Spatial variability constraints to modeling soil water at different scales

Geoderma 85 Ž1998. 231–254

Spatial variability constraints to modeling soil water at different scales M. Seyfried

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USDA-ARS, Boise, ID 83712, USA Received 25 March 1997; accepted 9 October 1997

Abstract There is increasing interest in modeling soil water content over relatively large areas or scales. In general, the spatial variability of soil water content increases with scale, but it is not known how much or at which scales. High spatial variability constrains soil water models by reducing the accuracy of input parameters, calibration and verification data. It may also require representation of soil water in a spatially distributed manner. Soil water content data were collected at the Reynolds Creek Experimental Watershed at scales ranging from 12 m2 to 2.3 = 10 8 m2 to determine how scale affects spatial variability. We found significant spatial variability at the 12-m2 scale, which could be described as random in large-scale models. The increase of spatial variability with scale was controlled by deterministic ‘sources’ such as soil series and elevation-induced climatic effects. The satellite-derived, soil-adjusted vegetation index showed that spatial variability at the scale of Reynolds Creek Ž2.3 = 10 8 m2 . is not random, and may have abrupt transitions corresponding to soil series. These results suggest a modeling strategy that incorporates soil series characterized by random spatial variability nested within the larger, elevation-induced climatic gradient. The distinctions between soils and elevations is greatest early in the growing season and gradually diminish as the effects of differential precipitation and snowmelt timing are erased by evapotranspiration until late in the summer, when they virtually disappear. These conclusions are landscape-dependent, so that representation of spatial variability should be an explicit part of model development and application. q 1998 Elsevier Science B.V. All rights reserved. Keywords: spatial variability; soil water; scale; remote sensing; vegetation index

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Fax: q1 208 334 1502; E-mail: [email protected]

0016-7061r98r$19.00 q 1998 Elsevier Science B.V. All rights reserved. PII S 0 0 1 6 - 7 0 6 1 Ž 9 8 . 0 0 0 2 2 - 6

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1. Introduction In recent years, there has been increased interest in modeling and measuring soil water content Ž u , m3 my3 . across the landscape for a variety of applications. For example, u exerts a strong control on the rates of evaporation and transpiration, resulting in a major impact on the partitioning of incoming solar radiation which, in turn, provides an important feedback between the land surface and the atmosphere Ž Avissar, 1995. . Soil water content also has a major impact on hydrologic processes such as groundwater recharge, infiltration and overland flow; thus, there is great interest among hydrologists in watershed–scale u estimates ŽWood, 1995.. In pedology, u affects soil patterns across the landscape in numerous ways such as, for example, by influencing leaching rates and oxygen concentrations Ž Nielsen et al., 1996. . All these applications involve modeling and measurement of u at varying spatial scales. The term ‘scale’ has been used differently in cartography, hydrology and soil science. In the context of this paper, the following three components of scale are recognized: extent, spacing and support Ž Bloschl, 1996.. Extent refers to the size of the area of interest, spacing to the distance between samples, and support to the measurement area. In this paper, scale refers primarily to extent. That component is the primary impetus for this research, as we are interested in applications over relatively large areas. The distinction with the other components will be made where appropriate because the effects of changing scale depend, to some extent, on which component of scale is changed. Measurement and modeling techniques of u have traditionally focused on the point to roughly 1 m2 scale. Soil water content has been measured with techniques such as time domain reflectometry ŽTDR. , neutron probe, or gravimetric analysis, which can provide very accurate information with depth over time, but with a measurement volume Žsupport. that is less than 1 m2. Similarly, most soil water models have been developed to describe one-dimensional soil water dynamics at a point, often applied to experimental plots, over time. Application of these models to larger areas requires that they be extended or extrapolated, which has been called ‘upscaling’ by some Ž e.g., Bloschl and Sivapalan, 1995. . Upscaling soil water models would be straightforward, were it not for the large spatial variability of relevant soil properties observed in nature ŽNielsen et al., 1973; Jury, 1985.. The effect of high spatial variability, combined with a relatively small measurement support, is that model input values, such as soil texture or depth, as well as u itself, are known only with some considerable uncertainty. Uncertainty in parameter values is transmitted throughout the model to the final output Že.g., Rao et al., 1977.. Uncertainty in u limits model calibration and verification accuracy. Both of these uncertainties compromise the quality of soil water model outputs. Much more attention has been devoted

