteria" (BIC) (Schwartz, 1978). Both methods rely on the generation of log-likelihood functions for the estimated transition probabilities of the fitted Markov model. Given a two-state Markov model, the log-likelihoods for orders 0, 1, 2, and 3 are given by Wilks (1995, page 3 01) as:





51




Lo = ni ln(p) (4- 5a)
j=0

L = nii ln(pij) (4-5b)
i=O j=0
L2 = I -'nhij In(phii) (4- 5c)
h=O i=O j=O

L3 = -- ghij ln(pghij) (4-5d)
g=0 h=O i=O j=O



These log-likelihoods are then used to calculate the AIC and BIC scores; the order giving the lowest score is the most parsimonious (i.e., the most "efficient") model. The criteria are calculated using the following equations for an m-order, s-state model: AIC(m) = -2Lm + 2sm(s- 1) (4-6a)

BIC(m) = -2Lm + sm(ln (n)) (4-6b)

Given that this study will use two-state ("rain/no rain") models, these equations simplify to:

AIC(m) = -2Lm + 2m+1 (4-7a)

BIC(m) = -2Lm + 2m(ln (n)) (4-7b)



The BIC is generally considered the more conservative of the two criteria, and will often give more definitive results than the AIC. In this study, both AIC and BIC are calculated and compared, in order to determine the best order to be used for the Costa Rica data.

Most studies of daily rainfall (e.g., Katz, 1977, 1981; Coe and Stem, 1982) have found that a first-order model is sufficient to describe daily rainfall. This allows the





52


occurrence process to be represented by four parameters, of which two must be calculated:

pij = Pr{ Xj n I Xi = ml mn = 10,11 (4-8)

where


Xn I if precipitation occurred (4-9)
0 otherwise

and

Poo + P10 = 1 (4-1 Oa)
P10+P11 1 (4- 1 Ob)


The process of computing the occurrence statistics is illustrated in Figure 4-1. Each three-month sequence is passed through a computer program which tallies the sequences of wet and dry days. These totals are used to create the transition probabilities, for orders or 0, 1, 2, and 3. The AIC and BIC statistics are computed, and the resulting monthly time series of probabilities and information criteria are segregated according to whether the year-type was El Nifio, La Nifia, or neither. In addition, a tally is kept of all years, to be used as a baseline for comparison. An evaluation of the order results is made, and the order with the best (i.e. lowest) overall AIC/BIC is used for later analyses. In the case of mixed results, the higher order is chosen so as not to inadvertently hamper further analysis with loss of information.

Precipitation Modeling -- Magnitudes

Determination of the proper distributional forrn for a set of rainfall measurements is more problematic than modeling the occurrences. It is a computationally more intensive process, and sensitive to the available sample size due to the presence of lowprobability high-intensity events (Guttman, 1994). In addition, since it is very unlikely





53

that a particular distribution will perfectly model the entire data record (both temporally and spatially), some means of choosing the best-fit distribution from among the candidates must be found.

Previous research (Hosking and Wallis, 1987 ; Hosking, 1989; Harrison, 1995) has indicated that the use of L-moments is more effective than standard method-ofmoments and maximum-likelihood techniques for deriving the parameters associated with probability distributions, due to their demonstrated robustness and ease of computation. For this reason, L-moments are used in this research. The algorithms discussed in Hosking (1989) and Vogel and Fennessey (1993) are used to compute the Lmoments, using the same temporal data streams as were employed in determining the transition probabilities. The L-moments are composed of "probability weighted moments" (PWM). The rth PWM, br, which are computed using a descending-sorted data series xO) composed of n elements, as follows:


1tFr ___br nr ) (4-11)




The L-moments, X, are computed from the PWM as follows:


,, = -bk(-1)rk (4-12)
k=O k

Analogous to product-moment ratios, Hoskings (1990) defines the L-moment ratios:


r2= L-coefficient of variation (L-CoV) (4-13a)





54


=3 L-coefficient of skew (L-skew) (4-13b)


-= L-coefficient of kurtosis (L-kurtosis) (4-13c)


The probability density functions f(x), probability distribution functions F(x), and inverse distribution functions x(F) used in this study are listed in Table 4-1, along with the L-moment parameter estimations (Hosking, 1989).

Once computed, the L-moments are used for two purposes:

1. To select the proper distributional form that best represents the data. In the

interest of parsimony, the candidate distribution is limited to those that have five or fewer parameters, but also have the ability to represent large values, such as those found in rainfall records. Candidate distributions include the

exponential, generalized normal ("log-normal"), Pareto, generalized extreme value (GEV), logistic, Gumbel, gamma, Wakeby, lambda, and Pearson type III. These distributions have been extensively studied with regard to their Lmoment properties, and have been widely applied to climate and hydrology

problems.

2. To correlate with the atmospheric indices (e.g., SOI, NAO, Niflo SST's) that

describe aspects of the ocean-atmosphere system, with an aim towards

understanding how ENSO affects the statistics defining the rainfall models.

The following steps will be used to determine the optimal distributional form for the daily rainfall intensities:

1. The candidate distributions (eleven total) are evaluated using a standard Chi-squared

goodness-of-fit test, using five partitions (yielding 4 degrees of freedom). The results





55


will be used to reduce the number of candidates to approximately half the original

number.

2. Simulations employing the remaining distributions and the estimated p,, parameters

are run, and the resulting monthly rainfall totals recorded. The "best fit" distributions are to be selected based upon a visual comparison to the historic monthly totals. The

simulations are run 1000 times to insure an adequate statistical sample.

3. Further distinction between the remaining candidate distributions can be obtained by

running a "bootstrap simulation" in which the five highest magnitude rain events

from each of the three test stations are compared to the simulated maximum rainfalls.

This method is described in detail in Wilks (1993). The simulation is run 500 times

for each distribution, for each station, and for El Nifto and non-El Nifto years. The

purpose of this test is to examine the ability of each remaining candidate distribution

to accurately simulate the expected extreme rainfalls at the sites. The results will be plotted as "box-whisker" plots for two critical months, January and July, for each of the remaining distributions. The "best fit" distribution will be chosen on the basis of two criteria. First, it is desired that the actual rainfall value will closely correspond to

the median value of the simulated results for each of the five largest magnitude

events. Second, the variation displayed by the simulations (as illustrated by the size of

the boxes and whiskers) should be as small as possible; this is indicative of the

sampling variation of the simulated values (Wilks, 1993). The distribution with the highest correspondence between the simulated and actual values, coupled with the

lowest sampling variation, is the distribution of choice.





56


Relationship to ENSO Processes Segregated Analysis

The first step towards understanding the effect of ENSO on the occurrence and intensity patterns is to segregate the time series of calculated parameters into year-type. The segregation scheme is based on that described by Diaz and Kiladis (1992), and discussed in the previous chapter.

For each of the three stations, the data are segregated into El Nifto, La Nifia, and "other" years. Each year-type is then averaged on a monthly basis, resulting in a twelvemonth sequence of months (January to December) for each year-type. In addition, the months for all years are averaged as well, to provide a baseline for comparing the individual year-types. The results are then plotted to determine whether there is a discernible difference between the types of years, as well as a geographic pattern based upon the location of the three stations.

Relationship to Ocean-Atmosphere Processes Lag-Correlation Analysis

While the preceding analysis is useful in discriminating ENSO influences at a high level, it relies on the somewhat arbitrary division between ENSO and non-ENSO year-types, as well as forcing the imposition of the start and end of the calendar year upon a process that may not necessarily correspond to the cycle of geophysical processes. For this reason, it is desirable to perform a lag-correlation analysis on the calculated parameters against the time series of the atmospheric indices representing the processes of interest.

Each of the indices will be correlated against the monthly time series of

parameters, using a standard Pearson product-moment correlation procedure. Each monthly observation can be correlated separately, and the index time series lagged by





57


36 months. A correlation coefficient and a statistical significance level for that correlation will be calculated.

Examination of the correlation results will allow detection of possible connections between ocean-atmosphere processes and the rainfall parameters, with the goal of establishing the degree of causality that may be present between the time series. This exceeds the simple "El Niflo vs. non-El Niflo" analysis of the preceding section, and attempts to illustrate possible causal linkages between Costa Rican rainfall and tropical ocean-atmosphere conditions. The strength of the correlations may also give an indication of the ability to forecast the rainfall parameters based on the atmospheric conditions that have already occurred. This application is dealt with in the next section.

