Spatial coherence of rainfall variations using the Oklahoma Mesonet

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1 INTERNATIONAL JOURNAL OF CLIMATOLOGY Int. J. Climatol. 32: (2012) Published online 25 March 2011 in Wiley Online Library (wileyonlinelibrary.com) DOI: /joc.2322 Spatial coherence of rainfall variations using the Oklahoma Mesonet K. Margret Boone, a Renee A. McPherson, b * Michael B. Richman c and David J. Karoly d a Oklahoma Department of Environmental Quality, Oklahoma City, OK 73101, USA b Oklahoma Climatological Survey and School of Meteorology, University of Oklahoma, Norman, OK 72072, USA c School of Meteorology, University of Oklahoma, Norman, OK 72072, USA d School of Earth Sciences, University of Melbourne, VIC 3010, Australia ABSTRACT: Studies on precipitation patterns and the spatial distribution of precipitation are beneficial for many aspects of society, including agriculture, transportation, and business. In this research, data from over 100 Oklahoma Mesonet stations were used in a space-time decomposition of Oklahoma rainfall from 1 March 1994 to 31 December Spatially coherent patterns of annual, warm-season, and cold-season rainfall events were derived using principal component analysis. Because the Oklahoma Mesonet records rainfall every 5 min, relatively short events (e.g. 15 min or 3 h) could be examined. Moreover, rainfall events were split into warm season and cold season to better understand the spatial differences by precipitation type (e.g. stratiform or convective). The results were not sensitive to domain size or shape. For 24-h, 3-h, and 15-min rainfall accumulations, four similar coherent rainfall patterns were identified, located across NW, NE, SE, and SW Oklahoma. As expected, as the timescales considered became smaller, the spatial scale of the patterns, especially from the 24-h to the 15-min pattern, decreased slightly. The 15-min rainfall analysis also identified a fifth region of coherent rainfall in central Oklahoma that was not identified in the first four principal components (PCs) of the 24-h or 3-h rainfall. The associated PC scores verified the rainfall patterns described by the PC loadings. Warm-season and cold-season rainfall patterns also were calculated for the 24-h, 3-h, and 15-min rainfall accumulations. There was not much difference between the warm-season and cold-season rainfall patterns, both demonstrating coherent regions in the four quadrants of Oklahoma. Copyright 2011 Royal Meteorological Society KEY WORDS rainfall; rainfall pattern; principal component analysis; spatial coherence; Oklahoma Mesonet Received 6 April 2010; Revised 13 February 2011; Accepted 13 February Introduction As numerous states, regions, or countries plan for new mesoscale observing networks, a primary question continues to surface: Can a mesoscale observing network characterize the nature of rainfall? Toreti et al. (2009) noted that analysing precipitation requires a spatially dense network of meteorological stations with high-quality data that are temporally continuous. With the establishment of the Oklahoma Mesonet, a surface observing network comprising more than 110 stations across km 2, such data are readily available to analyse the spatial coherence of rainfall and to better understand the ability of a mesoscale network to characterize rainfall across the humid subtropical and semiarid climate zones of Oklahoma. In addition, as noted by Wickramagamage (2009), pattern analysis of highresolution rainfall data (in both space and time) is useful for both agricultural and water resource management. * Correspondence to: Renee A. McPherson, Oklahoma Climatological Survey and School of Meteorology, University of Oklahoma, 120 David L. Boren Blvd., Norman, OK 73072, USA. renee@ou.edu Likewise, Toreti et al. (2009) stated that rainfall spatial analyses also have implications in assessing climate change impacts and water resource availability. Hence, studying rainfall patterns and the spatial distribution of rainfall is beneficial for many aspects of society beyond the surface weather observation community. Previous spatial studies of rainfall have been focused on either the entire United States or large geographical regions, despite the vast geographical variation of rainfall. Rainfall observations across the United States generally are of limited frequency (e.g. daily) and typically are gathered by non-meteorologists. There are 600 automated surface observing station (ASOS) sites that provide hourly observations. These are typically located at large airports. Because of the temporal and spatial distribution of rainfall measurements, the patterns identified by these somewhat limited data sets may not be physically representative of the actual rainfall. Additionally, the previous work conducted on rainfall patterns focused primarily on rainfall accumulations over 1-day, 3-day, 7-day, or monthly time periods. These times reflected synoptic conditions during the rainfall events. As noted above, because of the distribution of the ASOS sites and Copyright 2011 Royal Meteorological Society

