Soil water content interpolation using spatio-temporal kriging with external drift

Transcription

1 Geoderma 112 (2003) Soil water content interpolation using spatio-temporal kriging with external drift J.J.J.C. Snepvangers*, G.B.M. Heuvelink, J.A. Huisman Institute for Biodiversity and Ecosystem Dynamics (IBED), Centre for Geo-Ecological Research (ICG), Universiteit van Amsterdam, Nieuwe Achtergracht 166, 1018 WV Amsterdam, The Netherlands Received 19 November 2001; accepted 14 October 2002 Abstract In this study, two techniques for spatio-temporal (ST) kriging of soil water content are compared. The first technique, spatio-temporal ordinary kriging, is the simplest of the two, and uses only information about soil water content. The second technique, spatio-temporal kriging with external drift, uses also the relationship between soil water content and net-precipitation to aid the interpolation. It is shown that the behaviour of the soil water content predictions is physically more realistic when using spatio-temporal kriging with external drift. Also, the prediction uncertainties are slightly smaller. The data used in this study consist of Time Domain Reflectometry (TDR) measurements from a 30-day irrigation experiment on a m grassland in the Netherlands. D 2002 Elsevier Science B.V. All rights reserved. Keywords: Geostatistics; Space time interpolation; Soil hydrology 1. Introduction Sampling and monitoring often put a heavy load on the budget of environmental studies. Techniques that can increase the insight in the spatio-temporal (ST) distribution of an environmental variable, without increasing the measurement effort, are therefore valuable. Geostatistics offers a variety of techniques to make optimal use of measurement information for interpolating variables in space (S). However, many branches within the earth and environmental sciences deal with variables that vary not only in space but also in * Corresponding author. Present address: Netherlands Institute for Applied Geosciences, P.O. Box 80015, 3508 TA Utrecht, The Netherlands. addresses: j.snepvangers@nitg.tno.nl (J.J.J.C. Snepvangers), g.b.m.heuvelink@science.uva.nl (G.B.M. Heuvelink), s.huisman@science.uva.nl (J.A. Huisman) /02/$ - see front matter D 2002 Elsevier Science B.V. All rights reserved. PII: S (02)

2 254 J.J.J.C. Snepvangers et al. / Geoderma 112 (2003) time (T). For instance, within soil science, there are numerous dynamic spatial attributes, such as soil water content, infiltration rate, water pressure and solute concentration. Kyriakidis and Journel (1999) presented a thorough review of the status of ST geostatistical techniques. Most of the techniques mentioned in their review only make use of measurements of the variable of interest itself, so-called primary information. In many studies, however, secondary information is available as well. For example, other variables are measured that show a (strong) relationship with the variable under study. Using such relationships in interpolation, by co-kriging or kriging with external drift, may decrease the prediction uncertainties (Goovaerts, 1997). Interpolation in the full ST domain offers new possibilities for these techniques, as dynamic relationships can also be taken into account. The aim of this study is to show how a dynamic physical relationship between soil water content (h) and net-precipitation can be used to improve h interpolations. h depends on the amount of water leaving and entering the soil and on soil hydraulic properties. Netprecipitation is an important characteristic in this system as it largely determines the soil water fluxes at the top of the system. As compared to the other fluxes in the system (the fluxes to the groundwater and the horizontal fluxes), the net-precipitation flux is often relatively large. Furthermore, net-precipitation can be measured relatively easy. We compare ST ordinary kriging (ST-OK), which ignores secondary information, with ST kriging with external drift (ST-KED), which employs net-precipitation as secondary information. We demonstrate how KED can be used in the ST domain and analyze the advantages and disadvantages of ST-KED compared to ST-OK. The data for this study were obtained from an irrigation experiment on a m grassland in the south of the Netherlands, which we will refer to as the Molenschot dataset. 2. Spatio-temporal geostatistics Extending S interpolation techniques to the ST domain is not simply adding another dimension, as there are some fundamental differences between the space and time domain (Christakos and Vyas, 1998; Kyriakidis and Journel, 1999; Rouhani and Myers, 1990). Space represents a state of coexistence, in which there can be multiple dimensions (or directions) and interpolation is usually of main interest. Time on the contrary represents a state of successive existence, a clear ordering (nonreversible) in only one dimension is present and extrapolation is usually of main interest. Moreover, the origin of the S and T variation can be different. For example, in the case of h, one can imagine that the T behaviour is dominated by net-precipitation and drainage, whereas S variation in h depends more on soil texture, soil physical properties and vegetation. The difference in origin of variation can lead to strong anisotropic behaviour, both geometric and zonal. In recent years, progress has been made in building ST geostatistical models in several scientific disciplines, for instance, in environmental science (e.g. Buxton and Pate, 1996; De Cesare et al., 1996, 2001a,b; Angulo et al., 1998; Christakos and Vyas, 1998; Kyriakidis and Journel, 2001), agronomy (e.g. Stein et al., 1994; Hoosbeek, 1998), meteorology (e.g. Handcock and Wallis, 1994; Bogaert and Christakos, 1997a; Cressie and Huang, 1999; Bechini et al., 2000), hydrology (e.g. Rouhani and Myers, 1990;