Creation of Statistical Forecast Models of Rainfall Parameters

Two types of forecast models will be examined in this study, and will be tested by creating hindcast predictions for 1991, the last year of available daily rainfall data for Sanatorio Duran and Villa Mills. These hindcasts are compared to the historic parameters. For all stations and parameters, the two model types are evaluated on the basis of the degree of divergence from the actual data time series.

Multiple regression models. The first model type employs the results of the previous section to create multiple linear regression models for each parameter, based upon the highest levels of lag-correlation between the parameter and the four indices being used. Since the purpose is to test whether the parameters can be accurately forecast, only those correlations with synchronous or negative lags will be used in the creation of the models. The multiple regression model for n observations of a dependent variable y takes on the general form:





58


y = Bo + Blxi +- B2X2i + ... + Bpxpi + ui i = 1...n (4-14) where B0, B l, ..., Bp are constants referred to as the "model partial regression coefficients" (or simply "regression coefficients") for p independent variables x, ... Xp, and ui represents the random error term (Chatterjee and Price, 1991). Individual regression models will be produced for each month, based upon a visual inspection of the lag-correlation results for each parameter.

ARIMA forecast models. The second model type creates ARIMA models based upon the time series of the individual parameters. This type of model does not employ the ocean-atmosphere indices, but rather assumes that the parameter time series contains all the needed information for accurate model creation. The basic form of the ARIMA model with K autoregressive parameters k, and M moving average parameters 0r, ,is: K M
x,+ I-p = --' (x-k -,U) + + I+ 10rn &-n, + (4-15)
k=1 m=1

where (xt+l i) is the predicted anomaly for the variable as a departure from the mean value p xt+k-1 represents the previous K autoregressed values, and tm+I represents the weighted moving average over the previous m values (Wilks, 1995). Since the process of manually computing the parameters is very complex, the Statistica computer package will be used to create the models (Statsoft, 1995). The autoregressive and moving average parameters of the models will be determined based upon the autocorrelation analysis derived during the model-creation process.









Table 4- 1. Distributions used in this study.*


Density Function Distribution Inverse Distribution
Name RX) Function Function L-moment Estimators'
F(x) x(F)
(2n)"'2exp(+x [L)/CY]2/2) (:r-P)/Cr 1 = k,
Normal (2;T) ff (t)dt NAF CY = 7t 1/2 ?12
_W
Exponential (x_1exp[-x/oc] -exp[-x/(x] -cclog(l -F) (X = X,

I CC k2/log 2
Gumbel cc exp[-(x- )/oc]expl-exp[-(x-4)/a]I exp[-expj-(x-4)/aj] 4-(xlog(-Iog F) 4 ?,, Y(X
y = Euler's constant
Gamma [x'-'exp(-x/P)]/P"F((x) Y((X, x/p)/F((X) NAF NAF
where F(x) = gamma function (see Hosking (1989) for help)
(27E)'1/2(x-'exp(k YY2 /2) Y
Generalized Normal where y = -k-'Iog[ I -k(x-4)/a] k # 0 (2z) ff (t)dt NAF NAF
Y = (x-4)/(x k=O
CC (X2k)-'(l -2-') F(l A)
OL"Iexp[-(l -k)y]exp[-exp(-y)] )k 4 ?,, oc{l-r(l+k)j/k
Generalized Extreme where y=-k-'log[l-k(x-4)/a] k#O exp[-exp(-y)] 4+ajl+logF )/k k#O k 7.8590c + 2.955C2
Value y = (x-4)AX k=O 4 + a log (-log F) k = 0 where
c [(2bj-bo)/(3b2-bO)]
[log 2/log 3]
a-iexp[-(I-k)y]/(I+exp(-y)' 4 + (x(l-((l-F)/F Ik J/k k#O k = -T3
Generalized Logistic where y = -k-1log[l-k(x-4)/a] k # 0 1 /[1 + exp(-y)] cc log [(] -F)/F] k = 0 a kAFO-k) 1-(I+k))
y = (x-4)/cc k = 0 4 ?., cc(l-F(l-k) F(I+k))/k
Pearson Type III ---f(x--4-f-Texp(-(x-4)/P)I/Pa F((X) y(a, x/P)/F(cc) NAF NAF
where F(x) = gamma function (see Hosking (1989) for help)
Wakeby NAF NAF 4+af I -F) 0)/p yf 1-(I-F)-')/ NAF
(see Hosking (1989) for help)
Entries which no explicit analytical form exists are marked "NAP; these must be solved by numerical iteration (see Hosking (1989) for details)
** bn = n, PWM







Daily time series
Day 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 State W D D W D W W W D D W D W D D ***
"01" "11" "00" "10"
today
dry wet
("0") ("1") Sum noo = number of "dry-dry" days
6d" 1.0 n1 = number of "wet-wet" days
I dry ( 1.0"
SPoo = noo/nox Pol = nol/nox
wet ("1") Po j- = 1.0 Plo = no/n i Pl = nl/nix
"transition matrix" (for each month)
(for each month)

Monthly transition probability time series
Jan Feb Mar Apr May Jun
POO .15 .23 .21 .65 .51 .15
P11 .58 .51 .47 .26 .58 .77 ***
Figure 4-1. Markov model construction of daily rainfall occurrence.
















CHAPTER 5
RESULTS


The identification of the proper statistical models for daily rainfall follows in a hierarchical fashion, beginning with the determination of basic model parameters and ending with the validation of the model results. This chapter will discuss the results of the model creation process, model validation, data simulation and forecasting procedures. Only the specific results are reported; more general observations and their related conclusions are discussed in the next chapter.

Model Identification

Markov Model Order

A necessary precursor to the production of Markov model parameters is the

determination of the optimal order for the model. The accepted method is to examine the models using an "information criteria." Those used are the Akaike Information Criteria (AIC) and the Bayesian Information Criteria (BIC), discussed respectively in Akaike (1974) and Schwartz (1978).

Figures 5-1 through 5-3 illustrate how frequently (as measured by the total

number of station-months on the ordinate axis) a model of particular order gives the most efficient modeling results as judged by the AIC and BIC. The data were first modeled in aggregate, and then segregated into El Nifto and non-El Nifio years, according to the classification of Diaz and Kiladis (1992), as summarized in Table 4-4.




61






62


The graphs show a definite seasonal trend, clearly differentiating between

summer and winter. During the winter, a first order model dominates using both the AIC and BIC, while summer reverts to a zero-th order model. This pattern persists through both El Niflo or non-El Niho years. In all cases, the third order models are the least efficient, while the second order models in general perform less well than the first order models in terms of overall efficiency. The transition from a winter regime to a summer regime (and vice-versa) is quite rapid, requiring only a month in each case (March and November, respectively). Both the AIC and BIC indicate that the dominance of the zeroth order model rises rapidly as the summer rains resume in June, reach a maximum in October, and then falls off rapidly in early winter (November/December). The zero-th and first order models predominate over higher-order models throughout the year, with the BIC presenting a greater degree of separation between the orders, especially in the summer.

On the surface, there appears to be little to distinguish between the order of models selected in ENSO versus non-ENSO years. In fact, when viewing the winter season results on the AIC graphs, there is virtually no difference whatsoever between the segregated year types. However, the BIC tends to show a greater distinction between years in terms of order preference. During ENSO years, January becomes especially problematic, with very little to distinguish between zero-th and first orders; other winter months, however, remain strongly biased towards a first order model, using both the AIC and BIC.

The summer season is more distinctively biased towards a zero-th order model. This trend lasts for the entire length of the summer rainy season, and is especially






63


pronounced in the BIC. An interesting feature of both the AIC and BIC El Nifio and nonEl Nifo graphs are the months surrounding and including July, the time of the veranillos. The zero-th order tally during July decreases in a subtle, but definite, fashion, which is more marked during non-El Niflo years than in El Niflo years. First Order Markov Model Results Segregated Years

The first-order Markov model parameters can be segregated based on whether the year-type is classified as El Nifio, La Nifia, or "other" years. Monthly estimates of p00 and pll are shown in Figures 5-4 to 5-6, using stations Sanatorio Duran (073011, east), Villa Mills (073033, transition), and Nagatac (080005, west).