2 844 K. M. BOONE et al. the non-automated rainfall measurements, shorter duration rainfall events are more difficult to study. Advances in technology and measurement techniques motivated this new rainfall study that benefited from higher spatial and temporal resolutions of rainfall measurements. The Oklahoma Mesonet provides an excellent opportunity to study mesoscale phenomena, including rainfall. Mesonet rainfall data have a temporal resolution of 5-min and a spatial resolution of 30 km. Previous studies have been conducted using Oklahoma Mesonet data. There have been spatial analyses of temperature, pressure, and other variables (e.g. Elmore and Richman, 2001; Brotzge and Richardson, 2003; Fiebrich et al., 2003; McPherson et al., 2004). However, spatial distributions of Mesonet rainfall have not been studied extensively. Section 2 overviews Oklahoma s rainfall climatology, while Section 3 describes the Oklahoma Mesonet, methods to ensure data quality, and the PC analysis methods used in this study. Section 4 discusses the results of the study, and Section 5 summarizes and concludes this study. 2. Background During , the normal annual precipitation across Oklahoma showed a large gradient from about cm per year in the southeast to cm per year in the northwest (Figure 1). Because of the location of Oklahoma with respect to the Gulf of Mexico and frequent southerly and southeasterly flow, moisture advects northward from the Gulf of Mexico to modulate the amounts of rainfall across central and eastern Oklahoma. In contrast, most of western Oklahoma experiences southwesterly and westerly winds from the high plains and deserts of the Southwest United States and Mexico. Therefore, western Oklahoma climatologically is much drier than eastern Oklahoma. The annual precipitation gradient is not necessarily representative of that for each individual month. Because of changes in the intensity and location of the jet stream, and fluctuations in moisture gradients (e.g. dryline, vegetation gradients), the monthly normal precipitation gradient can vary substantially from the warm season (April through September) to the cold season (October through March). For example, August normal precipitation is more uniform than the normal annual precipitation across the state (Figure 2(a)). Because the jet stream is typically located over Canada during August, summer rainfall in Oklahoma generally results from weak frontal zones or from daytime heating of buoyant air. The pattern of December normal precipitation (Figure 2(b)) is similar to the annual normal precipitation, but the amount of precipitation is much less. In December, the jet stream is closer to Oklahoma, allowing for more intense low-pressure systems and their associated fronts to dominate. 3. Data and methods 3.1. Data from the Oklahoma Mesonet Jointly owned by the University of Oklahoma and Oklahoma State University, the Oklahoma Mesonet is a state-of-the-art network that measures several surface hydrological, agricultural, and meteorological variables at 5-min intervals since 1994 (Brock et al., 1995; McPherson et al., 2007). Each of the automated observing stations of the Oklahoma Mesonet monitors and gathers over 20 meteorological variables, including temperature, relative humidity, incoming solar radiation, soil temperatures, wind speed and direction, and rainfall, resulting in a data archive of more than 4 billion observations as of May There is at least one station per county in Oklahoma (77 counties total). The Oklahoma Mesonet measures rainfall at each site using a 30.5-cm diameter MetOne 380C tipping-bucket rain gauge, with the orifice located 0.6 m above ground (McPherson et al., 2007). The gauge is unheated and measures frozen precipitation when it melts and enters Figure 1. Map of the normal annual precipitation (in centimetres) across Oklahoma for Precipitation data courtesy of the Cooperative Observer Network of the National Oceanic and Atmospheric Administration. This figure is available in colour online at wileyonlinelibrary.com/journal/joc Figure 2. Monthly normal precipitation (in centimetres) for (a) August and (b) December for the years This figure is available in colour online at wileyonlinelibrary.com/journal/joc

