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Agronomy Journal 92:706-714 (2000)
© 2000 American Society of Agronomy

SPATIAL VARIABILITY

Assessing Spatial Variability in an Agricultural Experiment Station Field

Opportunities Arising from Spatial Dependence

D.K. Cassela, O. Wendrothb and D.R. Nielsenc

a Dep. of Soil Science, North Carolina State Univ., Raleigh, NC 27695-7619 USA
b Institut für Bodenlandschaftsforschung, ZALF, Eberswalder Str. 84, 15374 Muncheberg, Germany
c Dep. of Land, Air, and Water Resources, Univ. of California, Davis, CA 95616 USA

keith_cassel{at}ncsu.edu


   ABSTRACT
TOP
 NOTES
 ABSTRACT
INTRODUCTION
 Methods and materials
 Results and discussion
 Conclusions
 REFERENCES
 
Spatio-temporal field soil and crop processes are important for site-specific farming. The objectives of this study were to spatially evaluate selected soil physical and chemical properties and their relationship to wheat (Triticum aestivum L.) yield, and to discuss stochastic approaches to help identify processes underlying yield variability in heterogeneous field sites. Modified grid sampling included 330 sites including a primary transect. Soil properties measured for the Ap, E if present, and upper B horizons at each site included pH, P, Zn, Cu, exchangeable cations, percentage base saturation, cation exchange capacity, bulk density, soil water contents at -10, -33, and -1500 kPa, texture, and humic matter content. Wheat grain and straw were hand-harvested on 1- by 2-m plots centered on each site. Soil water content on the primary transect was determined by neutron attenuation on nine dates. Field and primary transect means and semivariograms for a given soil or plant parameter were similar. The range of spatial dependence or autocorrelation of soil parameters ranged from 10 m for Ap horizon depth to 100 m for -1500 kPa water content of the Ap. Base saturation and available water storage capacity were cross-correlated with grain yield to a distance of ±15 and 12.5 m, respectively. State-space analysis was used to develop a grain yield model using these two variables. Spearman rank correlation of the soil water content data suggests that the temporal stability of soil water storage is less for shallow than for deeper soil layers.

Abbreviations: ANOVA, analysis of variance • CCF, crosscorrelation function • CEC, cation exchange capacity • EM, expectation–maximization • NNA, nearest neighbor analysis


   INTRODUCTION
TOP
 NOTES
 ABSTRACT
 INTRODUCTION
Methods and materials
 Results and discussion
 Conclusions
 REFERENCES
 
HISTORICALLY, AGRONOMIC FIELD RESEARCH utilized classical randomized block and similarly designed experiments using different treatments thought to have an impact on crop growth. In order to avoid influences of spatial variability, replicated treatment plots were established in a location as homogeneous as possible. A widely accepted assumption was that existing field soil variability could be compensated for by a large number of replicated plots (Davis, 1986). Accordingly, agricultural research stations were generally established on land comprised of soils typical of a particular region, and of sufficient area to provide adequate homogeneous sites for an anticipated number of experiments. Operationally, before a new experiment was laid out and conducted on a long-established research station, particularly where previous plots had been treated with different treatment levels, the common practice was to crop the site uniformly for one or more years to "remove" the previous treatment effects.

Within the past 15 yr, technologies have been developed to help farmers better manage each of their fields utilizing spatially varying prescription intensities based on localized plant growth requirements or deficiencies (Robert et al., 1995; Mulla and Schepers, 1997; Stafford, 1997). Sophisticated technologies include crop yield monitoring systems (Eliason et al., 1995); site-specific fertilizer applicators (Persson and Bangsgaard, 1999); remote sensing from satellites, aircraft, and tractors; and sophisticated crop and soil sensors (Viscarra Rossel and McBratney, 1997). Recently, agricultural consultants have imported spatial and temporal field data (e.g., crop yield, indirect measures from remote sensing, analyses of soil for plant nutrient availability, etc.) into geographic information systems to produce spatial distribution maps for improved on-farm management alternatives (Mulla and Schepers, 1997). Using those technologies, soil and crop scientists face a dilemma when they wish to conduct research on a field station originally developed for and presently devoted to small, randomized plot experiments. Their dilemma stems from the fact that crop response varies within a field because the underlying crop growth processes and their response to concomitant soil processes are variable in space (Nielsen et al., 1999) and time (Stafford, 1999). Instead of varying treatment levels on replicated small plots, several attributes of the soil as well as the crop may be sampled and observed at different scales of space and time to better understand the local covariance structures of plant and soil attributes (Webster and Cuanalo, 1975; McBratney and Webster, 1983). Spatial domain considerations range from small scales within the root zone of a single plant to those of a small plot, or to much larger farm scales. This requirement causes a reconsideration of management practices on traditional agricultural research stations.

