A spatial regression analysis of the influence of topography on monthly rainfall in East Africa

Transcription

1 INTERNATIONAL JOURNAL OF CLIMATOLOGY Int. J. Climatol. 31: (2011) Published online 15 June 2010 in Wiley Online Library (wileyonlinelibrary.com) DOI: /joc.2174 A spatial regression analysis of the influence of topography on monthly rainfall in East Africa S. L. Hession* and N. Moore Department of Geography, Michigan State University, East Lansing, MI 48823, USA ABSTRACT: Precipitation in Kenya is highly variable and dominated by a variety of physical processes. Statistical studies of climate patterns have historically focused on application of ordinary least squares (OLS) regression to test hypotheses related to multiple predictive variables, perhaps in an attempt to better understand the physical mechanisms that drive precipitation, or on use of spatially explicit models, typically kriging- or spline-based analyses, for the purpose of improving predictions. Each of these approaches may be individually useful; however, they all possess limitations. OLS approaches have yielded biased results in the presence of spatially autocorrelated data. Kriging- and spline-based studies often focus on providing improved predictions rather than understanding. Here we use spatial regression, a method not commonly used in analysis of climate data, to assess the role of predictive variables while explicitly incorporating spatial autocorrelation in parameter estimation and hypothesis testing. This approach can yield a better understanding of relationships between precipitation and multiple predictive variables with improved statistical rigour. Using spatial regression, we show that topographic variables such as elevation and slope strongly influence rainfall during the long rains and short rains, which are vital for Kenyan agriculture. Outside these seasons, we find that smaller (mesoscale) variations in elevation are statistically significant. Further, we demonstrate the shortcomings of automated selection procedures such as stepwise regression through the identification of spurious results due to confounding. Copyright 2010 Royal Meteorological Society KEY WORDS spatial regression; geostatistics; precipitation; topography; Kenya; East Africa Received 28 May 2008; Revised 29 April 2010; Accepted 29 April Introduction Knowledge of spatial and temporal patterns in precipitation is fundamental for detailed and comprehensive environmental assessments; however, this knowledge is often limited when precipitation data are collected in sparse locations and/or intermittent time intervals that may not fully represent the region of interest. Furthermore, large amounts of variability over space and time and the local nature in which many rainfall events occur, along with data scarcity in the developing world, complicate the modelling of patterns in precipitation. Many factors play a role in the formation of precipitation across various spatial and temporal scales, and knowledge of these factors can be used to inform modelling efforts, e.g. land/sea breezes, orographic lifting, and other mesoscale processes. For example, mesoscale processes occur over regions ranging from a few kilometres to approximately one hundred kilometres in diameter (Ahrens, 2007), including land/sea breezes and orographic uplifting over mountainous terrain. Synoptic scale processes impact on the spatial distribution of precipitation over areas ranging from 100 to 1000 km 2. * Correspondence to: S. L. Hession, Department of Geography, Michigan State University, 1405 S. Harrison Road, East Lansing, MI 48823, USA. hessions@msu.edu These processes include high and low pressure areas and associated weather fronts. In East Africa, a region of variable topography, synoptic features like the presence of low pressure further south in January brings northeasterly winds [north of the intertropical convergence zone (ITCZ)] coming in from the Indian Ocean (cf Figure 2, Nicholson, 1996). In July, the low pressure cell moves to the northeast, over Asia and India, resulting in the summer monsoon season there, and south to southwesterly winds in East Africa. Global-scale processes such as the Indian Ocean Dipole and the ITCZ also play a role in rainfall patterns. In the tropics, the position of the ITCZ and its associated bands of rainfall move northward and southward over the year, dominating the spatial patterns of precipitation. In practice, the effects of mesoscale, synoptic scale, and global-scale processes can be modelled through the use of predictive or independent variables in a statistical correlation and regression analysis. Variables that are highly correlated with precipitation and for which data are more readily available, such as elevation and its derivatives (measures of mesoscale processes), geographic measures of location (proxies for synoptic and global-scale processes), and season-specific analysis (allowing for variation due to synoptic and global-scale processes), can be useful in predicting precipitation at unsampled locations. Techniques that utilize covariate Copyright 2010 Royal Meteorological Society