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toward determining how processes should be represented in soil water models than to how the spatial variability and distribution of those processes should be represented, although both are critical. This is partly because information concerning the effect of scale on spatial variability is rare. Two kinds of spatial variability have been widely recognized, deterministic and stochastic ŽPhilip, 1980; Rao and Wagenet, 1985; Seyfried and Wilcox, 1995.. Deterministic variability has also been labeled systematic Ž Wilding and Drees, 1983. and organizational Ž Bloschl and Sivapalan, 1995. . It conveys that spatial variability is known. This may be expressed in the form of a map or mathematical relationship with spatial data, such as the distance from a stream or elevation. Stochastic variability is random. It may be further subdivided into spatially dependent and spatially independent variability. Spatial dependence is generally quantified by a variogram, and indicates that samples spatially closer to one another are more similar than those farther apart. Spatially independent variability includes or is indistinguishable from, measurement imprecision Ž assuming no bias., which is usually relatively small, small-scale Žrelative to the measurement scale. variability and other ‘unexplained’ variability. Where the deterministic component can be described as a function of position or is constant, the relationship between these parameters is summarized with the following equation: Z Ž x . s mŽ x . q ´ X Ž x . q ´ Y Ž x .

Ž1.

where x is the position in 1, 2 or 3 dimensions, ZŽ x . is the parameter value Že.g., u ., mŽ x . is the deterministic component, e X Ž x . is the stochastic, spatially dependent component, and e Y Ž x . is the spatially independent component ŽBurrough, 1993.. Other deterministic variability, such as that described by maps, may not be described as a mathematical function of x as in Eq. Ž1., but may be thought of as being partitioned as in Eq. Ž 1. . Although it has been generally observed that the spatial variability of soil properties increases with scale ŽBeckett and Webster, 1971; Wilding and Drees, 1983., neither the amount of increase nor how it is partitioned among the three terms in Eq. Ž 1. has been widely reported in the literature. It is generally expected that e Y Ž x . will increase with scale because the number of interactions or unexplained processes probably does so Ž Wilding and Drees, 1983. . It has been proposed that, at some scale, termed the representative elementary area, this term is dominant, and spatial variability may be viewed as strictly random ŽWood et al., 1990. . At this scale, spatial data Ždata tied to spatial coordinates. is not required, and spatial variability may be portrayed by statistical parameters such as the mean and standard deviation Ž s .. The magnitude of e X Ž x . is also expected to increase with scale, although this will be highly dependent on the description of mŽ x . and the sample spacing Že.g., Russo and Jury, 1987.. The deterministic component may also change considerably with scale and location

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ŽSeyfried and Wilcox, 1995. . When this component is relatively large, spatial data is required, and distributed modeling approaches should be used. Some have expressed the view that the application of new technologies, particularly the merging of remote sensing and geographical information systems, will supplant the need for upscaling models. While these technologies can provide very useful information that greatly facilitate upscaling, soil water models are still required to produce reliable u estimates. Remote sensing has the great attribute of providing synoptic, spatially distributed information. This can be in the form of model input parameters, such as leaf area index ŽPrice, 1992., or a direct measure of u . The direct measurements may be useful, but are, at best, surface Ž0–2.5 cm. measurements subject to a variety of interferences that are difficult to quantify Ž Jackson et al., 1996.. Also, remote sensing can provide only a temporal snapshot, whereas temporally continuous information is much more useful. Thus, even with these technologies, models are needed to temporally interpolate data and extend it to more meaningful depths. This study is part of a larger project aimed at modeling soil water content, stream flow, and ultimately plant production, on semiarid rangelands. These lands, by virtue of relatively low productivity per unit area, require aerially extensive management. The specific objectives of this paper were to: Ži. quantify spatial variability of u over a range of spatial scales, Žii. describe the partitioning of deterministic and random spatial variability as it is affected by scale, and Žiii. relate the implications of the results from the first two objectives to modeling u at different scales. The focus is on the variability of u , as opposed to the parameters used to calculate it, such as hydraulic conductivity or soil water–tension relationships, because the spatial variability of u is less well known, and because it represents a more fundamental, model-independent, constraint. In describing the partitioning of spatial variability changes with scale, we consider the primary distinction to be that between random and deterministic variability, partly due to data limitations and partly due to the apparent nature of spatial variability at the watershed.