3 Bogaert and Christakos, 1997b) and soil science (e.g. Comegna and Vitale, 1993; Heuvelink et al., 1996). Basically, the aim of these studies is the same, namely to predict an attribute z={z(s, t)jses, tet} defined on a geographical domain SoR 2 and a time interval ToR, at a space time point (s 0, t 0 ), where z was not measured. The prediction is to be based on n measurements at n ST points (s i, t i ), i =1,..., n. To predict z(s 0, t 0 ), it is assumed that z is a realization of a ST random function (ST-RF) Z, and Z(s 0, t 0 ) is predicted conditional on the measurements z(s i, t i ). The ST-RF model Z consists of a trend component representing some average behaviour of the ST process (m) and a zero-mean residual component (e): Zðs i ; t i Þ¼mðs i ; t i Þþeðs i ; t i Þ i ¼ 1;...; n: ð1þ 2.1. The trend component J.J.J.C. Snepvangers et al. / Geoderma 112 (2003) The simplest way to model the trend component m(s, t) is to assume an unknown constant mean. Interpolation can then be carried out using ST ordinary kriging (ST-OK). When the assumption of a constant mean is not realistic, a trend must be taken into account. A simple option is to detrend the data beforehand, after which ST-Simple Kriging can be used for interpolation (Angulo et al., 1998; De Cesare et al., 2001b). However, uncertainties in the detrending procedure are not taken into account in further analysis. This causes the interpolation uncertainty to appear lower than it is. It is also possible to model the trend component as a linear trend function, consisting of the sum of products of some known base-functions and some unknown coefficients: mðs; tþ ¼ Xp i¼1 f i ðs; tþb i : ð2þ In the simplest case, the base-functions are the coordinates (x,y,t) (universal kriging). When secondary information is available, then this may also be used to define the basefunctions (kriging with external drift). Bogaert and Christakos (1997a), for example, use altitude as secondary information in their ST study of thermometric data The residual component The residual in Eq. (1) can be characterised by the ST semivariogram, c(s i, s j, t i, t j ): cðs i ; s j ; t i ; t j Þ¼ 1 2 E ðeðs i ; t i Þ eðs j ; t j ÞÞ 2 : ð3þ Under appropriate stationarity assumptions, an estimate of the ST variogram may be obtained from the measurements by computing the experimental semivariogram ĉ(h S, h T ): ĉðh S ; h T Þ¼ 1 2Nðh S ; h T Þ Nðh X S ;h T Þ i¼1 ½eðs; tþ eðs þ h S ; t þ h T ÞŠ 2 ð4þ

