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DIVISION S-6-SOIL & WATER MANAGEMENT & CONSERVATION

Calibrating an Electromagnetic Induction Instrument to Measure Salinity in Soil under Irrigated Cotton

^a Australian Cotton Cooperative Research Centre, Dep. of Agricultural Chemistry and Soil Science, The University of Sydney, Sydney NSW 2006, Australia
^b CSIRO Mathematical and Information Sciences, Private Bag 10, Clayton MDC, Clayton, Victoria 3169, Australia

johnt{at}acss.usyd.edu.au

	ABSTRACT

TOP ABSTRACT INTRODUCTION Materials and methods Results and discussion Conclusions REFERENCES

 
Various calibration approaches have been proposed for determining profiles of apparent soil electrical conductivity (EC_a) or soil electrical conductivity of a saturated soil paste extract (EC_e) using an EM38 instrument. One of the most promising of these, the established-coefficients approach, was selected for calibration of the EM38 to some irrigated cotton (Gossypium hirsutum L.)-growing soil located in the Edgeroi district of the lower Namoi valley, northern NSW, Australia. However, the fitted salinity profiles were locally erratic, although global trends could be recognized. This method was compared with simple linear regression applied to the data from each depth increment separately. This was also unsatisfactory, and in fact, was not obviously different from the established-coefficients method on the data. An alternative and statistically more rigorous approach was used in which a logistic curve is fitted to each calibration hole (the logistic profile model). This mixed random and fixed effects nonlinear model predicts EC_e from measurements generated by an electromagnetic (EM) instrument (EM38) held in the horizontal mode of operation EM_0,H. The logistic profile model fitted the data well, with no obvious patterns in the residuals, and depended on far fewer parameters than the two alternative methods. In addition, unlike the established-coefficients approach, the logistic model provided meaningful prediction errors.

Abbreviations: EC, electrical conductivity • EM, electromagnetic

	INTRODUCTION

TOP ABSTRACT INTRODUCTION Materials and methods Results and discussion Conclusions REFERENCES

 
INVESTIGATIONS of soil salinity provide important information about the impact that farm practices, such as irrigation, soil, and crop management, have on both the field and regional scale. Electromagnetic measurements have proved very useful for these purposes because they are closely related to soil salinity. Other important reasons include robust construction of the instrumentation, its compact design, ease of use, and, most significantly, the noncontacting nature of the measurement. By comparison with direct electrical methods and time domain reflectometry, no physical insertion of probes is required, and hence less time is spent obtaining a measurement at each location. Consequently, a larger number of survey measurements can be made.

Previous applications of EM instruments for soil salinity assessment include Cameron et al. (1981), Rhoades et al. (1989), Williams and Baker (1982), Williams and Fiddler (1983), Wollenhaupt et al. (1986), McKenzie et al. (1989), Williams and Arunin (1990), Slavich and Petterson (1990), and Lesch et al. (1992). The success of these studies depended on the establishment of soil profile calibration equations relating soil electrical conductivity of a saturated paste extract (EC_e), soil water extract (EC_1:5), or bulk soil electrical conductivity (EC_a) to EM measurements. Several calibration approaches have been proposed, including multiple regression coefficients (Rhoades and Corwin, 1981), simple depth weighted coefficients (Wollenhaupt et al., 1986; Cameron et al., 1981), established-coefficients (Corwin and Rhoades, 1982, 1984), modeled coefficients (Slavich, 1990), and mathematical coefficients (Cook and Walker, 1992). These methods were developed for a Geonics Ltd. instrument (EM38, Geonics Ltd., Mississauga, ON, Canada) designed to measure salinity in the agriculturally significant part of the soil (i.e., root zone, typically to a depth of 1–1.5 m).

Of these approaches some of the more successful calibrations have included the multiple coefficients approach of Rhoades and Corwin, (1981). Unfortunately, this approach requires a relatively large number of EM measurements to be made at various heights above the soil surface as well as in the vertical and horizontal modes of operation. These measurements, taken at calibration test holes along with direct measurements of EC_a, are used to build a linear predictor of EC_e; this may be used to estimate EC_a at various depths from EM measurements at new locations. However, the numerous measurements mean that this method can be time consuming. Another drawback is that the prediction of EC_a at each depth uses the same EM predictors, but the predictive equations are derived independently, so the prediction curve may appear discontinuous with depth.