A long-standing requirement for a valid analysis of variance of small plot data is that the observations are independent of each other. With greater recent emphasis placed on regionalized variable analyses, the choice of plot size and sample volume used in a field experiment must not be arbitrary, but rather must be based on measured variance and spatial correlation structure (Warrick et al., 1986; Oliver, 1999). This is in addition to the usual considerations such as farm equipment availability, number of treatments or subtreatments, irrigation system, and land area available for the experiment because of the original layout of the research station. Plot arrangement should also be based on magnitudes of the spatial correlation lengths associated with observations of the soil and plant attributes being investigated.

Our objectives were (i) to evaluate the spatial variability of wheat grain and straw yields and selected soil chemical and physical properties 1 yr following land renovation for expansion of an agricultural experiment station, and (ii) to discuss stochastic approaches to help identify processes affecting the underlying observed yield variability in a heterogeneous field site.


   Methods and materials
TOP
 NOTES
 ABSTRACT
 INTRODUCTION
 Methods and materials
Results and discussion
 Conclusions
 REFERENCES
 
Field Renovation
The Oxford Tobacco Research Station of the North Carolina Department of Agriculture is located in the Piedmont province in Oxford, NC (36°17' N, 78°47' W). The land is gently rolling, well drained, highly erodible, and the soils are dominated by Hapludults and Kanhapludults. Expansion of the research program at the station in the late 1970s and early 1980s required acquisition of additional land to conduct field research. Before and after purchase, but before land renovation, grasses, tobacco (Nicotiana tabacum L.), and small grains were grown on different parts of the tract. Results from several field experiments conducted on the newly acquired 18-ha tract indicated that the land was not as uniform as desired and drainage was inadequate for experimental research plots.

The decision was made in the early 1980s that this tract should be renovated if the area was to be used for field research. With land surveys indicating the depth of cut and fill at various locations, the renovation began in 1982 and was completed in 1983. Renovation included realignment of unpaved roads, land shaping, installation of grass borders and waterways, and installation of underground irrigation lines. Special care was taken to preserve the natural sequence of soil horizonation. For example, when soil was removed from the higher-elevation areas, the A horizon material was stockpiled, soil below the original A horizon removed, and the A horizon material replaced in its original position. Likewise, when soil material removed from areas of higher elevation was used as fill, the topsoil at the site to be filled was removed and stockpiled, the proper depth of fill material added, and the original A horizon material replaced.

Field Experiment
Dolomitic lime (0.5 to 1.0 Mg ha-1) was incorporated in the Ap horizon by moldboard plow on 29 Sept. 1983. Nitrogen, P, and K at 40, 40, and 80 kg ha-1, respectively, were applied and disked in. Wheat, `Coker 747', was drilled at a rate of 67.2 kg ha-1 beginning 19 October and continuing for several days. Nitrogen (31 kg N ha-1) as NH4NO3 was applied on 2 Apr. 1984.