2. Materials and methods This work was conducted in the Reynolds Creek Experimental Watershed in southwest Idaho, USA ŽFig. 1. . The watershed exhibits a range in climatic and soil conditions that are typical of many rangelands in the western USA at a scale consistent with the management of those lands. In particular, the elevation range and the associated variability in climatic, vegetative and soil conditions are of interest. The watershed is 234 km2 in extent, and ranges in elevation from 1097 m to 2237 m. The geology at the upper elevations is a mixture of basalt, granite and tuff. At lower elevations, large areas of lacustrine sediments are also

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Fig. 1. Reynolds Creek Experimental Watershed. Elevation contour intervals are 200 m. The boxes indicate the location of the six sampling sites used for the overall Reynolds Creek sampling. Reynolds Creek runs northward roughly through the middle of the watershed.

present. Different species of sagebrush Ž Artemesia spp.. dominate the vegetation throughout the watershed. These are associated with saltbush Ž Atriplex confertifolia. and greasewood Ž Sacrobatus Õermiculatus. at lower elevations, and Douglas Fir Ž Pseudotsuga menziesii . and quaking aspen Ž Populus tremuloides . at the higher elevations. Mean annual precipitation increases with elevation from 229 to 1107 mm. Most of the precipitation in the higher elevations comes as snow. Summers are very dry at all elevations ŽFig. 2. . 2.1. Soil water content measurements The data reported were collected as part of three independent studies designed to examine u at different scales. The smallest scale study is referred to as the plot, the intermediate scale studies were at Lower Sheep Creek Ž LSC. and Upper Sheep Creek Ž USC., and the largest scale study was for all of Reynolds Creek ŽRC.. Each of these studies is described in detail below. The location and methodologies are summarized in Table 1.

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Fig. 2. Average monthly distribution of precipitation within Reynolds Creek at three locations representing different elevations. Elevations above mean sea level are: Reynolds Mountain, 2097 m; Democrat, 1649 m; and Flats, 1184 m ŽFig. 1.. Democrat is located near the 1600 m contour line just west of Reynolds Creek.

2.2. Plot Shrubs at Reynolds Creek rarely cover more than 60–70% of the soil surface and more commonly cover 40–50%, the remainder being largely bare ground with a scattering of grasses and forbs. The shrubs are more or less uniformly distributed and separated by interspace areas, both having spatial dimensions of about 1 m2. The plot measurements were designed to determine the difference in soil water content and soil temperature between soil within the shrub canopy and bare soil in adjacent interspace areas. It has been widely observed that infiltra-

Table 1 Experimental conditions associated with u data collection

Extent Žm2 . Sampling Number Sample dates Measurement depth Žcm. Method

Plot

Lower Sheep Creek

Upper Sheep Creek

Reynolds Creek

12 Selected 12 37 0–10 TDR

1.3=10 5 Grid 83 5 0–30 TDR

2.6=10 5 Grid 104 5 0–30 TDR

2.3=10 8 Transect 176 4 0–5 Gravimetric

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tion rates under the shrub canopy is much greater than in interspace areas ŽBlackburn, 1975; Johnson and Gordon, 1988; Seyfried, 1991. , and that soil temperature fluctuations are modified considerably under the shrub canopy ŽPierson and Wight, 1991.. Here, we describe the soil water data that illustrate the effect of those processes on spatial variability of u at a relatively small scale. The plot site is located within the Lower Sheep Creek subwatershed, which is described in Section 2.3 Ž Figs. 1 and 3. . Soil water content measurements were

Fig. 3. Lower Sheep Creek subwatershed. Elevation contours are at 2 m intervals. The TDR grid shows the locations of the LSC–scale u measurements. The SAR transect lines indicate the location of the gravimetric sampling at LSC as part of the overall Reynolds Creek sampling. The plot location is also indicated.

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made with a ‘Trase 1 ’ time domain reflectometry ŽTDR. unit. Six, three-rod waveguides were placed under different shrubs, and six more were placed in adjacent interspace areas. The rods were 20 cm long and inserted at a 458 angle to make an effective measurement depth of 0–10 cm. All the measurement sites were located within a 2-m radius. Monitoring was done 36 times, about every two weeks, from October 1, 1993 to April 13, 1995. The effective soil dielectric constant– u relationship for this soil was shown to conform closely to standard calibration equations Že.g., Topp et al., 1980. in a recent study Ž Seyfried and Murdock, 1996. ; therefore, no special calibration was performed. This relationship was also applied to TDR readings at other sites. 2.3. Lower Sheep Creek The Lower Sheep Creek subwatershed is 13 ha in extent and ranges in elevation from 1588 to 1658 m with an average slope of 8.68. There is one soil series mapped in the subwatershed Ž Stephenson, 1977. , the Searla silt loam Žloamy–skeletal, mixed, frigid Calcic Argixeroll. . The vegetation is dominated throughout by low sage Ž Artemesia arbuscula.. Measurements were taken on an 83-point, 30 = 60 m grid ŽFig. 3. on five dates ŽApril 17, 1990; May 14, 1990; July 19, 1990; April 2, 1991; August 6, 1991., resulting in three spring and two summer dates. These dates were selected to encompass a wide range of soil water conditions. Readings were made with TDR using two stainless steel rods inserted vertically 30 cm into the soil. 2.4. Upper Sheep Creek The Upper Sheep Creek subwatershed is approximately 26 ha in extent with an elevation range of 1848 to 2023 m and an average slope 15.68 ŽFig. 4. . The soils map contains two soil series differentiated by slope and coarse fragment content into five different mapping units Ž Stephenson, 1977. . Harmehl silt loam Žfine–loamy, mixed Argic Pachic Cryoboroll. is deep Ž) 2 m., silt loam textured, with very low coarse fragment content. Gabica gravelly loam Ž loamy– skeletal, mixed, frigid Lithic Argixeroll. is shallow with a high coarse fragment content. The Gabica soil is found on ridges and west facing slopes, the Harmehl on north facing slopes and near the channel. The three vegetation types apparent in the subwatershed are low sage, mountain sagebrush Ž Artemesia tridentata Õasayena. and quaking aspen. 1