2 INFLUENCE OF TOPOGRAPHY ON MONTHLY RAINFALL IN EAST AFRICA 1441 information can make use of relationships between these variables and precipitation to more accurately estimate precipitation patterns over space and time. Many researchers have relied on establishing a correlation between precipitation and elevation to improve our understanding of precipitation patterns (Spreen, 1947; Daly et al., 1994; Pardo-Iguzquiza, 1998; Goovaerts, 1999a, 1999b, 2000; KiefferWeisse and Bois, 2001; Marquinez et al., 2003; Kyriakidis et al., 2004; Arora et al., 2006). Relatively high levels of correlation have been identified; the relationship is generally an increasing one (Spreen 1947; Arora et al., 2006): as elevation increases, precipitation increases. This is mainly due to the orographic effect of the mountain terrain. However, precipitation can have a very complex relationship with elevation (Arora et al., 2006); analysing this relationship can be complicated by station distribution and other factors (Hulme and New, 1997). Although some researchers have focused on using elevation as a single predictive variable, many other factors have been found to play a role in the distribution of rainfall. Other researchers have expanded the list of potential predictive variables to include derivatives of elevation and other variables summarizing geographic location (Spreen, 1947; Hutchinson, 1998b; Kieffer Weisse and Bois, 2001; Marquinez et al., 2003; Diodato, 2005; Oettli and Camberlin, 2005). For example, Spreen (1947) found that distribution of rainfall also depends on variables such as slope, exposure, and orientation. Similarly, Keiffer Weisse and Bois (2001) found that topographic variables were correlated with heavy rainfall events (e.g. 10- and 100-year events), particularly when measured at short-time steps (i.e. less than 3 h). Regional topographic variables (e.g. distance to the Mediterranean, characterization of the general shape of the Alps, distance to corresponding features of the Alps) were found to influence heavy rains, whereas local measures of topography (e.g. altitude, slope, or azimuth) were less influential. When longer periods of data were evaluated (e.g. several months or years), statistical regression analyses have been conducted using monthly or seasonal precipitation as the dependent variable to account for temporal patterns (Daly et al., 1994; Marquinez et al., 2003; Diodato, 2005; Oettli and Camberlin, 2005). In this approach, the data were stratified temporally, and independent regression analyses run for each time period. Daly et al. (1994) also stratified rainfall data over space into individual topographic facets to account for varying topography over large regions (e.g. the entire western United States), instead of developing one multivariate regression model to represent spatial variation. Independent regression analyses of precipitation versus elevation for each facet were then conducted. Many of the above-referenced studies relied on ordinary least squares (OLS) regression, however, and did not explicitly account for spatial autocorrelation. Recently-used statistical methods that relate precipitation to elevation and other predictive variables are summarized in Table I. Statistical methods vary widely from simple OLS regression to spatially explicit geostatistical methods. Most of the spatially explicit studies incorporate only one or possibly two predictors of rainfall, generally including elevation, and compare interpolation techniques ranging from Theissen polygons and inverse-distance weighting methods to more sophisticated geostatistical methods such as kriging with an external drift and cokriging (both of which incorporate independent variables or covariate data such as elevation to improve predictions of precipitation). The focus of the spatially explicit studies is often to develop the most accurate estimates of precipitation without necessarily understanding the role of various predictors or formally testing hypotheses about them. Schabenberger and Gotway (2005) noted that the use of predictive variables is not the primary focus of many of these types of studies: predictive variables were often used to account for a spatially varying mean and to avoid bias. In many cases, there was no intention of interpreting the relationships between the predictive variables and the dependent variable, or their significance. Conversely, as shown in Table I, studies designed to test hypotheses regarding multiple predictors and their influence on rainfall rely heavily on OLS regression. However, OLS regression does not account for spatial autocorrelation and, therefore, yields biased results when used to evaluate spatially autocorrelated data. Another group of methods has been used to estimate the spatial distribution of rainfall: methods such as Laplacian smoothing splines (in the two-dimensional case, these are also known as thin plate smoothing splines) were used to smooth and interpolate precipitation data over space using location coordinate information (Hutchinson and Bischof, 1983; Hutchinson, 1998a, 1998b). Rather than explicitly incorporating spatial autocorrelation in the data into parameter estimation, however, these studies utilized a systematic data thinning procedure to avoid problems with short-range correlation over space. Hutchinson (1998a, 1998b) evaluated spatial patterns in 1 day of rainfall data. Hutchinson and Bischof (1983) studied long-term mean seasonal (i.e. precipitation data were stratified into seasons) and annual precipitation. Additionally, process-based physical models have been used to predict the spatial distribution of precipitation in mountainous areas. For example, Barros and Lettenmaier (1993) modelled the advection of moisture over topographic barriers utilizing a 4D Lagrangian model. This model simulated orographically induced precipitation at a scale sufficient to resolve dominant topographic features. Although methods utilizing splines or process-based physical models were used to interpolate rain data over space, they were not used to conduct formal hypothesis testing of relationships between precipitation and possible explanatory variables. Hutchinson (1998b) and Sharples et al. (2005), however, informally evaluated the influence of explanatory variables on model results. For example, Sharples et al. (2005) demonstrated that the addition of an elevation term to the model improved results by significantly reducing interpolation errors.