After careful observation of the wheat crop early in March 1984, 330 sampling sites (plots) on a modified regular grid were selected to measure wheat yield and selected soil chemical and physical properties. The coordinates of each site (Fig. 1) were measured with a tape. Three transects with sampling sites spaced 10 m apart ran in the north–south direction. One of these transects is denoted as the primary transect and was later instrumented to monitor temporal changes in soil water content (see Fig. 1). Running in the east–west direction were four transects with sampling sites spaced 10 m apart. Areas with missing points in these transects were occupied by roads or pieces of land that could not be used for small plot research. Additional sites at 20-m intervals were selected between the two sets of transects. The small area in the northwest corner of the tract had one main transect running northeast–southwest and four small transects normal to it. Plots (1 by 2 m) for wheat harvest were centered at each of the 330 sites. The long plot dimension was oriented north–south.



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  		Fig. 1 Locations of sampling sites on the renovated Oxford Tobacco Research Station field. Broken lines indicate the location of old unpaved roads. Plots in the primary transect, in which soil water content was measured with the neutron probe, are indicated by solid circles

 
One soil core was taken at the center of each plot using a truck-mounted hydraulic probe. Each 33-mm-diam. soil core was 42 cm or longer and included the Ap horizon, the E horizon if present, and the upper portion of the B horizon. Soil cores were separated into A, E, and B horizon material, placed in plastic bags, and taken to the laboratory. Each subsample was air-dried and analyzed for selected chemical properties by the North Carolina Agronomic Services Laboratory. Phosphorus, exchangeable cations, and Zn and Cu were extracted with Mehlich-3 extractant (Mehlich, 1984a); exchangeable acidity at pH 6.6 was determined by the method of Mehlich (1976); and humic matter was determined as described by Mehlich (1984b). Humic matter content is a measure of the active organic matter fraction and is typically less than the organic matter content of a soil. Soil pH was measured at a 1:1 soil-to-water mass ratio. Cation exchange capacity was determined by summing the cation concentrations and buffer acidity. The following soil physical properties for each soil core were measured: thickness of the A horizon; depth to the top of the B horizon; sand, silt, and clay content of each horizon by the hydrometer method (Gee and Bauder, 1986); and soil water content of each horizon at soil water pressures of -10, -33, and -1500 kPa (Klute, 1986). Soil bulk density for each horizon was not measured directly, but was estimated as the dry mass per unit volume of air-dried soil material required to fill a small scoop of known volume.

During 13 to 15 June 1984, wheat in each 1- by 2-m plot was harvested to a nominal distance of 5 cm above the soil surface with a hand sickle. Plant material was bagged and placed in a dryer for several days. The grain was threshed and the weights of grain and straw determined. Yields of grain (adjusted to 140 g kg-1 moisture) and straw (oven-dried) were calculated on a per-hectare basis. Clearly, the support size and shape of the soil and plant parameter measurements are different. This is common in field studies.

After wheat harvest, neutron probe access tubes were installed at each site along the north–south primary transect indicated in Fig. 1. A single calibration curve (Van Es, 1987) was used, knowing that the soil variability would introduce a source of error. This is justified in the fact that it is rare to use more than one calibration curve in a field as small as 18 ha. Moreover, the soil water content data were not used in a strict physiological or physical sense, but rather as an indirect measure of spatial differences that are highly associated with other soil differences. Soil water content was determined by neutron attenuation at nine times (Date 1, 11 Aug. 1984; 2, 29 Aug. 1984; 3, 17 Sept. 1984; 4, 4 Oct. 1984; 5, 18 Oct. 1984; 6, 12 Dec. 1984; 7, 8 Jan. 1985; 8, 21 Feb. 1985; 9, 13 Mar. 1985) at depths of 15, 23, 30, 46, 61, 76, 91, and 120 cm.

Statistical Analyses
The structure of spatial variance between observations was derived from the sample semivariogram that was calculated as follows:

(1)
where Z(xi + h) and Z(xi) are observations at positions xi + h and xi, respectively, h is the distance between observations, and N(h) denotes the number of pairs of observations separated by the same lag distance h. The semivariogram is the average variance between neighboring observations spatially separated by the same distance (lag).