Strong seasonal trends continue to manifest themselves when examining the Markov model results for the three stations. Differences in parameter values related to ENSO are more pronounced during summer. Two notable exceptions are evident. First, Villa Mills show a distinctly higher pl1 throughout the winter months during La Nifia years, combined with a marked early winter (November/December) decrease in p00. In addition, the El Nifio early winter p00 for Villa Mills is distinctly higher than all other year types.

The separation between ENSO states is more subtle at Sanatorio Duran. During the early winter, the p, for La Nifia years is somewhat elevated compared to other year types, while in late winter, the trend reverses, with El Nifio years becoming higher. The p00 parameter remains nearly the same for all year types throughout the mid- and latewinter.

Overall transition patterns. The summer patterns for all stations are more

complex, due in part to the numerous precipitation-generating mechanisms that come into






64


play during this time. The most apparent trend is for a dramatic increase overall in the p, 1, coupled with a synchronous decrease in p00. Within this framework, however, more subtle changes can be noted. For all stations, the veranillos period in July/August is clearly visible as an increase in the p00 and a decrease in the pi, giving an indication of the overall reduction of rainfall during this period. The timing and magnitude of these changes is of interest from a geographical viewpoint. The change in pll at Villa Mills is slight compared to the other stations, which are located at opposite extremes of the study transect.

Transition patterns ENSO-based variations. There is a marked summer variation in transition parameters between El Nifio and La Nifia years, though the separation between year types seems to be more pronounced at Nagatac and Villa Mills than at Sanatorio Duran. In addition, there are indications that the type of response has a definite geographical dependence, in that the different sides of the transect region respond in opposing fashions depending on year type.

Nagatac (west) overall shows a much higher p00 (and lower pi 1) response in El

Nifo years during the summer compared to La Nifia years, while Sanatorio Duran shows a much reduced differentiation between year types. Another feature which is clearly visible in the Nagatac Pxx graphs is the presence of the veranillos during July and August, and persisting into September; p00 increases during El Nifto years, and the p1 decreases. Sanatorio Duran shows less interannual variability; the Pxx values are remarkably stable throughout the summer, with a somewhat greater degree of separation for p00 in October. Villa Mills is intermediate to the east and west stations in terms of response, but seems to show a more consistent differing response to La Nifia years, with p00 substantially lower






65


during late summer, and p1 1 higher in mid-summer. One more feature of interest in the Nagatac results concerns the timing of the maximum p, and minimum po0 values: while Figure 3-3 clearly shows that the month of maximum rainfall is October for all year types, pi reaches its maximum (and p00 its minimum) during September for all year types except El Nubt P00. This would seem to indicate that the increased monthly rainfall in October must be accounted for in the magnitudes, rather than the frequencies, of daily rain events.

L-Moment Model Results Segregated Years

The characteristics of daily rainfall magnitude (Figures 5-7 to 5-12) of the three stations under consideration correspond reasonably well to the Markov chain parameters discussed in the previous section and the total monthly precipitation characteristics illustrated in Figures 3-1 to 3-3. Increases in precipitation magnitudes seem to be associated with increased p, and decreased poo, as well as with an increase in total monthly precipitation.. However, several interesting deviations from this expected pattern arise, which will be discussed in this section.

Sanatorio Duran. The average daily rainfall amount (reflected in the first Lmoment, L I) for Sanatorlo Duran (Figure 5-7) shows a nearly exact correspondence between total monthly precipitation and daily rainfall magnitudes. One important difference is in the early summer changes between El Nifio to La Nii'ia years. During this period (May/June), the La Nifia year rains have a lower magnitude than other years, while El Niflo years show higher magnitudes; during this same period, the total monthly precipitation (Figure 3-1) shows the years to be nearly identical. This apparent contradiction can be resolved by examining the poo values for the same period; El Nifio






66


years experience a greater value than La Nif'ia years, meaning that while the daily amounts increase during this period for El Niiio years, rainfall occurs less frequently.

The relationship between El Nifio and La Nifia years reverses later in the summer. The La Niiia L I values are substantially elevated from July to October, while the P00 values are lower; this indicates that rainfall is occurring more frequently, and in heavier amounts, during La Nifia years in late summer, which is reflected in the total average monthly precipitation graph.

It is interesting to note that, for most of the year, there does not appear to be a great deal of difference in the higher order L-moments (T2 and T3, the coefficient of variation and coefficient of skew, respectively). The most substantial change occurs in March. During this period, the p T2 and T3 moments are lower in La Nii'ia years than in El Nifio years, while the L, moment is elevated somewhat. Since the total precipitation amount remains nearly unchanged, it can be concluded that the increase in individual precipitation amounts is countered by a decrease in precipitation frequency. The change in the higher L-moments can be attributed to the upward shift in the L, coupled with the decrease in rainfall frequency; fewer data points with a higher mean value would result in lower variability and skewness calculations.

Villa Mills. This station displays a more consistent L-momnent seasonal shift than Sanatorio Duran. In this case, the total monthly precipitation graph (Figure 3-2) shows La Nifia years experience higher summer amounts, an observation supported by the L I graph (Figure 5-9), with La Nifia averages being uniformly higher than El Niiio years, particularly during the veranillos, and in October. During the same period, the T2 and T3 (Figure 5-10) values remain fairly consistent for El Niiio and La Nifia years, as does the






67


p11 parameter. Both the T2 and T3 parameters show a slight elevation during the July veranillos, which is consistent with the lower overall precipitation during this period; lower daily averages would allow more sporadic higher rainfall amounts to drive up the variability and skew of the underlying distributional form.

Nagatac. The L-moments associated with the Nagatac daily record (Figure 5-11) represent a clear "Pacific" pattern for the region. The winter daily rainfalls are relatively modest and widely spaced, which leads to relatively large readings for T2 and T3 (Figure 5-12). This is especially apparent during El Nifto years, when it is not uncommon to find completely dry months (e.g., March).

A clear veranillos pattern emerges during mid-summer. El Niflo year rainfalls are consistently lower than all other year types starting in late June, with a very marked reduction in L, values during July. La Nifia years, on the other hand, show a substantial increase in daily precipitation values, reaching their maximum at the height of the rainy season in October. These rainfall differences can be linked to the ENSO-based changes in the Trade winds during mid-summer, and the commensurate attenuation of the northward migration of the ITCZ during ENSO events in the late summer.

Time Series Correlation of Transition Probabilities

The results of lag-correlating the po0 and pi1I parameters against the oceanatmosphere indices discussed in "Methodology" are presented in Figures 5-13 to 5-36. Each figure shows the calculated Pearson correlation coefficient for each month, and lags from -36 to +36 months (a negative lag indicates that the index value leads the pxx, in time; thus, a lag of -i would mean that the pxx was correlated with index values which






68


occurred one month before), along with the degree of statistical significance of the calculated correlation for the same lag-month pairs.

Sanatorio Duran P,,, time series correlation. The most notable feature in Figures 5-13 to 5-20 is the overall lack of significant correlation during the summer months for nearly all the indices used. The winter months, especially, December and January, tend to display groups of lags which are more significantly correlated with the p,,,, Many of these groupings occur during positive lag, in which the p,, seems to be leading the index conditions by some months.

The SOI tends to be strongly associated with poo and p, I only during very

narrowly defined periods and lags, notably for December (lag +2 months, +8 months, negative correlation) and January (-16, negative; 1, positive) for poo, and for December (+l, positive; +16, +20, +24, negative) and January (-6, -1, negative) for p, 1. There is also a p I I summer association for June (- 16, positive), and for August (- 1, negative).

The only significant lag-correlation which occurs with respect to the NAO is a negative association at lag -2 for January. Most other features with reasonably strong significance occur at substantially high lags, and therefore probably represent statistical artifacts (i.e., associations which are related to physical factors other than NAO).

The NIN034 index shows very significant associations with pxx during the winter months, and as with SOI, most of the significant correlation is found during early to midwinter (November to January), with some strong correlations reaching into February and early March. The poo has a strong positive correlation in December for lags -2 through + 12, with the strongest associations appearing in lags + 10 through + 12; strong negative associations with poo occur at lags -12 to -18. The lag 0 trend tends to carry over into






69


January through March as advancing lags, culminating with a significant March poo positive correlation at lag -7 months. This corresponds to a negative lag-correlation during the same time period for p I,, which seems consistent.

