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The relationships among irrigation regime, salt leaching and water uptake, and their effect on yields of bell pepper (Capsicum frutescensvar. ‘Maor’) and tomatoes (Lycopersicon esculentumvar. ‘Naama’) were studied in a gypsiferous desert soil. Trickle irrigation treatments (two irrigation frequencies: daily irrigation and irrigation every 4 d; and four different amounts of water, ranging from 0.6 to 2.2 times the Class A pan evaporation) were used to establish different leaching regimes. The changes in soil water content within the root zone were monitored by the neutron probe technique, and the vertical flux at the bottom of the root zone was determined from the hydraulic gradients (using tensiometers) and hydraulic conductivity values. Soil solution within the root zone was sampled by soil suction probes and was analyzed for electrolytic conductivity (EC), Cl−, and SO2−4. The soil exchangeable sodium percentage (ESP) as well as the crop yields were also measured. Results showed that the different irrigation regimes produced different soil water profiles and different leaching fractions ranging from 0.09 to 0.53 which, in turn, affected the soil salinity profiles. Analysis of the soil solution ionic composition showed that with the higher levels of leaching, the excess chloride was essentially removed, whereas considerable sulphate concentration was maintained as a result of gypsum dissolution which was dominated by the changes in the soil's ESP. Both water uptake (evapotranspiration) and crop yields were closely correlated with the soil water and salinity profiles. This indicated that the effect of both salinity stress and water stress is to reduce crop water uptake and, concurrently, to reduce crop yield.

Field data of bell pepper (Capsicum frutescens‘Maor’) yield (Y), soil water pressure potential (P), and salinity (EC) obtained under trickle irrigation, and of two soil hydraulic parameters (K1and α), were used to analyze spatial variability of these properties and its impact on relationships between crop yield and soil properties. Both conventional statistical analysis methods and spatial structure analysis methods were used. With the conventional methods, observations of each property were considered as statistically independent regardless of their spatial position in the field. Results of multiple regression analysis suggest the variability in K1and α is responsible for about 25% of the variabilities in both Pand EC, which in turn, are responsible for < 25% of the variability in Y, when experimental data from the entire field are considered. With the spatial structure methods, a variogram was used to investigate spatial structure of different properties, and a cross‐variogram was used to investigate spatial dependence of P and EC on K1and α and spatial dependence of Yon Pand on EC. Results show field spatial structure affects the relationships between different properties over distances which were related to the respective ranges of direct variograms of pertinent properties. The characteristic length of field spatial structure (in terms of integral scale, J) varied between J= 25 m (K1) and J=78 m (α) with an average of J=43 m. The direct variograms of properties were used to calculate kriging estimates of pertinent properties, which in turn, were used for two subsequent analyses. In the first analysis, kriging estimates obtained for a square domain (43 by 43 m in size) moving over the entire field were used to calculate the correlation between different properties at different parts in the field. The analysis showed that in some parts of the field variability in K1and α was responsible for about 90% of the variabilities in both Pand EC, which in turn, were responsible for about 90% of the variability in Y. In the second analysis, kriging estimates of Y, Pand EC obtained for regions in the field which are associated with a constant saturated zone in the vicinity of the trickle emitter (in terms of its radius ru= ru(K1, α; Q); Qbeing trickle emitter discharge) were used to analyze the dependence of Yon Pand on EC. The analysis showed that in regions which were associated with relatively small values of ru(ru< 0.05 m) the variability in Pand EC is responsible for > 0.8 of the variability in Y. In regions which were associated with relatively large values of ru(ru>0.08 m) and a relatively large leached zone, it was not soil water pressure or salinity, but other factors that most influenced yield.