3 SPATIAL COHERENCE OF RAINFALL VARIATIONS 845 the water table. A previous study showed the MetOne 380C to be accurate within ±5% in high winds (Duchon and Essenberg, 2001). In this study, almost 10 years of Oklahoma Mesonet rainfall data, representing over one million 5-min observations per site, are utilized. When this study began, the rainfall data set available ranged from 1 March 1994 to 31 December Since March 1994, several Mesonet sites have been commissioned and decommissioned. Therefore, to ensure that the data are as complete as possible, only 101 Oklahoma Mesonet sites were utilized of 120 possible sites (Figure 3) Quality assurance of the data set Rigorous quality assurance (QA) procedures were performed for all meteorological variables gathered at Oklahoma Mesonet sites (Shafer et al., 2000; Fiebrich et al., 2006). Basic QA procedures included the following: (1) laboratory testing and calibration, (2) intercomparisons on-site, (3) manual inspections, and (4) automated routines. Both static and dynamic tests were conducted on the MetOne 380C tipping-bucket rain gauge in the Fred V. Brock Standards Laboratory before the gauge was installed in the field. Meteorologists then compared 24-h accumulated rainfall to the 24-h radarestimated rainfall total for any given site. If the rainfall estimate was at least 0.5 in and the corresponding Mesonet site reported less than 25% of that estimate, then a report was sent to the QA manager for manual inspection. This manual QA procedure was completed on daily, monthly, tri-monthly, and yearly timeframes. Besides the basic QA procedures performed, the rainfall data used in this study also were scrutinized using a double-mass analysis. The double-mass analysis compared the total number of tips from a Mesonet rain gauge and the totals for the nearest five Mesonet sites. The double-mass analysis revealed that more than observations of over 100 million were not correct and consequently were removed from the data set (Reader, 2004). In addition, all frozen precipitation events were removed from this study. According to Richman et al. (2008), the worst-case scenario for replacing the missing data was a casewise deletion. That study found linear regression superior Figure 3. Map of Oklahoma Mesonet sites used in this study. Other Mesonet sites did not have a continuous record from 1 March 1994 to 31 December This figure is available in colour online at wileyonlinelibrary.com/journal/joc to deletion for inputing missing data cases; hence, it is used in this study. For each Mesonet site, the two sites that were best correlated to the given site were used to calculate the coefficients of the linear regression. The rainfall amount at the best-correlated site was used for the independent variable in the regression equation. If that site was missing the same observation, then the second highest correlated site was used. Using this technique, all missing data points were replaced with estimated values. If all the Mesonet sites surrounding the missing observation recorded no precipitation, then instead of using the linear regression method, the missing observation was replaced with zero PCA methodology One method for studying spatial coherence of rainfall data is principal component analysis (PCA), a method that determines the important features of a multivariate data set in this case, a single variable, rainfall data that vary in space and in time. PCA has been utilized as a statistical tool in the atmospheric sciences since the 1950s (Fukuoka, 1951; Lorenz, 1956). The equations are well documented in the study by Richman (1986). One advantage of PCA is that, as an eigenvector method, it promotes the understanding of the time variation among the variables through analysis of the two displays that are mapped in geographical space and time, the loadings and scores, respectively. For rainfall data, PC loadings describe the regions of coherent rainfall in a study area; the scores indicate how much rain fell across the associated spatially coherent region for any given time period (i.e. for a given time period, the score multiplied by the loading equals the standardized rainfall anomalies assuming that all the eigenvectors are retained). For example, Wickramagamage (2009) used PCA to analyse monthly rainfall means from 646 observing stations across Sri Lanka. From the PC scores identified in the study, the spatiotemporal rainfall patterns across Sri Lanka could be described as of the following four modes: weak southwest monsoon, strong southwest monsoon, strong northeast monsoon, and a mixed mode that resembled a transition to the northeast monsoon, with these modes being divided temporally for March April, May October, December February, and November, respectively. Although dominant patterns were discovered in his analysis, Wickramagamage (2009) did note that it would be more logical to use a finer temporal resolution to better identify distinct spatiotemporal rainfall modes. The choice of the type of similarity matrix used (e.g. correlation, co, or cross-products) depends on the type of data being analysed. A correlation matrix describes the standardized variations among a given set of input variables. The mean and of each variable do not contribute to the relationships. The principal diagonal is a uniform value of 1. The co matrix describes the anomalies or deviations from the means of each variable to form a

