Monitoring for Temporal Changes in Soil Salinity using Electromagnetic Induction Techniques

struments have received the most attention for fieldscale agricultural applications, particularly the EM-38. Electromagnetic induction surveys are often used in practice to In 1992, Dı́az and Herrero discussed the monitoring estimate field-scale soil salinity patterns, and to infer changing salinity of soil salinity conditions with time in two fields using conditions with time. We developed a statistical monitoring strategy EM-38 survey data. In their study, both EM-38 and that uses electromagnetic induction data and repetitive soil sampling sample soil salinity data were collected at multiple to measure changing soil salinity conditions. This monitoring approach requires (i) the estimation of a conditional regression model that is points in time within each field. The focus of the study capable of predicting soil salinity from electromagnetic (EM) survey was to examine various ways that one might use the data, and (ii) the acquisition of new soil samples at two or more multistage EM-38 survey data to monitor and predict previously established survey sites, so that formal tests can be made the time-dependent changes occurring in the field-scale on the differences between the predicted and observed salinity levels. soil salinity conditions. Their data sets are rather unique We examined two test statistics in detail: a test for detecting dynamic in that they represent one of the few published data spatial variation in the new salinity pattern and a test for detecting sets where both EM-38 and soil salinity data have been a change in the field median salinity level with time. We applied this acquired at multiple points in time from within the monitoring and testing strategy to two EM survey–soil salinity data same fields. sets collected at multiple points in time from the saline irrigation A comprehensive statistical methodology for the predistrict of Flumen, Spain. Our results demonstrate that this monitoring diction of soil salinity using EM signal data was sugapproach was successfully able to quantify the temporal changes in gested by Lesch et al. (1995a,b). This prediction apthe soil salinity patterns occurring within these two fields. proach was based on the development of field-specific multiple linear regression models that could be used to predict soil salinity levels from EM-38 survey data. T use of electromagnetic induction sensors for These researchers also suggested that the regression the assessment and monitoring of soil salinity condimodeling methodology could be employed to monitor tions has received considerable attention in the soil scichanges in the soil salinity conditions with time, proence literature (Lesch et al., 1995a,b; Dı́az and Herrero, vided additional soil samples were acquired at one or 1992; Hendrickx et al., 1992; Rhoades, 1992; Rhoades more survey sites in the future. and Corwin, 1990; Rhoades and Miyamoto, 1990; SlavThere is a clear need for the development of costich, 1990; Williams and Baker, 1982; McNeill, 1980). effective, quantitative salinity monitoring techniques. These sensors can generally be classified into one of The initial diagnosis of the soil salinity conditions within three types: (i) four-electrode sensors, including either a field typically represents just the first step in a longsurface array or insertion probes, (ii) remote EM inducterm reclamation project or salinity management protion sensors, such as the Geonics EM-31, EM-34, and cess. Periodic monitoring of the evolving salinity condiEM-38 (Geonics Ltd., Mississauga, ON)1, and (iii) time tions is just as essential when identifying the most worthdomain reflectometric sensors (Rhoades, 1992; Dalton, while reclamation or management strategies, and for 1992). Of these three sensor types, the remote EM indeveloping meaningful cost–benefit analyses. Furthermore, this type of information is often needed for the calibration and testing of various types of dynamic salin1 Mention of trademark or proprietary products in this manuscript ity transport models. does not constitute a guarantee or warranty of the product by the U.S. Department of Agriculture and does not imply its approval to Unfortunately, there appears to be no generally acthe exclusion of other products that may also be suitable. cepted, quantitative statistical monitoring strategy sugS.M. Lesch and J.D. Rhoades, USDA-ARS, U.S. Salinity Lab., 450 gested in the soil science literature that can incorporate W. Big Springs Rd., Riverside CA 92507; and J. Herrero, Soils and repetitive EM survey and soil sample data into some Irrigation Dep., SIA, Government of Aragon, P.O. Box 727, 50080 Zaragoza, Spain. Received 15 Nov. 1996. *Corresponding author Abbreviations: EM, electromagnetic; ANOVA, analysis of variance; ([email protected]). ECe, electical