An interesting set of results involves the correlation between poo and p, 1 and the equatorial stratospheric zonal winds (represented by the QB050 index). The quasibiennial nature of the QB050 is clearly evident during the winter/summer transition period in April (for poo), as well as the summer/winter transition period in October and November (for pi 1). In both cases, periodicities of approximately 27 to 30 months appear in the strong positive and negative correlation pattern for both parameters. For pi 1, negative correlation exists for October/November for lags of -8 and + 16 months, and positive correlation exists for the same months for lags of -25 and +5 months. For poo, the more significant associations are confined mainly to April; lags of -17 and + 12 months show the greatest positive results, and -6 and +24 months give the strongest negative results. There is also a nearly synchronous (lag -2 months) negative association for poo during the July/August period, which corresponds to the veranillos.

Villa Mills p,,, time series correlation. When viewed in aggregate, there is a somewhat stronger lag-correlation between the Villa Mills p (Figures 5-21 to 5-28) than with Sanatorio Duran for some of the better-understood indices, especially for the NIN034 SST measurements. It is interesting to note, however, that the SOI continues to have few overall significant correlations with the Villa Mills p,,,, which suggests that Pacific conditions geographically closer at hand (e.g., NIN034 SST) have a more immediate influence over rainfall frequencies than more distant broad-scale pressure system variability. For the most part, significant SOI positive correlations with p, I appear






70


during the early winter months, and are nearly synchronous, with lags of -3 to +5 months. A time of somewhat weaker significance appears during August, approximating the same lags and also positively correlated. The poo shows virtually no period of sustained highly significant correlations.

The NAO (Figures 5-23 and 5-24) shows a similarly weak overall relationship with the px measurements at Villa Mills. The few significant correlations (for poo, June,

- 19 months lag; November, 16 months; for p 11, January, +2 months) are fairly low (ranging between +/- .20 or less. From this it can be deduced that the NAO is of minimal importance to rainfall frequency at Villa Mills, both from an explanatory viewpoint and as a forecasting tool.

NIN034 SSTs show a more coherent signal. The p, 1 shows a stronger overall relationship than poo, with the pOO/NIN034 resulting in virtually no correlations with significance exceeding 80% (Figure 5-25). However, the p, I correlations (Figure 5-26) show several areas of relatively strong significant correlations: during May (positive correlation, lags +5 to +8), July and August (negative, lags -4 to +6), and December and January (negative, lags -4 to + 14).

In direct contrast to Sanatorio Duran, Villa Mills exhibits a much weaker

relationship with the QB050 index, both for poo and p, 1. In addition, the few significant correlations which do appear are for the most part out of phase with those seen at Sanatorio Duran. As might be expected, the strongest relationships for a given month appear to be spaced at an integer multiple of the approximate length of the QBO cycle (27-30 months); the significant February poo negative correlations (Figure 5-27) are at lags of -29 and +28 months, thus showing two complete QBO phases. A similar situation






71


arises for the p I correlations (F igure 5 -2 8) for May (negative, +4 and 3 2 months; positive, -9 and +22 months; one cycle each) and August/September/October (-30 and +29 months, 2 cycles).

Nagatac: px, time series correlations. Nagatac is located farther to the west, and is therefore more likely to experience a stronger relationship with the indices that describe the state of the Pacific ocean-atmosphere system. The correlations between pxx, and the SOI (Figures 5-29 and 5-30) are much more pronounced than at either of the other stations, particularly during the summer months. The near-synchronous negative relationships for p00 are most significant starting in July, and extending into the late summer. For each month, lags from -4 to +6 months are most important for poo; in addition, positive correlations appear for May lags of -14 to -22 months. For mid-to-late summer (August/September), a strong positive correlation between SO! and p, I is shown, from 0 to +6 months; the synchronous correlation corresponds to the time frame of the maximum northerly migration of the ITCZ, and can help explain the decrease in rainfall during ENSO events for stations on the Pacific slope. The positive correlation exists for a lag of 5 months demonstrates a relationship between the August p11I measurement and the following January SOT, and it is important to note that this is when the SOI reaches its maximum negative value during ENSO events.

The westerly location of Nagatac helps to explain the lack of significant

correlations between the px and the NAO (Figures 5-31 and 5-32). Such a location would allow Pacific effects to dominate those of the Atlantic basic. While correlations are evident, they tend to be statistically weak, and are therefore of very limited utility for either explanation or forecasting.






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The NIN034 correlations (Figures 5-33 and 5-34) provide greater insight into the relationship between px and ENSO at Nagatac. Strong positive correlations for poo exist for August (lags -4 to 8) and March/April (lags +6 to 12), and negative poo associations appear in March/April (lags -8 to -2). Strong negative p11 correlations exist for March/April (lags +8 to + 12 and August/September (centered on lag 4), and positive p1I1 correlations manifest themselves during March/April (lags -6 to 14) and August/September (lags -14 to -24). In each case, the relationships are strongly suggestive of a linkage between the onset of warm water into the NIN034 zone during ENSO events and the migration of the ITCZ during the boreal summer. The correlations during the winter months is probably most closely tied to the effect of ENSO on the nortes; however, firm statements are difficult to make due to the scanty nature of the winter rains in western Costa Rica.

There appears to be little in the way of strong relationships between the QB050 and the p,, for Nagatac (Figures 5-35 and 5-36). As with the other stations, the quasibiennial nature of the cycle is clearly evident, but the resulting correlations are fairly weak.

Time Series Correlation of L-Moments

Geographic distinctions appear when the L-moments are lag-correlated with the atmospheric indices previously discussed. As a general statement, it is apparent that the amount of daily precipitation occurring on wet days is more strongly correlated to the indices in the eastern part of the transect than in the central and western areas. This is reflected in the L, lag-correlation diagrams for the three stations contained in Figures 537 through 5-72.






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One important finding is common to all three stations: the NAO shows very

significant correlations for any of the parameters examined. The conclusion to be drawn, which is discussed in more detail in the next chapter, is that the status of the North Atlantic is of minimal importance in characterizing the region's daily rainfall.

Sanatorio Duran. Sanatorio Duran is more sensitive than the other stations to changes in the conditions in the equatorial Pacific. This seems to be true in both the summer and winter seasons, particularly the late winter. Figure 5-37 shows the nearsynchronous positive correlations between the SOI and Li from July through September, indicating that during El Nifio episodes, the quantity of rainfall falling on wet days is reduced. This seems to correspond to Figure 3-1, which shows that during El Niflo periods the total monthly precipitation is reduced. The reverse would be expected as well; La Nifia periods would be expected to experience higher intensity daily rain events according to Figure 3-1, and this is supported by Figure 5-37. This trend is supported by the L, correlations found for NIN034 (Figure 5-40), in which the synchronous July to September correlations are negative, indicating that warmer Pacific waters are negatively associated with rainfall magnitudes for Sanatorio Duran.

Another feature which appears during the summer months is the association

between L I and the period preceding it by 12 months (lag -12). During this period, the SOI correlation is negative, while the NIN034 correlations are positive. This is strongly suggestive of an annual "see-saw" effect coming into play, in which the effects of ENSO on the rainfall of Sanatorio Duran reverses itself during the years surrounding the actual event; this supports the findings of previous research which have noted this phenomenon at annual time scales (Waylen et al., 1996).






74


The QBO (Figure 5-46) has little association with the rainfall magnitudes during the summer months, despite the strong association between p, I and the QBO during the late summer months. This peculiar combination of factors seems to point towards the QBO having less impact of storm strength than on storm frequency.

The early-winter correlations show a distinct relationship with the index values preceding the rainfall by six to ten months. This especially important, because it would indicate that the strength of the norte events in eastern Costa Rica is correlated to the occurrence of ENSO events which begin the previous summer. In this case, the correlation is negative, which translates to heavier late-winter rain events during warmPacific events initiating the previous summer (i.e., during El Nifto years).