To increase efficiency in collecting information about the spatial distribution of soil salinity across the field, Corwin and Rhoades (1982, 1984) developed the established-coefficients approach that requires only the vertical and horizontal EM measurements taken at the ground surface (denoted by EM_0,V and EM_0,H, respectively). In order to construct good predictions from these limited data, the method also uses the theoretical EM depth response functions. However, the method is still partly empirical, requiring EM measurements obtained at the ground surface and EC_a data collected at various depths for a set of calibration holes. The resulting predictor is still linear in EM_0,V and EM_0,H, but the coefficients are very different for normal (salinity increasing with depth) and inverted (salinity decreasing with depth) profiles. This was adopted as the reference method in our study.

More recently, Lesch et al. (1992) provided a more robust universal calibration approach that does not depend on profile shape. Alternatively, Lesch et al. (1995a, 1995b) developed and applied multiple linear regression calibration models capable of producing multiple types of soil salinity estimates, including EC_e point estimates, conditional probability estimates, and field average estimates. Their approach covers preliminary data analysis, selection of calibration and validation sites, spatial model building, and map construction.

We propose a new method based on a logistic profile model for calibrating EM38 measurements. The method is demonstrated on profiles of Belar soil in the Edgeroi district of northern NSW, Australia, where there are extensive irrigated cotton farms (Triantafilis and McBratney, 1993). This new calibration approach is compared with two more traditional methods, the established-coefficients approach (Corwin and Rhoades, 1982) and simple linear regression (see Rhoades and Corwin, 1981). In addition, it is shown how errors can be attached to the prediction using this approach.

	Materials and methods

TOP ABSTRACT INTRODUCTION Materials and methods Results and discussion Conclusions REFERENCES

 
Study Area
The study area is located at the northern center of the Namoi region, situated 450 km northwest of Sydney, New South Wales, Australia. More specifically, it is located in two corporate irrigated cotton farms (i.e., Togo Station and Auscott), with a combined irrigated area exceeding 10000 ha. Vertisols are the predominant soil type in the area.

Site Selection
Recently, soil sampling protocols for the calibration of EM38 for soil salinity assessment (Rhoades et al., 1989; Lesch et al., 1992) have been developed for field studies. In this study, the spatial distribution of soil salinity within the area appears to be due to larger-scale geomorphological processes, so field-scale sampling would be too detailed. A larger-scale soil sampling protocol was therefore required in order to determine the location and magnitude of a moderately saline subsoil layer identified by Triantafilis and McBratney (1993).

To assist in establishing calibration sites, EC_1:5 data available on a 2.8-km equilateral triangular grid of the Edgeroi data set was used (McGarry et al., 1989). The survey covered a 50 by 30 km area. The depth intervals 0.70 to 0.80 and 1.20 to 1.30 m showed greater concentrations of soluble salts than topsoil depths (0–0.10, 0.10–0.20, 0.30–0.40 m). It was apparent from interpolated maps that salinity within the area steadily increased from the Namoi River in the south to Galathera Creek in the north.

Owing to this strong gradient, a calibration transect area running from the northwest to southeast corner was defined. Figure 1 illustrates that, for the most part, the area selected intersects Galathera Creek on Togo Station and ends near the southern part of land farmed by Auscott.

View larger version (22K): [in this window] [in a new window]   Fig. 1 Location of EM38 calibration soil profiles on Togo Station and Auscott including Galathera Creek

The EM_0,V and EM_0,H data generated at each of the calibration sites are presented in Fig. 2 ; a good linear relationship exists between EM_0,V and EM_0,H for all profiles except Hole 27. This site was exceptional in that EM_0,H was larger than EM_0,V. Corwin and Rhoades (1984) suggest that in such instances the distribution of soil salinity is inverted, that is, decreasing with depth, although for Hole 27 the profile was uniformly saline. On examining the site history, it was found that the area had been used as a stock dam prior to development for irrigation. Water had been obtained from an underground source and was of poor quality. Salinity appears to have built up within this profile and adjacent areas, probably because of lack of any significant deep drainage. Therefore Hole 27 was omitted from the calibration.