4 256 J.J.J.C. Snepvangers et al. / Geoderma 112 (2003) where h S and h T are the S and T lags and where N(h S, h T ) is the number of pairs in the ST lag. Fitting a model to the ST experimental semivariogram has some additional problems over conventional semivariogram modelling, due to the distinct differences between the S and T variation. One way of coping with these problems is to use completely separate S and T semivariance structures and to model the total ST semivariance as the sum of these structures. Although this approach facilitates the structural analysis, it has some important drawbacks that are caused by the strict separation. For instance, the ST separation means that knowing the attribute value at three corners of a rectangle in the ST domain completely determines the attribute value at the fourth corner (Heuvelink et al., 1996). This implies that the S behaviour must be the same for all time points and that the T behaviour must be the same for all space points. However, this is not what we see in practice, where different spatial patterns emerge at different times and where time series at different locations show different behaviour. Complete ST separation is therefore unrealistic from a physical perspective. Furthermore, when measurements at all four corners are collected, a singularity problem will be encountered (Rouhani and Myers, 1990). Separate product structures, such as suggested by Rodriguez-Iturbe and Mejia (1974), Bogaert (1996) and De Cesare et al. (1996), may overcome the singularity problem. However, they still do not model space time interaction. Consequently, these structures are severely limited in their ability to fit the data well (Cressie and Huang, 1999). From a mathematical statistical standpoint, a variety of more advanced permissible nonseparate semivariance structures, which do not suffer from the above drawbacks, have been proposed (Cressie and Huang, 1999; De Cesare et al., 2001a; De Iaco et al., 2002). Although these structures are mathematically correct, they often lack physical support and are somewhat artificial. Therefore, environmental scientists often do not feel comfortable with them. Bilonick (1988) presented a simple nonseparate model form. He proposed an extension of the separate-sum models using geometric and zonal anisotropy to solve the problems arising from the differences in S and T variability. In the Bilonick model (Eq. (5)), the residual component is divided in three parts. These are an S part e S (s), a T part e T (t) and an ST part e ST (s, t) that only comprises geometric anisotropy and no zonal anisotropy: eðs; tþ ¼e S ðsþþe T ðtþþe ST ðs; tþ: ð5þ Assuming these three parts to be second-order stationary and mutually independent, the semivariogram of e(s, t) is the sum of three components: cðh S ; h T Þ¼c S ðh S Þþc T ðh T Þþc ST ðh ST Þ: ð6þ ThepST lag h ST is obtained by introducing a geometric anisotropy ratio a: h ST ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi h 2 S þ ah2 T. The advantage of the Bilonick model is that it has S, T and ST components that can be fairly easily interpreted in a physical sense (Heuvelink et al., 1996). The disadvantage is that estimation of the model parameters is not easy. Also, by introducing the space time anisotropy ratio a, it is assumed that distances in space and time can be reduced to a single space time distance. This may not be very realistic in all practical situations. Model

5 building is all about providing a sufficiently realistic description of the real world while still being able to identify the model parameters and apply the model. In balancing these two concerns, the level of complexity of the Bilonick model seems appropriate for the case study investigated here Spatio-temporal kriging J.J.J.C. Snepvangers et al. / Geoderma 112 (2003) When models for the trend and the residual are obtained, ST kriging can be carried out; either ST-OK or ST-KED. The equations for kriging in the ST domain are exactly the same as the standard S kriging equations. One should be aware, though, of the consequences of kriging in the T domain. Future measurements influence present predictions just as much as past measurements, because they are both weighed using the same semivariogram. This may lead to physically unrealistic results, especially when sudden inputs in the system occur. One may choose to use only past measurements for interpolation, but this causes a loss of information. ST-KED can reduce the unrealistic effects without ignoring information because the sudden inputs can be incorporated in the base-functions of the linear trend. For the OK and KED equations and their derivation, we refer to geostatistical handbooks, for example, Goovaerts (1997). 3. Molenschot dataset In the summer of 2000, we carried out an irrigation experiment on a grassland (60 60 m) located in Molenschot, the Netherlands (51j35VNand4j52VE). The soil was classified as a Plaggept on sandy loam (US soil taxonomy; USDA, 1975). Sprinklers with different ranges and intensities created an S pattern of h on the grassland on two occasions in a 30-day monitoring period (August 16 September 14, 2000). Drying and re-wetting by natural precipitation caused the S pattern to change over time. We chose to do this type of irrigation experiment with a distinct irrigation pattern on a relatively small field, as we needed a strong S and T structure in h to make semivariogram modelling worthwhile and to obtain a good insight in the net-precipitation information used by ST-KED. We monitored h with vertically installed 10-cm Time Domain Reflectometry (TDR) probes. TDR measures the propagation velocity of an electromagnetic wave along parallel metallic rods inserted in the soil. The velocity depends on the permittivity (K a ) of the soil. Permittivity can be translated to h, as the permittivity of water ( F 80 at 20j) is much larger than that of the other soil constituents (air: 1, solids: F 4 8) (Topp et al., 1980).We used a site specific p calibration equation to convert the permittivity to h. The calibration curve, h ¼ 0:1116 ffiffiffiffiffi K a 0:1543, has an R 2 of 98.7% and a standard error of m 3 /m 3. We collected TDR measurements both manually and automatically. For the manual measurements, a probe was manually installed at each of 229 locations at every measurement time (Fig. 1). It is impossible to reinstall a probe at exactly the same location, and therefore, we chose to reinstall within 15 cm of the exact measurement