3 1442 S. L. HESSION AND N. MOORE Table I. Recent studies of spatial patterns in precipitation. References Methods Application Predictors Treatment of precipitation data Results Goovaerts (1999a, 2000) OLS regression, Thiessen polygons, Inverse squared distance, OK, SKlm, KED, CK Goovaerts (1999b) OLS regression, SKlm, KED, CK Pardo-Iguzquiza (1998) Thiessen polygons, OK, CK, KED Estimation Elevation Averaged on a monthly and annual basis Estimation Elevation (average of values at four discrete points in a 1-km 2 cell) Erosivity data were averaged on a monthly and annual basis Estimation Elevation Mean annual rainfall over 20-year period SKlm outperformed KED, CK, and univariate methods (cross validation); OK outperformed OLS regression when r<0.75. CK outperformed other methods (cross validation) KED outperformed others methods (cross validation) Arora et al. (2006) OLS regression Hypothesis testing Elevation Averaged on a seasonal and Different models generally Distance to lowest station annual basis; divided by mountain range, windward and leeward side resulted for different mountain ranges and for windward/leeward sides Diodato (2005) IDW, OK, CK Estimation Elevation Averaged on a seasonal and Highest correlation between Smoothed elevation (DEM) annual basis topographic index and average Vegetation cover factor annual precipitation Topographic index (r 2 = 0.542) Kyriakidis et al. (1) time series at stations Estimation Elevation Daily Applicability of the method (2004) (2) spatial regression of Large-scale specific humidity from was demonstrated; the method coefficients with elevation and National Centers for was able to reproduce the atmospheric data Environmental Prediction (NCEP)/ spatiotemporal characteristics (3) CK of residuals National Center for Atmospheric of observed rainfall (4) reconstruction of trend Research (NCAR) reassessment measurements coefficients by adding predicted residuals (5) conditional stochastic simulation Marquinez et al. OLS regression (backwards Hypothesis testing Distance from each station to Dry season Adjusted correlation (2003) stepwise regression) coastline; Distance from each Wet season coefficients ranged between station to relative west; Elevation; Annual 0.58 and 0.67 Daly et al., 1994 OLS regression over individual topographic facets Oettli and Camberlin (2005) OLS regression (forward stepwise regression) Average elevation per sub-basin; Average slope per sub-basin Estimation Elevation Averaged on a monthly and annual basis Hypothesis testing and estimation Topo. principal components Averaged on a monthly and annual basis Cross validation For 35 different scaling windows: Calculation of estimates at Average and median elevation gridpoints Standard deviation, amplitude, PRISM exhibited the lowest cross-validation bias and absolute error when compared to kriging, detrended kriging, and cokriging See text

4 INFLUENCE OF TOPOGRAPHY ON MONTHLY RAINFALL IN EAST AFRICA 1443 Table I. (Continued). References Methods Application Predictors Treatment of precipitation data Results Keiffer Weisse and Bois, 2001 Calculation of residuals at skewness, and kurtosis gridpoints with stations Slope Kriging of residuals (cubic Geographical locators (lat, long, interpolation) distance from Lake Victoria) Multivariate OLS regression(forward stepwise regression) with kriging of residuals Estimation Regional variables such as X and Y coordinates, distance to the Mediterranean; local variables such as elevation, smoothed elevation, exposure parameters, and slope Heavy rainfall amounts (10-year and 100-year rainfall events) at time steps ranging from1hto24h Multivariate coefficients of determination ranging from 0.77 for hourly data decreasing to 0.57 for daily 100-year data Hutchinson and Laplacian smoothing splines Estimation Latitude Averaged on seasonal and Analysis is objective and Bischof, 1983 Longitude annual basis explicit; prior Elevation standardization of records not required; each fitted surface is valid for entire catchment; surfaces are consistent apart from data points; method provides percentage predictive error estimates Hutchinson, 1998a Thin plate smoothing splines Estimation X and Y coordinates One day of rainfall data Estimates show good agreement with withheld data; short-range correlation was partially overcome by removing one point from close data pairs; square-root transformation of the data improved estimates; higher order splines were found to perform less well. Hutchinson, 1998b Thin plate smoothing splines Estimation X and Y coordinates One day of rainfall data Analysis confirmed the Elevation importance of incorporating Barros and Process-based physical approach Lettenmaier, 1993 utilizing a 4D Lagrangian model East and north components of the topographic variables unit normal vector to represent slope and aspect Estimation Seasonal and annual runoff data Average areal precipitation Point estimates of monthly was reproduced with errors precipitation from snow in the range of 10 15% courses and low-elevation precipitation gauges IDW, inverse-distance weighting; OK, ordinary kriging; SKlm, simple kriging with varying local means; KED, kriging with an external drift; CK, cokriging.

5 1444 S. L. HESSION AND N. MOORE Figure 1. Meteorological station locations and the study area boundary are shown above. The region of East Africa is shown on the lower right corner with a depiction of the study area location within the region. This figure is available in colour online at wileyonlinelibrary.com/journal/joc When the goal of an analysis is formal hypothesis testing, spatial regression thus provides a solution that allows for formal statistical testing of hypotheses related to multiple independent variables while also accounting for spatial autocorrelation. Schabenberger and Gotway (2005) identify spatial regression as a form of data analysis where the focus is on modelling and understanding the mean function. [Emphasis added.] Understanding is gained when significant variables that play a role in precipitation patterns are identified. Schabenberger and Gotway (2005) go on to note that, inareversalfrom [kriging-type methods], the covariance parameters may, at times, take on the role of the nuisance parameters. This study aims to re-examine work completed by Oettli and Camberlin (2005), hereafter referred to as O&C 2005, and extend their work