Spatial patterns from EOF analysis of soil moisture at a large scale and their dependence on soil, land-use, and topographic properties

Advances in Water Resources 30 (2007) 366–381 www.elsevier.com/locate/advwatres

Spatial patterns from EOF analysis of soil moisture at a large scale and their dependence on soil, land-use, and topographic properties Summer D. Jawson, Jeffrey D. Niemann

*

Department of Civil Engineering, Colorado State University, Campus Delivery 1372, Fort Collins, CO 80523-1372, USA Received 1 August 2005; received in revised form 20 May 2006; accepted 23 May 2006 Available online 24 July 2006

Abstract We hypothesize that the spatial and temporal variation in large-scale soil moisture patterns can be described by a small number of spatial structures that are related to soil texture, land use, and topography. To test this hypothesis, an empirical orthogonal function (EOF) analysis is conducted using data from the 1997 Southern Great Plains field campaign. When considering the spatial soil moisture anomalies, one spatial structure (EOF) is identified that explains 61% of the variance, and three such structures explain 87% of the variance. The primary EOF is most highly correlated with the percent sand in the soil among the regional characteristics considered, but the correlation with percent clay is largest if only dry days are analyzed. When considering the temporal anomalies, one EOF explains 50% of the variance. This EOF is still most closely related to the percent sand, but the percent clay is unimportant. Characteristics related to land use and topography are less correlated with the spatial and temporal variation of soil moisture in the range of scales considered.  2006 Elsevier Ltd. All rights reserved. Keywords: Empirical orthogonal functions; SGP97; Soil moisture

1. Introduction Soil moisture is an important variable in hydrology and many related disciplines. It influences runoff production from precipitation events as well as groundwater recharge and evapotranspiration rates between events. For example, Jacobs et al. [20] utilized remotely sensed soil moisture data to improve the Natural Resources Conservation Service (NRCS) curve number method of runoff estimation. Timbal et al. [38] demonstrated the importance of soil moisture variability in the prediction of seasonal rainfall and other climatic variables. Soil moisture has also been shown to play an important role in the prediction of rainfall on shorter timescales, as in the case of the 1995 South Ticino flash flood [3]. Liu et al. [25] analyzed the spring dust storm frequency in northern China and found that soil moisture *

Corresponding author. Tel.: +1 970 491 3517; fax: +1 970 491 7727. E-mail addresses: [email protected] (S.D. Jawson), [email protected] (J.D. Niemann). 0309-1708/$ - see front matter  2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.advwatres.2006.05.006

is an important link between antecedent precipitation and the subsequent frequency of dust storms. Soil moisture is also an important variable in perpetuating droughts and wet periods through precipitation recycling [9,11]. As a result of these roles in hydrology, soil moisture affects flood warning systems, land management practices, and irrigation scheduling, among other applications. Despite their practical importance, only limited soil moisture observations are available (see [34] for a summary of available data). Soil moisture varies significantly by geographical location, time, and depth in the soil. Rainfall events and infiltration processes distribute soil moisture in a highly variable pattern throughout a watershed, and soil moisture is further redistributed during inter-storm periods through evaporation, transpiration, lateral flows, and groundwater recharge. Ground-based soil moisture measurement techniques include the use of impedance probes and time domain reflectometers (TDRs). Typically, manual soil moisture measurements allow for ample spatial coverage over a field, but they are not practical for capturing the

S.D. Jawson, J.D. Niemann / Advances in Water Resources 30 (2007) 366–381

variability of soil moisture through time. Stationary probes can provide a detailed picture of the soil moisture dynamics in a soil column, but they cannot practically provide enough information to characterize the spatial variability throughout a watershed. Available ground-based soil moisture datasets include the Illinois dataset [15], the Little Washita experimental watershed [19], and the Tarrawarra catchment [41]. Airborne and satellite-based sensors observe near-surface soil moisture over a 0.64–2500 km2 footprint. Although this remotely sensed data can provide ample spatial coverage, the information is too coarse for most practical applications. Most remote sensing datasets are from research-oriented field campaigns such as the Washita 1992 [19], the Southern Great Plains 1997 (SGP97) [18], and the Southern Great Plains 1999 (SGP99) [13] campaigns in Oklahoma (see [36] for partial review of these field campaigns). Yoo [45] analyzed the use of ground-based measurements to validate remotely sensed measurements using the Washita 1992 data. In order to overcome the disconnection between available soil moisture observations and the needs of practical applications, numerous downscaling or interpolation methods have been developed. These methods range from strictly statistical approaches to methods based on physically-based modeling of soil moisture. Thattai and Islam [37] evaluated the use of kriging to characterize and estimate soil moisture variations. Kim and Barros proposed a fractal interpolation method based on contraction mapping [22], which uses the observed scaling invariance of soil moisture to interpolate observations. This method generates unique fractal surfaces and includes spatially and temporally varying scaling functions. Pellenq et al. [32] developed a physically-based method to downscale remotely sensed soil moisture data for use in hydrologic modeling. Their method employs a simple soil–vegetation–atmosphere transfer model coupled with Topmodel to simulate both near-surface and deep soil moisture. Wilson et al. [43] developed an interpolation method by examining the changing influence of topographic and regional characteristics on soil moisture through cycles of wetting and drying. Their method is based on a dynamic multiple linear regression for soil moisture that requires prior knowledge of the relationships between soil moisture and regional characteristics. Margulis et al. [26] devised an ensemble Kalman filtering procedure to estimate soil moisture from microwave measurements, ground-based measurements, and model predictions. They tested the method using the SGP97 data. Heathman et al. [14] used data assimilation by direct insertion of near-surface soil moisture and a model to estimate soil moisture throughout the soil column (0–30 cm). Development of reliable downscaling and interpolation methods requires a sound understanding of the variables that control soil moisture patterns over the relevant range of spatial scales. Hu et al. [16] found that the distribution of soil moisture from the Washita 1992 experiment was well approximated with a normal distribution. Niemann