Kriging techniques for interpolating spatial phenomena have recently been applied by soil scientists to map soil properties in heterogeneous field. A fundamental component of these techniques is the variogram which has to be estimated experimentally. A method to design an optimal sampling network for estimation of the variogram is proposed. Using a constant number of pairs of points (Np) per lag class, the criterion for selecting the location of the sampling points is the uniformity of the values of the lag vector, h, within a given lag class, for each of the lag classes which cover the field domain. For a given sample size, N, the method provides a set of scaling factors which in turn is used to calculate the new locations of the sampling points by an iterative procedure. Using the aforementioned criterion, the best set of the sampling points is selected. Analysis of the results showed that by using the proposed method, the variability of h within a given lag class, as well as the variability of h among the lag classes, was considerably reduced relative to the situation where the original locations were used. The method was found to be sensitive to both Nand Npas well as to the initial locations of the points. The effect of the method on the estimation of the variogram was tested using simulated realizations of a two‐dimensional stationary and isotropic field. The results for fifty different realizations showed that the variograms which had been estimated from data generated on the coordinates of the sampling points provided by the method were smoother and fitted the theoretical variograms better than those which were estimated from data generated on the original coordinates of the sampling points. When only the smaller lag classes were considered, the differences between the two sets of the variograms become smaller.

Results of integral scales and coefficient of variations (C.V.) from Part I are applied to evaluate errors in estimating average values of several hydraulic properties. Based on the Ergodic Hypothesis and on standard statistical theory, an equation is derived to calculate the total field area and number of measurements needed to estimate (for a prescribed error) ensemble averages of a given hydraulic property. For a 0.8‐ha field performing 90 measurements to estimate mean values of various hydraulic properties, the errors are in the range between 7 and 46%. Various combinations of number of integral scales (I) and number of measurements (N) are used to estimate mean values of a given hydraulic property with a given error. An “optimal” I‐Ncombination is derived. With this approach, 4 to 156 integral scales and 2 to 190 measurements are needed to estimate mean values of the various hydraulic properties with 20% error. The kriging technique and fictitious point method are used to evaluate number and locations of additional observation points needed to estimate mean value of the saturated hydraulic conductivity (Ks) over a finite field with a given accuracy. To obtain the C.V. of Ksof 2.5%, nine additional points are required.

Using actual field variability data and the stochastic approach, effects of spatial variability of soil hydraulic properties on the predictions of water and salt flow distributions, and of correlation structure, are evaluated. The governing water and salt flow equations are solved numerically using actual field data of soil hydraulic conductivity and differential water capacity. These data, which had been obtained previously from measurements made in 30 different locations in a field, were the same as those obtained from a Monte‐Carlo simulation that considered the correlations between the parameters of the hydraulic functions. Simulated values of pressure head (h), water fluxes (q), wetting front positions (Zf), and solute concentration (c) for each of the 30 locations in the field were used to estimate distribution types (normal or log‐normal) and its moments, for h, q, Zf, and c. The two‐point autocorrelation functions of pressure head, wetting front, and solute concentration have been calculated and used to estimate the integral scales of h, Zf, and c. Distances in the field in which the horizontal flow components were negligibly small compared with the vertical flow components were calculated and found to be in the order of 10 cm and larger. Field dispersion of solutes was calculated from the simulated salt distribution results. Values of field scale dispersivity were estimated to be larger than the pore scale dispersivity and to increase with soil depth. An “equivalent” uniform porous medium, which was defined at the soil surface, exists for large infiltration time when a unit hydraulic gradient may be assumed.

Determinations of the sorptivity (S) and five parameters describing the hydraulic conductivity [K(h)] and soil water retentivity [θ(h)] functions have been used to analyze the spatial distribution of the hydraulic properties in an experimental field. The parameters are: saturated hydraulic conductivity (Ks); water entry value (hw); saturated (θs) and residual (θr) water contents; and a constant β characterizing the pore size distribution of the soil. For a given depth, each of these parameters is described as a realization of a stationary two‐dimensional isotropic and random process. These stochastic processes are characterized by truncated normal or log‐normal probability density functions independent of the spatial position, and by autocorrelation functions between any two spatial points in the field which depend solely on the size of the vector separating the two points. The spatial variability of each of the six parameters has a structure that is characterized by a characteristic length—the integral scale, J, representing the largest distance for which the parameter is correlated with itself. Values of J, which are calculated from the autocorrelation functions for each parameter generally decrease with depth. On the average over depth the calculated values of Jare 21, 44, 55, 25, 35, and 15 meters for the parameters Ks, hw, θs, θr, S, and β, respectively. The spatial variability of the hydraulic functions are described by the probability density functions and the autocorrelation functions. Since the integral scales of K(h) and θ(h) vary with both hand depth, the characteristic length of the field has been chosen as the integral scale of the weighted mean diffusivity which is 18 m.