The spatial relation between two variables x and y is determined with the crosscovariance function Cxy(h), defined as

(2)
where and are mean values. The normalized crosscovariance function, i.e., the crosscorrelation function (CCF), {gamma}xy(h), used to visualize the spatial correlation between two variables was calculated as follows:

(3)
with the standard deviation {sigma}x and {sigma}y of x and y, respectively. The CCF describes the relationship between two variables where one variable, the tail variable, lags behind the head variable by the lag distance h. For instance, in our case, the CCF between wheat grain yield and base saturation in the Ap horizon indicates over which spatial range a wheat grain yield observation on the average is linked to Ap horizon base saturation.

For the purpose of identifying the spatial association between two or more variables, state-space analysis was applied. In the way it was used here, state-space analysis (Shumway, 1988) is a special kind of autoregression. In the analysis, the state of a variable or a set of variables, a system's state at location i, is considered with respect to the system's state at location i - h, where h = 1, 2, 3, ... n - 1. Soil temperature and water content series (Morkoc et al., 1985), crop yield and soil N status (Wendroth et al., 1992), and lake water storage (Assouline, 1993) processes are now commonly modeled with state-space approaches. The basic equation, the state equation, is

(4)
where Zi is the state vector, i.e., a set of p variables at location i, {Phi} is a p x p matrix of state coefficients indicating the measure of spatial regression, and {omega}i is the uncorrelated zero mean model error (Nielsen et al., 1994). Unlike common autoregressive approaches, in state-space models, the true state of the variable or of the state vector is embedded in an observation equation

(5)
with Yi being the observed vector, related via an observation matrix Mi to the state vector Zi, plus an uncorrelated mean zero observation error vi. In other words, measured values can be considered as an indirect measure of true quantities plus an error term. In the state-space model, a matrix of transfer coefficients is optimized together with measurement and model noise via Kalman filtering using the expectation–maximization (EM) algorithm (Shumway and Stoffer, 1982). These transfer coefficients are an important part of the information that we want to obtain. From the transfer coefficient matrix, we can derive which variables are helpful in describing the spatial process. The measurement and model noise or variance are relevant in the Kalman filter (updating of prediction), as they both determine the relative weight that is given to the prediction or to the observation. If the measurement variance is known to be high while the model variance is small, more weight is put on the prediction, and in cases where the confidence in the data is high, the model may not fully be reflecting the processes. The basic idea is that the model description of the series process is not dominated (or is falsely affected) by noisy data if the noise is due to measurement uncertainty and not to signal. Besides deterministic relations between different state variables, the impact of local conditions that cannot be measured in detail is stochastically integrated with the autoregressive term. It is desirable to identify state variables that explain the process of, for example, crop yield even in cases where we do not have an observation available, and the prediction cannot be updated in the Kalman filter. For numerical reasons, the different variables used in the state-space analysis should be the same order of magnitude. Therefore, base saturation data were divided by 20. For further details, see Shumway (1988).

Cospectral analysis provides an opportunity to learn whether cyclic patterns in two or more soil properties across a field are spatially related to each other. Using the crosscorrelation function from Eq. [3], the cospectrum Co(f) was calculated as

(6)
where f2 is the frequency equal to p-1 and p is the period. This equation is not based on the sample, but on the theoretical crosscorrelation function rxy(h). The period is a characteristic length of waves over which the data fluctuate. Short periods can be due to management operations; long periods, to geologic impact. Coherency, {Gamma} 2(f), a measure of the consistency within designated probabilities at which the two sets of observations are related at different spatial frequencies, provides a statistic limit analogous to the coefficient of determination of a simple linear regression between two variables. Coherency (0 < {gamma}2 < 1) is defined as

(7)
where Q(f) is the quad spectrum defined by Eq. [6] when cosine is replaced by sine, and Sx(f) and Sy(f) are the spectra of the x and y series calculated from Eq. [6] when rxy is replaced by rxx and ryy, respectively. Using Eq. [6] and [7], we applied cospectral analysis to the soil water content data measured along the north–south primary transect at different times and depths by neutron attenuation.

In order to design a sampling regime that allows one to sample soil water content as effectively as possible and still maintain the information about the spatial variability of water content, the temporal stability of spatial soil water content variations can be determined with a set of tools described by Vachaud et al. (1985). First, the relative difference in water content was calculated as follows:

(8)
where at a respective time j, {theta}ij is an individual soil water content measurement at location i, and j is the average soil water content. Moreover, the Spearman rank correlation coefficient was calculated as an indication of temporal stability of the spatial pattern. For further details, see Vachaud et al. (1985).