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and Eltahir [30] found that the spatial distribution of Illinois soil moisture was well described by an erlang or gamma distribution while the temporal distribution of the spatial average soil moisture was well described by a beta distribution. Rodriguez-Iturbe et al. [35] demonstrated that the variance of soil moisture patterns from the Washita 1992 experiment exhibited a power law relation as a function of area, which is a signature of scaling invariance. Hu et al. [16] examined the higher moments of the Washita data and concluded that the type of scaling invariance was multi-fractality. Western and Blo¨schl [40] examined the spatial scaling of soil moisture at the catchment scale using the Tarrawarra dataset. Similarly, Green and Erskine [12] considered soil moisture at the field scale in a semi-arid climate and found that it lacked a clear correlation length, which is consistent with scaling invariance. Cosh and Brutsaert [5] examined the correlation scales and stationarity of soil moisture patterns from Washita 1992. Because data for many topographic, soil, and land-use characteristics are widely available, an understanding of the dependence of soil moisture on these characteristics could be particularly useful for interpolation and downscaling. Western et al. [42] investigated the relationships between soil moisture and various topographic characteristics at a relatively small spatial scale (the Tarrawarra catchment). Their research indicates that soil moisture on moderately wet days exhibits a high degree of organization and is best predicted using the natural log of specific drainage area among the terrain characteristics considered. Conversely, soil moisture on dry days exhibits little spatial organization and is best predicted using the potential solar radiation index. Florinsky et al. [10] also found varying relationships between soil moisture patterns and topographic characteristics at a site in the Canadian prairie. However, soil moisture patterns have a weaker relationship with topographic properties at this site, perhaps because the topographic relief is small. Kim and Barros [21] examined the relationship between soil moisture and regional soil and land-use characteristics using the SGP97 data. Their results indicate that soil moisture is most strongly connected to soil texture attributes, specifically percent sand and percent clay. Similarly, Cosh et al. [6] used the Washita 1992 data to examine the basic statistics of the spatial and temporal variability of soil moisture. They found that the skewness and kurtosis were not significant and that the coefficient of variation was relatively constant. Regressions against soil texture, vegetation density, and time since the last precipitation event account for 55% of the variability, and persistent patterns, which might be tied to topography, explained 31% of the variability. Yoo et al. [47] considered the impact of rainfall on soil moisture variability using a modeling approach and found that rainfall primarily affects soil moisture patterns during the rainfall event, while soil properties are more important during inter-storm periods. One way to analyze the spatial patterns of soil moisture and their connection to regional characteristics is through

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empirical orthogonal function (EOF) analysis. EOF analysis decomposes a dataset into a series of orthogonal spatial patterns. These patterns can be correlated with regional characteristics to identify whether the characteristics have an influence on the most important tendencies of the data. EOF analysis has been used to analyze soil moisture patterns at a variety of scales in the past. Yoo and Kim [46] analyzed soil moisture patterns at the field scale (over an area of 0.64 km2). Their research used gravimetric soil moisture data from two areas in the SGP97 study region. By analyzing one month of daily soil moisture readings, they were able to determine a small number of patterns (or EOFs) that explain a large amount of the observed variance. The most important EOF for each site explains over 60% of the variability in that field. The correlations between these patterns and elevation, slope, permeability, porosity, and NRCS curve number were calculated. At one field site, the two most important EOFs are most highly correlated with elevation. At the other field site, the most important EOF is most highly correlated with slope, and the second most important EOF is most highly correlated with elevation and NRCS curve number. At the basin scale (144 km2), Verhoest et al. [39] applied EOF analysis to a wintertime sequence of eight raw remotely sensed synthetic aperture radar images. Through this method, they were able to distinguish soil moisture information from other backscatter information and noise in the satellite images. Kim and Barros [21] used EOF analysis to investigate remotely sensed soil moisture patterns from SGP97 across 10,000 km2. They combined one day of soil moisture data with vegetation water content, elevation, and percent sand and used EOF analysis to determine the pattern that explains the largest amount of variance between these datasets. No matter which day of soil moisture data was selected, the most important EOF was most closely represented by percent sand. However, the second most important EOF was most similar to soil moisture on wet days and most like vegetation water content on dry days. Wittrock and Ripley [44] utilized EOF analysis to examine a 34 year dataset of annual soil moisture measurements in the Canadian prairie. Their dataset includes 23 locations that stretch across the agricultural region of Manitoba. They determined that the most important EOF, which represents 34% of the variance in the dataset, is a response to remote forcing (e.g., sea surface temperature), while the second and third EOFs seemed to be responses to more local forcing (e.g., surface temperatures and vegetation characteristics of the Great Plains). Liu [24] utilized a coupled version of EOF analysis to study monthly to seasonal soil moisture and precipitation variability throughout east Asia. The two fields were generated using the National Center for Atmospheric Research (NCAR) regional climate model. Liu’s analysis demonstrated the importance of spatial patterns of soil moisture on precipitation prediction. In this paper, we examine soil moisture patterns across a 10,000 km2 region using the remotely sensed observations

from SGP97. We use the same dataset as Kim and Barros [21], but our objectives are different. Here, we use EOF analysis to identify a small number of spatial structures that capture a large portion of the soil moisture variability, and we compare these spatial structures to available regional characteristics to determine if the characteristics are useful predictors of the spatial structures. The roles of the regional characteristics are evaluated across a range of spatial scales and for both wet and dry conditions. Our analysis is similar to that of Yoo and Kim [46], but we examine soil moisture patterns at a much larger spatial scale. The following section (Section 2) describes the dataset used in this analysis, and Section 3 describes the EOF methodology in more detail. Section 4 describes the results at the fine spatial resolution when all days of data are considered. Section 5 discusses how the results change as the spatial scale of observation changes, and Section 6 discusses how the results change between wet, average, and dry days. Section 7 summarizes our main conclusions. 2. SGP97 dataset The SGP97 field campaign was a collaboration between the National Aeronautics and Space Administration (NASA), the United States Department of Agriculture (USDA), other governmental agencies, and several universities. The goal of this campaign was to characterize soil moisture dynamics across a range of scales. From June 18 to July 17, 1997, satellite, airborne, and ground-based measurements were collected for a variety of atmospheric, hydrologic, and surface variables. The SGP97 region stretches from the northern border of Oklahoma to nearly the southern border and includes the Little Washita experimental watershed [1]. The region is dominated by silt loam soils and is primarily agricultural lands and grasslands. Our analysis uses soil moisture estimates derived from the airborne electronically scanned thinned array radar (ESTAR). The soil moisture estimates apply to the soil over a depth of 0–5 cm [18,20]. ESTAR measurements were taken throughout the one month period along seven flight paths. Problems with instrumentation and weather limited the results to 16 days of soil moisture typically covering an area of about 10,000 km2. Here, we consider only the subregion that has data available on all 16 days, which is 9463 km2. The data are available at a 800 m grid resolution, so the region includes 14,787 grid cells. The grid resolution clearly restricts our consideration of soil moisture variability to relatively large spatial scales, which is expected to differ from the variability at smaller scales [8]. The reliability of the soil moisture estimates was investigated by Mohanty and Skaggs [28] using three sites in the Little Washita experimental watershed. Their results indicated that the estimates are well calibrated for sandy loam soils in rolling topography, while improvement could be made for silty loam soils on rolling or flat topography. Precipitation was observed at three gages in the study region [18]. One of the gages was in the northern part of the