Mention of manufactures is for the convenience of the reader only and implies no endorsement on the part of the author or USDA.

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Fig. 4. Upper Sheep Creek subwatershed. Elevation contours are at 12 m intervals. The TDR grid shows the locations of the USC–scale u measurements. The SAR transect lines indicate the location of the gravimetric sampling at USC as part of the overall Reynolds Creek sampling.

Soil water content measurements were made at 104 points with 30 cm long TDR rods in a 30 = 60 m grid, which covered the lower portion of the subwatershed ŽFig. 4. . Each of the sampling dates immediately followed those at LSC. 2.5. Reynolds Creek The largest area sampled was the entire Reynolds Creek watershed. Samples were collected from the following sites: Summit, Flats, Nancy’s Gulch, Lower

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Sheep Creek, Upper Sheep Creek, and Reynolds Mountain Ž Fig. 1. . The sites were selected to provide a range of soil water conditions in conjunction with an investigation of the application of synthetic aperture radar Ž SAR. to mountainous terrain. Samples were collected from a depth of 0 to 5 cm, placed in a can, sealed and transported to a lab, where they were weighed before and after oven drying. Soil water content was converted to a volumetric basis using the bulk density. Sampling was done in 200 to 400 m long transects with 10 to 20 m sampling interval depending slightly on the location. The transects at Lower Sheep Creek ŽFig. 3. and Upper Sheep Creek ŽFig. 4. are included as an illustration. Samples were collected on four dates, September 10, 1989, March 19, 1990, May 3, 1991 and April 19, 1994, resulting in three spring and one late summer collection dates. The Reynolds Mountain site was sampled only once ŽSeptember 10, 1989., because deep snow prevented sampling on the other dates. 2.6. Remote sensing A Landsat Thematic Mapper image, acquired August 1, 1993, with a spatial resolution Žsupport or pixel size. of 30 = 30 m, was used to provide a measure of vegetative density distribution throughout Reynolds Creek. Soil water content, or at least seasonal soil water availability, is directly linked to vegetation in this predominantly semiarid climate because available water is the primary edaphic factor limiting plant growth Ž Sneva and Hyder, 1962; Le Houerou et al., 1988.. Patterns of vegetation density are temporally stable in this area. Vegetative density, or ‘greenness’, as quantified by the leaf area index, has been successfully measured with remote sensing using vegetation indices Ž Huete, 1988; Price, 1992.. We used the soil adjusted vegetation index Ž SAVI. , developed by Huete Ž1988. specifically for regions of sparse vegetative cover to account for the soil signal. It is defined as: SAVI s

Ž NIR y RED. Ž NIR q RED q L .

Ž L q 1.

Ž2.

where NIR is the thematic mapper near infrared band, RED is the red band and L is a constant. We used L s 0.5 as recommended by Huete Ž1988. . The SAVI was calculated using apparent reflectance values and a correction for the solar angle. 2.7. Statistical description of spatial Õariability Elementary statistics were used to illustrate how spatial variability of u at different scales affected measurement precision, which for these examples, approximates the measurement accuracy, since the bias should be very small.

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This affects modeling via the calibration and verification accuracy. Probably the most straightforward statistic for normally distributed, random populations, is the confidence interval about the mean Ž CI., which is defined as: s CI s x " t a, n Ž3. 'n where x is the estimated mean, ta , ny1 is the value of the t-statistic at the Ž1 y a . confidence level with n y 1 degrees of freedom, n is the number of samples measured, and s is the estimated standard deviation. The product to the right of the " will be referred to as the CI and related to the measurement precision or accuracy. The a used is 0.05, so that the confidence interval is for 95%. An alternative expression used in sampling design is to consider the number of samples required to achieve an estimate within a specified difference from the mean given a known or estimated population s, is: n s Ž ta2 , ny1 s 2 . rD 2

Ž4.

where D is the difference from the mean considered acceptable. Eqs. Ž3. and Ž 4. are interchangeable when D equals the confidence interval and the same t and s are used. In both Eqs. Ž3. and Ž4. , the measure of dispersion, or spatial variability, is the s. This is often modified by dividing by the mean to produce the coefficient of variation ŽCV.: CV s

s

ž x / 100

Ž5.