6 258 J.J.J.C. Snepvangers et al. / Geoderma 112 (2003) Fig. 1. Locations of the TDR and meteorological measurements. Shades of grey represent the irrigation pattern. location. The displacements were taken into account in the geostatistical analysis. In total, there were 19 manual measurement rounds (Fig. 2). The probes for automatic measurements remained installed throughout the whole experimental period at 34 locations (Fig. 1). They were connected to two computercontrolled measurement systems (Heimovaara and Bouten, 1990). The two systems, A and B, caused clustering of the automatic probes as the quality of the TDR measurements decreases with cable length (Heimovaara, 1993). Automatic measurements were carried out every 15 min. The measurements started at August 12 for system A and at August 18 for system B. There is a large difference between the number of automatic and manual measurements (Table 1). To balance the number of automatic and manual measurements and their distribution over the T and S domain, we drew at random 6% of the automatic measure-

7 J.J.J.C. Snepvangers et al. / Geoderma 112 (2003) Fig. 2. Course of the daily average NP natural during the experiment, displayed with the irrigation and manual TDR measurement times. ments for further analysis and omitted the other 94%. Table 1 shows that the data reduction has a negligible effect on the summary statistics. We also carried out meteorological measurements to gain information on the netprecipitation. We distinguished between natural net-precipitation, NP natural, and total netprecipitation, NP total. NP natural is the input to the topsoil from precipitation P(t) minus the output from the topsoil through actual evapotranspiration ET a (t): NP natural ðtþ ¼PðtÞ ET a ðtþ: ð7þ We assumed that NP natural was constant over space. We further assumed that the potential evapotranspiration (ET p ) calculated with the Penman equation was a satisfactory estimate of ET a, since during the monitoring period, no water shortage occurred in the field. For obtaining ET p, we measured air temperature, relative humidity, wind velocity, and net-radiation in the southwest corner of the field (Fig. 1). Table 1 Statistical summary of the TDR measurements N Minimum Maximum Mean Standard deviation All Manual Automatic System A Automatic System B All Automatic All % Automatic + All Manual

8 260 J.J.J.C. Snepvangers et al. / Geoderma 112 (2003) Table 2 Number, type and characteristics of sprinklers used for the S irrigation pattern Shape Size Number Estimated area a [m 2 ] Average intensity a [mm/h] Number of measurement cups Round Large Round Medium Round Small Square Medium With number of raingauges, we mean the number of measurement cups used to estimate the S pattern of a sprinkler type. The total net-precipitation NP total is NP natural plus irrigation I(s, t): NP total ðs; tþ ¼NP natural ðtþþiðs; tþ: ð8þ Due to the irrigation, NP total varied not only in time but also in space. We irrigated approximately one third of the field with 52 sprinklers of four different types (Table 2) at two dates: August 17 (day 230) and September 1 (day 245). At both days, we started the irrigation early in the morning, respectively, at 5.42 and 4.00 AM, to prevent evaporation during irrigation. Both irrigations lasted 4 hours. To obtain the S distribution of the irrigated water, we assumed that all sprinklers of one type had the same irrigation characteristics. This allowed us to measure the distribution of irrigated water around one sprinkler per sprinkler type and translate this to a total irrigation pattern. In Fig. 1, the irrigation patterns for the two irrigation dates are shown. In Fig. 2, the course of the daily natural net-precipitation is shown together with the irrigation dates and the manual TDR rounds. 4. Application of spatio-temporal interpolation to the Molenschot dataset 4.1. Spatio-temporal ordinary kriging The first step in the analysis was to carry out ST-OK. The only prerequisite for ST-OK is a model of the ST semivariance structure The ST semivariogram Fig. 3 (top graph) shows the experimental semivariogram for the ST-OK case. There are clear differences in the behaviour of the semivariance in the space and time directions. In the S direction, there is a strong increase in semivariance until 5 m and a less pronounced increase up to 10 m. In the T direction, a periodicity with a period of approximately 15 days stands out. This periodicity can be attributed to the fact that both the irrigation days and the heavy rainstorms occurred with intervals of approximately 15 days (Fig. 2). The largest semivariance in the time direction can be found at lags of about 9 days. In the marginal semivariograms for ST-OK (Fig. 4), the differences in the semivariance behaviour in the space and time directions are more clearly visible. In addition to the

9 J.J.J.C. Snepvangers et al. / Geoderma 112 (2003) Fig. 3. Experimental semivariograms for ST-OK (a), ST-KED (linear) (b) and ST-KED (logarithmic) (c).