conductivity of the saturated soil extract; MLR, multiple linear regression; MSE, mean square error. Published in Soil Sci. Soc. Am. J. 62:232–242 (1998). LESCH ET AL.: MONITORING TEMPORAL CHANGES IN SOIL SALINITY 233 a Geonics EM-38 meter. Both horizontal (EMH) and vertical type of formal statistical “test” (for detecting a change in (EMV) readings were acquired at each site, and then temperasoil salinity conditions with time). For example, suppose ture corrected to 258C using the correction coefficients given that an EM survey is conducted across a given field, U.S. Salinity Laboratory Staff (1954). The P parcel was sursoil samples are acquired at some of these survey sites, veyed twice (March 1988 and January 1989), and the M parcel and a regression model is estimated from this data. In was surveyed three times (May 1988, January 1989, and this example, assume that the regression model can be April 1990). used to convert the EM survey data into a predicted A limited number of soil samples were acquired during soil salinity level at each survey site across the field. each survey in both fields. The sampling sites that were seNow, at some point in the future suppose one wishes lected were chosen to both (i) span the observed range in the to formally test for a change in the spatial salinity pattern EM-38 signal data, and (ii) provide reasonable (i.e., approximately uniform) coverage across each parcel. The numbers or average salinity level in this field. Then how should and locations of these sample sites varied from year to year this be done? Should one conduct a new EM survey, (see Fig. 1). Soil samples at each sample site were acquired acquire new soil samples, and fit a new regression in 0.25-m intervals down to depths of 1.5 and 1.0 m in the P model? Or should one only acquire new soil samples and M parcels, respectively. Both electrical conductivity of without resurveying the field; or perhaps only conduct the saturated soil extract (ECe) and electrical conductivity of a new survey without collecting any new soil samples the 1:5 soil water extract were determined using standard (and in either case, somehow use this data in conjunction laboratory methods on each soil sample (U.S. Salinity Laborawith the old regression model). Additionally, does it tory Staff, 1954). matter if the two survey grids or sets of soil sample sites Some additional information concerning the 1988 P parcel are collocated? and 1988 and 1989 M parcel EM survey and soil salinity data We examined these questions, and developed a cohercan be found in Dı́az and Herrero (1992) and López-Bruna and Herrero (1996). However, in the analysis that follows, we ent statistical monitoring methodology for use with the typical, regression-based EM survey techniques commonly applied in practice. We first developed an appropriate regression equation by incorporating the ideas behind the more traditional, mixed linear analysis of variance (ANOVA) model into the regression modeling assumptions. These modeling assumptions in turn determine how the overall survey should be carried out; i.e., when and where the EM survey and soil salinity data should be acquired. Next, we developed two statistical tests based on these modeling assumptions. The first test can be used to determine if the salinity pattern has changed in a spatially variable manner, and the second test can be used to determine if the average salinity level across the entire field has changed with time. We then used the previously mentioned survey data (Dı́az and Herrero, 1992) to demonstrate this monitoring methodology, and used these statistical tests to quantify changes occurring in soil salinity patterns with time. MATERIALS AND SURVEY METHODS The data to be analyzed come from two 0.5-ha salt-affected parcels in the irrigation district of Flumen (Aragon, Spain). The first parcel (P, 0.54 ha) contains soil classified as a fine, mixed (calcareous), thermic Oxyaquic Torrifluvent. At the time of sampling, it was slightly leveled and had been regularly cropped with rice (Oryza sativa L.) duirng the previous 25 yr. The second parcel (M, 0.40 ha) is more texturally heterogeneous. Approximately 55% of the parcel contains soil classified as a coarse-silty, mixed (calcareous) Xeric Torriorthent. The remaining 45% of the parcel is classified as (i) a loamy, mixed (calcareous) thermic shallow Lithic Xeric Torriorthent (15%); (ii) a coarse-loamy, mixed (calcareous), thermic Xeric Torrifluvent (15%); and (iii) a coarse-loamy, mixed, thermic Xeric Haplocalcid (15%). In 1987 the parcel was laser-leveled and a buried tile drain system was installed. Thereafter, the M parcel was used for maize (Zea mays L.) production. Electromagnetic induction surveys and soil sampling were carried out in both parcels in 1988 and 1989, and in the M Fig. 1. Locations of EM-38 survey sites and soil