Villa Mills. A more subdued pattern emerges for Villa Mills. In the case of L, for all of the major indices (Figures 5-49, 5-55, 5-58), there is very little correlation found for any season. This is in contrast to the results for Sanatorio Duran (LI) and the p,,, transition findings for Villa Mills. An important observation, however, is in the behavior of the higher order moments, T2 and T3, which reflect the variability and skew of the rainfall distributions, respectively. For the mid- to late-winter period for both NIN034 (Figure 5-56 and 5-57), there is a strong positive correlation at lag = -10 months, as well as a strong negative correlation for lag = +6 to +8 months. This corresponds well to the observed behavior of the notes during and following an El Nifto event; the +8 month negative lag shows that higher rainfall values are depressed during the year of an ENSO event, whereas the year following (-10 month lag) the more intense rainfalls resume. In addition, the NAO shows extremely few high-significance correlations, which parallels the finding at the other stations.






75


The QB050-L I relationship is more difficult to interpret. The only clear

interpretation that can be made is that the 27-30 month cycle of the QBO is clearly evident in the LI, T2, and T3 graphs (Figure 5-58 to 5-60). The correlations are strongest for the T2 and T3 moments; early winter positive associations with the QBO result with a lag of 16 to 12 months, during early summer (May) for near synchronous correlations, and during the veranillos (July) period for lags of -9 to 12 months.

Nagatac. The Nagatac L I correlations (Figure 5-61) show few similarities with the poo and p, I correlations. The three lag-regions which show significant correlations are March (negative, lag 0 to +10), July (positive, lag +I), and October (negative, lag -8 to 12). These same regions appear as highly correlated in the NIN034 with the opposite associations; the March region is strongly positive, July is strongly negative, and October is positive. None of these areas are significant on the p,,,, lag-correlations, which seems to suggest that changes in the western Pacific pressure dipole, and associated shifts in warm water across the equatorial Pacific, manifests themselves in different ways for different months. The relatively stronger significance measurements in the NIN034 results seems to indicate that the region responds more readily to alterations in the Pacific SST's rather than with changes in the pressures over the central and western Pacific.

The QB050-L, correlation results (Figure 5-67) show a much stronger association between the zonal stratospheric winds and the intensity of rainfall in western Costa Rica. The quasi-biennial oscillation is clearly evident in the spring months (March/April), and even more strongly from October to December. This strongly suggests that a physical association is present; unfortunately, this area of research is in a developing state, and few conclusive studies have been made concerning the linkage between climate






76


anomalies and stratospheric wind flow. Nevertheless, the associations present in the record indicate that QB050 could represent an excellent forecasting variable even if a physical linkage cannot be found.


Model Choice, Validation, and Rainfall Simulation Initial Model Choice. The first step in validating a particular statistical intensity model is to identify the candidate distributions to be tested. As stated in the "Methodology" chapter, I I candidate distributions were identified, which range in complexity from a single parameter exponential model to a more complex five-parameter Wakeby distribution. Two months (January, July) were chosen for each station, and yeartypes segregated into "ENSO" and "non-ENSO" years based upon the Kiladis and Diaz classification used for the earlier tests. The results of the Chi-squared initial elimination process are shown in Table 5-4.

The results clearly show a bias towards the higher-parameter distributions. The exception is the choice of the gamma distribution, which exceeded both the Generalized Logistic and Generalized Extreme Value distributions in choice likelihood based on the Chi-squared tests. It is interesting to note that the Gamma distribution is the only twoparameter distribution to include scale and shape as the parameters, rather than the location and scale parameters for the Normal and Gumbel distributions.






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Table 5-4. Tallied results of distributional Chi-squared tests.

distribution January January July July TOTAL
(# of params.) ENSO years non-ENSO ENSO years non-ENSO COUNT
years years
Exponential 1 0 1 0 2
(1)

Gamma (2) 3 0 3 3 9*


Gumbel (2) 0 0 0 0 0


Normal (2) 0 0 0 0 0

Generalized
Extreme 3 2 1 1 8
Value (3)
Generalized 4 2 0 1 7
Logistic (3)

Generalized 2 3 2 3 10 *
Normal (3)

Generalized 3 3 3 3 12 *
Pareto (3)

Pearson 111 (3) 2 2 3 3 10 *


Kappa (4) 3 3 3 3 12 *


Wakeby (5) 2 3 2 2 9 *

indicates that the distribution was selected for "final six" consideration



Rainfall simulations. The next step in deciding upon a proper distributional form is to use the calculated parameters (both intensity and frequency) and perform simulations based upon the candidate distributions and the Markov model parameters. The simulations are run for each station/distribution/year-type/month combination, with






78


1000 iterations being used to calculate the total monthly precipitation. The results are plotted in Figures 5-73 to 5-78.

It is apparent that most of the distributions give a very close correspondence to the actual average monthly totals for all stations. Sanatorio Duran and Villa Mills show the greatest overall correspondence, with Nagatac monthly totals being the most difficult to replicate. This is especially true during the veranillos period of El Nifio years, and for the latter half of summer during La Nifia years. It is noteworthy that, in general, all the distributions seem to underestimate the actual monthly totals during the surnmer at Nagatac; it is likely that this reflects the very high totals that routinely fall in the western part of Costa Rica during this period. The underestimation during El Nifto years is most significant during the veranillos, when the estimated values are about half of the actual amounts; this is probably attributable to the sudden decrease in daily rainfall amounts during the onset of the increased Trade winds, making predictions at the monthly scale highly variable.

As mentioned, nearly all of the distributions do an adequate job of modeling the precipitation at the monthly scale. The notable exception is the Kappa distribution, which radically underestimates amounts for both Villa Mills and Nagatac during the spring months (March/April). This is sufficient evidence to lead to the conclusion that the Kappa distribution is not a preferred model for daily precipitation, and can be removed from further consideration.

Extreme value analysis. Since the monthly rainfall simulation failed to distinguish among the candidate distributions, it is necessary to consider the effect that each of the models has on simulating the extreme daily precipitation events. The "bootstrap"






79


technique outlined by Wilks (1993) is employed, as described in the Methodology chapter. The data are segregated as in the Chi-squared test, and 500 bootstrap simulations are performed. The distribution of choice will be the one which provides the best fit between the median value and actual value ("median-to-actual fit"), as well as the least sampling variability (i.e., the "tightest" fit). The results are shown in Figures 5-79 to 584. In each of the bootstrap diagrams, the five largest daily rainfall events are shown on each distribution graph as asterisks, along with a box-whisker plot showing the 5%, 25%, 75%, and 95% boundaries for the 500 sample calculations representing the rainfall estimate for that value.

Several notable observations can be seen in the results. Overall, it is apparent that the distributions with the largest number of parameters do not provide the best fit to the extreme values in any of the cases. In fact, in many cases, the best fit is accomplished by the distribution with the fewest parameters (Gamma) gives the best overall fit in terms of median-to-actual fit, as well as minimizing the sampling variability. In addition, both the Gamma and Pearson III models seem to mirror the others responses; this is probably due to the fact that the Pearson III distribution is nearly identical to the Gamma with the addition of a location parameter.

Sanatorio Duran shows a tendency towards consistent overestimation as well, but the departures are minimized by using either a Gamma of Pearson III distribution. (In fact, the Gamma shows a slight tendency to underestimate, rather than overestimate, the largest values.) The results are similar for both January and July, and for El Niho and non-El Niho months. In addition, the Gamma and Pearson III models show the lowest






80


amount of sampling variation. For all cases, it is clear that the Generalized Logistic, Generalized Normal, Kappa, and Wakeby distributions can be discounted.

Villa Mills during July experiences at least one very large rainfall in both ENSO and non-ENSO years, and the resulting distributional fits are quite imprecise. The January estimates are much more consistent, in that it is possible to find models with a reasonable degree of fit to the data. In this case, the Pearson III model provides the highest degree of correspondence to the actual amounts as well as the second-lowest sampling variability (behind the Gamma distribution). The Pearson III and Gamma distributions also seem to give an adequate fit to the extreme values when the largest value is disregarded.

The Nagatac extreme values in July have a very difficult time being fit by any of the model choices, despite the relative consistency of the actual amounts. The degree of deviation from the actual amounts is greatest for ENSO years, when there is a clear tendency to overestimate the largest values. The degree to which the deviation occurs is minimized with the Gamma and Pearson III models, both in terms of the actual-tomedian difference and among the sampling variability of the bootstrap samples. The Kappa model shows the greatest amount of sampling variation, especially during January, when the rain events seldom occur.