The effects of mixed Na‐Ca solutions on soil water diffusivity and unsaturated hydraulic conductivity were tested for a loamy soil from Gilat, Israel. Soil water diffusivity functions, D(θ), were determined by the horizontal infiltration method. Unsaturated hydraulic conductivity functions, K(θ), were determined both directly (by vertical infiltration for suction heads of 0–50 cm H2O) and indirectly (from diffusivity functions and soil water retention curves). Prior to determinations, soil samples were equilibrated with solutions containing a given concentration and composition of Ca and Na. The results show that D(θ) and K(θ) are independent of solution concentrations in a Ca‐saturated system. In mixed Na‐Ca systems, hydraulic conductivity and soil water diffusivity functions are highly dependent on the composition and concentration of the soil solution, in addition to their dependence on soil water content, θ. For any given θ, both K(θ) and D(θ) decrease as either soil solution concentration decreases or the sodium fraction in the soil solution increases. The negative effect of a combination of high Na‐to‐Ca ratio and low soil solution concentration on K(θ) is directly related to the degree of water saturation of the soil. Low values of θ can compensate for the negative effects of high Na‐to‐Ca ratio and low solution concentration.

A model is developed to diagnose the effect of Na/(Ca)1/2ratios and soil solution concentrations on the hydraulic conductivity for an unsaturated soil. The model takes into account the porous nature and the electrical properties of the soil. Interactions between ions in the exchange phase and in the liquid phase are expressed quantitatively by the diffuse double‐layer theory. Experimental data on the effect of soil solution and soil‐water pressure on clay‐platelet organization and structure were considered in the model. A test is made of the sensitivity of the model to variations in the various input parameters. The model is found to be sensitive to the relationship between the number of platelets per clay particle, and the ESP of the soil. The diagnostic model is compared with experimental data for three soils of different texture. The model approximates the hydraulic conductivity function for many practical combinations of Na/(Ca)1/2ratios and soil solution concentrations for the ranges of soil water content in which most of the water flow occurs.

The covariance function of the stochastic component of various soil properties plays an essential role in the estimation and interpolation of spatial phenomena using kriging techniques and in the solution of transport problems in heterogeneous media using stochastic continuum transport modeling. In this study the effect of the sampling network on the estimates of covariance functions of two‐dimensional, isotropic, second‐order stationary processes, characterized by different underlying correlation scales, was analyzed. Estimates of the covariance function based on a standard systematic sampling network were compared with estimates of the covariance functions based on a modified sampling network. Results of these comparisons showed that the modified sampling network was better suited than the systematic network to resolving the statistical properties of the underlying stochastic process at small separation distances. This may result in an improvement in structure identification. Analyses of the effect of the sampling network used for estimating the covariance function on the results of conditional inference procedures such as kriging showed that the kriging estimates were essentially insensitive to the geometric configuration of the sampling network. Uncertainties about these estimates were reduced when using the modified sampling network, however. In the case of unconditional inferences, such as with stochastic analyses of transport problems in heterogeneous media, predictions of the head variance provided by using estimates of the parameters of the covariance function of the log saturated hydraulic conductivity based on the modified sampling network fit the theoretical head variance better than those based on the systematic sampling network. The effect of the sampling network used for estimating the covariance function on the predicted head variance was significant, particularly when the underlying correlation scale was relatively small, when the porous medium texture became finer, and when the hydraulic conductivity variations were treated as being three‐dimensional.