View larger version (19K): [in this window] [in a new window]   Fig. 2 Plot of EM0,V vs. EM0,H measured at the calibration sites, plus the fitted regression line where a is the slope of the line and b is the y intercept

A comparison between the two methods of EC determination showed a strong positive linear relationship. The equation of the fitted line relating EC_e to EC_1:5 values was given by:

(1)

This relationship was then used to convert EC_1:5 to EC_e for all samples. This equation is similar to the relationships obtained by Talsma (1968), Loveday et al. (1972), and Shaw (1981) for Vertisol-type profiles. Selected profiles are presented in Fig. 3 . In general, all profiles exhibit normal salinity profiles in which salinity increased with depth. It is apparent that there is a saline bulge around the 1.0- to 1.5-m depth. Apart from Hole 27, salinity within the topsoil (i.e., 0–0.30 m) ranged between 0.5 to 2.5 dS m^-1. These values are considered nonsaline.

View larger version (20K): [in this window] [in a new window]   Fig. 3 ECe distributions of a number of calibration profiles

The other factors that could affect the instrument's response, such as clay content, mineralogy, and moisture content, can be discounted here. The clay content in these soil profiles is 60% and the parent material is all alluvial; hence, mineralogy (i.e., montmorillonite clay) is similar. The moisture content differences are negligible because profiles were only selected if irrigation had occurred a few days prior to sampling. Therefore, the above assumption of EC_e accounting for most of the difference in EC_a is acceptable.

Methods of Calibration
Linear Regression
The linear regression approach is a special case of Rhoades and Corwin (1981) and in its simplest form consists of fitting a linear regression model

(2)

to the calibration data at each depth z. The errors _z are assumed to be normally distributed with mean 0 and variance ² and to be independent from hole to hole. Fitting the regression model (Eq. [2]) to the calibration holes yields least-squares estimates _z, _z, _z of the unknown parameters _z, ß_z, _z. The predictive equations can then be used to predict EC_a at locations where only EM_0,H and EM_0,V are measured. They should only be used at locations similar to the calibration sites. In particular, the relationship between EM_0,H and EM_0,V should conform to that of the calibration holes.

One potential problem is that Eq. [2] is fitted to the data at each depth separately. However, the data from adjacent depths within each hole are likely to be correlated. In principle, one can overcome this problem by fitting a multivariate regression, with 20 responses (one EC for each depth) and two predictors (EM_0,H and EM_0,V) per hole and correlated errors within holes. Following the usual formulation of the multivariate regression model, separate regression coefficients may be fitted for each depth, allowing for a completely general correlation structure for the within-hole errors, but assuming that the errors between holes are independent. It transpires that the fitted multivariate regression equations are identical to the univariate regression equations fitted separately to the data from each depth (Mardia et al., 1979, Theorem 6.2.1; Greene, 1993, Section 17.2.1), although prediction errors are, in general, different.

It is important to decide if both EM_0,H and EM_0,V are needed as predictors. This depends on the context. In some circumstances, statistical testing may suggest that either EM_0,H or EM_0,V can be dropped. Of course, EM_0,H and EM_0,V would only be discarded if they could be dropped for all depths; it would not be sensible to include both EM_0,H and EM_0,V as predictors at some depths and only one of them at other depths. Multivariate analysis of variance (Mardia et al., 1979, Chapter 12) allows us to test, simultaneously for all depths, whether both are needed.

Established Coefficients
The established-coefficients approach of Corwin and Rhoades (1982) was developed in order to provide a more rapid calibration procedure than the multiple-coefficients approach of Rhoades and Corwin (1981). The equations depend on the assumption that the theoretical depth response curves of the EM38, as outlined by McNeill (1980) for homogeneous media, hold equally well for nonhomogeneous media. The approach results in equations for predicting EC_a in various depth ranges. All predictors take the linear form

(3)

where z represents a depth range, c_z is derived theoretically, and a_z and b_z are derived empirically using calibration data from the site being investigated. The linear form (Eq. [3]) is not immediately obvious from the paper of Corwin and Rhoades (1982), but may be verified by substituting the calibration relationships in their Fig. 3 into the appropriate equations in their Table 1 . In this evaluation of their approach, EC_e was used as a surrogate for EC_a, and the calibration coefficients a_z, b_z, and c_z were calculated accordingly.