Parameter Forecasting

The one-year forecast results based upon the creation of individual month-bymonth regression models for each parameter, as well, as the ARIMA forecast results, are shown in Tables 5-1 to 5-9, and graphically illustrated in Figures 5-85 to 5-87, and the ARIMA results (with 95% confidence intervals illustrated) are shown alone in Figures 5-






81


88 to 5-90. The results show that, in general, the ARIMA model performs better in tracking the actual parameter values. This is especially evident when the month-to-month transitions are relatively "smooth," although this is not exclusively the case; it appears that, because the multiple regression models are individually created for each month without regard for the predictors used in adjacent months, large inter-monthly variations can occur.

One of the most striking observations concerning the multiple regression models is the frequency with which the NAO index is used as a predictor. This is in stark contrast to the individual monthly time-series correlations discussed earlier, where it was found that the NAO had very low significant correlations with the parameters. From this result, it may be inferred that that the strong showing of the NAO in the prediction models indicates there is a significant amount of interaction (both statistical and physical) between the NAO and the processes reflected in the SOI, NIN034, and QBO indices.

A further observation about the multiple regression forecast models concerns the apparent lack of a relationship between the multiple-R and adjusted-R 2 measurements and the ability of the models to accurate forecast the parameters. The R and adjusted-R 2 values seem to be more related to the number of model predictors, and a strong R appears to give no guarantee that the model will perform adequately as a forecast tool. In fact, it was often the case that the models gave impossible values as forecasts (i.e., transition probabilities greater than one or less than zero, or L, values less than zero). The ARIMA model showed no such tendency, indicating it is a more "stable" prediction scheme.






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Table 5-1. Forecast model results for Sanatorio Duran LI.

Multiple Regression predicted actual
Model* R adjusted R2 (mult. reg.) (ARIMA) value

-1.6431
JAN + 2.017 NIN8 + 0.745 NAO18 0.892 0.753 3.449 4.324 4.254
0.457 NAO7 0.307 NAO12

FEB 2.0475 0.749 0.521 1.843 2.511 0.780
0.475 NAO 18 + 0.772 S0120

MAR 2.N53 0.694 0.481 2.514 1.853 0.400
0.719 NAOI17 0.816 S016
2.6U3
APR 0.129 QB09 + 2.113 S0126 0.754 0.506 -0.899 2.335 1.565
2.914 S0124
9.373
MAY + 2.530 S0122 1.559 SOI6 0.755 0.489 12.096 9.385 7.879
+ 2.424 SO 11 + 0.776 NA024

+1.627 NA035 2.649 SO1l5
JUN 0.927 0.814 8.303 5.974 8.271
0.224 QBO20 + 0.190 QBOI6 0.075 QB036 + 0.465 NA025
-3b6
JUL + 1.447NAO-0.063QB033 0.912 0.778 6.115 6.065 5.308
+ 3.911 NIN9 + 0.462 NA022 2.314 NIN6 + 0.535 NA036
9.'328
AUG 2.077 SOil2 + 1.208 NAO9 0.679 0.387 9.074 6.984 8.069
+ 0.730 NAOI t

SEP + LO3 0.436 0.159 10.650 9.357 5.400
+ 1.030 NA012
11.122

OCT -2.678SO19+0.852NAO17 0.815 0.584 14.311 10.775 8.500
+ 0.919 NA027 + 0.775 NAOi I
0.056 QB023
tu..)U)
0.105 QBO8 0.525 NAO8
NOV -0.677NA030+ 1.115 S0133 0.947 0.841 9.180 8.500 5.700
-2.492 S0131 -1.927 S017
0.063 QBO35 0.511 NAOI2
+ 0.082 QBO7
-84.367
DEC + 3.296 NIN7 0.048 QB09 0.755 0.514 6.603 6.025 3.810
+ 1.234 SO17

The number of months of lag applied to atmospheric indices as a model predictor is indicated by the number following index (e.g., "NAO 10" indicated the NAO lagged by 10 months)






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Table 5-2. Forecast model results for Sanatorio Duran poo.

Multiple Regression I__________predicted actual Model R adjusted R- (ARIMA) value
(mult. reg.)
AN0.791
JAN 0.09 1 0.708 0.458 0.795 0.805 0.709
+ 0.054 SOh 8 0.002 QBO 3
U.849
+ 0.070 SOI16 + 0.034NAO24
FEB 0.035 NAO6 0.077 S0123 0.962 0.887 0.958 0.853 0.788
+ 0.044 S0127 0.010 NAO30 0.001 QBOI4 + 0.007 NA033
3.055
MR- 0.028 NAO8 + 0.029 NAGOI4
MAR 0.034 NA08 + 0.081 N14 0.851 0.653 1.049 0.837 0.880
+ 0.034 NA033 0.081 NIN32
0.027 S0125
0.8 18
APR + 0.089 S0121 + 0.043 NA014 0.804 0.595 1.056 0.799 0.417
+ 0.036 NAO 18
0.531
MAY + 0.133 S0134 0.067 NA022 0.849 0.667 0.692 0.360 0.000
+ 0.093 SOhlo 0.003 QBO8
0.M0Z
+ 0.007 QBOI2 0.142 S0126
JUN + 0.067 NA01 + 0.058 NA126 0.879 0.716 0.605 0.569 0.462
0.067 NA08 + 0.058 NA035
+ 0.057 S0135
3.11)
JUL+ 0.086 NA027 + 0.080 NAOi 0.911 0.787 0.582 0.293 0.286
+ 0.108 S017 0.103 NIN21
0.082 S0116
U.453
0.152 NAO35 0.070 NA027
AUG + 0.041 NA032 0.056 S0120 0.940 0.829 0.667 0.274 0.400
0.006 QBO21 0.055 NA028 + 0.006 QBO16 0.111 S0124
0.345
SEP+ 0.087 NA2 + 0.077 NA33 0.838 0.634 0.139 0.321 0.600
+ 0.055 NA036 + 0.055 NA023
+ 0.036 NA035
0.339
OCT .2 0.746 0.520 0.221 0.217 0.200
0.215 S0123 + 0.085 S0120
0.519
NOV + 0.086 S0116 0.061 NA032 0.763 0.506 0.512 0.467 0.481
0.099 S019 + 0.027 NA022
0.100
DEC +. S0U 0.508 0.229 0.725 0.689 0.571
+ 0.049 SOI 171 1






84


Table 5-3. Forecast model results for Sanatorio Duran P, .

Multiple Regression ______ _predicted actual
Model R adjusted R2 (mult. reg.) (ARIMA) value

-2.306
+ 0.005 QBO6 + 0.050 NAO18
JAN 0.052NA036 +0.109NIN16 0.954 0.876 0.477 0.645 0.543
+ 0.037 NAO31 + 0.026 NAO35
+ 0.016 NAO8
2.364
FEB 0.041 NAO30 0.038 NAO9 0.810 0.588 0.326 0.575 0.458
0.029 NAOI 3 0.072 NIN7
0.213
MAR 0.043 SOI15 + 0.108 NAO21 0.849 0.647 0.361 0.379 0.600
+ 0.091 NAO1 0.065 NAO3o
U.367
APR 0.072 NAOu5 + 0.007 QB028 0.750 0.475 0.165 0.395 0.529
+ 0.072 NA034 + 0.005 QBOI 8
'.235
MAY 0.041 NA028 + 0.038 NA033 0.820 0.610 0.787 0.683 0.893
+ 0.040 NAO9 0.056 N1N26
0.130
JUN 0.022 NAO6 + 0.055 SOl 5 0.739 0.459 0.610 0.833 0.625
0.023 NA034 0.022 NAO21
01.642
0.143 SOI36 +0.143 S0I34
JUL 0.04 S0136 + 0.057 N0134 0.859 0.655 1.330 0.774 0.783
-0,084 NAOi9 + 0.057 NAO15 + 0.041 NA024 + 0.030 NAOI7
0.164
AUG +.004 0.659 0.385 0.804 0.756 0.880
+ 0.005 QB025 0.058 NA025
0.724
0.040 NA032 0.073 NA036
SEP +0.003 QB034+0.057NAOI6 0.888 0.715 0.610 0.768 0.958
+ 0.103 S0127 0.035 S0120
+ 0.027 NAO3o
3.937
+ 0.008 QBO2o + 0.052 NA033
OCT 0.026 NAO12 0.116 S016 0.929 0.802 0.500 0.826 0.800
0.117 NIN7 0.021 NAO2i 0.003 QBO6 0.002 QB022
U.954
+ 0.005 QB026 0.027 NAOI3
NOV+ 0.019 NA6 +0.039 NA033 0.951 0.853 0.861 0.747 0.778
0.015 NA022 + 0.052 NINI8
0.064 S0125 0.057 NIN3o
+ 0.036 S0129
0.638
+ 0.628 NA033 + 0.033 QB027
DEC + 0.028 NAO6 + 0.086 S016 0.892 0.720 0.560 0.672 0.769
0.032 NA024 0.042 S0129
+ 0.024 S0133







85


Table 5-4. Forecast model results for Villa Mills L1.