4 846 K. M. BOONE et al. matrix where the anomaly cross-products divided by the number of degrees of freedom form the off-diagonal elements, while the of each variable forms the principal diagonal. Hence, this matrix is sensitive to differences in the s of the variables (Wilks, 2006). Calculation of the cross-products matrix does not involve removal of the mean of each variable; hence, it is sensitive to the magnitude of each variable. The co matrix tends to create patterns that are centred on areas of maximum. This may be a problem when the gradient of variability is large across a given region (Richman, 1981). Unlike the co matrix, the advantage of using a correlation matrix is that the input observations are weighted equally, not biasing the position of the pattern centres. If comparing patterns, however, the results would be a function of shape and not intensity due to isolines of the patterns being nondimensional (Richman, 1981). The cross-products matrix is rarely used, but is a useful tool because it is an uncentred matrix and does not draw information from the anomalies of the observations. For example, Molteni et al. (1983) chose cross-products as the input matrix because they wanted a direct analysis of rainfall instead of an analysis of the rainfall deviations. Although many rotational algorithms exist, two examples of rotation methods include VARIMAX and Promax. VARIMAX criterion, described by Walsh and Richman (1981), Horel (1981), Richman (1981), and Richman and Lamb (1985), is a linear transformation to maximize a simplification criterion that gives a PC position that maximizes the magnitude of the PC loadings on a subset of variables. The VARIMAX rotation is a rigid, orthogonal rotation in contrast to the Promax rotation, which is an oblique, non-rigid rotation. Although similar to VARI- MAX, the Promax rotation often achieves a higher degree of localization through relaxing the orthogonality constraint. An oblique rotation, such as Promax, is used most appropriately when the major modes of variation are not orthogonal and, thus, are correlated (Easterling, 1991). In some cases, results from orthogonal and oblique rotation methods are similar (e.g. results from Toreti et al., 2009, for annual and seasonal precipitation in Italy from 1961 to 2006). Generally, most PCA studies that invoke an orthogonal rotation use the VARIMAX criterion. When an oblique rotation is used, no one algorithm is selected in the majority of studies; however, the simulation study by Richman (1986) suggests that Promax yields accurate results over a wide range of correlation configurations. To reveal the number of patterns in the rainfall data (i.e. number of PCs to retain), the scree test (Wilks, 2006; Wickramagamage, 2009) was used with the eigenvalue separation rule of thumb (North et al., 1982). The scree test indicated graphically where the plotted eigenvalue magnitude changed slope and, hence, separated the signal from the noise. North et al. (1982) enhanced the scree test with the rule of thumb method that accounted for sampling errors by adding confidence intervals to the plot of eigenvalues. The area where the eigenvalues and their sampling errors extensively overlap (i.e. a degenerate multiplet) indicated the PCs that are intermixed and, hence, affected the uniqueness of patterns found by PCA (i.e. were not physically representative). Retaining too many PCs ( overfactoring ) or not retaining enough ( underfactoring ) might distort the rainfall patterns; thus, the associated PCs might not be physically representative of the rainfall region (Richman, 1981). Walsh et al. (1982) conducted a PCA of 70 years of monthly precipitation using 61 stations spaced across the entire United States. Their analysis used the VARIMAX orthogonal rotation and described f = 9 regions or subsets of stations with spatial coherence as those areas with loadings greater than 0.5 when the loadings were plotted on maps. The resultant nine patterns identified regions of synoptic-scale precipitation that were correlated to sealevel pressure patterns to identify a potential method to enhance precipitation forecasting. Richman and Lamb (1985) created a PCA to study the patterns of 3- and 7-day summer rainfall across much of the central United States. They discussed a range of eigenvector methods that could be applied to a rainfall data set consisting of 402 stations across the central portion of the United States from 1949 to In additional to regionalization, they compared the unrotated and rotated PCs. The study also found the rotation of PCs to be the most effective method for interpreting the results. Using the VARIMAX orthogonal rotation, Richman and Lamb (1985) discovered that the 3- and 7-day rainfall patterns tended to be spatially coherent (loadings 0.4). Several of the coherent patterns found in their study encompassed regions of Oklahoma, as seen in Figure 4. (Note that region 5 highlighted eastern and central Oklahoma and region 6 included western Oklahoma.) They found that regions identified as spatially coherent by the unrotated PCs did not produce the same results as those