extract electrical parcel in 1990. Electrical conductivity readings were acquired conductivity (ECe) sample sites in parcels P and M for each survey date. at the soil surface across each field on a 10 by 10 m grid using 234 SOIL SCI. SOC. AM. J., VOL. 62, JANUARY–FEBRUARY 1998 will consider only the ECe data, since this data is generally of estimated model parameters including the intercept, and s2 is the estimated mean square error of the regression model considered a more reliable measurement of soil salinity. Additionally, to simplify the analysis, we have chosen to only relate (Myers, 1986). In Eq. [1a] and [1b] the residuals, e, are assumed to be the EM-38 signal data to the average of the ECe data within the 0to 1-m sampling depth in each field. normally distributed with homogeneous variance and spatially uncorrelated. All of these assumptions must be verified before the model can be used for prediction purposes. The assumpSTATISTICAL THEORY tion of residual spatial independence can be examined by The prediction of soil ECe from EM measurements requires using a Moran residual autocorrelation test (Brandsma and that a model be developed that relates the two sets of data Ketellapper; 1979, Lesch et al., 1995a). The remaining residual to each other. Numerous researchers have suggested various assumptions should be verified using the standard residual deterministic (theoretically based) or statistical ECe–EM prediagnostic plots or tests (Myers, 1986; Atkinson, 1985; Weisbdiction models (López-Bruna and Herrero, 1996; Lesch et erg, 1985). al., 1995a,b; Yates et al., 1993; Rhoades, 1992; Slavich, 1990; A final assumption intrinsic to the regression modeling apWilliams and Baker, 1982; McNeill, 1980). The approach we proach concerns the X matrix, which may be considered fixed used was to develop field-specific, multiple linear regression or random. When the X matrix is considered fixed, one implic(MLR) models that can predict soil ECe levels at each survey itly assumes that this matrix contains all the information about point from the acquired EM data (Lesch et al., 1995a). Therewhich any inference will be made. In practice, this means that fore, a review of the modeling assumptions intrinsic to this one is actually restricting any inference (i.e., model predicregression approach will be given first. We will then briefly tions, statistical tests, etc.) to the EM survey grid, rather than review the modeling assumptions behind the more traditional, the entire field. For example, this is the approach taken in mixed linear analysis of variance model. This is the most comLesch et al. (1995a,b). mon type of statistical model one would typically use to test An alternative approach would be to consider the X matrix for changing salinity conditions if no EM covariate data were to be random, rather than fixed. By this we mean that the available. Finally, we will show how these ANOVA assumpEM information associated with the finite number of survey tions can be incorporated into a conditional regression model, sites observed during the survey process actually represent and demonstrate how such a model can be used to test for a just a subsample of the potentially infinite number of survey change in soil salinity with time. sites that could have been observed. This type of regression equation is commonly referred to as a conditional regression model, since the model is conditional on the observed (ranMultiple Linear Regression Model Definitions dom) X data matrix. Such a model usually arises when the and Assumptions covariate data (i.e., the X matrix) comes from either observaIn the MLR modeling approach defined here, we assume tional studies or designed experiments with both fixed and that there exists a linear relationship between the natural log random treatment effects (Montgomery, 1984; Ryan, 1997). (ln) transformed soil ECe levels and the ln-transformed EM Under this approach it is still possible to treat the X matrix readings. We also assume that additional “trend surface” paas though it were fixed, provided that conditional distribution rameters may need to be included in the MLR model to of the observed response data, given the observed X matrix, account for spatial drift or lateral trends in the EM signal data is normal with constant variance, and that the distribution of unrelated to the soil salinity levels. For example, when two X does not depend on either the b parameter vector or the EM-38 signal readings are acquired at each survey site, a error variance (Neter et al., 1989). However, the inference MLR salinity prediction model with first-order trend surface space in this approach could include the entire field, rather parameters would be defined as than being exclusively restricted to just the survey grid. ln(ECe) 5 b0 1 b1z1 1 b2z2 1 b3u 1 b4v 1 e [1a] The ANOVA (Mixed Linear