in East Africa by incorporating spatial regression as a more innovative approach to hypothesis testing which explicitly incorporates spatial autocorrelation in the data. Our goal is to improve modelling of precipitation in East Africa by accounting for spatial dependence in the data, thus avoiding the pitfalls common to OLS regression analysis of spatially autocorrelated processes: understatement of uncertainty in model parameters and overstatement of the reliability of model predictions. Additional goals are to differentiate between regional and localized effects through consideration of multiple spatial scales, and to evaluate the effects of seasonality on precipitation in East Africa. This analysis will furthermore allow for the independent evaluation of the dominant mechanisms of generating precipitation throughout the year. 2. Study area and data 2.1. General description of study area Our study area ranges from 34.5 to 38 E longitude and 2 S to0.6 N latitude, falling mainly in the East African country of Kenya and slightly overlapping into northern Tanzania (Figure 1). The study area contains Mount Kenya to the northeast, and much of the Kenya Highlands to the west. Elevations in this study area range from 522 to 5825 m, and average 2174 m with a standard deviation of 992 m. Substantial differences in local terrain occur across the region, from Mount Kenya, the second highest peak in Africa, to the Great Rift Valley, which cuts through the centre of the region, and Lake Victoria, which borders the western edge of the region. Equatorial East African rainfall seasonality is dominated by the long rains (March through May) and short rains (October through December) associated with the strong atmospheric convergence of the ITCZ (Hastenrath et al., 1993; Stock, 2004). This seasonal pattern in rainfall is illustrated in Figure 2. The long rains provide more

6 INFLUENCE OF TOPOGRAPHY ON MONTHLY RAINFALL IN EAST AFRICA ITCZ overhead ITCZ to the south ITCZ to the north ITCZ overhead Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec Average precipitation Figure 2. Monthly precipitation (tenths of millimetre) averaged over all meteorological stations within the study area. The position of the intertropical convergence zone (ITCZ) relative to the study area is labelled at various points in the year. This figure is available in colour online at wileyonlinelibrary.com/journal/joc rainfall than the short rains and have a lower interannual variability (Camberlin and Okoola, 2003), but the short rains start is more predictable. Outside of the long and short rains coincident with the passing of the ITCZ, precipitation is typically localized convective rainfall or, in the highland areas, stratiform rainfall during the cooler months (Ng ang a, 1992). Thus, the spatial scale of precipitation varies from large scale during the long rains and short rains to mesoscale during the drier months, driven by processes at their respective scales. Higher rainfall occurs at higher elevations in part due to forced convection, Lake Victoria s moisture, and the mountains acting as barriers to moisture-bearing winds. The slopes of the surfaces interact with elevation (and also the direction of the moisture-bearing winds) such that steeper slopes extract more precipitation. Easterly flow from the Indian Ocean is the dominant source of moisture for this region, particularly during the short rains (Black et al., 2003). Since the relatively dry northeasterly and southeasterly monsoonal winds are weaker at these transition times, onshore moisture transport is stronger (Nicholson, 1996). Blocked by the Rift Valley slopes, the western parts of East Africa receive moist westerly flows from the Congo basin. Thus, generally wetter conditions occur near Lake Victoria and the Indian Ocean coast. While Lake Victoria acts as a moisture source for local convective rainfall in the surrounding highland areas, the coastal climate differentiates itself from the highland climate due to small-scale diurnal convection from the land/sea breeze (Camberlin and Planchon, 1997). This study area approximates the northern portion of that used by O&C Limitations to our precipitation data set from Tanzania (missing data in some places/times, lack of continuous data at all sites for all months) precluded exact replication of their study domain; however, the study area used is sufficiently large for analysis: the region is large enough to observe spatial variability in precipitation and to allow for the evaluation of predictive variables at multiple scales. Furthermore, the sample size of 166 precipitation stations is adequate for statistical analysis Data Monthly precipitation is the dependent variable in this analysis. Monthly precipitation data (in tenths of mm) from 166 meteorological stations obtained from the Department of Meteorology, Government of Kenya, were used in this analysis. Meteorological station locations are shown in Figure 1. Meteorological stations are generally located in areas of higher population; therefore, areas of high elevations (i.e. greater than 2800 m) are not well represented. Since the availability of station data varies over time for each station, data from 1926 to 1998 were combined to calculate average monthly precipitation for each station. This approach yielded a larger data set for analysis: if data from smaller time periods were used, it would be more likely that a given station would be excluded from the analysis due to missing data. Stations with large data gaps (i.e. 10 years or more) or stations with less than 5 years of data were omitted from the analysis and are not included in the 166 stations (omitted stations are not shown in Figure 1). Independent variables included longitude (decimal degrees; Long) and latitude (decimal degrees; Lat) of the meteorological stations; a digital elevation model (DEM) at an approximate 1-km resolution; and averaged elevation (m), standard deviation (m), and slope (percent rise) at four different scales (9 km 9km, 39 km 39 km, 123 km 123 km, and 213 km 213 km). The four spatial scales above ranging from 9 to 213 km were chosen to provide a broad coverage of spatial scales at which mechanisms of precipitation may operate. Furthermore, the work of O&C 2005 showed significant relationships between topographic variables and precipitation at scales up to 123 and 213 km.