MonthMultiple Regression predicted actual
M Model R adjusted R2 (mult. reg.)value


JAN 1.597 SO12 0.245 QB023 0.793 0.564 2.610 2.410 5.541
0.059 QB032 + 0.168 QB024
-43.858
FEB 4.858 0.583 0.289 3.440 1.395 1.858
+ 1.707 NIN9 + 0,531 SOh I
2.99"2
MAR +0.520S0125 0.416 0.140 3.564 1.617 1.064
--/"-11.8/76
+ 13.418 NlNii + 2.122 NA34
APR 0.927 0.818 18.476 5.765 5.000
+ 1.918 NA025 + 0.110 QB028
5.472 NINio- 0.740 NA016
I L.U4
+ 1.190 NAOI7 + 1.162 SO14
MAY 1.545 NA025 1.727 SO16 0.892 0.718 7.236 10.237 11.842
+ 1.077 S017 + 0.824 NA032
+ 0.543 NA029 + 0.436 NAOi i
11.934
JUN 2.299 SO13 + 1.148 NAOio 0.855 0.689 5.980 10.495 15.124
0.082 QB033 + 0.904 NA018

0. 137 QBOi i + 1. 146 NAOI 3

JUL + 0. 1 032 + 1.33 S0111 0.949 0.854 4.176 8.201 5.292
+ 0.934 S0130 1.433 SOl 15
+ 0.612 NAO12 + 0.038 QB025
0.485 NA025

4.894 SO116 + 2.313 NAO18
+ 1.728 S0129 1.914 S017
AUG + 5.215 S0112- 1.077NA027 0.974 0.921 20.781 12.323 13.075

-1.019 NAO26 + 3.502 MNi I 0.505 NAO8 + 0.031 QB028

SEP 1.617NA036-0.121 QBO14 0.869 0.717 8.597 12.948 14.258
2.089 SO17 0.084 QBO3o
LU5.) IU
1.780 S0129 5.784 SO13
OCT 7.249 MN14 0.138 QB014 0.946 0.850 16.491 14.430 12.748
+ 1.051 NA026 + 0.875 NAO9
+ 0.859 NA023 1.698 S016
9.03O
.114 BO3- .02QB2
NOV -0.114QB013-0.092QB026 0.864 0.691 8.148 10.509 7.507
0.561 NAO6 0.658 NAO3i
0.686 S0120

-0.112 QB09 0.154 QB022
DEC + 0.711 NA033 0.075 QB029 0.927 0.803 8.755 6.580 3.508
+ 0.491 NAO8 + 0.469 S0126
-0.481 NAO17+0.115QB023







86


Table 5-5. Forecast model results for Villa Mills poo.

Multiple Regression predicted predicted actual
Model R adjusted R2 (mult. reg.) (ARIMA) value

1.952
JAN + 0.032 NA026 0.046 NIN16 0.722 0.462 0.692 0.699 0.731
0.025 NAOI9
1.121
FEB 0.041 NAO20 + 0.026 NA027 0.844 0.665 0.682 0.673 0.724
0.002 QBO7 0.038 NIN12

MAR 0.049 NAO21 0.003 QBO8 0.818 0.611 0.807 0.589 0.750
0.032 NAO17 0.053 NIN31
0.654
APR + 0.048 NA032 + 0.003 QBOI6 0.800 0.576 0.820 0.564 0.625
+ 0.081 S0127 0.074 S0129
0.263
MAY +0.109SO134+0.014QB018 0.685 0.403 0.039 0.141 0.000
0.011 QBO20
U.1 /6
JUN 0.057 NA027 + 0.196 S0124 0.821 0.633 -0.041 0.058 0.800
0.121 S0125
0.022
0.006 QBO9 0.006 QB022
JUL 0.037 NAO7 0.005 QB034 0.860 0.653 -0.200 0.110 0.000
+ 0.024 NAO18 0.040 NA023
0.029 NAOI3
0..30
AUG 0.389 0.117 0.414 0.164 0.143
0.047 NAOi5
0.124
+ 0.077 S017 + 0.009 QB023 0.008 QB027 0.050 NA022
SEP + 0.007 QB034 + 0.074 S0130 0.982 0.930 0.391 0.170 0.000
+ 0.061 NA033 + 0.029 NA025 0.061 NA033 + 0.029 NA025
+ 0.032 NAOI7 0.037 S019
-! .190
OCT 0.052 NA028 + 0.035 NA034 0.875 0.706 0.078 0.097 0.000
0.026 NAO7 0.073 NIN3 1
-2.406
+ 0.054 NAOi5 + 0.024 NA029
+ 0.115 NIN22 0.065 NAO6
NOV 0.006 QB023 0.008 QB2 0.988 0.956 1.038 0.541 0.478
+ 0.037 NAOI2 + 0.002 QBO16
0.053 S0135 + 0.039 NAO28 + 0.031 S0133 + 0.033 S0120
+ 0.002 QB026
4.0M
DEC 0.058 NAOi9 0,050 NA017 0.860 0.683 0.705 0.666 0.545
0.125 NIN9 0.028 NAO18
+ 0.021 NAO31






87


Table 5-6. Forecast model results for Villa Mills p, .

Multiple Regression 2_Predictedpredicted actual Month Mdlpredicted
Model R adjusted R- (mut. reg) (ARIMA) value

0.579
+ 0.0 14 QB032 + 0.027 NAOI9 0.029 NA025 0.027 NAO28
JAN -0.024 QBO31 + 0.0188 QBO30 0.984 0.947 0.594 0.726 0.632
0.002 QBO8 + 0.016 NAOi + 0.009 NAO23 0.012 NA029
0.013 NA033
-1.796
+ 0.065 NAO7 0.048 NA036
FEB +0065NA 0 1 0.048 0.852 0.667 0.625 0.638 0.531
0.0 18 NA023 + 0.115 SOIl17
0.087 NIN16
0.474
MAR 0.354 0.091 0.315 0.617 0.500
+ 0.066 NAO35
U. 9U
APR + 0.041 NA035 0.129 S0121 0.689 0.383 0.629 0.668 0.538
+ 0.110 S0123 0.033 NAO9
4.266
MAY -0.125N1N14-0.026NA021 0.763 0.534 0.797 0.872 1.000
+ 0.044 S0136
0.011I
JN- 0.027 NAO33 + 0.034 NIN9
JUN 13 NAO3 -004 NA01 0.814 0.592 0.882 0.970 0.958
-0.0 13 NA06 0.011 NAOI18
+ 0.001 QBO6

+ 0.046 NAOI2 0.008 S0133
JUL + 0.04Q6 +2 0.002 S03 0.874 0.700 0.983 0.890 0.800
+ 0.004 QB036 + 0.002 QB023
0.060 NIN17 0.016 NAO6
U.913
AUG + 0.003 QB036 + 0.028 NAOI4 0.737 0.463 0.980 0.810 0.739
0.039 S0134 0.001 QBO8
-U.829
SEP 0.002 QB033 + 0.064 NIN28 0.883 0.734 0.821 0.888 0.840
+ 0.030 S013 0.022 S017
0.001 QBO21
U.939
OCT -.027 0.585 0.317 0.901 0.948 0.929
-0.027 S0136
0.864
NOV 0.014 NAO3o + 0.019 NA017 0.645 0.345 0.903 0.886 0.806
+ 0.002 QBO3o
0./I!
DEC+ 0.004 QB3 + 0.024 NAO8 0.869 0.689 0.911 0.806 0.711
0.030 NA024 + 0.028 NAOi 1
0.002 QBO7 0.012 NA023







88


Table 5-7. Forecast model results for Nagatac L I.