This is a dimensionless statistic that is used for comparing different kinds of measurements or to examine the amount of variation relative to the mean. In this paper, the CV was used to relate our results to other studies and variables.

3. Results and discussion The s for the plot, LSC, USC and RC, is plotted as a function of the average soil water content at each scale Ž uavg . to illustrate the effects of scale on measurement precision ŽFig. 5.. The use of uavg for the x axis is convenient for comparing data collected on different dates, while still giving an indication of seasonal trends, which will be discussed shortly. Standard deviation, as opposed to other measures, particularly the CV, was used because it displays clear trends that are directly related to the measurement precision as defined by the CI. The CI was not presented because the sample size varied considerably among scales ŽTable 1..

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Fig. 5. Spatial variability, as measured with the standard deviation, at the four different experimental scales as a function of uavg .

An obvious trend is that s decreased with uavg . This held at all scales, sites and sampling depths. The same trend has been observed at other locations ŽWood, 1995. under very different conditions. It should not, however, be regarded as a universal trend, since others have observed the minimum variability at high uavg ŽDunin and Aston, 1981.. This may be most apparent where high u , characteristic of poorly drained soils, is common Že.g., Seyfried and Rao, 1991.. Another obvious trend is that s increased with scale. This observation is complicated by the fact that measurements at the differing scales were made at different depths. Ideally, the sampling depths would be identical for all scales. They were not because each study was conducted for a different purpose. In general, it might be expected that spatial variability would increase with decreasing sampling depth, although there does not appear to be any literature to that effect. In order to determine if there was a sample depth effect in the data, we compared the standard deviation of u as a function of uavg for samples taken from different depths at LSC and USC. At those locations, the extent, spacing and support were similar, so that the only difference was the sampling depth. At LSC, for a given uavg , the s of the shallow Ž5 cm. measurements was slightly higher than that of the deeper Ž 30 cm. measurements, except during the driest sampling date, when they were the same. At USC, the shallow measurements had a lower s than the deeper ones when in the same uavg range. From this, we concluded that the standard deviation of the different depths is approximately equal, and it is valid to make cross-scale comparisons.

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3.1. Plot Soil water contents in the shrub and interspace areas were generally in close agreement throughout the year ŽFig. 6.. The low u values in the summer months corresponded with the dates of the lowest s values. The s of the lumped interspace and shrub measurements at the plot scale ranged from 0.0039 to 0.0477 m3 my3, which corresponds to a CV range of 7 to 35%. These CV values are similar to what have been reported for some physical properties such as sand content or u at a given soil water tension, and lower-than-physical properties related to water movement, such as the hydraulic conductivity for a soil series ŽWilding and Drees, 1983.. The 95% CI for the median s Ž 0.019 m3 my3 . was "0.0013 m3 my3, and that for a relatively high s Ž0.03 m3 my3 . was 0.019 m3 my3. Thus, one can expect to measure u with "0.02 m 3 my3, in a 12-m2 area using 12 measurements. About half of that variability may be attributed to measurement imprecision Ž Seyfried and Murdock, 1996. . Stratifying this variability by cover type Žshrub or interspace. did little to reduce the CI. The large difference in the means of the two cover types in December and January of 1993 was accompanied by high spatial variability, and the u for the two cover types was significantly different Ž a s 0.05. on only one of the 36 sampling dates. Under proper snow cover and air temperature conditions, the freeze–thaw dynamics and surface runoff generated under the cover types are quite different ŽFlerchinger and Seyfried, 1997. . The interspace areas, lacking the insulation of shrub cover and receiving direct solar radiation, tend to thaw and lose water to runoff during short warming trends. Since TDR

Fig. 6. Soil water content, as stratified by position relative to the shrub cover over a 19-month measurement period at the plot scale.