7 1446 S. L. HESSION AND N. MOORE Table II. Summary of independent variables used in stepwise regression analysis (n = 166). Independent Variable Scale (km) Abbreviated Name Min Median Mean Max Latitude NA Lat Longitude NA Long Elevation (m) 1 1 GT GT GT GT GT Std Dev (m) 9 9 ST ST ST ST Slope 9 9 SL (percent rise) SL SL SL Sin(Aspect) 9 9 SINASP (unitless) SINASP SINASP SINASP Cos(Aspect) 9 9 COSASP (unitless) COSASP COSASP COSASP Abbreviated names are shown as well as descriptive summary statistics for each independent variable. Values shown represent measurements at station locations. Elevation data (Figure 1) were obtained from the U.S. Geological Survey s 30 resolution GTOPO30 DEM (Gesch and Larson, 1996). Near the equator, a 30 resolution equates with approximately 1 km. These data were used to obtain average altitude measurements (m) at the various scales listed above as well as the elevation derivatives at these scales. Averaged elevation at each scale was calculated using the FOCALMEAN function in ArcMap (ESRI, 2006). At each scale, average elevation was calculated for each 1-km pixel utilizing the surrounding pixels within 9, 39, 123, or 213 km. Consequently, each averaged elevation map had a 1 km resolution, but was increasingly smoothed for coarser scales. Elevation data within a large buffer of approximately 250 km surrounding the study area were included to avoid edge effects or biased results within the study area. Aspect was also derived at each spatial scale and transformed using sine and cosine functions. Sin(aspect) yields a measure of east/west exposure (+1 represents due east, 1 represents due west) and cos(aspect) yields a north/south exposure (+1represents due north, 1 represents due south). Data for each of the independent variables were extracted from the grid cells overlapping meteorological station locations and combined with precipitation data at each respective station. Table II lists the independent variables and provides summary statistics for each using data for each station location Hypotheses Elevation and its derivatives are expected to reflect mesoscale processes related to orographic precipitation. Increasing precipitation is expected to occur with increasing elevation (Spreen, 1947; Arora et al., 2006). Since steeper slopes provide stronger orographic lifting, increasing slope or standard deviation is expected to be associated with higher rainfall (locally, at least; Buytaert et al., 2006). It is anticipated that the spatial scale at which these processes occur will vary month to month due to different synoptic forcings, wind patterns, and other seasonal variations (e.g. long rains and short rains) (Ng ang a, 1992). Synoptic and global-scale processes will be indirectly modelled through the use of the latitude and longitude as well as sine and cosine of aspect. It is also expected that the role of these factors will vary according to month and/or season. Latitude is expected to serve as a proxy for the position of the ITCZ. The coefficient for the latitude term is expected to be positive in months when the ITCZ is north (since increasing rainfall is expected with increased latitude at this time) and negative in months when the ITCZ is south of the study area (since the opposite is expected during these months). The significance of the latitude term is expected to vary during the year. For example, latitude is not expected to be significant in April, when the ITCZ is overhead; however, it is expected to be significant when the ITCZ is in its extreme northerly (July) and southerly (December and January) positions. Longitude is more likely to represent

8 INFLUENCE OF TOPOGRAPHY ON MONTHLY RAINFALL IN EAST AFRICA 1447 a complex interaction of different local, regional, and mesoscale processes that include the sea breeze near Lake Victoria in the western portion of the study area and orographic effects in the highland areas (Ng ang a, 1992; Camberlin and Planchon, 1997). 3. Methods 3.1. Model selection Average monthly precipitation data (January through December) were evaluated in 12 independent analyses over space. First, a forward stepwise regression analysis (Section 3.1.1) was completed for each month using average monthly precipitation as the dependent variable and all of the independent variables identified in Section 2.2. Next, spatial regression modelling (Section 3.1.2) was conducted using the subsets of predictive variables identified by the forward stepwise regression Forward stepwise regression To identify a subset of explanatory variables for use in the spatial regression analysis for each month, OLS regression was completed in a forward stepwise process. This process involves identifying the best explanatory variable or predictor and incorporating it into the model first, then iteratively identifying the next best predictor until the model can no longer be improved within certain constraints. Two criteria were used in the selection of best predictors: tolerance values and statistical significance (represented by p-values). Tolerance values, which represent the degree to which independent variables are correlated, were used to evaluate the underlying assumption of independence between explanatory variables in OLS regression. Tolerance values close to one indicate independence; tolerance values near zero indicate multi-collinearity. Tolerance values less than 0.6 were interpreted as evidence of non-independence. Statistical significance was evaluated using p-values, which represent the probability that the observed data and associated test statistics would be observed if the null hypothesis were true. In this case, the null hypothesis for each independent variable was that the regression coefficient is zero. When the p-value fell below a prespecified cut-off value, the null hypothesis was rejected in favour of the alternative hypothesis, that the regression coefficient is not zero. A maximum p-value of 0.15 was used to identify variables that were significant at a 15% level of significance. This relatively high level of significance was used as a conservative (i.e. less restrictive) measure for selecting preliminary independent variables for use in spatial regression. Tolerance values and p-values were calculated for each independent variable at each step in the process. Independent variables with associated tolerance values greater than or equal to 0.6 and p-values of less than or equal to 0.15 were entered into the model in a stepwise fashion. OLS model: OLS regression functions were of the form: y = Xβ + ε (1) where y is a vector of dependent variable observations, X is a matrix of observations of the independent variables, β is a vector of coefficients for the regression model, and ε is a vector of independent and identically distributed error terms following the standard normal distribution. Once obtained, OLS results were evaluated to confirm the following underlying assumptions of OLS regression: (1) independence