MnhMultiple Regression predicted_ predicted actual

-4.80 odlR adjusted R-2(ut e. (ARIMA) value


JAN 2.255 NA017 0.233 QB025 0.948 0.867 6.390 4.203 2.743
+ 2.167 NIN8 + 0.150 QB029
+ 0.093 QBOI18
11.737
FB+ 0.316 QB09 2.400 NA06 0.9 097 310 441 212
FB-3.080 NA014 1.270 NA033 0.9 097310 441 212
1.641 NA09 -0.727 NA025
21.3tfl
MAR + 0.464 QB07 + 3.301 NA08 0.999 0.999 -1.740 5.640 5.300
+ 0.009 QB030

-0.814 QB012 4.498 NA022 + 5.095 S0113 + 3.818 NA032
APR 3.269 NA034 -2.105 NA024 099 099 1.0 .8 .1
+ 0. 155 QB0 8 + 0.976 NA013 099 099 1.0 .8 .1
0.058 QB024 + 1.194 S0123 0.233 S0127 0.005 QBOio
0.007 NIN34

MAY + 2.340ONAOi i 5.822 S0124 0.813 0.589 23.620 12.862 16.375
-2.23 0 NA021 2.240 NAOI131

JU 21.42A +.19A2 0.700 0.442 13.110 11.199 29.308

JUL -03 A010.468 0.183 9.710 9.300 14.150
16.:9U
7.095 S0113 + 8.742 S0114
AUG 1.447 NA028 2.053 S0128 095 095 1.8 152 3.8
+ 1.572 NA031 + 1.627 NA027 095 095 1.8 152 3.8
+ 1.393 NA023 1.796 S0118
+ 2.069 S0126 -0.664 S0131
13.669
4.226 NAOI13 + 2.041 NA031
SEP 0.331 QB0I3 + 10.485 S0116 0.944 0.838 16.522 9.324 23.447
0.237 QB028 2.798 S016
+ 1.602 NA07

OCT 12U.2826II .8 01 0.673 0.396 15.300 18.807 40.121
15.597
NOV + 2.528 NA036 0.130 QB035 0.806 0.577 23.601 14.115 27.433
-1.364 NA032 3.960 SOI 1
91.2o0
DEC + 3.634 NIN34 + 1.698 NA028 0.807 0.575 8,027 9.270 11.770
-6.548 NIN7 1.941 S019







89


Table 5-8. Forecast model results for Nagatac poo.


MnhMultiple Regression predicted_ predicted actual

.93 odlR adjusted R- Ml. e. (ARIMA) value


U.931

JAN 0.0 15 NA031 0 .0019NA024 0.706 0.456 1094 0.990 0.900


+ 0.008 NA027 0.0 16 S0114 0.001 QB033 0.001 QB08
0.007 S0135
U.944
MAR 0.005 QB020 + 0.0 17 NA07 0.761 0.499 0.860 0.975 0.750
+ 0.0 15 NA029 + 0.004 QBOI19
01.9u 1
APR 0.040 NAOI6 +0.027 NAO6 0.748 0.499 0.960 0,956 0.8 10
1+ 0.022 NA020
4.133
MY+ 0. 173 S0124 0.022 QB028 0.927 0.824 -0.030 0.648 0.000
MY+ 0.0 15 QB030 0.130 NIN33
0.055 NA013
U).204
JUN 0.005 QB020 + 0.058 NAO28 0.742 0.489 0.077 0.521 0.333
0.068 NAO33
U.448
JL+ 0.066 NAO 18 + 0.075 NA0340.1 073 042 068 050
JL+ 0.081 NAO26 + 0.054 NAO270.1 073 042 068 050
0.05 8 NAO 12 0.041 NAO 14
0.490
AUG + 0.007 QB0I3 + 0.05 1 NA06 0.680 0.411 0.2 10 0.455 0.500

SEP +10.12784-0.6 A 2 0.730 0.471 -0.090 0.532 0.000
+. 127 0A2. .07N0 72
OCT U.12072 0.468 0.230 0.309 0.000
0. 007 QBO22 0.172 S01 17
U. /26
NV+ 0. 196 S0136 0.095 S0130 0.3 084 052 074 084
NV+ 0.038 NAO 12 0.028 NA036 0.3 084 052 074 084
+ 0.054 NIN36
DEC + .2 A2 008S1 0.681 0.419 0.908 0.8591 0.915






90


Table 5-9. Forecast model results for Nagatac P, .

Multiple Regression 2_predictdpredicted actual Month Mdlpredicted
Model R adjusted R e (ARIMA) value
I__ U .2 9 4 (m u lt. re g .)
1 0294
0.011 QBO9 0.011 QB027
JAN + 0.0 15 NA032 + 0.459 NIN30 0.968 0.904 0.496 0.062 0.286
+ 0.220 S0128 + 0.007 QB028 0.072 NINI4 + 0.027 NAO21
0.343
FEB 0.080 NAQii 0.065 NA032 0.976 0.926 -0.093 0.169 0.000
0.056 NAO8 0.053 NAOI9 0.041 NAOI2 0.003 QB036
0.389
0.189 NAOI0 + 0.199 S0130
MAR 0.999 0.999 0.824 0.325 0.000
+ 0.107 NA09 0.001 QBOII
+ 0.001 S0116
U.396
APR + 0.096 NA027 0.073 NAOI3 0.771 0.536 0.298 0.482 0.500
0 .-*42
MAY+ 0.073 NA033 + 0.044 NAOi 0.901 0.754 0.550 0.626 0.667
0.048 NAO7 + 0.126 S0121
+ 0.005 QBO31 0.029 NA028
-0.509
JN- 0.03 8 NAO28 + 0.070 S0I25
JUN 0.026 NA028 + 0.04 N125 0.888 0.723 0.750 0.682 0.826
+ 0.026 NA020 + 0.048 NIN24 0.014 NA029 + 0.014 NAOio

JUL+ 0.216 NN36 0.078 0117 0.899 0.761 0.565 0.689 0.833
+ 0.068 S0130 0.066 S0128
+ 0.063 S0121
U.68I7
AG- 0.128 S0il6 + 0.057 NAOI4
AUG 0.039NA16 + 0.044 NA014 0.963 0.903 0.968 0.730 0.885
-0.039 NAO 18 + 0,044 NA023 + 0.020 NAOio + 0.016 NAOi
0.824
SP- 0.076 S0119 + 0.032 NAO12
SEP 0.06NA1+0.025NA1 0.912 0.785 0.898 0.662 1.000
+ 0.040 NAO 15 + 0.02 5 NAO 16
+ 0.051 S0122
2.U4b
OCT + 0.076 S0126 0.045 NIN36 0.776 0.543 0.785 0.648 0.893
+ 0.026 NAO30

NOV 0.040 NA034 0.141 NIN6 0.733 0.475 0.558 0.553 0.810
0.099 NIN26
DEC + 0.074 NAO367- 0.051 NAO3 0.675 0.406 0.343 0.451 0.650






91




Results Overview

The results indicate that the daily rainfall intensity and occurrence characteristics of the region are quite complex, both from a temporal and spatial standpoint. The simulation results show that, despite the inherent complexity of the processes, the behavior can be modeled, and that many of the mechanisms that underlie the system can be adequately modeled using the candidate distributions described. Attempts to apply these concepts to forecasting using multiple regression and ARIMA methods results in considerable divergence from the withheld data, which reiterates the underlying complexity of the system.







92





AIC -- All Year Types 120 ......




100








zero-th order 60 -first order
0-- second order ..-third order 40




20




0i
JAN FEB MAR APR MAY JUN JUL AUG SEP OCT NOV DEC
month



BIC All Year Types 120




100




80 .


--- zero-th order firstt order
6 0 ... ..

--- second order
-)K- third order 40




20




0 X--X
JAN FEB MAR APR MAY JUN JUL AUG SEP OCT NOV DEC
month




Figure 5-1. Monthly AIC and BIC statistics, all years







93





AIC El Nifio Years 30





25




20 ..


zero-th order
15 ...first order
0
E.-- second order
-third order








10






JAN FEB MAR APR MAY JUN JUL AUG SEP OCT NOV DEC
month



BIC --El Nitro Years 30




25




20


zero-th order
--15 --first order
E second order
)K thid order


10






5


0 \
JAN FEB MAR APR MAY JUN JUL AUG SEP OCT NOV DEC
month


Figure 3-2. Monthly AIC and BIC statistics, El Nifio years




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