In general, the shapes of the salinity profiles are essentially sigmoidal. As a result, the logistic model, popular in biometry and ecology, was selected to model these salinity profiles, although other statistical cumulative distribution functions (such as a Weibull model) could have been used with equal success. The logistic model has the general form

(4)

where y is the response EC_e and x is depth. Mathematically, the model has four parameters. The first parameter ₁ is the lower asymptote to the curve, ₂ is the distance between the lower and upper asymptotes, ₃ is the depth at which the predicted EC_e is half-way between the upper and lower asymptotes , and ₄ is related to the slope at ₃:

(5)

Parameters ₁, ₂ and ₃ are illustrated in Fig. 4 .

View larger version (13K): [in this window] [in a new window]   Fig. 4 Parameters of a logistic curve

	Results and discussion

TOP ABSTRACT INTRODUCTION Materials and methods Results and discussion Conclusions REFERENCES

(6a)

(6b)

(6c)

For these 26 holes, EM_0,V and EM_0,H are highly correlated, suggesting that one may be dropped as a predictor. It was decided to include EM_0,H before EM_0,V, and the significance of each predictor was assessed using multivariate analysis of variance. The Wilks statistic for EM_0,H is 0.026; this corresponds with an F statistic of 7.38 on 20 and 4 degrees of freedom, which is significant at the 3.3% level. Similarly, the Wilks statistic for EM_0,V is 0.224, corresponding with an F value of 0.692 on 20 and 4 degrees of freedom, which is significant at the 74.4% level. This is much higher than 5%, suggesting EM_0,V is not required.

Refitting the equations with only EM_0,H as a predictor yields

(7a)

(7b)

(7c)

Prediction and Validation
The fitted equations (Eq. [6] or [7]) do not demonstrate the method's predictive capability. For this purpose, Holes 1 to 5, 7, 8, 12, 13, 17, 18, 20, 21, and 23 to 26 were included in a development set. A new set of equations was obtained from these 17 profiles and was used to predict the profiles on the remaining nine holes (called the validation set). The prediction curves are compared with the observed data in Fig. 5 for four of the nine validation holes. The allocation of holes to development and validation sets was made prior to any data scrutiny or analysis.

View larger version (18K): [in this window] [in a new window]   Fig. 5 Comparison of data and predictions for four of the nine validation holes and Hole 27. In each panel, the measured ECe values are represented by solid points. The heavy solid line is the prediction from the logistic profile model, with the corresponding 95% pointwise confidence limits shown as a pair of short-broken lines. The light solid line is the prediction from the established-coefficient method, and the long-broken line is the prediction from linear regression. The prediction curves have been derived from the calibration data in the set of development holes (1–5, 7, 8, 12, 13, 17, 18, 20, 21, and 23–26)

Established-Coefficients
Calibration
To test this method of calibration, a set of equations was derived, using the approach of Corwin and Rhoades (1982), to a depth of 2.0 m. The equations were evaluated using the EM38 data generated in the field and EC_e determined in the laboratory for the depth increments 0.0 to 0.1, 0.1 to 0.2, and so on.

The predictive equations derived from the 26 calibration data are

(8a)

(8b)

and so on for the lower depths. These appear to be different from the linear regression equations, suggesting that an objective comparison is needed.

Prediction and Validation
As a means of validating this approach, a new series of coefficients and equations were obtained from the 17 profiles in the development set. These coefficients differed mildly from those given above. The regression relationships obtained were then applied to the measurements of EM_0,V and EM_0,H made at each validation site. Estimates of EC_e at successive 0.1-m depth increments and hence salinity distributions were obtained by using the resulting equations. The results, illustrated in Fig. 5, indicate the quality of the predictions of salinity distribution within these profiles. A major disadvantage of this method is that no prediction errors are available.