10 262 J.J.J.C. Snepvangers et al. / Geoderma 112 (2003) Fig. 4. Marginal experimental semivariograms for ST-OK, ST-KED (linear) and ST-KED (logarithmic): S: c(h S,0) (a) and T: c(0, h T ) (b). already mentioned differences, it can be seen that the T behaviour is much smoother than the S behaviour. Besides this, a substantial nugget effect is present in the space direction, whereas it is absent in the time direction. This is caused mainly by small-scale S variation due to texture, vegetation differences, and animal activity (among others molehills). The marginal semivariograms were used to obtain the model forms of the S and T model parts of the Bilonick model. The S part was modelled with a nugget model plus an exponential model. The T part was modelled with solely an exponential model. The T periodicity was not taken into account as it hardly influences the interpolation, because the period is large in relation to the observation density. An idea of how the ST part should be modelled cannot be obtained using marginal semivariograms, but because the other two components showed exponential behaviour, we decided to model the ST part with an exponential model as well. This resulted in a total semivariance model for the ST-OK case: cðh S ; h T Þ¼p S Nugð0Þþq S expðr S Þþq T expðr T Þþq ST expðr ST Þ ð9þ with p S being the S nugget value, q S, q T and q ST being the sill parameters of the S, T, and ST model parts and r S, r T and r ST being the range parameters of these parts, respectively. Recall that the ST semivariance model also contains the ST-anisotropy parameter a.

11 J.J.J.C. Snepvangers et al. / Geoderma 112 (2003) Fitting the model to the experimental data is difficult due to the fact that eight parameters must be estimated. We used a weighted least squares method minimizing: nrxof lags i¼1 w i ðĉðh Si ; h Ti Þ cðh Si ; h Ti ÞÞ 2 ð10þ where the weighing factor w i is the quotient pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi of the number of pairs in the lag N(h Si,h Ti ) and the square root of the semivariance ( ĉðh Si ; h Ti Þ). To prevent the iteration to get stuck in a local minimum due to the initial choice of parameters, parameters were fitted with 500 randomly chosen initial parameter sets. The optimization algorithm we used is a standard Matlab subspace trust region algorithm based on the interior-reflective Newton method (Coleman and Li, 1996). Mean values and standard deviation bands for the eight parameters are visualized in Fig. 5. All parameters, except for the temporal range, are estimated rather accurately, suggesting that the influence of local minima is small. The large standard deviation in the temporal range is explained by the periodicity in the marginal temporal semivariogram (see Fig. 4b). This causes the optimized ranges to follow a bimodal distribution. In some cases, the optimization algorithm yields a range of about 8 days, in other cases, a range of about 18 days Spatio-temporal interpolation We examined the behaviour of the ST-OK h predictions at different times around the irrigation times. We used S-block/T-point kriging with 11 m S-blocks, as we were not interested in the variability of h at a smaller S support than 1 m 2, but we were interested in interpolation at exact time points. The top row of Fig. 6 gives ST-OK interpolations at Fig. 5. Parameters for the Bilonick model for ST-OK, ST-KED (linear) and ST-KED (logarithmic) (displayed for each parameter in that order; y: average, -: one standard deviation bands).

12 264 J.J.J.C. Snepvangers et al. / Geoderma 112 (2003) Fig. 6. h Interpolations using ST-OK, ST-KED (linear) and ST-KED (logarithmic) for three time points on day 230 (August 17, 2000): 2.30 AM, AM and 8.00 PM. three time points on August 17. Due to local dryer and wetter areas, caused by small-scale spatial variation, all three maps show some spotting. The overall pattern is clear, though. Most striking is that the pattern of irrigation (see Fig. 1) is already visible (as a darker area in the north-east) at 2.30 AM in the morning (left), three hours before the irrigation started. This is because future measurements influence predictions as much as past measurements in kriging, as was mentioned before. At AM (middle) and 8.00 PM (right), it can be seen that the irrigation caused a strong increase in h in a large part of the field and that this area stayed wet. The dry corner in the north-east of the field is caused by a large tree, 5 m outside our study area. To examine the T behaviour, a test location (x = 31.55, y = 19.00; Fig. 1) was selected to test how well measured h time series could be reproduced by ST-OK. The automated TDR measurements of system B were not included in the interpolation as the coverage in the T domain is very high around automated TDR locations, leaving little challenge for interpolation. As the comparison here was against point measurements, we used ST point kriging. In Fig. 7 (top graph), measured and predicted h at the test location are displayed. At times without sudden precipitation, measurements and ST-OK predictions are within the uncertainty limits of plus or minus one standard deviation, but close to irrigation times and heavy rainstorms, the measurements are much higher than the predictions. This is due to the smoothing effect of kriging. In Fig. 8, the kriging standard