from rotated PCs. Because the unrotated PCs were affected by the domain shape and the closeness of the eigenvalues, there were discrepancies between sets of unrotated PCs. Therefore, Richman and Lamb (1985) advised that rotated PCs should be used to identify physically interpretable regions of spatial coherence. Richman (1981) discussed the benefits of using an oblique rotation instead of a rigid orthogonal rotation. Because the oblique rotation did not constrain the orthogonality of the vectors, the resultant patterns better reflected the original data set. By constraining the vectors to orthogonality, there was a possibility that the patterns might be distorted. Although Richman (1981) did not analyse rainfall when studying the oblique rotation, 3-h sea-level pressure data from June, July, and August and June and July of 1975 were used to study synoptic-scale phenomena. Once the rotation is conducted, the associated PC scores are obtained. The following equation describes the method for obtaining the PC scores: F N f = Z N M A M f (A T f MA M f ) 1, (1)

5 SPATIAL COHERENCE OF RAINFALL VARIATIONS 847 Warm (cold) season was defined as April through September (October through March). Because Oklahoma experiences both convective and stratiform rainfall, splitting the rainfall data set into warm season and cold season was important to determine if the number and patterns of rainfall change between the seasons, similar to Molteni et al. (1983) and Richman and Lamb (1985), who analysed particular seasons. 4. Results Figure 4. The ten rainfall patterns identified by Richman and Lamb (1985). Two regions (5 and 6) included portions of Oklahoma, and two regions (9 and 10) bordered Oklahoma. where F is a matrix of standard scores known as the component scores matrix, Z represents the original data set in standardized anomaly form, A represents the component loadings matrix, and f indicates the number of PCs retained. For rainfall data, recalling that is accrued as squared deviations from mean, the scores indicate those events that most influence the rainfall patterns and therefore are used to reference those patterns. For this study, the advantages of using the correlation matrix (e.g. observations weighted equally) made it the preferred choice for the input matrix. An S-mode PCA was conducted with the Mesonet rainfall data set. S- mode is a PCA where one variable (e.g. precipitation) is analysed through space (treated as variables) and time (treated as cases), leading to an M M similarity matrix that is decomposed into M eigenvectors and M eigenvalues (Dyer, 1975; Molteni et al., 1983; Richman and Lamb, 1985; Easterling, 1991). The rainfall data were placed into an N M rainfall matrix, where N represented the number observations at any site and M represented the number of Mesonet sites. The rainfall data were analysed separately in three different time accumulations: 24-h, 3-h, and 15-min totals. Therefore, N changed with the varying timescale: N = 3589, , and non-overlapping observations for the 24-h, 3-h, and 15-min timescales, respectively. The M value stayed the same and represented the 101 Mesonet sites used. Using the interstation correlation, obliquely rotated (Promax) and orthogonally rotated (VARIMAX) PCs were calculated to examine the impact of the orthogonal rotation transformation on the analysis. The VARIMAX rotation criterion was used only for comparison with the Promax rotation criterion in the 24-h rainfall accumulations. The 24-h rainfall accumulations are highlighted in this section. Only a brief discussion of the results of the 3-h and 15-min accumulations is presented because of their similarities to the results of the 24-h accumulations. Where appropriate, the results from the different techniques (e.g. oblique vs orthogonal rotation) are also described. Finally, seasonal differences in the rainfall patterns are noted h annual rainfall accumulations Promax (oblique) rotation Using the scree test (Wilks, 2006) and North et al. (1982) rule of thumb, four PCs were retained (Figure 5) from the eigenvalues. Those four vectors of PC loadings were rotated using the promax function of the R statistical language (R Development Core Team, 2006) for both the annual and seasonal data sets. These four PCs explained approximately 58.5% of the in 24-h rainfall accumulations. Table I illustrates the explained and accumulated for each PC. The s explained in Table I may appear small, but the values indicate that there is good stability through each PC. As is always the case, the first PC (PC1) explained the greatest in the data set (17%). It highlighted a region of spatial coherence across the majority of northeastern Oklahoma (Figure 6(a)), where loadings greater Figure 5. Variance explained (%) by the first ten unrotated principal components of the 24-h annual rainfall, with North et al. (1982) rule-of-thumb (using effective degrees of freedom) shown (triangles). Four principal components were retained.