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region, one was in the central part, and one was in the southern part. During SGP97, rain events occurred between the observations on June 25 and 26, June 29 and 30, and July 10 and 11. The first two rainfall events were most significant in the northern portion of the region. The third included the entire region but was concentrated in the south [18]. Fig. 1 shows the soil moisture data (total volume of water divided by total volume of soil). Before the first rain event (June 18–25), the soil is relatively wet in the north and center of the study region and dry in the south. The first storm causes the north to become even wetter (June 25–26), but this area subsequently dries between the first and second storms (June 27–29). During the second storm (June 29–30), the soil becomes wet again in the north, and after the second storm (July 1–3) there is drying throughout the entire region. The third storm (July 10 and 11) produces wetter soils throughout the region. Finally, after the third storm (July 12–16), there is drying in the north with persistent wetness in the southwest. In this analysis, we will examine soil moisture in both spatial and temporal anomaly forms. To calculate the spatial anomalies, the mean soil moisture for each day is subtracted from the local soil moisture values. Thus, the spatial anomaly at location i and time t can be found from: m 1 X xi ðtÞ ¼ si ðtÞ  sj ðtÞ ð1Þ m j¼1 where xi(t) is the spatial anomaly, si(t) is the soil moisture at the same location, j is an index of locations, and m is the number of locations in the dataset (m = 14,787). Thus, the spatial anomalies describe the deviation of the local soil moisture from the regional mean. The spatial anomaly pat-

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terns are identical to the soil moisture patterns shown in Fig. 1 except that the mean of the spatial anomalies is the same for each day (zero). To calculate the temporal anomalies, the temporal mean soil moisture for each grid cell is subtracted from the daily soil moisture values. Mathematically, the temporal anomaly at location i and time t can be written: n 1X si ðsÞ ð2Þ y i ðtÞ ¼ si ðtÞ  n s¼1 where yi(t) is the temporal anomaly, s is an index of observation times, and n is the number of observation times in the dataset (n = 16). Temporal anomalies do not identify wet or dry locations in the region. Instead, they identify locations that are wet or dry relative to their long-term tendency. Fig. 2 shows the temporal anomaly patterns. The temporal anomaly data initially show above average soil moisture values across the entire region with relatively uniform drying between events. After the third rain event, the temporal anomalies show persistent wetness in the central and southern areas. Numerous regional characteristics have been observed for the SGP97 region including: percent sand, percent clay, vegetation water content (VWC), bulk density, and surface roughness (Fig. 3). Percent sand and percent clay were estimated from soil texture categories from the coterminous United States multi-layer soil characteristics dataset (CONUS-SOIL) [27]. VWC represents the mass of water stored in the vegetation per square meter of land area. It was estimated by classifying the vegetation in each grid cell and using the normalized difference vegetation index (NDVI) to describe the density or greenness of the

Fig. 1. Remotely sensed soil moisture observations from the SGP97 field campaign. The timing of rain events is noted by the word ‘‘Rain’’ in the figure, and the north, center, and south portions of the region are also labeled. The soil moisture values represent the volume of water divided by the total volume of soil.

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Fig. 2. The temporal anomalies of the SGP97 soil moisture data. The values at each location are the deviations of the soil moisture from the temporal average at that location.

Fig. 3. Soil, vegetation, and topographic attributes of the SGP97 region.

vegetation. VWC is assumed to be constant during the one month period when soil moisture was observed [18]. Bulk density is the mass of soil and water per cubic centimeter

of soil. Soil samples were taken throughout the SGP97 region, and these samples were used to estimate bulk density for each land cover category. The surface roughness

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describes the variance of the surface elevation relative to the wavelength of the radar used [4]. For SGP97, the surface elevation variance was estimated from land-use information (e.g., bare agricultural areas, fields with corn, urban areas) [18]. Each of these characteristics is available on the same 800 m grid as soil moisture as part of the SGP97 dataset [18]. It should be noted that all of these characteristics were used in the procedure to estimate soil moisture from the ESTAR measurements [18]. However, the use of these characteristics in deriving the soil moisture values does not necessarily imply that the characteristics are important to the variability of the spatial or temporal anomalies, which will be examined later. In order for these characteristics to be important to the anomaly patterns, the characteristics must exhibit significant spatial variability within the region, and this variability must affect the spatial or temporal anomalies of soil moisture. In addition, we have calculated numerous topographic properties from a digital elevation model (DEM) available from Pennsylvania State University at a 100 m resolution. This DEM was translated to UTM zone 14, aggregated by simple averaging to an 800 m resolution, and masked to match the soil moisture data for the SGP97 site. From the 800 m elevation data, the natural log of contributing area, the slope, the wetness index, and the curvature were calculated (Fig. 3). Contributing area is the geographical area that would contribute flow to a grid point in a rain event if water always flowed in the direction of steepest descent between grid points. In many cases, the area that contributes flow to a point extends beyond the limits of the SGP97 region. While we extended our topographic analysis well beyond the boundaries of the SGP97 site to quantify these areas, some contributing areas could not be determined. These areas correspond to three major rivers that traverse the SGP97 region from west to east and are excluded from this analysis. The contributing area varies over several orders of magnitude, so we analyze the natural log of contributing area, which is more likely to exhibit a linear relationship with soil moisture. Slope is defined as the steepest downward slope between a grid point and the eight surrounding grid points. Wetness index is defined as the natural log of the ratio of contributing area per unit contour length, which is also called the specific contributing area, to slope [2]. In this analysis, the specific contributing area was calculated simply by dividing the contributing area by the linear size of the grid cell. Curvature is calculated as the rate of change of slope in the x-direction plus the rate of change of slope in the y-direction. 3. EOF methodology Consider a dataset that describes a spatial pattern at m locations, and each location is observed n times. This dataset is fully described by an m · n matrix. The matrix can alternatively be thought of as m points in n dimensional space, where we have a dimension or coordinate axis asso-

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ciated with each observation time. As an example, Fig. 4 shows a simple case with observations at many locations but only two times. Each point in Fig. 4a represents the observation at a single location. The x-coordinate indicates the value at one time, and the y-coordinate indicates the value at the second time. Instead of using the values at the two times as the coordinate axes, one could alternatively define two new axes that describe the data more efficiently. In particular, the first axis could be aligned with the main trend of the data. This particular axis would explain the maximum variance possible. If the second axis is constrained to be orthogonal to the first axis, the second axis would be identified as shown in the figure. If the dataset had more than two observation times (i.e. more than two dimensions), each additional axis would be oriented in the orthogonal direction that explains the most remaining variance. Notice that the selection of the new axes is equivalent to a particular rotation of the data as shown in Fig. 4b. Once the new coordinate axes have been defined, each data point can be described by coordinate values on the new axes. The procedure we have described is a simple example of EOF analysis. The coordinate transformations can be described with unit vectors that have a value for each observation time. Thus, the new axis can be thought of as time series. While terminology is inconsistent in the literature, we refer to the time series as principal components (PCs). In this example, the PCs describe how aligned the transformed axes are with the variability at each time. Each time series has an associated spatial pattern that represents the projections of the original data onto the transformed axes. We refer to the spatial patterns as EOFs. The EOFs describe the observations in terms of the transformed axes. In practice, the PCs and EOFs can be found from an eigenanalysis of the covariance matrix of the dataset. First, consider the spatial anomalies of the SGP97 soil moisture data. If the spatial anomalies are contained in a matrix X (m · n), then the relevant covariance matrix Rs (n · n) can be calculated: Rs ¼