13 J.J.J.C. Snepvangers et al. / Geoderma 112 (2003) Fig. 7. Measured versus predicted h time series at location (x,y)=(31.55,19.00) for ST-OK (a), ST-KED (linear) (b) and ST-KED (logarithmic) (c). The dashed lines show the + or one standard deviation bands. deviation is displayed at the test location. Close to the manual measurement times the ST-OK kriging standard deviation is small, but further away, the uncertainty quickly increases.

14 266 J.J.J.C. Snepvangers et al. / Geoderma 112 (2003) Fig. 8. Kriging standard deviation at location (x,y)=(31.55,19.00) for ST-OK, ST-KED (linear) and ST-KED (logarithmic) Spatio-temporal kriging with external drift The next step in our analysis was performing ST-KED. This requires a linear trend model to incorporate the dynamic secondary information and a semivariance model of the residuals The linear trend model We assumed net-precipitation, here NP total, to be the major influence on the inputs and outputs of water from the soil and therefore on h. It was difficult to decide though, how long in the past net-precipitation information is still informative for the h state at any given point in time t 0. We therefore calculated the cumulative amount of NP total over several intervals with different lengths back in time: NP p ðt 0 Þ¼ Z t0 p t 0 NP total ðtþdt ð11þ with time interval length pa(0.5, 1, 2, 3, 6, 9, 12, 18, 24, 36, 48, 72, 96, 120, 144) in hours. After calculation of the individual NP p s, stepwise multiple regression with the h data being dependent on the NP p s was carried out. In this way, the most informative netprecipitation delay periods were selected. The rule we used for selection is a significant (95%) increase of at least 1% in explained variance (R 2 ). Selection of several netprecipitation delay periods makes it possible to obtain a weighted NP total function with recent NP total weighing stronger than less recent NP total. By doing so, we assume that the relationship between h and NP p is linear. When we look at an example scatter plot of NP 96 versus h (Fig. 9), it becomes clear that assuming linearity is unrealistic because the natural maximum and minimum h values, respectively, the saturated h and the residual h, are ignored when assuming a linear model.

15 J.J.J.C. Snepvangers et al. / Geoderma 112 (2003) Fig. 9. Example of the cumulative net-precipitation (NP 96 ) h relationship and three models describing this relationship: linear, sigmoidal and logarithmic. A Bolzmann sigmoidal curve may resemble the relationship of NP p with h more closely. This model has four parameters: mðs; tþ ¼A þðb AÞ 1 : ð12þ 1 exp C NPpðs;tÞ D Parameter A represents a minimum value, B represents a maximum, C represents a horizontal shift and D represents the slope between the levels A and B. The model can be rewritten in a linear trend function with two base-functions with f 1 =1, b 1 = A, 1 f 2 ðs; tþ ¼ 1 exp C NPp D and b 2 = B A. A disadvantage of this model is that it is hard to estimate parameter A, the residual h, as this is far outside the measurement range. Furthermore, the model cannot be written in a linear form where all four parameters are to be estimated, as parameters C and D are integrated in base-function f 2, which should be a known factor. An alternative way to model the nonlinear relationship between NP p and h is to use a logarithmic form: mðs; tþ ¼E þ F lnðnp p ðs; tþþgþ ð13þ where E, F and G represent the vertical shift, the steepness and the horizontal shift of the model, respectively. This model can also be rewritten in linear form with two basefunctions f 1 = 1 and f 2 (s, t)=(ln(np p + G)), and coefficients b 1 = E and b 2 = F. As before, parameter G cannot be isolated as it is integrated in base-function f 2. However, this is less problematic than with the sigmoidal model because the horizontal shift is not a very sensitive parameter, in contrast to the combination of slope and horizontal shift in Eq. (12). The horizontal shift can be estimated beforehand based on the minimum NP p. Problems do occur with the logarithmic model at low h values, where minus infinity can be reached. For the range of h values in this study (Table 1), this is of minor importance.