of the predictive variables (i.e. lack of correlation between independent variables); and (2) normality of the error terms with constant mean of zero and a constant variance of one. As previously described, the first assumption was evaluated using tolerance values. Diagnostic checking of the error terms of the OLS models included plots of error terms versus fitted (predicted) values and normal probability (or normal Q Q) plots of error terms. For months in which heteroskedasticity was evident (i.e. unequal variance among error terms), rainfall data were square-root transformed for all subsequent modelling. Transformations of the dependent variable are commonly used to stabilize variance and improve normality (Neter et al., 1990; Gibbons, 1994; Hutchinson, 1998a, 1998b; Schabenberger and Gotway, 2005; Sharples et al., 2005), particularly for variables that exhibit asymmetrical distributions, such as precipitation. Precipitation data are more often right-skewed in distribution since the range of possible values is restricted by a lower bound of zero. This issue may be exacerbated during dry months, when precipitation amounts are lower and more often equal to zero Spatial regression Three different modelling approaches were evaluated in an effort to best approximate the relationship between the dependent variable, monthly precipitation (or squareroot precipitation), and the independent variables selected through the forward stepwise process. The three models considered were: the OLS model described in Section 3.1.1; a spatial lag model, also known as a spatial autoregressive (SAR) model; and a spatial error model (SEM). As previously noted, the OLS model assumes spatial independence of the dependent variables and the error terms or residuals of the model. The SAR model accounts for the presence of spatial autocorrelation in the dependent variable, but assumes spatial independence of the error terms. The SEM allows for spatial dependence of the error terms. The SAR and SEM models are summarized below. More detailed descriptions are provided by Anselin and Bera (1998) and Anselin (2006). SAR model: y = ρwy + Xβ + ε (2) where y is a vector of dependent variable observations, ρ is the autoregressive coefficient, W is a weights matrix,

9 1448 S. L. HESSION AND N. MOORE Wy is a spatially lagged dependent variable, X is a matrix of observations of the independent variables, β is a vector of coefficients for the regression model, and ε is a vector of independent and identically distributed error terms. SEM model: y = Xβ + ε and ε = λw ε + u (3) where y is a vector of dependent variable observations, X is a matrix of observations of the independent variables, β is a vector of coefficients for the regression model, ε is a vector of spatially autocorrelated error terms, λ is the autoregressive coefficient for the error terms, W is a weights matrix, Wε is a spatially lagged error term, and u is a vector of independent and identically distributed error terms. The spatially explicit SAR and SEM models were considered as alternatives to OLS models because of their ability to incorporate spatial autocorrelation through added terms: ρwy in the SAR model and λw ε in the SEM model. Without these terms, precision of the OLS estimates tends to be overstated in the presence of spatial autocorrelation, resulting in biased estimates of the OLS coefficients and values such as the coefficient of determination (R 2 ). The weights matrix W was calculated for each month using distance-based spatial weights, in which the definition of neighbour was based on the distance between points (Anselin, 2005). In this study, points were defined by meteorological station locations. Station pairs within a given distance threshold were identified as neighbours and coded with a non-zero weight in the weights matrix W. Pairs of points further apart than the distance threshold were coded as zero in the weights matrix, and were consequently excluded from the spatially lagged variables in Equations (2) and (3). Typically, the distance threshold used to identify neighbours has been established on an ad hoc basis (Walker et al., 2000; Anselin, 2005). In this study, however, selection of distance thresholds was informed through the use of semi-variogram analysis of OLS residuals. Spherical semi-variogram models were developed for each month using the respective error terms from the OLS analyses; values used to model the range of the semi-variogram (i.e. the distance over which spatial autocorrelation occurs) were used to establish distance thresholds for the calculation of weights matrices. Schabenberger and Gotway (2005) warn about the problematic undertaking when computing semivariograms using OLS error terms because: (1) the empirical semi-variogram represents autocorrelation present in the estimated residuals rather than the true error terms of the model, and (2) the estimated residuals are not secondorder stationary. The alternative courses of action they describe were not undertaken in this study but will be pursued in future work. We felt that it was reasonable to proceed with the semi-variograms of the OLS residuals for this study since we used them to estimate the distance over which spatial autocorrelation was present rather than to estimate the covariance structure of the error terms. Furthermore, the range of the semi-variogram should provide a more informed criterion for selecting a distance threshold than has previously been used. Most importantly, our final results were obtained through appropriate spatially explicit methods that did not depend on the variography of OLS error terms Selection of an appropriate spatial model The decision process recommended by Anselin (2005) was used to select the best model for analysis of the data. This process involved running an OLS analysis and calculating diagnostics for spatial dependence. OLS results should not be relied upon until diagnostic testing has shown that the underlying assumptions of OLS have been met. First, diagnostic statistics to evaluate the need for either a spatial lag or a SEM were calculated using OLS results and tested for significance. A significance level of 5% was used for all diagnostic testing. If neither diagnostic statistic was significant, the OLS results were used and no further modelling was completed. If one of the diagnostic statistics was significant, the corresponding model type was used to model precipitation data (e.g. if the diagnostic statistic for only the spatial lag model was significant, the spatial lag model was selected). If both of the statistics were significant, robust diagnostic statistics were tested for significance and the appropriate model selected in a similar manner. To further evaluate the fit for each of the abovedescribed models, the following were considered (Anselin 2003, 2005): (1) Akaike information criterion (AIC), (2) log likelihood value, (3) Schwarz criterion, and (4) likelihood ratio test for spatial dependence. Values of the first three criteria all decrease in absolute value as model fit improves. The likelihood ratio test for spatial lag or spatial error were significant (p-value <0.05) if the SAR or SEM model, respectively, were appropriate. In cases where one spatially explicit model clearly outperformed the other according to these criteria, that model was chosen as the best model. Using the spatially explicit approach described in this section, we should improve upon the statistical rigour of O&C 2005 by specifically incorporating spatial dependence into the modelling of precipitation, yielding more accurate estimates of uncertainty and reliability in model parameters and predictions. 