   Results and discussion
TOP
 NOTES
 ABSTRACT
 INTRODUCTION
 Methods and materials
 Results and discussion
Conclusions
 REFERENCES
 
Mean and Standard Deviation
The mean and standard deviation of the wheat grain and straw yields and selected soil physical and chemical properties based on all 330 measurements throughout the entire field are shown in Table 1 . The mean grain and straw yields were similar, although the standard deviation for straw was 0.31 Mg ha-1 greater than for grain. Mean thickness of the sandy loam Ap horizon was 28 cm, or about 3 cm deeper than typical Ap horizon thickness of soils in the Piedmont region. Water retained by the Ap horizon at soil water pressures between -10 and -1500 kPa was used to estimate plant available water (Cassel and Nielsen, 1986) and averaged 0.14 kg kg-1. When converted to the volume basis, using the mean bulk density value of 1.31 Mg m-3, the plant available water was 0.18 m3 m-3. The observed humic matter content, which corresponds to an organic matter content of about 1% using the modified Walkley-Black method (SCS, 1972), is in the range of values typical for these soils. The CEC of the Ap horizon is sufficient to retain cations from excessive leaching.


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  		Table 1 Field and primary transect statistics for yield and selected soil chemical and physical properties

 
Comparison of data for the whole field with that collected in the north–south primary transect (see Fig. 1) is listed in Table 1. The mean of each parameter using the 47 observations in the transect is essentially the same as the mean for all 330 observations. Compared to the field as a whole, the straw yield was 0.34 Mg ha-1 less than for the transect, the clay content in the upper B horizon was 5 percentage points greater for the transect, and the CEC was 0.5 cmolc kg-1 less for the transect. The variances of the yield parameters for the 2-m2 plots along the transect were slightly greater than for the variance of all 2-m2 plots on the regular grids. In general, the variances of the soil properties were slightly less for the transect. These differences might be due to spatial anisotropy.

Although knowledge of the means and variances of these properties is of great importance in a uniformity trial of this nature, of even more importance is the spatial variance structure of the soil and perhaps yield parameters, and the relationship of yield parameters to other state variables.

Auto- and Crosscovariance Function
Comparisons of the global and directional north–south spatial variance structure for grain yield, two soil physical properties, and two chemical properties are presented in Fig. 2 . The spatial variance for grain yield is similar for both cases (Fig. 2A and 2B) and increases with lag distance. The slope is steeper for the directional semivariogram.



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  		Fig. 2 Sample semivariogram values plotted vs. log distance for wheat grain yield, depth of Ap horizon, -1500 kPa water content, and cation exchange capacity (CEC) and pH of the Ap horizon. Plots A, C, E, G, and I are global, and plots B, D, F, H, and J are in the north–south primary transect

 
The depth of the Ap horizon for the entire field is not correlated beyond a distance of 10 m (Fig. 2C). For the north–south direction, measurements of Ap horizon depth seemed to be correlated to a similar range, although semivariance values increased up to a lag distance of 50 m. The semivariograms for -1500 kPa water content indicate structured variance up to distances of 100 m or greater. The directional semivariogram is not as smooth as the one for the whole field, because fewer points were involved in computing each semivariance value.

The range of spatial dependence for CEC in the Ap horizon is about 50 m (Fig. 2G). A second increase in spatial variance at a lag distance of 140 m indicates the presence of one or more unidentified factors causing a trend. The corresponding structure for CEC exhibited by the directional semivariogram is similar to that for the field, with the trend beginning at a separation distance of 140 to 150 m (Fig. 2H). The semivariogram for pH in the Ap horizon increased with lag distance for both the global and directional cases (Fig. 2I and 2J), but a higher spatial variance was found in the directional plot at greater distances.