The effect of irrigation water quality (salinity) and quantity on the yields of sweet corn (Zea mayscv. Jubilee) and cotton Gossypium hirsutumL. cv. Acala SJ‐2) was studied in a sandy loam (Typic Torrifluvent) soil. For each of the irrigation water salinities of Co= 1.8, 3.6 and 6.7 dS/m (for corn) and Co= 3.6, 6.7, and 10.5 dS/m (for cotton), 10 amounts of irrigation water (Q) ranging from 0.15 to 2.7 times the class A pan evaporation (Eo) were applied via trickle irrigation. Root zone soil‐water matric potential, water content, and salinity were monitored, and corn and cotton yield components were measured. Time‐averaged root zone soil water content (θ) and soil salinity (EC), as well as crop yield components (Y) were affected by both Qand Co. Results of the analyses of the crop‐water production functions suggest that increasing the amount of irrigation water may compensate only in part for the adverse effects of the salinity of the irrigation water. For corn, using irrigation water of Co= 3.6 and 6.7 dS/m, maximum ear yields were reduced by 8 and 27%, respectively, relative to the maximum ear yield obtained using water of 1.8 dS/m. For cotton, using irrigation water of Co= 6.7 and 10.5 dS/m, maximum seed cotton yields were reduced by 6.5 and 30%, respectively, relative to the maximum seed cotton yield obtained using water of 3.6 dS/m. Analysis of the Y(θ, EC) relationship for each of the corn and the cotton yield components suggested that, due to the mutual effect of θ and EC on Y, the crop response to salinity curve is affected by θ and, in general, should be described by a nonlinear concave expression rather than by the piecewise linear expression of Maas and Hoffman (1977).

When a deterministic irrigation regime is applied to a heterogeneous field, soil‐water content, soil salinity, and crop yield will vary in space because of the inherent spatial variability in the soil properties. It is important, therefore, to analyze their spatial distributions in the field in order to suggest improved irrigation management schemes for spatially‐variable fields. A stochastic approach was used to analyze the spatial variability of four parameters of the soil‐water retention curve as well as that of the initial soil‐water content and soil salinity, using actual measured field data. These properties were used as inputs for a simplified crop yield‐soil‐water model which, in turn, was coupled with the conditional simulation method to analyze the spatial distribution of the calculated soil‐water content (θ), soil salinity (C), and relative crop yield (Y) in an irrigated field, for various combinations of irrigation water quantity and quality (salinity). Analysis of the results showed that both irrigation water quantity and quality affected the spatial variability in θ, C, and Ybut essentially did not affect the spatial dependence of these properties. Although the average (spatial) response of the field was similar to that of a fictitious equivalent homogeneous field, the response of different sites in the field was quite different. For some combinations of irrigation water amount and salinity, the relative difference between relative crop yield obtained at different sites in the field exceeded 100%.

In a heterogeneous field in which the soil properties vary, under a deterministic irrigation regime, the crop yield might also differ from place to place in the field. These differences may, in turn, reduce the average (over the field) yield relative to that which would be obtained if the soil properties were uniform throughout the field. The simplified crop yield‐soil response model coupled with the conditional simulation method, presented in part I of this study, was used here to evaluate the spatial distribution of the amount of irrigation water required to maintain a prespecified relative crop yield, uniform throughout the field, for a given irrigation water salinity. By considering the water irrigation amounts at different sites in the field, rather than that of the entire field, it was shown that the amount of irrigation water required to obtain a given relative crop yield (for a given irrigation water salinity) can be reduced by 7 to 34%. The amount of water that could be saved increased as the level of the required crop yield increased and as the salinity of the irrigation water decreased. The spatial distributions of the amount of irrigation water required to maintain a given relative crop yield were combined with economic considerations in order to evaluate the spatial distribution of the optimal amounts of irrigation water for a given irrigation water salinity. It was shown that the optimal amounts of irrigation water can be reduced by 6 to 29%, when the responses of different sites in the field is considered rather than the average response of the entire field. Irrigation schemes based on imposition of engineering requirements on the spatial distributions of the optimal irrigation water amounts, increased the net benefit (price of the crop yield minus cost of the irrigation water) by 5.5 to 42% relative to that obtained under deterministic conventional water application schemes. The differences between the two schemes decreased as the irrigation water salinity increased.