4. Results Forward stepwise OLS regression analyses were completed for each month. Initial diagnostic plots were generated as described in Section Example diagnostic plots are shown in Figure 3 (March) and Figure 4 (September). Figure 4 depicts evidence of heteroskedasticity in the initial OLS results for September, with improved results after square-root transforming precipitation data. Square-root transformations were used for

10 INFLUENCE OF TOPOGRAPHY ON MONTHLY RAINFALL IN EAST AFRICA 1449 Residuals vs Fitted Normal Q-Q Residuals Standardized residuals Fitted values Theoretical Quantiles Figure 3. Residuals versus fitted values and normal probability plots for average monthly precipitation in March. Residuals and fitted values are in units of tenths of mm. Units on the normal probability plots are in terms of standard deviations from the mean (zero). The plot on the left shows residuals of March OLS regression analysis plotted against fitted values. Since the plot illustrates a generally random scatter, it was concluded that residuals are homoskedastic. The normal probability plot to the right does not indicate substantial deviations from a normal distribution. Consequently, average monthly precipitation amounts for March were not square-root transformed. This figure is available in colour online at wileyonlinelibrary.com/journal/joc Residuals Residuals vs Fitted Standardized residuals Normal Q-Q Fitted values Residuals vs Fitted Theoretical Quantiles Normal Q-Q Residuals Standardized residuals Fitted values Theoretical Quantiles Figure 4. Residuals versus fitted values and normal probability plots for average monthly precipitation in September (top row) and square-root transformed precipitation (bottom). Residuals and fitted values are in units of tenths of mm. Units on the normal probability plots are in terms of standard deviations from the mean (zero). The plot on the upper left corner shows the residuals of September OLS regression analysis plotted against fitted values. A distinct increase in variation is evident as fitted values increase, indicating heteroskedasticity. Additionally, the normal probability plot in the upper right corner reveals deviations from a normal distribution, particularly at the tails of the distribution. The plots generated after square-root transformation and reanalysis of the data (bottom row) show an improvement in residual variance, although problems at the tails of the distribution are still apparent. This figure is available in colour online at wileyonlinelibrary.com/journal/joc the following months to address non-normality and heteroskedasticity in the error terms: June, July, August, September, and November. June through September are generally the driest months within the study region; therefore, it is more likely that their distributions would be asymmetrical due to the zero-lower bound to precipitation measurements. Consequently, transformations of the dependent variable, precipitation, are more likely to be necessary for these months. Precipitation values in November were not influenced by the zero-lower bound

11 1450 S. L. HESSION AND N. MOORE based on inspection of the data; however, the data were somewhat skewed to the right, indicating relatively fewer locations with more extreme rainfall in November. In addition to heteroskedasticity, Figure 4 shows indications of non-normality in the error terms in the September analysis even after transformation; however, OLS is somewhat robust to non-normality, particularly in the absence of skewness (Neter et al., 1990), so no further transformations were sought. Diagnostic testing for spatial dependence was completed for each month as described in Section To calculate these diagnostics, weights matrices were established in which distance thresholds were estimated via semi-variogram analysis of OLS error terms (i.e. using the modelled semi-variogram range). Figure 5 shows the empirical semi-variograms and modelled semivariograms for each month. Table III summarizes the distance thresholds used to calculate each weights matrix. The type of model (OLS, SAR, or SEM) selected for each month is summarized in Table IV with selected performance measures comparing OLS to the spatial model, where selected. The performance measures shown include the familiar adjusted coefficient of determination (R 2 ) for the OLS regression, the pseudo-r 2 value from the spatial regression, and the AIC for both OLS and spatial models. The pseudo-r 2 cannot be interpreted in the same manner as the OLS R 2 and is, therefore, not directly comparable. Consequently, the AIC values are the more appropriate performance measures for comparing OLS and spatial regression results. The AIC for the spatially explicit model was lower in every case where a spatial regression model was used, indicating an improved fit of the spatial model over the OLS model. As shown in Table IV, the OLS model was retained for January; diagnostic testing did not indicate that a spatially explicit model was necessary for the January model. The OLS-adjusted R 2 is lowest for this month (0.395), suggesting that other variables than those included in the analysis may be influencing January precipitation patterns. In general, the SAR model was selected for months in the middle of the calendar year (i.e. June, August, and September). OLS and spatial pseudo R 2 values were highest during this period of time, indicating that these models explained more of the variability in precipitation than models for other months. As a result, there was less variability in the error terms of these models. Furthermore, since the SAR model was selected in lieu of the SEM, it was concluded that there was less spatial autocorrelation remaining in the error terms of these models and that the autocorrelation was in the dependent variable itself. For the remaining months, SEMs were selected. OLS and spatial pseudo R 2 values were generally lower in these months, with the exception of July, and error terms were more variable as a result. Since SEMs were chosen in these months, it would indicate that spatial autocorrelation remains present in the error terms for these models. July is an exception in that both model types improved on the OLS results, SEM only slightly more than SAR. More detailed results of the forward stepwise and spatial regression analyses are summarized in Table V. The following information is provided: (1) independent variables identified by the forward OLS stepwise regression, (2) OLS regression coefficients, (3) OLS levels of significance, (4) spatial regression coefficients, and (5) spatial regression levels of significance. 