Model) Approach where the (u,v) variables represent the spatial coordinates of each survey site, z1 and z2 are defined as z1 5 ln(EMV) 1 To clarify how one can use a conditional regression equation for monitoring purposes, it will be helpful to first briefly review ln(EMH), z2 5 ln(EMV) 2 ln(EMH), and e represents the stochastic, residual error component. Note that the adding and the modeling assumptions inherent in the more traditional, mixed linear ANOVA model. This is the model one would subtracting of the EM signal components (inherent in the z1 and z2 definitions) is simply done to reduce the effects of typically use if no EM covariate data were available; i.e., all monitoring activities were to be based only on the repetitive signal multicollinearity (Myers, 1986). It is generally more convenient to write Eq. [1a] in matrix collection of soil samples. For this discussion, define yijk as the observed natural-lognotation. Suppose that there are salinity data from i 5 1, 2, ..., n sample sites, where these “calibration” sites form a subset transformed salinity level from the jth sample site during the ith time frame, where i 5 1 or 2. Suppose also that we acquire of a larger set of N EM survey sites. Define y 5 [ln(ECe1), ln(ECe2), ..., ln(ECen )], xi 5 (1, z1i, z2i, ui, vi ), X 5 (x1, x2, ..., two salinity samples from each site (for example, by taking two soil cores arbitrarily close together), and let the k 5 1, 2 xn ), b 5 (b0, b1, b2, b3, b4), and e 5 (e1, e2, ..., en ), where T represents the matrix transpose symbol. Then Eq. [1a] can be subscript represent these “replicate samples.” Then a traditional mixed linear model can then be written as expressed as y 5 Xb 1 e [1b] yijk 5 u 1 ti 1 bj 1 (tb)ij 1 eijk [2a] for i 5 1, 2, j 5 1, 2, ..., n, k 5 1, 2, and with bj z iid N(0, Likewise, the predicted natural-log-transformed salinity at the ith survey site may be written as ŷi 5 xi b, where b represents sb), (tb )ij z iid N(0,stb), and eijk z iid N(0,s). Under these assumptions, it is well known (i.e., Montgomery, 1984) that the estimated b parameter vector. From standard regression theory, a 100(1 2 a)% prediction interval for a ln(ECe) sample the expected mean squared error for ti, (tb)ij, and eijk are E{MSt} 5 s 1 2stb 1 2n(t1 1 t2), E{MStb} 5 s 1 2stb, and E{MSe} 5 acquired at this site would be ŷi 6 t(a/2,n2p21) s(1 1 xi (XX)xi ), where t is the t-distribution, p 1 1 is the number s. Thus the ratio of MStb/MSe can be compared with an F LESCH ET AL.: MONITORING TEMPORAL CHANGES IN SOIL SALINITY 235 distribution to test for stb 5 0, and the MSt/MStb ratio can y1 | X1 5 X1b 1 e1 [4] be used to test for t1 2 t2 5 0. where e1 z N(0,s I ) and I represents the identity matrix. In a general ANOVA model, the eijk variance component Furthermore, assume that a suitable model for the second typically represents sampling error, but the (tb )ij interaction time frame is variance component can at times be difficult to interpret. When analyzing spatial data, however, this interaction variy2 | X1 5 X1b 1 d0 1 h 1 e2 [5] ance component has an obvious meaning. When stb . 0, one where d0 5 [d0, d0,... ,d0], h z N(0,uI ), e2 z N(0,sI ), and the h, can conclude that changes in the natural-log-transformed sae1, and e2 random error components are mutually independent. linity levels are spatially variable (i.e., different from site to Hence, (y2 | X1) 2 (y1 | X1) 5 d0 1h 1 e2 2 e1, which is simply site). Hence, testing stb 5 0 is equivalent to testing for spatially Eq. [3] written in matrix format. Therefore, in Eq. [5], d0 dynamic change across the field, while the t1 2 t2 5 0 test represents the shift in the average natural-log-transformed represents a test about the average shift with time in the mean salinity level between the two time frames and the additional natural-log-transformed salinity level across the entire field. error term h represents the dynamic variability component. We can therefore use these two statistical tests to determine After acquiring the first set of n calibration samples, supwhich one of the following four scenarios seems most likely, pose Eq. [4] is estimated as ŷ1 | X1 5 Xn b, were ŷ1 represents given the observed data: the vector of predicted natrual-log-transformed salinity levels Corresponding computed using Eq. [4], Xn represents the EM covariate matrix Scenario hypothesis associated with the n sites, and b represents the estimated 1. No change with time stb 5 0 and t1 2 t2 5 0 parameter vector. Then, from standard linear regression the2. Static (spatially constant) stb 5 0 and t1 2 t2 ≠ 0 ory, the prediction error associated with a new set of k prechange with time dicted sites located on the grid would be distributed as multi3. Dynamic (spatially variable) stb ≠ 0 and t1 2 t2 5 0 variate normal with a mean of 0 and a variance–covariance change with time matrix of s(Ik 1 Hk ), where Hk 5 Xk (Xn Xn )Xk, Xk repre(global shift not sents the matrix of EM covariate data associated with the k statistically significant) prediction sites, and Xn is the matrix of covariate data associ4. Dynamic (spatially variable) stb ≠ 0 and t1 2 t2 ≠ 0 ated with the n calibration sites (Graybill, 1976). In other change with time words, (y1 2 ŷ1) | X1 z N [0,s2 (Ik 1 Hk )]. However, note that (global shift is statistically these prediction errors are only valid for new samples acquired significant) during the first time frame; i.e., only for new samples acquired at the same time as the calibration samples. When no replicate samples are available, then the two error Now, suppose m new samples located on the grid are accomponents become confounded together and hence the ANquired during the second time frame. Let y2 represent this OVA model becomes vector of sample observations, ŷ1 represent the corresponding vector of predicted levels computed from Eq. [4] at these m yij 5 u 1 ti 1 bj 1 §ij §ij 5 (tb)ij 1 eij [2b] sites, and define Hm 5 Xm (Xn Xn )Xm, where Xm represents for i 5 1, 2 and j 5 1, 2, ..., n. In practice, when one computes the matrix of EM covariate data associated with these m prea paired t-test, one is actually using Eq. [2b] shown above, diction sites from the first time frame. Then Eq. [5] implies but respecified as that the prediction error associated with these sites would be (y2 2 ŷ1) | X1 z N [d0, u Im 1 s(Im 1 Hm )]; i.e., the observed y2j 2 y1j 5 dj 5 d0 1 hj 1 e2j 2 e1j [3] (Time 2) minus predicted (Time 1) differences will contain for j 5 1, 2, ..., n. In Eq. [3], d0 5 t2 2 t1, hj z iid N(0, stb), two sources of error, and may no longer be equal (on the eij z iid N(0, s2), hj and eij are assumed uncorrelated, and average) to 0. typically one assumes that stb 5 0. Note that the vector of differences d 5 y2 2 ŷ1 is observable (once the samples from the second time frame have been acquired), and that under our modeling assumptions, its distriThe Conditional Regression Approach bution is known. Furthermore, d implicitly contains information about d0 and u. Hence estimates and tests concerning It is possible to formulate a conditional regression model both d0 and u2 are derivable from these observed differences. using the mixed linear modeling assumptions just described. To motivate these derivations, assume that Eq. [4] has been Define y1j and y2k as the observed natural-log-transformed estimated as ŷ1 | X1 5 Xn b, and let s2 represent the calculated salinity levels from the jth and kth sample site acquired during model mean square error with n 2 p 2 1 degrees of freedom. the first and second time frames, where j 5 1, 2, ..., n and k 5 Additionally, suppose that d 5 y2 2 ŷ1 has been observed, 1, 2, ..., m. Let y1 represent the vector of observations from where the vector d 5 {d1, d2, ..., dm}. Define the calculated the first time frame, and y2 represent the observations from sample mean and variance of these observed differences as the second time frame. Additionally, define X1 as the matrix u and w2, where u 5 (1/m)(d1 1 d2 1 ... 1 dm ) and w2 5 (grid) of EM covariate signal data observed during the first [1/(m 2 1)][(d1 2 u)2 1 (d2 2 u)2 1 ... 1 (dm 2 u)2]. Clearly time frame. For this discussion, suppose that a survey grid of u represents a conditionally unbiased estimate of d0. Furthersize N (N . n,m) of representative EM covariate data has more, given the previously stated modeling assumptions, the been acquired during the first time frame only, and that the following three results can be derived: (i) an F test for den and m sample sites (from the first and second time frames, termining if u2 . 0, (ii) a method of moments estimate of u2, respectively) are chosen from this grid. Note that the two sets and (iii) an approximate t-test for determining if d0 5 0. These of sample sites need not be collocated. Furthermore, assume results are given below: that the conditional distribution of y given the observed X1 matrix is normal with constant variance, and that the distribu1. An F test for determining if u2 . 0 can be computed as tion of X1 does not depend on either the b parameter vector f 5 (d 2 u)To21(d 2 u)/(m 2 1)s2, where o 5 (I 1 Hm ), or the error variance. Assume that a suitable model for the and where f is compared to an F distribution with m 2 1 and n 2 p 2 1 degrees of freedom. first time frame is 236 SOIL SCI. SOC. AM. J., VOL. 62, JANUARY–FEBRUARY 1998 2. The expected value of w2 is u2 1 s2(1 1 l1 2 l2), with error are confounded together. Additionally, the two sets of soil samples no longer need to be collocated. (This is a distinct l1 5 (1/m )o hii and l2 5 {1/[m(m 2 1)]}oo hij ~ i ≠ j (where hij represent the ith, jth diagonal element of the advantage if the coring operation used to acquire the soil samples also disturbs the surrounding soil. For