6 848 K. M. BOONE et al. Table I. Principal components (PCs) of 24-h, 3-h, and 15-min annual rainfall accumulations and their associated percent total and percent accumulated. PC # 24-h annual rainfall accumulations Percent of total Percent accumulated h annual rainfall accumulations PC # Percent of total Percent accumulated min annual rainfall accumulations PC # Percent of total Percent accumulated than 0.8 existed across portions of far northeastern Oklahoma. The second PC (PC2) indicated a gradient of spatial coherence from southeast to northwest across the state, with a region of spatial coherence (loadings >0.4) across the southeast quarter of Oklahoma (Figure 6(b)). The region of greatest spatial coherence (loadings >0.8) extended across the five southeastern-most counties of the state. There were noted regions where the rotated PC loadings were negative. The negative loadings represented regions of either lesser amounts of rainfall (below the mean rainfall) or no rainfall in comparison to the regions with loadings >0.4. Interestingly, a combination of the first two PCs looked similar to region 5 of Richman and Lamb (1985; Figure 4) that encompassed all of eastern Oklahoma. The distinction between PC1 and PC2 in this study and region 5 in Richman and Lamb (1985) could be an artefact of using a denser surface-observing network in this study (e.g. the Oklahoma Mesonet) or the finer time interval examined (e.g. 1 vs 3 days). The third Promax-rotated PC denoted a coherent region across much of southwestern Oklahoma. The region of greatest spatial coherence (>0.8) was centred on far southwestern Oklahoma (Figure 6(c)). PC4 focused on a region across northwestern Oklahoma, including the entire Oklahoma Panhandle, with loadings greater than 0.7 extending from the eastern Panhandle to some northwestern counties (Figure 6(d)). Similar to PC2, PC4 exhibited a region of weak coherence across central Oklahoma, where the loadings were slightly negative. Notably, if the coherent regions of the third and fourth PCs were combined, the encompassed area would be similar to that of region 6 found by Richman and Lamb (1985; Figure 4): the majority of western Oklahoma. Note that for each PC, the loadings >0.4 were concentrated in a single region. In addition, with the exception of a minor overlap, the regions were mutually exclusive between the four PCs. Figure 6 illustrates that the Promax-rotated PC loadings were physically interpretable and not reminiscent of Buell patterns (i.e. reflective of the domain shape). If the Figure 6. Maps of the loadings of the first four Promax-rotated principal components for the annual 24-h rainfall accumulations: (a) PC1, (b) PC2, (c) PC3, and (d) PC4. The contour interval is 0.1, and the 0.4 contour is highlighted in white. This figure is available in colour online at wileyonlinelibrary.com/journal/joc

7 SPATIAL COHERENCE OF RAINFALL VARIATIONS 849 Figure 7. Examples of composite 24-hour rainfall patterns and values (in centimetres) from the events associated with the ten highest scores for (a) PC1, (b) PC2, (c) PC3, and (d) PC4. The associated Promax-rotated principal components are shown in Figure 6. This figure is available in colour online at wileyonlinelibrary.com/journal/joc patterns were similar to Buell patterns, the magnitudes of negative and positive loadings would be comparable and generally in a range of ± (Richman, 1981; Richman and Gong, 1999). Likewise, the negative loadings present in Promax-rotated PCs can be treated as in the hyperplane (i.e. values near zero) and therefore physically insignificant. The associated PC scores indicate the specific rainfall events that most influenced each particular PC. The top ten rainfall events for PC1 PC4 were examined to help confirm that the loading patterns were physically representative (Figure 7). The events occurred during all months except July, the summer minimum statewide for daily rainfall. Five of the 35 events corresponded to a top ten score for two PCs (e.g. the fifth highest score for PC1 and the sixth highest score for PC3 occurred on 23 September 1997). Not surprisingly, most of the top ten events were associated with frontal passage (typically cold fronts) and 850-hPa dew point temperatures of 10 C or greater near the rainfall region. Dew points at 700-hPa were 0 C or greater near the rainfall region for about half of the 35 events. Remnants of two tropical storms Dean in 1995 (Atlantic) and Fausto in 1996 (Pacific) resulted in the highest score for PC3 and second highest score for PC4. Days with significant (F2 or higher) tornadoes accounted for only two of the events (i.e. 9 June 1995 and 4 May 1999), according to the storm events database of the National Climatic Data Center (available online at VARIMAX (orthogonal) rotation To compare the different rotation methods, a PCA also was conducted using the VARIMAX rotation an orthogonal rotation on the 24-h rainfall. Similar to the oblique rotation, the first four PCs were retained. The same four patterns that were identified using the previous technique oblique rotation resulted from the VARIMAX rotation (Figure 8). The main difference between the loading patterns was that the oblique loadings exhibited more realistic gradients and larger maximum values. The small negative loadings indicated regions that exhibited less or no rainfall in comparison with areas of heavier rainfall. On the other hand, the VARIMAX-rotated