1 T X X m

ð3Þ

where the superscript T denotes the transpose of a matrix. Note that capital letters are used to denote matrices. Eigenanalysis is based on the fact that for almost any square symmetrical matrix Rs one can write: Rs Es ¼ Ls Es

ð4Þ

where Es (n · n) is a matrix whose columns contain the eigenvectors of Rs and Ls (n · n) is a diagonal matrix containing the associated eigenvalues. Because the covariance matrix has dimensions equal to the number of observation times (n), each eigenvector is a time series, which we call a PC. The associated spatial patterns (EOFs) can be found by projecting spatial anomaly matrix X onto the PC matrix Es:

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Value on PC2 axis

Value at Time 2

10

5

0

−5

5

0

−5

a −10 −10

−5

0

5

10

b −10 −10

Value at Time 1

−5

0

5

10

Value on PC1 axis

Fig. 4. A hypothetical demonstration of the EOF coordinate transformation when observations are available at two times and many locations.

F s ¼ XEs

ð5Þ

where the spatial patterns are the columns of the Fs (m · n) matrix. The matrix dimensions for Es and Fs indicate that the EOF analysis of the spatial anomalies produces n = 16 pairs of EOFs and PCs. The original anomalies can be reconstructed from the PCs and EOFs using: X ¼ F s ETs

ð6Þ

The portion of the variance explained by an EOF/PC pair is equal to its corresponding eigenvalue divided by the sum of all the eigenvalues. EOF/PC pairs are usually ranked according to the amount of variance that they explain. The primary EOF/PC pair explains the largest portion of variance, the second EOF/PC pair explains the second most, and so on. North et al. [31] suggested a test for the statistical significance of each EOF/PC pair based on the eigenvalues and the number of independent observations used to calculate each value in the covariance matrix. In this method, error bars are established for the eigenvalues associated with each EOF/PC pair using a normality assumption. An EOF/PC pair is considered significant if there is no overlap of its error bars with those of the next EOF/PC pair. Error bars extend a distance ds above and below each eigenvalue, so the matrix of error bar values Ds can be found from: Ds ¼ ð2=m Þ

1=2

Ls

ð7Þ

where m* is the number of independent observations used to calculate each covariance value. With this expression, we expect that the true eigenvalues will fall within the error bars 68% of the time. Because of the normality assumption, a coefficient of 2 can be added to Eq. (7) to determine the error bars associated with 95% probability, or a coefficient of 0.67 can be used to determine the error bars associated with 50% probability. Only a subset of locations in the SGP97 dataset can be considered independent observations in Eq. (7). To evaluate the spatial dependencies, the spatial anomaly data from each day were detrended using a multiple linear regression against the two horizontal coordinates, and an autocorrelation analysis was performed on

the detrended spatial anomaly data. The results of this analysis indicates that the anomalies have correlations above 0.2 for separation distances less than 10.4 km (13 grid cells) on every observation day. When the separation distance exceeds 10.4 km, the correlations tend to oscillate between about 0.2 and 0.2. If we arbitrarily assume that points separated by at least 10.4 km are effectively independent, then the SGP97 dataset includes approximately m* = 87 independent sample locations. A similar EOF analysis can be conducted using the temporal anomalies of the SGP97 soil moisture data. In this case, one could simply exchange the spatial and temporal dimensions in the analysis. If the temporal anomaly matrix is Y (m · n), then the relevant covariance matrix Rt (m · m) is: 1 Rt ¼ YY T n

ð8Þ

Eigenanalysis is now based on the relationship: R t E t ¼ Lt E t

ð9Þ

where Et (m · m) contains the eigenvectors and Lt (m · m) contains the eigenvalues. In this case, the eigenvectors are the spatial patterns, which we still call EOFs for simplicity. Projecting the original data onto Et produces Ft (m · n): F t ¼ Y Tt Et

ð10Þ

The columns of Ft are time series, which we call the PCs. Although the EOF analysis of the temporal anomalies produces m EOF/PC pairs, the covariance values in Rt were determined using observations for only n days where n < m. Thus, one can show that only n  1 of the Lt values will be non-zero (i.e. the rank of the Rt matrix is n  1) [17]. As a result, only n = 15 EOF/PC pairs will be necessary to reproduce the temporal anomaly observations. For the SGP97 dataset, it is impractical to calculate the PCs and EOFs for the temporal anomalies as described above because the covariance matrix is too large (recall that m = 14,787). However, the EOFs and PCs can also be found by analyzing the dispersion matrix of the temporal anomalies [17]. The dispersion matrix D is defined as:

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D ¼ Y TY

ð11Þ

and is only n · n where n = 16. D is not a covariance matrix because we have not divided by the number of locations and we have not subtracted the spatial mean (we subtracted the temporal mean). If one performs an eigenanalysis using Eq. (11), then one obtains a matrix E* (n · n), which contains the eigenvectors as columns, and L* (n · n), which contains the eigenvalues along the diagonal. Similarly, the projection of the dataset can be calculated as F* = YE* where F* is m · n. The benefit of this approach is that the n  1 meaningful values of Et and Ft (plus one EOF/PC pair with an eigenvalue of zero) can be calculated from: Et ¼ F  L1=2

ð12Þ

and F t ¼ E L1=2 

ð13Þ

where the exponents on L* refer to the exponents on individual values in the L* matrix. Because the eigenvalues for the meaningful EOF/PC pairs are the same for either procedure, the test of significance for the EOF/PC pairs can be written: D ¼ ð2=n Þ1=2 L

ð14Þ

where n* is the number of independent dates. Because the SGP97 dataset contains only 16 observation dates over a period of about one month, we assume that each date is independent (i.e. n* = 16). The values of both m* and n* are the highest ones that can be reasonably selected and