In Fig. 3 , values of measured grain yield (A), base saturation (B), and available water storage capacity of the Ap horizon (C), respectively, are shown along the primary transect. The available water storage capacity was calculated as the difference between volumetric water content measured at -10 and -1500 kPa soil water pressure, multiplied by the layer thickness in cm of the Ap horizon.



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  		Fig. 3 (A) Wheat grain yield, (B) base saturation, and (C) available water storage capacity of the Ap horizon across the primary transect (see Fig. 1)

 
In the crosscorrelation function (Fig. 4) , the crosscorrelation coefficient at lag equal to zero is the same as the classical correlation coefficient between two variables. Unlike classical statistical analysis, where that coefficient is the only measure of relation between variables, analysis of crosscorrelation between a variable and another variable that is sampled at neighboring locations with increasing distance provides insight into the spatial covariance structure. This information can help to spatially interpolate and estimate a variable by knowing the magnitude of the neighborhood observations much better than by classical regression only (Davis, 1986; Shumway, 1988). Considering the 95% confidence levels shown in Fig. 4 by the broken lines, base saturation and available water storage capacity of the Ap horizon are crosscorrelated with grain yield over distances of 30 and 25 m, respectively. Although grain yield, base saturation, and available water storage capacity were measured for different support areas or volumes, their ranges of crosscorrelation suggest that the sampling regimes were adequate to study their spatial associations. Other soil properties summarized in Table 1 were also crosscorrelated amongst themselves as well as with grain yield over various other distances. An exhaustive analysis of all of these data is not necessary to illustrate the spatial association of grain yield, base saturation, and available water storage capacity of the Ap horizon using the following state-space analysis.



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  		Fig. 4 Crosscorrelation functions of (A) grain yield vs. base saturation and (B) grain yield versus available water storage capacity for observations taken across the primary transect

 
State-Space Analysis
Having found that grain yield and both base saturation and available water storage capacity were spatially crosscorrelated, we used this information to describe an autoregressive process of two or more variables in space. In our case, this particular process is examined using a state-space analysis that provides a model describing how the state of yield at some location i is related to the system state of yield, base saturation, and available water storage capacity at location i - 1. For the case where all yield observations were considered for the parameter estimation, the 95% confidence interval of estimated yield is shown in Fig. 5A . The quality of estimation decreased when every other yield observation was not considered in the model (Fig. 5B). Especially in the latter part of the transect, from 450 to 550 m, the 95% confidence interval almost never included the observations. Although the number of measured locations is fairly low for this kind of analysis, grain yield was well described with the autoregressive state-space model. For the calculation shown in Fig. 5A, the measurement variances obtained from the EM algorithm optimization for grain yield, base saturation (divided by 20), and available water capacity were 0.09, 0.07, and 0.93, respectively. These optimized values indicate an accuracy of 0.30 Mg ha-1 for yield, 0.26 (multiplied by 20)% for base saturation, and 0.30 cm for available water storage capacity. Except for the available water capacity, model noise exceeded the measurement noise in the calculation shown in Fig. 5B. As an error statistic, the average of the squared deviations between measured and calculated yield was 1.36 and 2.75 for the models underlying Fig. 5A and 5B, respectively.



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  		Fig. 5 State-space model for grain yield (A) considering all observations and (B) considering every alternate grain yield observation

 
Cospectral Analysis
Water content measured by the neutron probe exhibited strong spatial variability and distinctive patterns across the transect within the soil profile during the 1984 to 1985 season. Because water contents were not measured for as many as five consecutive locations, cospectral analysis was made of the water content data for a 420-m segment of the primary transect having only three locations without measurements. On any sampling date, the water content of the topsoil along the transect varied more than 0.15 m3 m-3 with both regional trends and local variations (Fig. 6) . Because the shapes of these four soil water content distributions appeared to be similar to each other, we examined the coherency of the data along the transect for all nine sampling dates using cospectral analysis.