loadings did not exhibit a negative region, making it difficult to determine regions of less or no rainfall. Therefore, using an oblique rotation to analyse the coherent rainfall regions produced gradients that best represented the original rainfall data set h annual rainfall accumulations Similar to the 24-h rainfall accumulations, the first four PCs were retained for the 3-h rainfall accumulations. These four PCs, however, explained only 39% of the as compared with 58% for the 24-h interval (Table I). The regions of spatial coherence (loadings >0.4) were mutually exclusive, and each PC was associated with a single, dominant centre. The pattern for PC1 was similar to its counterpart (PC1) from the 24-h rainfall accumulations. The coherent region was smaller in the 3-h PC1 and shifted slightly westward. The greatest spatial coherence was 0.7 and covered only a small portion of northeastern Oklahoma a decrease in both size and magnitude from the 24-h loadings. In addition, more defined areas of negative loadings were apparent in the 3-h PC1 results than those of the 24-h PC1. The 3-h PC2 pattern was similar

8 850 K. M. BOONE et al. Figure 8. First four principal components for the annual 24-h rainfall accumulations, as rotated using the VARIMAX orthogonal rotation (left) and the Promax oblique rotation (right). The contour interval is 0.1. This figure is available in colour online at wileyonlinelibrary.com/journal/joc to the pattern described by the 24-h PC2. The coherent region for the 3-h pattern decreased, and the amount of greatest spatial coherence decreased. However, the greatest 3-h PC2 coherence (loadings >0.7) encompassed a larger region than PC2 of the 24-h accumulations. A region of small negative loading values, similar to the region described by the 24-h PC2, was also noticeable in the 3-h PC2. The third and fourth PC patterns were similar to 24-h PC3 and PC4 patterns. The region of coherence in PC3 decreased in size from the 24-h to the 3-h accumulations, and the greatest coherence diminished to encompass a four-county region. The amount of highest coherence reduced from >0.7 to >0.6.The size of the 3-h PC4 decreased from the 24-h PC4 pattern. The area of greatest coherence extended across northwest Oklahoma and the amount of spatial coherence (loadings >0.6) was reduced. Similar to the 24-h annual rainfall accumulations, the 3-h rainfall events associated with the largest ten scores for PC1 PC4 were examined. Typical synoptic conditions associated with these events were not distinctly different from those representing the highest scores of the 24-h PCs. The events occurred on 34 days, three of which were associated with high scores for two different PCs and two of which were associated with high scores for the same PC (e.g. the fourth, fifth, and ninth highest 3-h scores for PC4 occurred on 22 September 1997). All but two of the event days for the highest 3-h scores occurred during the months of April, May, June, September, October, and November, consistent with the climatological peaks of convective rainfall across Oklahoma. Additionally, 11 of the days were associated with the highest 24-h PC scores (discussed briefly earlier), typically representing the same region of the state.

9 SPATIAL COHERENCE OF RAINFALL VARIATIONS min annual rainfall accumulations For the 15-min rainfall data set, the first five PCs were retained and explained roughly 44% of the total (Table I). Although the s explained decrease with the shortening timescales, this is a reflection of the increase in the convective component of the. PC1, PC3, PC4, and PC5 exhibited the same patterns as the 24-h and 3-h PCs, and their associated patterns were mutually exclusive for loadings >0.4. A new pattern arose in this data set, highlighting a coherent region across central Oklahoma. The area of greatest spatial coherence (>0.6) was located across central Oklahoma. This pattern might reflect an artefact of retaining the first five PCs or, as the accumulation time decreased, the finer details of the storms emerged. The patterns in northeast, northwest, and southwest Oklahoma generally decreased in size (Figure 9) and magnitude of spatial coherence as compared with their 24-h counterparts Warm season versus cold season A PCA was conducted on the warm-season month (April to September) and cold-season month (October to March) rainfall data sets for the 24-h and 3-h accumulations to determine if there were significant differences in the patterns of the resulting PCs. Using the scree test (Wilks, 2006) for both seasons and accumulation lengths, the first four PCs were retained. Figure 10 compares the first PC for the 24-h annual, warm-season, and cold-season rainfall patterns, demonstrating that the coherent regions were similar. The areal extent of the greatest spatial coherence ( 0.6) did not change appreciably for either the warm-season or coldseason precipitation patterns when compared with the 24- h annual pattern (Figure 11). Like the 24-h patterns, the locations of the 3-h and 15-min patterns did not vary significantly between seasons, and no new patterns emerged in the first four PCs. 5. Summary and discussion Four main rainfall patterns were identified in this study of 24-h, 3-h, and 15-min rainfall accumulations using data between 1 March 1994 and 31 December 2003 from 101 stations of the Oklahoma Mesonet. These coherent regions were across the four quadrants of Oklahoma (NE, SE, NW, and SW). As the timescales became smaller, the spatial scale of the patterns, especially from the 24-h to the 15-min pattern, decreased slightly. The 15-min rainfall analysis identified an additional region of spatial coherence (i.e. as retained by the scree test) across central Oklahoma. Events associated with the top ten PC scores for the 24-h and 3-h rainfall accumulations were dominated by cold front passages and substantial low-level moisture. In many cases, closed or cut-off lows at 250 and 500 hpa were located over the Rocky Mountains of the southern United States, resulting in both westerly flow that enhanced the development of a lee trough to the northwest of Oklahoma and the advection of low-level moisture from the Gulf of Mexico toward Oklahoma. As the upper-level low started to move northeastward, its surface low and associated frontal boundaries provided the lift necessary to initiate and sustain convective storms within the moisture-rich environment across Oklahoma. Figure 9. Composite map of the 0.6 loading contour for the main Promax-rotated principal components of the annual 24-h (thick, dashed line), 3-h (solid), and 15-min (thin, dashed) rainfall accumulations. Note that the 15-min pattern for PC5 does not display when using the 0.6 contour; a loading contour of 0.5 would reveal the fifth pattern in the centre of the state. Figure 10. The first principal component for the 24-h (a) annual precipitation, (b) warm season precipitation, and (c) cold season precipitation. The contour interval is 0.1 and the 0.4 contour is highlighted in white. This figure is available in colour online at wileyonlinelibrary.com/journal/joc

10 852 K. M. BOONE et al. Figure 11. Composite map of the 0.6 loading contour for the primary principal components of the 24-h annual (thick, dashed line), cold season (solid), and warm season (thin, dashed) rainfall accumulations. An interesting finding arose from analysing the differences in the s explained by the various PCs. The PC spatial pattern associated with northeast Oklahoma explained approximately 17% of the for the 24- h accumulations, 10% for the 3-h accumulations, and 7% for the 15-min accumulations. This substantial decrease in explained did not mean that the 24-h rainfall accumulations were more dominant. Instead, because isolated or widespread rainfall could accumulate over the region during a 24-h period, the 24-h pattern could be representative of either stratiform or convective rainfall. The 15-min accumulation pattern, however, could only be ascribed to heavy homogeneous rainfall, from either stratiform rain or a mesoscale complex. Therefore, the s explained that the 24-h pattern likely were more representative of convective rainfall (17% of the compared with only 7% for 15-min). The PC scores verified this result. On the other hand, the second PC explained about 16% of the for the 24-h accumulations, 11% for the 3-h accumulations, and 14% for the 15-min accumulations. The explained by this 15-min PC was higher than the s explained by the 3-h PC1 and PC2. The cause of this is unknown currently and should be researched more. Warm-season and cold-season rainfall patterns were calculated for the 24-h, 3-h, and 15-min rainfall accumulations. There was not much difference between the warm-season and cold-season rainfall patterns, both demonstrating coherent regions in the four quadrants of Oklahoma. In comparing the orthogonally rotated patterns to the obliquely rotated patterns for 24-h rainfall, the oblique rotation was more physically representative. This study illustrated the benefits of using PC analysis to understand the spatial distribution of precipitation across Oklahoma. Previous studies on spatial coherence grouped only half of Oklahoma in a given PC. By having the benefit of using a dense network (in both space and time) such as the Oklahoma Mesonet, more regions of spatial coherence were identified. When these new regions were combined, the original regions described by the previous studies were evident. Therefore, the rainfall patterns identified by the PCA over the 10-year data set used in this study demonstrated the spatial distribution of rainfall in Oklahoma. Further work can be conducted with this data set and the associated findings from this study. As this study progressed, the importance of including a data denial study became apparent. A data denial study could distinguish the minimum number of Oklahoma Mesonet stations that would be needed to maintain the same coherent regions. In addition, a PCA could be conducted using radar-estimated rainfall for each Mesonet location. The coherent regions found using the radar-estimated rainfall could be compared to those found in this study. Conducting a similar PCA with additional years of Oklahoma Mesonet rainfall data and at timescales longer than 24 hours may provide additional insight into rainfall characteristics for Oklahoma. Acknowledgements The taxpayers of Oklahoma fund the Oklahoma Mesonet through the Oklahoma State Regents for Higher Education and the Oklahoma Department of Public Safety. David Karoly is supported by an Australian Research Council Federation Fellowship (project FF ). Michael Richman is supported by a National Science Foundation Grant (AGS ). The authors are grateful to Andrew Reader for his additional quality assurance of the rainfall data set used in this study. 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