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will tend to estimate the maximum reasonable number of significant EOF/PC pairs for a given probability level. 4. Results at the finest spatial resolution 4.1. Spatial anomalies Analysis of the spatial anomalies of soil moisture yields a series of sixteen EOF/PC pairs. The patterns of the first five EOFs are displayed in Fig. 5. EOF1 has high values in portions of the north and center of the study area and low values throughout the south. This EOF resembles the soil moisture patterns that occur through July 3 (Fig. 1). EOF2 shows a very different pattern that is characterized by low values throughout most of the study region with one cluster of high values in the southwestern corner. This EOF is similar to the soil moisture patterns observed after the third rainfall event (July 12–16). EOF3 is highly variable throughout the region. This EOF highlights the above average soil moisture values that were observed in the central portion of the region on nearly all days. The eigenvalues associated with the EOF/PC pairs can be used to determine the amount of the total variance explained by each EOF/PC pair. Because there are 16 days of observation, the 16 EOF/PC pairs together explain 100% of the variability. Using the significance criteria from North et al. [31] with m* = 87, the first five EOFs are considered significant at both the 50% and 68% probability levels. At the 95% probability level, six EOFs are judged to be significant. The first five EOFs together explain 93% of the variance, the first three EOFs together explain 87%, and EOF1 alone explains 61% of the total variance (Fig. 5).

Fig. 5. The first five EOFs generated from the spatial anomalies of soil moisture and the variance explained by each EOF/PC pair.

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Weighted Principal Component Coefficient−

These results indicate that the seemingly complex patterns of soil moisture in the SGP97 field campaign can largely be explained by a very small number of underlying spatial structures. The fact that 61% of the total variability has been captured in EOF1 indicates that a single spatial structure can explain much of the overall soil moisture pattern. Although the relative importance of EOF1 on daily patterns of soil moisture waxes and wanes during cycles of wetting and drying (as discussed later), the spatial pattern of EOF1 is invariant in time. In their analysis of field scale soil moisture in the SGP97 region, Yoo and Kim [46] also found that a single spatial structure could explain 60–65% of the variability in their 25-day dataset, depending on the field being analyzed. However, the remaining spatial structures each explain less than 10% of the total variability. Here, EOF2 explains 18% of the variance. This difference may suggest that there is a more important secondary structure present in the larger scale data, but it is also possible that the role of this EOF would diminish if 25 days of observations were available in the remotely sensed data. Multiplying the PCs by the amount of the variance they explain gives the weighted PCs, which indicate the relative importance of the EOFs in describing the variance in each day’s soil moisture pattern (Fig. 6). The weighted PCs exhibit temporal variations that can be related to the occurrence of precipitation events and the subsequent drying periods. During the first several days, when the above average soil moisture values are concentrated in the northern and central portions of the region, EOF1 is dominant. As the soil dries before the first rain event, the influence of EOF1 decreases until the first three EOFs become almost equally important in predicting soil moisture. The role of EOF1 increases again in response to the rain events and decreases as each period of drying progresses. After the third rain event (on July 10 and 11), when above average soil moisture values are in the southwestern corner of the region, EOF2 becomes the most important pattern in

0.25 0.2 0.15 0.1 0.05 0

−0.05 −0.1 −0.15 −0.2 −0.25 6/17

PC 1 PC 2 PC 3

6/22

Rain

6/27

Rain

7/02

Rain

7/07

7/12

7/17

determining soil moisture while the importance of EOF1 becomes small. The consistency between the weighted PCs and the known periods of wetting and drying is important because it suggests that the EOF/PC rotation captures some physically-based tendencies in the underlying data. Yoo and Kim [46] also considered the weighted PCs of their soil moisture data. Their results showed small changes in the relative importance of the EOFs during the cycles of wetting and drying. However, their results did not show any changes in the dominant EOF. In many other EOF analyses, the first EOF is dominated by one positive anomaly over the entire domain, and the second EOF has positive and negative anomalies occurring in north–south or east–west pairs [33]. The EOFs derived from the SGP97 soil moisture data do not entirely conform to these common patterns. The first EOF is dominated by positive anomalies in the northern and central regions rather than a single positive anomaly throughout. The second EOF has a positive anomaly in the south and a negative anomaly in the north, which is more consistent with typical EOFs. We evaluated the possible effects of the extent of the region considered by dividing the study area into northern and southern halves. These halves were then analyzed individually by recalculating the spatial anomalies and repeating the EOF analysis. The new EOFs could be substantially different if the spatial average varies significantly between the two sub-regions or if the temporal correlations used to identify the EOFs occur primarily in one sub-region or the other. The resulting EOFs for the sub-regions are very similar to those found when analyzing the whole domain, but the variability explained by each EOF does vary between the sub-regions. For the northern region, EOF1 explains 73% of the variance, EOF2 explains 10%, and EOF3 explains 6%. For the southern region, EOF1 explains 54% of the variance, EOF2 explains 26% and EOF3 explains 6%. These differences are expected because the first two rain events were concentrated in the northern region, and EOF1 is most associated with the soil moisture patterns produced by those events. Overall, however, this simple analysis shows that the domain shape has little effect on the spatial patterns identified by the EOF analysis in this case. One of the main benefits of the EOF analysis is that we have now identified a small number of orthogonal spatial patterns that together explain a large portion of the total variability of the soil moisture data. We now examine how closely these underlying patterns resemble regional characteristics that might influence soil moisture. For this analysis, we use the correlation coefficient between the EOFs and the available regional characteristics. The correlation coefficient r can be calculated: Pm Pn 

  Aij  A Bij  B ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r ¼ r 2 Pm Pn  2  Pm Pn  i¼1 j¼1 Aij  A i¼1 j¼1 Bij  B i¼1

Fig. 6. The first three weighted PCs for the spatial anomalies of soil moisture. The vertical dashed lines indicate the approximate timing of the rain events.

j¼1

ð15Þ

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where A is an EOF and B is a regional characteristic and A and B are the associated means. A rank correlation analysis can also be performed between the EOFs and the regional characteristics. This approach does not rely on the Gaussian assumption that underlies the simple correlation analysis, but the results of the rank correlation analysis showed little difference from those of the simple correlation analysis. The results of the simple correlation analysis are shown in Table 1. Recall that some of the characteristics being considered (percent sand, percent clay, VWC, bulk density, and surface roughness) were used to derive soil moisture from the ESTAR measurements. Despite this fact, none of the correlations with these variables in Table 1 is particularly high. This suggests that much of the variability in the soil moisture pattern is introduced directly by the ESTAR measurements. It also suggests that some of the characteristics may be important to the soil moisture values but less important to the most important spatial patterns underlying the spatial anomalies. EOF1 has a relatively high correlation with percent sand and moderate correlations with percent clay and slope. This result indicates that some of the variability of the EOF1 pattern is related to soil type and slope. This relationship is intuitively satisfying because percent sand and clay, when coupled with slope, determine how quickly a soil will drain after a rain event. Sandy areas drain quickly and therefore tend to be drier than other locations. Conversely, clayey areas tend to hold water and be wetter than other locations. This result is also consistent with the plot of the weighted PCs in Fig. 6, which shows that the influence of EOF1 wanes as the soil dries. The second EOF, which has a cluster of high values in the southwestern corner, is most correlated with elevation and has moderate correlations with percent clay and vegetation water content. During the middle portion of the observation period, this EOF is multiplied by negative PC values, which tends to promote wetter soil moisture at lower elevations (Fig. 6). However, near the end of the observation period, when this EOF is most important, it is multiplied by positive PC values, which promotes wetness at high elevations. It is difficult to identify a physical process that would produce higher soil moisture values at