The leaching characteristics of a stony desert soil were studied. The functional relationships between pressure potential head and gravimetric water content for the different soil layers were determined in the laboratory under drying conditions using samples in which the stony fractions (> 2 mm) were excluded. These gravimetric water content values were converted to field volumetric water contents, using the field‐measured values of the field bulk density and the stone fraction (on a mass basis). For a given pressure potential head, volumetric water content decreased with depth. Unsaturated soil hydraulic conductivity functions for the different soil depths were determined by the transient drainage profile method, using tensiometers and the appropriate soil water characteristics functions. The saturated hydraulic conductivities were estimated from the steady‐state infiltration rate and the appropriate hydraulic head gradients. Measured saturated hydraulic conductivity values decrease with soil depth while for a given pressure potential head, the unsaturated hydraulic conductivity increases with depth. A leaching experiment was conducted in the field, by operating a sprinkler system at a rate of 7.6 mm/h. The time variations of the electrical conductivity (EC) of the soil solution as well as the concentrations of Na+, Ca2+, Mg2+, Cl‐, and SO2‐4were recorded by extracting soil solution samples from four soil suction probes located at a depth of 0.45 m and spaced 12 by 12 m apart. The initial average EC of 165 dS/m was reduced to 7.5 dS/m as 800 mm of water were applied to the soil. This final value of the EC of the soil solution is three times higher than the EC of the irrigation water, as a result of the presence of gypsum and relatively high ESP in the solid phase of the soil profile. Measured values of the EC of the soil solution were used to estimate the value of the dispersivity for longitudinal dispersion, λ. The resulting λ = 2.91 cm is in relatively good agreement with published field data of λ for another soil, but is much smaller than the value of the equivalent field‐scale dispersivity calculated for two different soils.

A procedure for repetitive determinations of soil hydraulic properties is based on the assumption that the functional relationships between soil water content (θ) and soil water pressure head (h), and between the hydraulic conductivity (K) and h, can be expressed mathematically. The parameters in the mathematical expression are: saturated hydraulic conductivity (Ks) and water content (θs), residual (or irreducible) water content (θr), water entry value of h(hw), and the parameters η and β, characterizing the pore size distribution of the soil. Values of Ksand hware measured by the air‐entry permeameter of H. Bouwer. The parameters η and β were calculated with measured values of Ks, hw, θs, θr, and the sorptivity, S, as measured by C. Dirksen's method. The variability in the determination of the K(θ) function is evaluated in the field, from 30 measurements of each of the six parameters made in a single 2 by 2 m site. This variability is smaller than the variability of K(θ) as calculated from a simulation of a transient drainage. The K(h) and θ(h) functions are rapidly obtained and thus suitable for performing large numbers of measurements for the determination of the probability density functions (and joint probability density functions) of the soil hydraulic conductivity, K(θ), and soil water retentivity, h(θ), functions.

Field data for soil water characteristics and hydraulic conductivity have been scaled by two methods. The set of scaling factors (αj) determined by one method depends on soil water content. A correlation of r2= 0.86 is obtained between scaling factors determined from soil characteristic data and those determined from hydraulic conductivity data. In the second method, the set of αjis normally distributed, and is highly correlated with αjobtained by the first method from water characteristic data at water contents near saturation. In one method the sum of squares, about an average curve, was reduced by 44 and 66%, for the water characteristics and hydraulic conductivities data, respectively, while 92 and 82.5% reduction was obtained by the second method, for the same respective data. The results suggest that in practice, a single stochastic variable may represent the hydraulic properties of a heterogeneous field.