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measures the liquid, unfrozen soil water content, this was reflected in the early u measurements. These differences were not evident during warm seasons, or when the freezing and thawing were more uniform Ž e.g., during very cold weather., such as was apparent in 1995. The high degree of uniformity observed during the summer months was achieved as transpiration reduced u everywhere to near the wilting point, regardless of the amount of water lost to runoff. These data show that there is considerable spatial variability and process complexity within a few meter area. It may be critical for models operating at scales on the order of m2 to explicitly describe this variability. However, for larger scale models, this level of detail is impractical. Most larger scale models utilize spatial units or cells on the order of 900 m2 Ž digital elevation model resolution. or larger. The variability introduced by shrub–interspace interactions would be unresolved at that scale, and part of the random Ž unexplained. variability component. This level of spatial variability in u has been observed elsewhere Ž e.g., Rajkai and Ryden, 1992. , so that it is reasonable to consider it to represent an approximate minimum. 3.2. Lower Sheep Creek The expected increase in s with scale was not observed when moving from the plot to LSC as both the magnitude of s and the dependence of s on uavg were very similar for the entire LSC subwatershed and the plot Ž Fig. 5. . This indicates that most of the variability within this soil occurs in relatively small Ž- 12 m2 . scales. This is not necessarily surprising, in that relatively large amounts of variability have been observed elsewhere at small scales for other soil properties ŽBeckett and Webster, 1971; Wilding and Drees, 1983; Burrough, 1991.. Examination of the spatial distribution of u indicated no clear pattern, although some trends with topography might be expected. Thus, variability of u within LSC can be represented with non-spatial data such as the mean and s. This is consistent with the soil map, which represented the subwatershed with a single soil series. The dates of the two lowest s values were the two summer measurement dates. The increase in s with uavg may result from differential water inputs due to patterns of snow accumulation and runoff that occur at small scales. 3.3. Upper Sheep Creek As in the previous examples, there was an increase in s with uavg in USC ŽFig. 5.. In this case, however, s diverged from the plot and LSC at higher uavg . The increase in s, roughly from 0.04 to 0.08 m3 my3, substantially reduced u estimate precision. If, for comparative purposes, a sample size of 12 and a of 0.05 are applied to Eq. Ž3., the CI doubles, from "0.025 to 0.051. The full

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confidence interval range, considering both sides of the mean, is 0.102 m3 my3, or about 1r3 of the annual range in u . Either a substantial increase in sampling intensity must take place Ža sample size of 42 leads to a similar CI., or the calibration andror verification accuracy must suffer. The analysis above assumes a random distribution of u . Consistent with Eq. Ž1. , the additional variability observed at USC may be attributed to a deterministic trend, spatially dependent variability, or additional random variability. At USC, two distinctive soil series are mapped, the Gabica and Harmehl. The average u for the Harmehl soil was significantly greater than for the Gabica soil on all three high uavg sampling dates Ž Fig. 7. . The resulting within-soil series s values were only slightly greater than for LSC and the plot, ranging from 0.035 to 0.057 for the three relatively high uavg dates. Thus, most of the increase in spatial variability from LSC to USC may be attributed to the addition of a contrasting soil series. A major reason the two soils have such different u ’s is that they receive considerably different amounts of input water Žsee Cooley, 1988. . Harmehl soils are found on leeward slopes, and receive much more snow, due to deposition by the wind, than the Gabica soils, which are found on windward slopes and tend to lose snow by wind scouring. When the highest u values were measured Žspring., snow had only recently melted off the Harmehl soils, but the Gabica soils had been snow-free for more than a month, and were therefore much drier. As the year progressed, greater transpiration rates from the Harmehl soils Ž see Flerchinger et al., 1996. led to an equalization of u between the two soils. Thus, as the growing season progressed, and uavg decreased, the amount of spatial

Fig. 7. Effect of soil series on u at USC. Vertical bars represent the 95% confidence interval about the average for both soil series ŽHarmeal and Gabica..

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variability decreased, rendering more precise overall u estimates. In addition, the kind of variability changed, from being strongly determined by soil series, to a more nearly random pattern. Note that u distribution, while being strongly related to topography, does not vary in the manner commonly assumed in topographic indices ŽBeven and Kirkby, 1979. . In this case, the wettest soils are in the highest snow deposition zone near the top of the leeward slope. Since soils were significantly different in terms of u , it is logical to consider them as a basis for modeling. There is ample precedent for this, but there is another important criterion; are the differences significant in terms of the processes of interest? In this case, do soil differences affect the amount of vegetation present? Field inspection and SAVI data Ž Fig. 8. demonstrate that there is a dramatic difference in vegetation on the two soils. This is consistent

Fig. 8. Soil adjusted vegetation index for Upper Sheep Creek. Lower values indicate sparse Ž - 40%. vegetative cover of small Žlow sagebrush. shrubs. Higher values represent nearly complete canopy closure of large shrubs or, in the highest examples, aspen. Note that the SAVI trends closely match the soil map, with lower values associated with the Gabica soil, and higher values associated with the Harmeal soil.