16 268 J.J.J.C. Snepvangers et al. / Geoderma 112 (2003) In Fig. 9, the NP 96 h relationship and the three fitted models are given. Clearly, the A parameter of the sigmoidal curve cannot be estimated very well. Based on this and the fact that the logarithmic model can more easily be rewritten in linear form without loss of important parameters, we chose to use the logarithmic model to describe the nonlinear relationship between h and net-precipitation. The G parameter was chosen to be the minimum of the NP p data per delay period rounded down to the closest integer. As with the linear model, stepwise multiple regression with the same selection rule was used to determine which net-precipitation delay periods best explained h. In order to judge whether using the nonlinear model really improves the interpolations, both models were used in further analysis. The former being referred to as ST-KED (linear) and the latter as ST-KED (logarithmic). The results of the multiple regression of ST-KED (linear) and ST-KED (logarithmic) are given in Table 3. It is remarkable that the selected net-precipitation delay periods are rather long for both models. For both models, the shorter periods were just outside the selection range. This shows that h in our study area has a long memory regarding netprecipitation The spatio-temporal semivariogram The residuals from the multiple regression were used in the semivariance analysis. First the experimental ST semivariograms were calculated (Fig. 3). As with the ST-OK case, we chose exponential models for the S, T and ST parts of the semivariogram model. The ST-KED model parameters and their standard deviations are again visualized in Fig. 5. Comparing the ST-OK case with the ST-KED cases (Figs. 3 5) makes clear that there is a strong decrease in T and ST sills, which is caused by the detrending procedure. For the S direction, the effect of detrending is hardly visible. For the S nugget, this can easily be explained because no information of the small-scale variability was used in the ST trend. With regard to the S sill, it must be concluded that the contribution of the irrigation pattern to spatial variation in h was not very strong. Apparently there are more important sources of spatial variation in h such as soil texture, soil physical properties and vegetation. However, since these were not known in a spatially exhaustive manner, they could not be incorporated in the trend. Table 3 Results of the stepwise multiple regression for ST-KED (linear) and ST-KED (logarithmic); order of included parameters with explained variance R 2 and increase in R 2 per step ST-KED (linear) R 2 R 2 increase ST-KED (logarithmic) R 2 R 2 increase NP NP NP NP NP NP NP NP NP NP The parameters included in the final linear trend models are in bold.

17 J.J.J.C. Snepvangers et al. / Geoderma 112 (2003) Spatio-temporal interpolation As with ST-OK, the application of ST-KED is shown by mapping in space and time. The middle and bottom rows of Fig. 6 give the results for ST-KED (linear) and ST-KED (logarithmic) interpolation, respectively. Most striking is the clear imprint of the irrigation pattern in the time points following irrigation. Before irrigation, at 2.30 h, the h map shows a fairly even h picture, this in contrast to the ST-OK map, but in accordance with measurements. In the middle and bottom graphs of Fig. 7, the predicted ST-KED (linear) and ST-KED (logarithmic) time series at the test location are shown. As the h measurements used for the interpolation are the same, the average behaviour of both ST-KED predictions closely resembles the ST-OK predictions, with a slight underestimation of the h at the test location. It is striking though, that ST-KED predictions show more h variations in between manual measurement rounds than the ST-OK predictions. At some time points, even the daily cycle of h becomes clear. The occurrence of this small-scale variation is the result of the small T measurement support (15-min) of the net-precipitation measurements. Fig. 7 also shows that the ST-KED predictions follow the sudden increases in h much better than the ST-OK predictions. When comparing the two ST-KED variants, we see that the ST-KED (logarithmic) predictions behaves best at the sudden increases. This is related to the incorporation of the nonlinearity in the ST-KED (logarithmic) model. By levelling off the relationship between h and net-precipitation close to the saturated h, with the ST- KED (logarithmic) model, a strong increase in h is possible at lower net-precipitation amounts than with the ST-KED (linear) model. The prediction standard deviation at the test location in Fig. 8 shows a different pattern for ST-KED and ST-OK. The main cause for this is that ST-KED considers both the uncertainty due to the ST semivariogram and the uncertainty in estimating the trend parameters. Therefore, the strong decrease in uncertainty close to ST measurement points disappears and the strong increase away from measurement points is lowered because netprecipitation information is available at all points. The strong increase in uncertainty of ST-KED (logarithmic) around day 240 is a local excess, probably due to a local lack of data to adequately fit the linear trend model. 5. Discussion and conclusions In this paper, we showed how the relationship between net-precipitation and h can be used in ST-KED. It can be concluded that this kriging technique, which uses dynamic secondary information, has some clear advantages over ST-OK. Even though the h measurement coverage over the ST domain was high, ST-KED resulted in a decrease in prediction uncertainty. Another improvement was the physically more realistic behaviour of the ST-KED predictions. Especially the ST-KED (logarithmic) variant, which takes the nonlinearity of the relationship between h and net-precipitation into account, better described the sudden changes and the daily cycle in h. Some less positive remarks have to be made here as well. First of all, the reduction of the prediction uncertainty was less pronounced than we expected at first. This is attributed to the crude assumptions we made to keep the trend model simple and manageable. To