5. Discussion With the exception of January, OLS regression coefficients shown in Table V are biased due to the presence of spatial autocorrelation; regression coefficients from the spatially explicit regression models incorporate this spatial autocorrelation. For the months of February, April, June, and July, some independent variables that were identified as significant in the OLS model were not significant in the spatially explicit model (Table V); this illustrates how erroneous results can be obtained using OLS regression for spatially autocorrelated data. The remaining discussion and conclusions focus on results of the spatial regression models, with the exception of January, for which an OLS model was retained. Figure 6 illustrates graphically the pattern of independent variables identified as significant predictors of rainfall for the various months of the year. Latitude is a significant predictor of precipitation in July and December. The significance of latitude in these months as well as the direction of the relationship (increasing/decreasing) is expected due to the position of the ITCZ during these months. The northerly position of the ITCZ in July results in a significant increasing trend in rainfall as latitude increases whereas the southerly position of the ITCZ in December corresponds to a significant decreasing relationship. Longitude serves as a significant predictor of rainfall every month of the year except April, July, and December. The sign of the regression coefficient is negative in all cases but November. These results indicate that, in general, rainfall increases moving from east to west through the study area for much of the year. This pattern may be due, in part, to the presence of Lake Victoria to the immediate west of the study area and/or moist westerly flows from the Congo basin (Camberlin and Planchon, 1997). Another contributor here is the extent of the domain; the eastern part of the box, at lower altitude, is generally warmer and drier savanna. Warmer conditions lead to more rapid evapotranspiration and lower soil moisture available for recycling as compared to forested areas. Thus, longitude s significance is partly a consequence of temperature. Elevation up to the 9-km scale is a highly significant predictor of rainfall from May through September, which is consistent with mesoscale convection and its associated precipitation processes; this is close to the optimal 5-km scale found by Sharples et al. (2005), although that study found an optimal elevation dependence for stratiform rainfall at around 8 10 km. Since our study did not look

12 INFLUENCE OF TOPOGRAPHY ON MONTHLY RAINFALL IN EAST AFRICA 1451 Figure 5. OLS residual semi-variograms and spherical variogram models. Distance shown in decimal degrees.

13 1452 S. L. HESSION AND N. MOORE Table III. Distance thresholds used in development of the spatial weights matrix W based on modelled semi-variogram ranges. Month Distance threshold (miles) January 76 February 52 March 52 April 23 May 35 June 76 July 55 August 118 September 69 October 31 November 21 December 52 specifically at 5 km, our analysis is likely detecting the elevation dependence at the nearest scale available, namely, 9 km. At these scales, precipitation increases with increasing elevation, evident from positive regression coefficients. Elevation averaged over 213 km is significant for the remaining months of the year with the exception of April, for which no elevation term was significant. The regression coefficient for elevation at the scale of 213 km is negative, however. This result is peculiar, since precipitation is generally expected to increase with increasing elevation. Additional evaluation indicated that elevation and latitude are positively correlated (R = 0.527) for this domain at all scales, particularly at 213 km. In other words, within this study region, elevation generally decreases southwards. This finding provides evidence of confounding between elevation and latitude in this study area. Since the ITCZ is south of the equator during the months in which the regression coefficients for elevation are negative, it is more likely that precipitation is increasing with decreasing latitude, rather than elevation, in these months. This latter interpretation is consistent with the conclusions of Sharples et al. (2005), who found that models incorporating elevation at spatial scales beyond 30 m did not show improved results over models including horizontal position only. Slope and standard deviation of elevation were significant at multiple scales, slope more so than standard deviation. As expected, the resulting regression coefficient is positive in nearly every case, indicating that rainfall increases with increasing slope or standard deviation. Sine and cosine of aspect were not significant at any scale for any month other than May, in which the regression coefficient was positive for sine of aspect calculated at the 123-km scale. A significant positive regression coefficient for sine of aspect at a scale of 123 km can be interpreted as an indication of increasing rainfall as the direction a slope face approaches due east at a scale of 123 km. Other studies (Hutchinson 1995, Buytaert et al. 2006) have found aspect to have marginal to significant impacts on rainfall; however, these studies used much finer-scale rendering of topography. Additional work will be done in an attempt to better describe the relationship between precipitation and aspect in this study area. The combination of this along with significance of elevation at 1 km in May suggests that some large-scale processes are interacting with some very small-scale (almost microscale) processes. An abundance of surface moisture in May, at the end of the rainy season, combined with reduced easterly flow may signify Lake Victoria being a significant source of localized instability, leading to convection. Alternatively, lingering instability from the ITCZ transit could provide enough of a trigger to generate localized convection, and maybe even mesoscale convective complexes, during May. Climate modelling would be required to distinguish which process is playing which role here. Table IV. Selected model types and performance. Dependent Variable n Selected Model type OLS Adj. R 2 Spatial OLS AIC Spatial AIC Pseudo R 2 Jan 166 OLS Feb 166 SEM Mar 166 SEM Apr 166 SEM May 166 SEM SqrtJun 166 SAR SqrtJul 166 SEM SqrtAug 166 SAR SqrtSep 166 SAR Oct 166 SEM SqrtNov 166 SEM Dec 166 SEM OLS, ordinary least squares; SEM, spatial error model; SAR, spatial autoregressive model.