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with other studies that have found that u has a direct effect on the vegetation measured using vegetation indices determined from remote sensing of semiarid terrain ŽGrist et al., 1997.. In addition to input differences, the Harmehl soil is deeper and much lower in rock content, increasing the water holding capacity, and therefore the seasonally available soil water. The SAVI image clearly shows a dramatic difference between the soils, indicating a range of vegetative cover almost equal to the range of that observed over the entire Reynolds Creek watershed. The change is quite abrupt at a 30-m scale, which is the size of the individual cells in Fig. 8. Such abrupt, irregular changes are best described in terms of discrete mapping units. It is notable that the SAVI for LSC, which has only one soil mapping unit, was virtually uniform Žrange of 0.135–0.213. . 3.4. Reynolds Creek Interpretation of the data collected for the entire watershed is hampered by the sampling design, which was neither random nor systematic, but rather from areas chosen to represent different conditions at the time of the SAR overflights. The highest elevations were usually not represented, because three of the overflights took place in the spring, when there was considerable snow cover at high elevations. The data are informative, however, because they give an indication of the magnitude of spatial variability introduced when the measurements extend to the much larger areas that managers typically must contend with. The magnitude of s was much greater for RC than for the other scales except when uavg was relatively dry Ž at the end of summer. , when it was similar to the other scales. Assuming that the variability is random and using a representative value of 0.17 m3 my3 for s, and a s 0.05 with 12 samples as before, the calculated CI is "0.108 m3 my3. This 0.21 m3 my3 range in values is close the entire range observed during the year. A constant estimate of about 0.22 m3 my3 would be within that CI virtually all the time. Practically, any model should meet that criteria! In order to bring the confidence interval back to "0.025 m3 my3, as at LSC, a sample size of about 175 would be required. Following previous reasoning, we might attribute the increased variability to either deterministic or random variability. An additional deterministic source of variability that must be considered at the Reynolds Creek scale is that of climate as influenced by elevation changes. In addition to the dramatic differences in precipitation Ž Fig. 2. , temperature also changes considerably within the watershed, and is warmest where it is driest, at the lower elevations. This was not an important effect at the other scales, which did not have enough elevation change to affect the climate. These climatic trends were reflected in the u . We found that, in general, u increased with elevation except on the driest sampling date Ž Fig. 9. , which is consistent with the overall trend of low s with low uavg . Although each site has a

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Fig. 9. The effect of elevation on u for each year sampled over Reynolds Creek.

different soil as well as elevation, the strong trend with elevation ŽFig. 9., and the consistency with other studies Ž e.g., Rawls et al., 1973. , points to the importance of elevation. For example, in a previous study, Rawls et al. Ž 1973. examined u variation with time at three sites, Summit, LSC, and Reynolds Mountain over a 5-yr period and found that the maximum soil water content at the Summit Žlowest elevation. occurred in February, while that at Reynolds Mountain Žhighest elevation. occurred in late May. The maximum value for LSC Žintermediate elevation. occurred between the other two. Thus, during spring, we expect an increase in u with elevation. Although each site reached minimum values at different dates, they were all at the minimum in mid-September, which is when the driest sampling occurred. These climatic considerations also explain why the spatial variability varies with uavg . In spring, when uavg is high, the variability is also high due to differential melting times, precipitation inputs and evaporative demand. After July and August, however, which are warm and dry throughout the watershed, these differences are eliminated. Soils are dry everywhere, and spatial variability is minimized. In some cases the added, larger-scale variability introduced with elevation may overwhelm the effects of smaller-scale variability Že.g., soil series. . For example, the amount and timing of streamflow at Reynolds Creek is very strongly dominated by elevation effects Ž Seyfried and Wilcox, 1995. . This does not appear to be the case for u and elevation, however. Extending the qualitative relationship between u and SAVI observed at USC to all of Reynolds Creek, we see that, although there is a significant correlation between the two Ž r 2 s 0.40., there is a great deal of scatter associated with the relationship Ž Fig. 10. . This is

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Fig. 10. Soil-adjusted vegetation index ŽSAVI. for each Landsat pixel in Reynolds Creek excepting irrigated, cultivated portions, plotted as a function of elevation.

further illustrated in Fig. 11, which shows that there is extremely high vegetative contrast at most elevations as was the case at USC Ž Fig. 8. . Note that the data in Figs. 10 and 11 do not include the cultivated, irrigated valley, which has a very high SAVI that is not directly related to soil or native vegetation. 3.5. Constraints to modeling These data clearly demonstrate that soil water modeling, whether mechanistic in nature or empirical, must, to some extent, spatially aggregate soils and the processes that affect u . It is essentially impossible to explicitly represent all the complexity of soil water distribution, as affected, for example, by shrub–interspace interactions Ži.e., infiltration and runoff. at the scale of Reynolds Creek. Even if the computing power were available, the data on shrub and interspace dimensions and distribution would be neither available nor useable. Therefore, the output of a soil water model, even for a 12-m2 area, represents an aggregation of processes, as well as u and will always be associated with some uncertainty. These data also show that the scale at which the aggregating process is carried out is not arbitrary, but that there are identifiable patterns that, to some extent, constrain our choices. An increase in scale of about 10 4 , from the plot to

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Fig. 11. The spatial distribution of the soil adjusted vegetation index ŽSAVI. for all of Reynolds Creek excepting the irrigated, cultivated areas Žhatched.. Mapping scale is 1:112,000, and the orientation is due north.