Table 1 Correlations between the EOFs of the spatial soil moisture anomalies and the regional characteristics

Percent sand Percent clay VWC Bulk density Surface roughness Elevation ln(contributing area) Slope Wetness index Curvature

EOF1

EOF2

EOF3

EOF4

EOF5

0.42 0.22 0.07 0.15 0.12 0.14 0.03 0.21 0.12 0.00

0.02 0.20 0.24 0.12 0.04 0.47 0.00 0.12 0.06 0.01

0.09 0.01 0.12 0.05 0.06 0.21 0.02 0.03 0.01 0.00

0.23 0.10 0.02 0.04 0.01 0.11 0.00 0.11 0.03 0.00

0.09 0.17 0.07 0.10 0.01 0.14 0.02 0.01 0.03 0.01

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higher elevations. It is possible that elevation is itself correlated to subtle variations in the soil properties, perhaps associated with spatial patterns of loess deposition by wind. However, EOF2 is most like the pattern of soil moisture after the third rain event (July 10 and 11), which was concentrated in the south. Thus, it is also possible that the correlation of EOF2 with elevation occurs simply because the rainfall happened to occur in an area of high elevation. For the remaining EOFs, the correlations are usually close to zero, but percent sand, percent clay, vegetation water content, and elevation are typically the most important characteristics. 4.2. Temporal anomalies The EOF analysis was also performed on soil moisture data in temporal anomaly form. Whereas the analysis of spatial anomalies identifies spatial patterns that are correlated in time, the analysis of temporal anomalies looks for modes of temporal variability that tend to affect the same locations. The analysis of the temporal anomalies also tends to highlight areas with changing values. For example, a location that is consistently wet during the observation period would have no temporal anomalies, but a location that transitions from wet to dry will have significant temporal anomalies. The analysis of the temporal anomalies produced very different results than those for the spatial anomalies. As described earlier, the temporal anomaly analysis produces 15 meaningful EOF/PC pairs because the rank of the associated covariance matrix is 15. If all 16 days of observations are considered to be independent, the first three EOFs are considered to be significant at the 50% probability level. At the 68% and 95% probability levels, none of the EOF/PC pairs are judged to be significant. The number of significant EOFs is lower for the temporal anomaly case because of the small number of observation times and because EOF1 and EOF2 explain relatively similar portions of the variance (50% and 28%, respectively), which means that the error bars on their eigenvalues easily overlap. Fig. 7 shows the first three EOFs, which together explain 87% of the temporal variance. The primary EOF explains 50% of the variance and highlights the soil moisture anomalies observed between July 12 and 16. This EOF has high values in the southwestern corner of the region and low values throughout the north. This pattern is similar to but more exaggerated than EOF2 of the spatial anomaly data. The second EOF for the temporal anomalies, which explains 28% of the variance, has high values over most of the study region, but the values are highest in the north. This pattern is most similar to the anomalies associated with the first and second rainfall events and their subsequent drying patterns. The third EOF explains 9% of the total variance and is dominated by high values over much of the study area. The differences between the spatial and temporal anomaly EOFs demonstrate that the most important patterns underlying the temporal variability

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Fig. 7. The first three EOFs generated from the temporal anomalies of soil moisture and the variance explained by each EOF/PC pair.

gests the temporal variability in soil moisture that is explained by this pattern tends to be associated with sandy locations. In contrast, clayey locations are not associated with the temporal variations associated with EOF1 (or EOF2 or EOF3). These results are expected because sandy soils can drain water more quickly after precipitation events than clayey soils. The results also indicate that locations with steeper slopes and higher elevations tend to be more dynamic. This result is consistent with Mohanty et al. [29] who found that slope position was the largest contributor to temporal soil moisture variability. Steeper slopes are expected to promote drainage and produce higher temporal variability. As suggested earlier, locations with higher elevations may exhibit subtle differences in their soil properties in comparison to locations with lower elevations. They also may be more likely to be disconnected from stable sources of moisture such as rivers or regional aquifers and thus be more dynamic. However, higher elevation points may be more dynamic in this analysis simply because the third precipitation event occurred over a region of higher elevations. 5. Results for varying spatial resolutions

are different from those underlying the spatial variability. Also, the variance is distributed differently among the temporal anomaly EOFs and the spatial anomaly EOFs. EOF1 for the spatial anomalies explains more of the variance than EOF1 for the temporal anomalies. However, EOF2 for the spatial anomalies explains less of the variance than EOF2 for the temporal anomalies. This comparison indicates that a secondary pattern of variability is more important to the temporal anomalies than it is to the spatial anomalies. A correlation analysis was performed between the temporal anomaly EOFs and the regional characteristics (Table 2). This analysis shows that EOF1 is most related to elevation and percent sand, but this EOF has relatively high correlations with several other characteristics including slope. Recall that percent clay had a relatively high correlation with EOF1 for the spatial anomalies, but it has little correlation with EOF1 for the temporal anomalies. The high correlation between EOF1 and percent sand sugTable 2 Correlations between the EOFs of the temporal soil moisture anomalies and the regional characteristics

Percent sand Percent clay VWC Bulk density Surface roughness Elevation ln(contributing area) Slope Wetness index Curvature