Logistic Model
Calibration
A logistic model (Eq. [4]) was fitted to each of the calibration EC_e profiles. Estimates ₁, ₂, ₃, and ₄ for each profile were then plotted against EM_0,H. The results of this are shown in Fig. 6 . Here ₁ showed a mild linear trend with EM_0,H, ₂ exhibited a strong linear trend, and ₃ and ₄ were constant.

View larger version (13K): [in this window] [in a new window]   Fig. 6 Estimates â1, â2, â3, and â4 (from fitting a logistic curve to ECe data from each calibration site) plotted against EM0,H. Each estimate is indicated by the number of its calibration site. The broken line is fitted by robust linear regression, and the solid line by a local robust smoother (lowess) recommended by Cleveland (1979). The fitted regression equations for the broken lines are , and .8561 (see Eq. [4] and [9])

(9)

For a given EM_0,H and given depth z of interest, the above logistic model can be used to predict the salinity. When using Eq. [9] at a new location, EM_0,H and EM_0,V should fall on the straight line in Fig. 2, so that the location is in the same domain as the test holes. However, the observed EC_e calibration profiles systematically deviate from the curve in Eq. [9] by small amounts.

The method of nonlinear mixed effects models was used to quantify this apparent variation (Jones, 1993). Hence a model of the form in Eq. [10] was used to replace the above model.

(10)

where i denotes the calibration hole, and j the depth increment. Additionally, y_ij = EC_z,ij to avoid the confusing number of subscripts. In this model there are six fixed-effects parameters µ = (a₁, b₁, a₂, b₂, a₃, a₄). There are also four random effects _i1, _i2, _i3, _i4. These parameters model how each observed logistic curve departs randomly as a curve from the deterministic curve

(11)

In Eq. [10] _ij models the measurement error and other sources of local random error associated with each measurement. In Eq. [10], the term EM_0,H was substituted with EM_0,H - 2.1 in order to achieve better numerical stability. Mathematically, the equation is not altered. Comparing the logistic-profile model with the original logistic (Eq. [4]), it is evident a₁ + b₁(EM_0,H - 2.1) + _i1 replaces ₁, a₂ + b₂(EM_0,H - 2.1) + _i2 replaces ₂, a₃ + _i3 replaces ₃, and a₄ + _i4 replaces ₄. The random effects _i1, _i2, _i3, _i4 are assumed to be Gaussian and independent between curves, but correlated within curves; they have mean 0 (a vector of length 4) and a 4 by 4 covariance matrix B with 10 unknown parameters. The errors _ij are assumed to be Gaussian with mean 0 and variance ², but, for a fixed hole i, the errors _ij may be correlated between depths. It is sometimes convenient to write Eq. [10] more succinctly in terms of a bivariate function f as

(12)

where '_i = (_i1, 0, _i2, 0, _i3, _i4) is the vector of random effects, µ' = (a₁, b₁, a₂, b₂, a₃, a₄) is the vector of fixed effects and is the vector of covariates.

The model (Eq. [10]) was fitted to the calibration sites by maximum likelihood using the nonlinear mixed-effects command in the statistical package S-Plus (Mathsoft Inc., 1998). The full model (Eq. [10]) was nearly singular, so _i4 was set to 0. A sequence of within-hole correlation models for the errors _ij was investigated: for independent errors the log-likelihood was -412.62; for autoregressive errors of order 1, the log-likelihood was -377.78; for autoregressive errors of order 2 it was -377.65; for moving average errors of order 1, it was -385.88; and for moving average errors of order 2, it was -375.62. By the Akaike information criterion interpreted according to Jones (1993), a model with autoregressive errors of order 1 is acceptable. This resulted in the following equation to predict EC_e with increasing depth:

(13)

The residual correlation between adjacent depth measurements within holes was estimated to be 0.43, which, although not large, could not be neglected in a well-fitting model.