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  		Fig. 6 Soil water content at the 15-cm soil depth measured along the primary transect on four different sampling dates

 
Coherency functions between water content of the topsoil measured along the transect on 11 Aug. 1984 and each of those measured on all nine sampling dates are given in Fig. 7 . The 95% significance level of {Gamma} 2 is 0.91. With a solid circle indicating each date and frequency for which {Gamma} 2 >= 0.91, there is a strong coherence for most periodicities (i.e., separation distances) between the first two sampling dates. Variations of water content occurring at the smallest sampling distances as well as those at much greater distances along the transect were similar for both sampling times during August. However, for the majority of the sampling dates, significant coherencies were manifested for periodicities greater than 50 m associated with variations of slope and soil type.



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  		Fig. 7 Coherency {Gamma} 2 (f or p) between the soil water content of the 15-cm soil depth along the primary transect on August 11 and those measured on all nine sampling dates. Solid circles indicate significance >=95%

 
Average values of soil water content during the season (the mean from all nine sampling dates) for four soil depths are shown in Fig. 8 . Similar to data for the topsoil shown in Fig. 7, the average soil water content for any depth along the primary transect varied more than 0.15 m3 m-3 with both regional trends and local oscillations. The coherency functions between average soil water content of the 15-cm depth (measured along the transect to a distance of 420 m) and the average soil water content at each of the other seven depths are given in Fig. 9 . It is evident that cyclic water content variations in the topsoil are significantly coherent with those at 23 and 30 cm for periodicities greater than 50 m. These results are consistent and help explain the success of estimating wheat yield using state-space analysis that includes available water of the Ap horizon, which has an average thickness of 26 cm along the transect.



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  		Fig. 8 Seasonal average soil water content measured along the primary transect at four different depths

 


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  		Fig. 9 Coherency {Gamma} 2 (f or p) between the seasonal average soil water content of the 15-cm depth along the primary transect and those measured at all depths. Solid circles indicate significance >=95%

 
Temporal Stability Analysis
The mean relative difference across the nine sampling dates and its standard deviation were calculated and shown in Fig. 10 for the 15-, 23-, and 30-cm soil depths. The smaller the standard deviation of mean relative difference, the more significant is the rank of observation. In this case, we could not only find that the rank of a certain observed point differed significantly from others, but throughout the different depths, the ranking of respective locations was similar. This may indicate that a few locations with extreme low, high, or medium ranks may be enough to be measured on a routine basis after the spatial pattern has been determined with a few spatially intensive measurement campaigns. This procedure would give a reliable basis for estimating the change of field water storage and its spatial pattern.



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  		Fig. 10 Temporal stability analysis of soil water content: mean relative difference and its standard deviation for three different depths

 
The Spearman rank correlation technique (Vachaud et al., 1985) was applied to calculate the preservation of spatial ranking with a number integrating the information shown in Fig. 10. In the upper soil depth, relative to the other campaigns, water contents measured during sampling campaigns 4, 5, and 6 were less correlated with those obtained from campaigns 7, 8, and 9 (data not shown). The magnitudes of the rank correlation coefficients for depths of 15, 23, 30, and 46 cm are very large, and generally increase with soil depth. A similar result was found by Kamgar et al. (1992). They suggested that temporal stability of soil water storage was less pronounced in shallow soil layers owing to the impact of crop root water uptake, whereas pedogenetically derived variations in the deeper layers conserve a rather stable pattern of spatial variation through time.


   Conclusions
TOP
 NOTES
 ABSTRACT
 INTRODUCTION
 Methods and materials
 Results and discussion
 Conclusions
REFERENCES
 
Results of spatial analysis provide an adequate basis to evaluate the dimensions of experimental plots. With 95% confidence, the crosscorrelation between grain yield and base saturation was a distance of 30 m, and that between grain yield and available water storage capacity was 25 m. These soil variables also exhibited significant autocorrelation. Hence, with plot width chosen about 25 m, soil properties determined in one plot are not independent of the same and other properties observed in neighboring plots. We also note that grain yield had a spatial trend that yielded an autocorrelation length greater than 200 m. Consequently, grain yields from 25-m plots would definitely not be independent. If observations from treatment experiments had to be analyzed using analysis of variance (ANOVA), a procedure like nearest neighbor analysis (NNA) (Mulla et al., 1990) would have to be applied prior to ANOVA in order to remove local trends, as was shown by Bhatti et al., 1991. Only if the variogram analysis yields a pure nugget effect after applying NNA are the assumptions underlying ANOVA met.