EOF1

EOF2

EOF3

0.39 0.06 0.25 0.21 0.06 0.44 0.02 0.25 0.12 0.00

0.23 0.01 0.15 0.17 0.08 0.35 0.03 0.19 0.11 0.00

0.02 0.09 0.15 0.04 0.07 0.12 0.02 0.02 0.01 0.00

It is possible that the observed soil moisture patterns and derived EOF patterns have information from different regional characteristics imbedded at different scales. Research has shown that the relationship between the mean soil moisture and the variability of the sample depends on the spatial resolution being observed [7]. Additionally, the variance and spatial correlation of soil moisture have been shown to follow power laws, which is an indication of scaling invariant processes [35]. To investigate the influence of regional characteristics on patterns of soil moisture at different spatial resolutions, the EOF and correlation analyses were repeated using the SGP97 data at different spatial resolutions. The soil moisture data were aggregated by identifying new grid cell sizes and arithmetically averaging the available data within the boundaries of each cell. This aggregation technique allows one to observe soil moisture from a finest resolution of 0.64 km2 (the original data) to a coarsest grid cell size of 256 km2. The largest cell size averages data from 400 pixels (20 · 20) to determine the new soil moisture value. Our aggregation scheme is similar to Kim and Barros [21] because we use some rectangular grid cells to gain more levels of aggregation. The number of grid cells included in the analysis ranges from 14,787 at the finest resolution to only 68 at the coarsest resolution. The regional characteristics were aggregated in the same way. After completing the aggregation procedure, the spatial and temporal anomalies were calculated and the EOF analysis was repeated for each level of aggregation. It is important to note that the signs determined for the EOFs and PCs in the eigenanalysis are arbitrary because the positive direction for an axis can be defined in either direction. Thus, the signs of the EOFs and PCs have the potential to change unpredictably at dif-

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ferent spatial resolutions. To simplify comparisons between spatial resolutions, our analysis in this section considers only the absolute values of the correlations. 5.1. Spatial anomalies The results of the correlation analysis for the spatial anomaly EOFs and the regional characteristics are shown in Fig. 8. For EOF1, most of the correlations increase as the grid cell size increases. This tendency may suggest that the EOF is more related to the regional characteristics at coarse spatial resolutions than it is at fine spatial resolutions. It may also suggest that the coarse scale representations have averaged out some local noise in the observations that obscure the correlations. For all grid cell sizes, percent sand remains the most highly correlated variable. As the grid cell size increases, bulk density overtakes percent clay as the variable with the second highest corre-

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lation. For EOF2 and EOF3, more of the correlations remain approximately constant across the cell sizes considered. It is interesting to note that the relative roles of topographic, soil, and land-use properties can change as we analyze the data at different resolutions. For EOF1, the correlations with land-cover dependent characteristics such as VWC and surface roughness increase with spatial resolution, but they remain secondary to soil properties such as percent sand and bulk density. For EOF2, the correlations with elevation and VWC drop slightly at large spatial resolutions, while the correlations with percent clay and bulk density become larger. In fact, the correlations with these soil properties become larger than the correlation with VWC. The differences in the correlations found by Yoo and Kim [46] at the field scale and those presented here at the regional scale support the proposition that different

Spatial EOF1 Correlation Coefficient

1 0.8 0.6

% Sand % Clay Vegetation Water Content Bulk Density Surface Roughness Elevation

0.4 0.2 0 10−1

100

101

102

103

102

103

102

103

2

Grid Cell Area (km )

Spatial EOF2 Correlation Coefficient

1 0.8 0.6

% Sand % Clay Vegetation Water Content Bulk Density Surface Roughness Elevation

0.4 0.2 0 10−1

100

101 2

Grid Cell Area (km )

Spatial EOF3 Correlation Coefficient

1 0.8 0.6

% Sand % Clay Vegetation Water Content Bulk Density Surface Roughness Elevation

0.4 0.2 0 10−1

100

101 2

Grid Cell Area (km )

Fig. 8. The correlations between the spatial anomaly EOFs and regional characteristics for a range of grid cell sizes.

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resolution, and this correlation eventually overtakes the correlation with elevation. Small increases in the correlations with percent sand and surface roughness are also observed at coarse spatial resolutions. Most of the other correlations remain fairly constant. For EOF2, the correlations with bulk density, percent sand, and surface roughness increase as the cell size increases. For EOF3, only the correlation with surface roughness shows a notable increase with increasing cell size. For EOF1, we see a change in the relative roles of topographic, soil, and land-use characteristics across spatial resolutions. At fine spatial resolutions, elevation was most correlated with this EOF, while at coarse spatial resolutions soil properties (bulk density and percent sand) become more correlated. At small spatial resolutions, high elevations are relatively good indicators of locations with dynamic soil moisture, perhaps because these locations are associated with ridge tops. However, regional average

characteristics influence soil moisture at different spatial resolutions. While we find that soil texture is most important at coarse spatial resolutions, Yoo and Kim [46] found that topographic characteristics are most correlated with the most important EOFs in their 0.64 km2 region. For one field site, they found that EOF1 and EOF2 are most highly correlated with elevation (0.47 and 0.44, respectively). For another field site, they found that EOF1 is most correlated with slope while EOF2 is most correlated with elevation (0.59 and 0.76, respectively). 5.2. Temporal anomalies To determine the properties that affect large scale dynamic areas, the correlation analysis across spatial resolutions was also completed for the temporal anomaly EOFs and the regional characteristics (Fig. 9). For EOF1, the correlation with bulk density increases with increasing grid

Temporal EOF1 Correlation Coefficient

1 0.8 0.6

% Sand % Clay Vegetation Water Content Bulk Density Surface Roughness Elevation

0.4 0.2 0 −1 10

0

10

1

2

10

10

3

10

2

Grid Cell Area (km )

Temporal EOF2 Correlation Coefficient

1 0.8 0.6

% Sand % Clay Vegetation Water Content Bulk Density Surface Roughness Elevation

0.4 0.2 0 10 −1

10 0

1

2

10

3

10

10

10 2

10 3

2

Grid Cell Area (km )

Temporal EOF3 Correlation Coefficient

1 0.8 0.6

% Sand % Clay Vegetation Water Content Bulk Density Surface Roughness Elevation

0.4 0.2 0 10 −1

10 0

10 1 2

Grid Cell Area (km )

Fig. 9. The correlations between the temporal anomaly EOFs and regional characteristics for a range of grid cell sizes.

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elevation values are not as reliable as regional soil characteristics in determining locations with more variable soil moisture. It is important to consider the issue of statistical significance as one increases the spatial resolution of observation because the number of data points is reduced as the grid cell size increases. A simple test of significance using a Tstatistic [23] was performed for this analysis to determine the correlation that must be observed at each resolution to be considered significantly different from zero. At the finest grid cell size, correlations above 0.02 were found to be significantly different from 0 with a 95% confidence limit. At the coarsest spatial resolution, all correlations above 0.24 were found to be significantly different from 0 with a 95% confidence limit. This test does not consider whether the differences between observed correlations at a given resolution are statistically significant, but the significance of these differences becomes more uncertain as the spatial resolution becomes more coarse.