It is possible that EM_0,V may be required as a predictor in addition to EM_0,H. This was investigated by fitting the model

(14)

which generalized Eq. [10] by including EM_0,V - EM_0,H as an extra predictor. This model has eight fixed-effects parameters (a₁, b₁, c₁, a₂, b₂, c₂, a₃, a₄) and four random effects _i1, _i2, _i3, _i4. As before, it was necessary to set . The log likelihood (with autoregressive errors of order 1 between depths within holes) was now -375.92. A log likelihood ratio test was used to check if this model is better. The test statistic should be compared with the 95% quantile of a ₂² distribution, namely 5.99, because there are two extra parameters in this model. Since 3.72 is well below 5.99, the simpler model (Eq. [10]) is adequate, and EM_0,V does not need to be used as a predictor in addition to EM_0,H.

The logistic profile model developed here provides a single statistically robust equation that can be used to predict EC_e at a particular depth of interest from EM_0,H, as opposed to the 20 semiempirical equations derived by adapting the method of Corwin and Rhoades (1982) or the 20 totally empirical linear regression prediction equations.

Prediction and Validation
Validation of the logistic mixed model was similarly attempted by refitting the model to data from holes in the development set, resulting in a new version of Eq. [10] with slightly different coefficients. The function was then used to make predictions every 0.10 m within each of the validation soil profiles. The results are presented in Fig. 5, which shows the measured EC_e at 0.10-m depth increments and the fitted logistic profiles.

It is apparent that the logistic profile model also captures the broad trends in salinity throughout each of the validation profiles.

Prediction Errors
It may be argued that any practical, useful calibration method must provide prediction errors. Approximate prediction errors can be calculated for the logistic profile model using local linearization. At a new location, only EM_0,H and EM_0,V will be measured. According to this fitted model, the prediction of EC_e at depth z will be given by (Eq. [13]). To calculate the prediction error, let

(15)

(16)

(17)

(18)

(19)

(20)

These are the derivatives of f in Eq. [12] with respect to a₁, b₁, a₂, b₂, a₃, a₄ evaluated at . Set and . Then statistical theory yields an (asymptotic) error of to be associated with ÊC_z (Vonesh and Carter, 1992). Here is the estimated variance–covariance matrix of the estimated fixed effects and is the estimated variance–covariance matrix of the random effects, both of which should be output by the statistical package used for fitting the nonlinear mixed-effects model. There are three components of variation in v_z: the first expresses uncertainty due to the estimation of the fixed effects, the second allows for variation due to the random effects ₁, ₂, ₃, ₄, and the third captures measurement and other local random error. The errors may also be estimated by simulating repeatedly from the fitted model; this avoids local linearization but is laborious.

A similar predictive equation to (Eq. [13]) was estimated from the 17 profiles in the development set, and point predictions and standard errors were calculated for each of the validation holes and Hole 27. The predictions Ê_z and approximate 95% confidence intervals Ê_z ± 2 are compared in Fig. 5. The agreement appears to be acceptable, except for Hole 27.

Comparison of Methods
The predictive capabilities of the three calibration methods may be compared qualitatively by inspecting Fig. 5. It is clear that the Corwin and Rhoades (1982) predictions and the linear regression predictions are almost identical in most holes, except possibly at the shallowest depth. Note that the Corwin and Rhoades (1982) predictions for the shallowest depth are obtained by direct calibration, whereas lower predictions involve differencing. The logistic model produces less erratic predictions than the other two methods. However, a more formal comparison is needed.

Accordingly, for each method and each validation hole i = 6, 9, 10, 11, 14, 15, 16, 19, and 22 (and also for Hole 27), the mean prediction sum-of-squares were calculated as

(21)

where Ê_ij is the value in hole i at depth interval j predicted from the model derived from the development set (Holes 1–5, 7, 8, 12, 13, 17, 18, 20, 21, and 23–26). The resulting values are presented in Table 1.

The mean prediction sums-of-squares within each row of this table are not independent, because they use the same values of EC_ij. Furthermore, the distribution of the prediction sum-of-squares is not likely to be straightforward, because of within-hole correlations between increments. However, it is valid to use rank methods, such as the Quade test or the Friedman test (Conover, 1980, Section 5.8), to analyze the summary data presented in Table 1. Such methods merely assume that the predictions are independent from hole to hole and that for a given hole the predictions may be ranked in some objective way. Thus, in the experimental layout in Table 1, holes are regarded as blocks and prediction methods as treatments, which may be ranked from one to three within blocks according to their mean prediction sums-of-squares. If the treatments are identical then each ranking within a block is equally likely. The Quade and Friedman tests both exploit this observation, albeit in slightly different ways. Conover (1980) recommends the Quade test over the better-known Friedman test when there are only three treatments, because it is then more powerful.