example, imHm matrix). Hence, a method of moments estimate of u is v 5 w 2 s(1 1 l1 2 l2). properly filled bore holes at the first-stage sampling sites can become preferential pathways for vertical water movement 3. An approximate t-test for d0 5 0 can be computed as c 5 u/g, where g 5 (1/m )v 1 2s[(1/m ) 1 hmu], hmu 5 during subsequent irrigations. Hence, one might want to avoid these sites during future monitoring activities.) Both of these xmu (XnXn )xmu, xmu 5 (1/m)(x1 1 x2 1 ... xm ), and where c is compared with a t distribution with n 2 p 2 1 degrees features result from the fact that the blocking parameters in Eq. [3] have been replaced by regression parameters in Eq. of freedom. Note that this test statistic assumes that the two sets of soil samples are not collocated. [4] and [5]. Fourth, it is critically important that both sets of soil samples Proofs for each of these derivations are shown in the Apare associated with the same larger set of N survey sites colpendix. lected during the first time frame. This requirement must be Lesch et al. (1995a) suggested a mean shift test statistic satisfied because the conditional regression model implicitly defined as u/t, where t 5 s[(1/m) 1 hmu], and the remaining assumes that the X1 matrix is known (i.e., observed without terms are the same as those shown in Result 4. Under our error). If the X1 matrix was either unknown or had to be modeling assumptions, this would not be a valid test for deestimated, then we could not use it to develop a valid regrestermining if d0 5 0, since the variance term is incorrectly sion equation. In practice, this means we are restricted to specified. What this statistic actually tests is only whether there sampling on the original grid, since these are the only sites is a statistically significant difference between the observed where the EM-38 signal levels are known a priori. and predicted mean natural-log-transformed salinity level Finally, although the ECe–EM regression relationship is across the m new monitoring sites; it is not a valid test for typically modeled on the ln–ln scale, it is generally desirable to inferring change across the entire field. “back-transform” the predicted change in the average naturalSome other important features about this conditional relog-transformed salinity level with time to a more meaningful gression model are worth highlighting. First, this model asestimate. In the conditional regression model, if we define ŷ sumes that there are two potential sources of error present to represent a predicted mean natural-log-transformed salinity during the second time frame; sampling error (which is also level, then exp( ŷ ) represents an unbiased estimate of the present during the first time frame) and dynamic spatial variacorresponding median salinity level. Since u is an estimate of tion. Hence, in order to test for a change in the mean salinity t2 2 t1, where t1 and t2 represent the field mean natural-loglevel with time, both errors must be accounted for. This means transformed salinity levels at Times 1 and 2, exp(u) 5 exp(t2)/ that both variance components must be estimated, which in exp(t1) and therefore a test of u 5 0 is equivalent to a test of turn means that we must acquire soil samples during both time exp(t2)/exp(t1) 5 1. Hence, 100[exp(u) 2 1] represents an frames. Second, we do not have to acquire a new grid of EMestimate of the percentage increase (or decrease) in the field’s 38 survey data during the second time frame to compute these median level with time. Likewise, the test of u 5 0 actually test statistics. Acquiring a second set of survey data is almost tests whether the salinity pattern in the field has changed always a good idea, however, because one can then estimate in a strictly proportional manner (i.e., constant change on a a new regression model (when necessary), which in turn can percentage basis). be used to estimate a new salinity map. This can prove to be very important, since Eq. [5] cannot be used to estimate the spatial salinity pattern during the second time frame, unless RESULTS there is no dynamic salinity variation (i.e., unless u 5 0). Third, under this conditional regression model, u (the dyPreliminary Data Analysis namic salinity variance component) can be estimated even The EM survey and soil sampling grids for both parthough only one sample is acquired during each time frame cels are shown in Fig. 1. Note that the number of survey at each site. Hence, this approach will yield more information points increased in the P parcel in 1989 (from 59 to 73 than the traditional paired t-test design, since in the latter design the treatment-block interaction error and the sampling sites), while the survey grid remained relatively constant Table 1. Distribution summary statistics for EM-38 survey and soil salinity data, by parcel and sampling date.