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data equally between the categories. Fig. 10 identifies the category for each day. The EOF analysis was repeated for the spatial anomaly data in each of the three categories, and the primary EOF for each category is shown in Fig. 11. All three primary EOFs are comparable to the primary EOF for the entire dataset because they have high values in the northern and central portions of the region. However, as one considers drier conditions, the high values in the EOF are distributed more evenly throughout the region and include the southwestern corner of the region. Table 3 shows the correlations between these primary EOFs and the regional characteristics. It is interesting to note that percent sand is most correlated with the primary EOF from wet days. As the EOF is generated from drier conditions, the correlation with percent sand decreases. This behavior occurs because percent sand is related to the ability of a soil to drain quickly. Thus, it is more important shortly after rain events when the soil is still wet. Conversely, percent clay is

6. Results for wet, average, and dry conditions It is expected that different processes are important in the vadose zone water balance depending on the overall amount of moisture. This could lead to stronger relationships between the soil moisture patterns and certain regional characteristics for wet or dry days. In order to assess this possibility, the soil moisture data were divided into wet, average, and dry days. To classify the data, the soil moisture measurements from all 16 days were combined, and the mean and standard deviation of the combined dataset were calculated (0.185 and 0.10, respectively). The mean for each day was then compared to the mean for the entire dataset. If the daily mean was at least one quarter of the standard deviation above or below the overall mean, then the day was labeled wet or dry, respectively. The remaining days were considered average. One quarter of the standard deviation was selected in order to divide the 30 Rain

Rain

Fig. 11. The primary spatial anomaly EOFs generated from wet, average, and dry days.

Rain

Soil Moisture

25

Table 3 Correlations between the primary spatial anomaly EOFs generated from wet, average, and dry days and the regional characteristics

20

15

10 Daily Mean Soil Moisture Mean of All Soil Moisture Mean of All Soil Moisture +/− 0.25*Standard Deviation

5 6/17

6/22

6/27

7/02

7/07

7/12

7/17

Fig. 10. Daily mean soil moisture and the days classified as wet, average, and dry.

Percent sand Percent clay VWC Bulk density Surface roughness Elevation ln(contributing area) Slope Wetness index Curvature

Wet

Average

Dry

0.44 0.22 0.10 0.18 0.11 0.23 0.04 0.22 0.12 0.01

0.43 0.27 0.08 0.10 0.15 0.19 0.05 0.25 0.14 0.00

0.32 0.34 0.04 0.03 0.19 0.11 0.05 0.17 0.11 0.01

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most correlated with the primary EOF from dry days, and the correlation drops as the EOF is generated from wetter conditions. Percent clay is an indication of the soil’s ability to retain water and therefore identifies areas with above average soil moisture for a longer period after rain events. The correlations between the topographic characteristics and the primary EOFs also exhibit some interesting tendencies. When the primary EOF is generated from wet conditions, elevation is the most correlated topographic characteristic. When the EOF is generated from dry conditions, slope is the most correlated topographic characteristic. The fact that elevation becomes much less correlated with the primary EOF as the soil dries is consistent with our interpretation that elevation appears important because the rain events happened to produce the most rain in a region with high elevations. Most of the topographic characteristics exhibit lower correlations when the primary EOFs are generated from dry conditions. This behavior is consistent with the idea that topography is more important in determining soil moisture patterns during moderate to wet periods than during dry periods [21]. For all three categories, slope has a higher correlation with the primary EOF than wetness index. The EOF analysis of wet, average, and dry days was repeated using a range of grid cell sizes. While these results are not shown in a figure for brevity, percent sand remains the most important variable at all spatial resolutions considered for wet and average conditions. For the dry periods, percent sand and surface roughness become more important than percent clay at coarser resolutions. For all three divisions of data, the role of elevation becomes less important relative to the other characteristics at coarse resolutions. 7. Conclusions We have utilized EOF analysis to determine spatial patterns that efficiently explain the variance in the SGP97 soil moisture dataset, and we have utilized correlation analysis to determine whether these patterns are related to regional characteristics. From this analysis, we can draw three main conclusions. First, a seemingly complex dataset of soil moisture can be well approximated by a very small number of underlying spatial structures, called EOFs. For the spatial anomaly data, one EOF explains 61% of the variance in the 16-day dataset. The large amount of variance explained by this EOF suggests that much of the observed spatial variability of soil moisture is fixed in time. The influence of this EOF is modulated by its associated PC time series, which reflects observed cycles of wetting and drying, but the spatial pattern itself does not change. The spatial patterns of the temporal anomalies are slightly more complex in the sense that a single EOF can explain only 50% of the total variance. In either case, the ability to capture such a large amount of the variability in a small number of patterns could be useful for downscaling large-scale remotely sensed data.

Second, this analysis identified soil texture as the physical characteristic most related to the EOF patterns. Percent sand and percent clay are the two factors most correlated with the primary EOF patterns for the spatial anomalies. Sandy locations tend to be associated with dry moisture conditions whereas clayey locations tend to be associated with wet conditions. As one considers soil moisture patterns at larger spatial scales, percent sand becomes even more correlated to the primary EOF, although characteristics related to land use (VWC and surface roughness) become increasingly correlated with the primary EOF as well. Percent sand also plays a more significant role on wet days, whereas percent clay becomes more important on dry days. For the temporal anomalies of soil moisture, percent sand is still the characteristic most related to the primary EOF, but percent clay becomes unimportant. This result suggests that percent clay is useful in identifying sites that are wet relative to other locations, but it is not useful in determining locations that are wet relative to their 16day average. Third, most of the topographic characteristics we analyzed are relatively unimportant in determining soil moisture patterns across the range of scales considered. High elevations tend to be associated with more temporally variable soil moisture. This result is possibly due to the greater occurrence of rain in areas with high elevations in the analyzed dataset. Slope has moderate correlations with the primary EOFs for both the spatial and temporal anomalies, whereas the natural log of contributing area and the topographic curvature are essentially unrelated to all of the EOFs. The roles of all of the topographic characteristics diminish as one considers drier conditions. These results contrast with those of Yoo and Kim [46] who found that topographic characteristics are most related to the EOFs they generated in the same region at a much smaller spatial scale. The contrast may occur because topographic characteristics influence soil moisture largely through lateral flows, which are not easily observed at the finest scales of our analysis. Acknowledgments We gratefully acknowledge financial support from the Terrestrial Sciences Program of the US Army Research Office. We also thank Mark Perry for preparing the topographic data and conducting the autocorrelation analysis of the soil moisture anomalies, and we thank three anonymous reviewers for several helpful suggestions. References [1] Allen PB, Naney JW. Hydrology of the Little Washita River Watershed, Oklahoma: data and analyses. USDA, ARS-90, 1991. [2] Beven KJ, Kirkby MJ. A physically based, variable contributing area model of basin hydrology. Hydrol Sci Bull 1979;24(1):43– 69. [3] Cassardo C, Balsamo GP, Cacciamani C, Cesari D, Paccagnella T, Pelosini R. Impact of soil surface moisture initialization on rainfall in

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