The Quade test statistic derived from the validation set (Holes 6, 9, 10, 11, 14, 15, 16, 19, and 22) is 3.71. To test if treatments are identical, this should be compared with 3.63, the 95% quantile of an F_2,16 distribution. The prediction methods are statistically different. Subsequent multiple comparisons (Conover, 1980, p. 297) suggest that the logistic model predictions are better than those from the established-coefficients method. The linear regression predictions are intermediate, but not significantly different from the other two methods.

Table 1 also sounds an obvious warning: the largest prediction errors will occur when a method is used outside its domain of application (Hole 27). In practice, it will be necessary to develop a quantitative procedure for deciding if a new site falls within the calibration domain. For example, the calibration EM_0,H and EM_0,V values may be modeled as bivariate normal (or log normal) and new EM readings may be compared with this distribution (the joint probability method). Alternatively, EM_0,H may be predicted from a newly acquired EM_0,V measurement using linear or nonlinear regression, and the acceptance criterion would be based on the difference between the observed and predicted EM_0,H values (the conditional probability method).

	Conclusions

TOP ABSTRACT INTRODUCTION Materials and methods Results and discussion Conclusions REFERENCES

 
The method of Corwin and Rhoades (1982) was shown to work as a calibration approach for predicting EC_e at 0.10 m to a depth of 2.0 m for the normal salinity profiles under investigation here. However, it predicts each increment separately, uses both EM_0,H and EM_0,V as predictors even when they are highly correlated, and involves a mix of theoretical and empirical coefficients in the predictive equations. Two parameters need to be estimated for each depth increment. This yields 40 unknowns in this investigation, which seems excessive. Further, prediction errors are not available.

A linear regression method (an adaptation of Rhoades and Corwin [1981]) was also used as a calibration method. It also predicts each increment separately. Prediction errors are available in principle, but to be realistic should take into account within-hole correlations between increments. However, the method requires the estimation of two unknown parameters per depth if EM_0,H is used as a predictor (or three if both EM_0,H and EM_0,V are used), so that 40 (or 60) unknown parameters need to be estimated for 20 depth increments. Thus, the linear regression method may be overparameterized, and the covariance matrix of the errors may not be estimable. An alternative approach was therefore developed.

The new methodology developed here involved fitting a mixed random and fixed effects logistic model to each of 26 calibration holes. The salinity profiles are therefore modeled as smooth curves, with deterministic and random components. Further, within-hole deviations may be serially correlated. It is possible to assign meaningful errors to the predictions at a new location where only EM_0,H and EM_0,V are measured. The method was shown to be slightly superior to that of Corwin and Rhoades (1982) on the data presented here and provides a smoother prediction profile than linear regression. It requires the estimation of only six fixed effects, so it is much more parsimonious than the alternatives.

Mixed nonlinear model fitting is a relatively recent development in the discipline of statistics, but commands for fitting such models have been incorporated in the latest versions of some major statistical software packages, such as S-plus (Mathsoft Inc.,1998). The nonlinear models are parametric, but current research is attempting to extend the method to nonparametric mixed models, resulting in a more flexible and widely applicable tool for researchers and practitioners. Hence, in the future, more complex salinity profiles (including inverted salinity profiles) than those described here could be modeled.

	ACKNOWLEDGMENTS

 
The authors are grateful to the Australian Cotton Research and Development Corporation for providing the senior author with a postgraduate scholarship and the provision of research moneys to carry out this work. We also acknowledge the very helpful comments of the referees.

Received for publication April 23, 1999.

	REFERENCES

TOP ABSTRACT INTRODUCTION Materials and methods Results and discussion Conclusions REFERENCES

 

• Cameron D.R., De Jong E., Read D.W.L., Oosterveld M. Mapping salinity using resistivity and electromagnetic inductive techniques. Can. J. Soil Sci. 1981;61:67-78.
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