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LSC scale resulted in a very small Ž- 25%. increase in s and corresponding CI. However, only a doubling of scale from USC to LSC produced a doubling of s. A further increase in scale of about 10 3, from USC to RC, yielded a further increase in s of 3- to 4-fold. The increase in s is therefore not continuous with scale but occurs in ‘jumps’. These jumps in s were accompanied by identifiable, deterministic ‘sources’ of variability. Information is lost when aggregates are made across these jumps in variability. This goes beyond the loss of precision described above. The resulting ‘picture’ of u distribution on the landscape can become completely distorted. Consider the implication of a random distribution of u on the landscape portrayed in Fig. 11. The resulting interspersion of high SAVI, forested cells with low SAVI desert scrub cells, independent of elevation or position on the slope, is completely unrealistic. This loss of information may be acceptable for some modeling objectives and scales, such as global or mesoscale climatic modeling. However, in the context of hydrologic and vegetation resource modeling at the scale of Reynolds Creek, it is not. It provides no information about when, where and how much vegetation is produced, or runoff is generated. Given these constraints, some basis is needed for spatially distributing andror lumping landscape properties to represent spatial variability. Barring extremely intensive sampling, which would be required to elucidate the patterns evident in Fig. 11, it is not clear how this should be done. We have shown that soil series are useful to this end under some circumstances, but soil maps suffer some problems. For example, the soil mapping scale may be inappropriate, so that functionally important features are lumped with unimportant features, or mapped pedogenic differences may not have functional significance, thereby creating numerous irrelevant delineations Ž we lumped mapping units into common soil series at USC.. For these reasons, other surrogate attributes have been used. Topographic indices, which are based on assumed relationships between slope, landscape position and u , have been widely used towards this end Ž Beven and Kirkby, 1979; Burt and Butcher, 1985. . In general, topographic indices may be expected to be most effective when soils are relatively moist, and when there are contrasting soil horizons parallel to the slope. These expectations are supported by observations ŽBurt and Butcher, 1985; Grayson et al., 1997. . Except at the highest elevations, these conditions are not common at Reynolds Creek, and current topographic indices do not appear to apply over most of the watershed, as noted at LSC and USC. We have shown, qualitatively, that SAVI may be useful in this context. It has at least four advantages in arid and semiarid landscapes. First, it can be directly linked to vegetative parameters, such as LAI, of interest. Second, it Žthe imagery. is available at a practical scale. Landsat is available at 30-m resolution, which is appropriate in terms of the aggregating scale of u . Third, it seems to be

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closely linked to soil properties, which is important because modeling will require soil information. Finally, SAVI patterns portray, at least qualitatively, if the changes in u are effectively discrete or continuous. Considered as a whole, the conceptual model of the scale–spatial variability relationship that emerges is as follows. There is a small, practically irreducible amount of spatial variability present at relatively small scales, which will be unresolved in large-scale models. This is approximately equal to the variability found within soil series. It appears to pertain to all soils in the area. There is an additional, elevational component, which becomes apparent at scales much larger than the soil series delineations, but well within the Reynolds Creek scale. The spatial variability of u can be effectively minimized in spatially distributed models that explicitly incorporate those components. The impact of these deterministic sources of variability is greatest in the spring, when soils are relatively moist. As the growing season progresses and the soils dry, both soils and elevational effects are reduced to the point that, by the end of summer, they are practically eliminated, and the entire watershed variability is similar to that of a single soil series. The discussion above pertains directly to the Reynolds Creek Watershed. Although it almost certainly applies to a much larger region, it is far from universal. Spatial variability–scale– u relationships are landscape dependent. This may be the most important constraint to modeling. Since no model will represent all of the complexity of nature, it must crudely represent those aspects most critical to the processes of interest. These will be different in different landscapes and scales. Therefore, model applicability depends not only on the processes simulated, but also the approach to incorporating spatial variability. This may explain why many models seem to work best in the landscape they are developed in. The representation of spatial variability should be an explicit part of model development and application.

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