18 270 J.J.J.C. Snepvangers et al. / Geoderma 112 (2003) reduce the prediction uncertainties further, one can try to add more information (e.g. soil texture, soil physical properties, vegetation) in the trend model. This would require an additional measurement effort, though. Also, including more and more additional information such as physical laws will at some point become extremely difficult due to the rigid structure of ST-KED. For incorporation of more advanced physical process information, a Kalman filtering approach will be much more convenient (e.g. Or and Hanks, 1992; Heuvelink and Webster, 2001; Bierkens et al., 2001), although it should be kept in mind that this requires a major investment in development time. Second, ST-OK also has some important advantages over ST-KED: it needs less data and it is a simpler technique. Although this may seem trivial, it is something one should keep in mind when selecting an interpolation technique. After all, the best technique is not necessarily the technically most advanced one. Acknowledgements NWO-ALW grants (J.J.J.C. Snepvangers) and (J.A. Huisman) financially supported this study. We thank B. Jansen, K. Raat, M. Van Der Velde, A. Visser, P. De Willigen and especially L. De Lange for assistance during the fieldwork period. The comments of P. Bogaert and an anonymous reviewer substantially improved the paper. References Angulo, J.M., Gonzalez-Manteiga, W., Febrero-Bande, M., Alonso, F.J., Semi-parametric statistical approaches for space time process prediction. Environmental and Ecological Statistics 5, Bechini, L., Ducco, G., Donatelli, M., Stein, A., Modelling, interpolation and stochastic simulation in space and time of global solar radiation. Agriculture, Ecosystems & Environment 81, Bierkens, M.F.P., Knotters, M., Hoogland, T., Space time modeling of water table depth using a regionalized time series model and the Kalman filter. Water Resources Research 37, Bilonick, R.A., Monthly hydrogen ion deposition maps for the northeastern U.S. from July 1982 to September Atmospheric Environment 22, Bogaert, P., Comparison of kriging techniques in a space time context. Mathematical Geology 28, Bogaert, P., Christakos, G., 1997a. Spatiotemporal analysis and processing of thermometric data over Belgium. Journal of Geophysical Research 102 (D22), Bogaert, P., Christakos, G., 1997b. Stochastic analysis of spatiotemporal solute content measurement using a regression model. Stochastic Hydrology and Hydraulics 11, Buxton, B.E., Pate, A.D., Estimation of joint temporal spatial semivariograms. In: Baafi, E.Y., Schofield, N.A. (Eds.), Geostatistics Wollongong. Kluwer Academic Publishing, Dordrecht, pp Christakos, G., Vyas, V.M., A composite spatiotemporal study of ozone distribution over eastern United States. Atmospheric Environment 32, Coleman, T.F., Li, Y., An interior, trust region approach for nonlinear minimization subject to bounds. SIAM Journal on Optimization 6, Comegna, V., Vitale, C., Space time analysis of water status in a volcanic Vesuvian soil. Geoderma 60, Cressie, N., Huang, H.C., Classes of nonseperable, spatio-temporal stationary covariance functions. Journal of the American Statistical Association 94, De Cesare, L., Myers, D.E., Posa, D., Spatio-temporal modelling of SO 2 in Milan district. In: Baafi, E.Y., Schofield, N.A. (Eds.), Geostatistics Wollongong. Kluwer Academic Publishing, Dordrecht, pp

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