The rather stable pattern of spatial variation of water content through time indicates systematic differences of soil properties across that field that have to be considered when the effects causing yield variability among plots are identified. Such systematic differences would tremendously complicate the design of adequate treatment experiments with replicated plots. Additionally, in such heterogeneous field soils, crop response to a set of soil properties often does not follow a unique function, but may change through space. This fact may prohibit successful application of classical multiple regression analysis. Instead, as our example of state-space analysis showed, stochastic approaches can be chosen that effectively help to identify processes underlying yield variability in heterogeneous field sites and that account with high flexibility for changing soil conditions causing spatially differing crop response. Stevenson and van Kessel (1996) developed a conceptual landscape-level–based model of the major processes (i.e., occurrence of weeds, diseases, available N, and water) that significantly affected the benefit of pea (Pisum sativum L.) in a pea–cereal compared with a cereal–cereal rotation. Their study showed that response of the crop to the impacts mentioned depended on the position within the landscape. If such natural boundary conditions can be included in the analysis, the transfer of knowledge to other field sites may be improved. Compared with traditional research utilizing replicated treatment plots, stochastic approaches allow as much deterministic information as necessary to be combined with the state of a greater and more diverse set of variables in relation to their local neighborhood at different scales of space and time.

The analysis of spatial variation of crop yield and soil state variables within this renovated field site showed that observations were highly auto- and crosscorrelated. The spatial autocorrelation reached ranges that are greater than the common dimensions of experimental plots. If results from treatment experiments must be analyzed, we suggest that data should be processed in order to remove local trends. However, analytical approaches were applied in order to investigate conceptual and practical alternatives for on-site data analysis. With these techniques, spatial association of variables was analyzed and described. Based on deterministic relations between variables and on an autoregressive state-space equation in combination with a stochastic filter, transition coefficients were optimized that described the spatial grain yield process. This description is not limited to stationarity of data. In future studies, the type of spatial equation, as well as the set of variables used, can be altered.

Frequency domain-based analysis of spatial soil water content series was conducted over both space and time based on the fact that regularly observed data series can be discretized into cyclic variation components. High values of squared coherency indicated a strong stability of variation at respective wavelengths for different sampling dates, and a lower coherence was found between different soil depths. These results coincide with the temporal stability that was identified by ranking analysis. Once such pattern stability over time can be observed, sampling programs may be designed more efficiently.

With the analytical tools applied here, inherent variability of field sites can be analyzed based on spatial dependence between observations, and the spatial pattern of, for example, crop yield, can be described by identifying underlying spatial processes. The set of analytical approaches applied in this study can be considered promising for on-site monitoring and data analysis in farmers' fields, especially when local variation of soil and crop response, and of crop demand and leaching potential, becomes a solid criterion for spatially varying management operations, such as site-specific application rates.Bhatti Mulla Koehler Gurmani 1991; Soil Conservation Service 1972


   ACKNOWLEDGMENTS
 
We acknowledge Ellis Edwards for his extensive field assistance and the analysis of soil physical properties. We also thank Don Eaddy, former director of the Agronomic Division of the North Carolina Department of Agriculture, for facilitating the soil chemical analyses.


   NOTES
TOP
 NOTES
ABSTRACT
 INTRODUCTION
 Methods and materials
 Results and discussion
 Conclusions
 REFERENCES
 
Contribution of the Dep. of Soil Science, North Carolina State University; Institut für Bodenlandschaftsforschung, Zentrum für Agrarlandschafts-und Landnutzungsforschung (ZALF); and Dep. of Land, Air, and Water Resources, Univ. of California, Davis.

Received for publication February 18, 1999.
   REFERENCES
TOP
 NOTES
 ABSTRACT
 INTRODUCTION
 Methods and materials
 Results and discussion
 Conclusions
 REFERENCES
 




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