In 1996, Norwegian authorities implemented mandatory reporting of all drillings related to groundwater in mainland Norway . The purpose of the legislation was to provide the society with relevant groundwater observations. The Geological Survey of Norway (NGU) manages the regulations and stores the data in the Granada well database. As a public service, the data are freely accessible for downloading . According to recent statistics, about 44 % of the recorded boreholes were drilled for the purpose of energy extraction . At the startup of this study the total number of recorded observations was 54 194 (Table ). Of these recordings, 48 628 were boreholes, 3740 wells were in unconsolidated sediments, and 1826 were probe drillings. Explicit documentation of D was not available for all Granada recordings. For boreholes, however, it is possible to derive D with quite high precision by using information of the casing length. A casing is necessary in locations with unconsolidated material to prevent sediments from entering the well. Because casing is a considerable cost, the casing length is usually reported. Based on the Granada recordings, the casing was on average drilled 2 m into the bedrock. Hence, in cases where only casing length was reported, D was set equal to the casing length minus 2 m. In the following, the Granada recordings are referred to as boreholes because this is the vast majority of the data material.

The secondary variable, L, was calculated from digital maps of unconsolidated sediments . The total areal extensions of different sediments are listed in Table . The sediments are represented in terms of polygons in a geographical information system (GIS). Sediments covered by water (lakes, rivers, and glaciers) are not included in Table . The total sum of land cover polygons is 307 104 km2, while the total area of mainland Norway is 323 781 km2 . The difference should in principle be identical to the areal extension of lakes, rivers and glaciers. Thus, according to the land cover polygons (Table ), water covers 5.2 % of mainland Norway. Updated figures from the Norwegian Mapping Authority, however, show that lakes (5.7 %), glaciers (0.8 %) and rivers (0.4 %) constitute together 6.9 % of mainland Norway . The difference (1.7 %) indicates the irreducible uncertainty for this kind of statistics. The relative uncertainty for individual categories is higher because positive and negative deviations cancel out each other. It is also important to keep in mind that the actual uncertainty, with respect to areal information, increases with decreasing size of the land category. This precaution is relevant when point information from one data source (Granada) is combined with areal information from another source (GIS maps).

Before explaining the primary screening of boreholes, a few words on the geological setting are required. The vast bulk volume of unconsolidated sediments in mainland Norway is from the last glaciation (Weichselian). More than 90 % of the glacial erosion products were deposited offshore, and exposed bedrocks or sparse covers of sediments characterize the Norwegian landscape . Here, in the current study, the term “exposed bedrock” includes polygons identified as uncovered bedrock (id. 130, Table ). In addition, polygons labeled as “exposed bedrock or very thin cover of soil or organic matter” were included (id. 100, 101 and 140, Table ). Exposed bedrock constitutes about 35 % of mainland Norway according to this definition. Patchy and thin till material covers about 20 % of the land area (id. 12, Table ), and include this category when they define areas classified as exposed bedrock. In that case exposed bedrock makes up 55 % of the land area. Peatlands cover 5 % of the country (id. 90, Table ). According to the average thickness of the continuous till is approximately 6 m. They did not include any further discussion on the estimation of sediment thickness based on recorded data. This issue will be elaborated further in the study presented below.

There is no mandatory method for recording of drilling coordinates as part of the Granada standard. Quality tags were therefore attached to the observations to identify the uncertainty of the geographical coordinates. Geographical precision is important to consider during inference on the statistical structure of the data material, and it is decisive for spatial resolution of the final estimates. Hence, for the purpose of the current study, observations with less precision than 10 m (18898) were excluded from further analysis. Wells located on unconsolidated sediments but without any information on D were also omitted (3090) from the analysis. The same was done for probe drillings without information about D (1186 locations). Finally, all boreholes or probe drillings located inside polygons classified as “exposed bedrock” (10588) were omitted from further analysis. In these areas D is by definition given as E[D|L=0]=0.

Summing up the excluded locations (numbers given in parentheses above), the primary screening reduced the number of recordings from 54194 to 20432. The locations of the remaining boreholes (Nt=20432) are indicated in Fig. . Some of these boreholes (750) also had recordings of D=0, and these wells were also excluded from the statistical analysis. Thus, the number of wells included in the further analysis was N=19682. For these wells, L>0 and D>0.

Figure  shows that both D and L deviate strongly from Gaussian (normal) probability density functions (pdfs). The same is also true for the logarithmic values (Fig. ). The mean value of D=5.5 m corresponds well to the value reported by , but 50 % of the recorded data had D<=2 m, which implies a positively skewed pdf. The average horizontal distance to the outcrop is L=832 m, while 50 % of the boreholes had L≤460 m.

Clustering of boreholes (Fig. ) can easily be seen on the Granada webpage . This uneven spatial sampling affects the inference of statistical moments and the spatial correlation structure.

The mean and standard deviation of D and L as a function of separation distance h are given in Fig.  for Δh=20 m and Δh=150 m. It should be noted that the highest values of the mean and standard deviation of D occur at small (h<100 m) separation distances. This is opposite to what is shown for mean and standard deviations of L, which are small for minor separation distances, and which increase to maximum values around h=2.5 km, and then decay towards h=10 km. From Fig.  it is clear that when the separation distance h to the nearest borehole increases, the number of low values of D increases. This feature might be caused by preferential sampling, which implies that there is a systematic overrepresentation of drillings that has minor D values. Thus, Fig.  indicates a bias in the observations of D.

The purpose of declustering is to compensate for uneven sampling. This was done by giving less weight to observations in areas of high sampling density and a relative increase in weights in areas of sparse sampling. For this case study, the weights were found by gridding of the sampling domain and counting the number of observations in each grid cell. The weights were set equal to the inverse of the number of boreholes in the corresponding grid cell. These weights, however, are grid dependent. Hence, the following procedure was implemented to minimize the grid dependency.

Decide the size for the grid elements Δu=(Δx,Δy) that constitute a uniform grid.

The declustering coefficients c in Eq. () imply that the total variance of the experimental data is reduced and the correlation length is increased. This effect is called regularization in geostatistical terminology. It means that the declustering coefficients also depend on the grid size Δu. Thus, the final step is to repeat (1) to (5) above, but with a different grid size. The grid size that minimizes the regularization effect should be employed.

Application of Gaussian interpolation methods implies that the estimated function Z belongs to a standard normal pdf N(0,1). In this case, the stochastic function X=(D,L) is not Gaussian (∉ N(0,1)), which means that a transformation is necessary. The normal score transform implies that the quantiles pk in the original cdf, F(X), correspond to the quantiles in a standard normal Gaussian cdf, G(Z), where Z∈N(0,1) : ZX(ui)=φ(X(ui))=G-1[F(X(ui)]=G-1[pk*], where pk* is the quantiles in the standard normal cdf, and φ denotes the transformation of X corresponding to the inverse Gaussian G-1 cdf of D or L. The transformation in Eq. () was done by linear interpolation (or extrapolation) from the table of regular sampled Z∈N(0,1) based on the ranked values (percentiles) of X(ui)∉N(0,1).

The normal score transform requires a monotonic function to be unique. This is a problem if the data are censored , which means that the true value is only observed within intervals. This is the case for the lower values in the current experimental data (D=[0.1;0.5;1] m), which indicate that the true depth is only roughly recorded. For the current study, the normal score transform was done on declustered data which “corrected” the observations and thus removed overrepresentation of some observations; thus, the transformation to N(0,1) was unique.

The spatial structure of the data Z was described by the experimental semivariogram function: γ^(h)=12N(h)∑i=1N(h)[ZX(ui)-ZX(ui+h)]2, where N(h) is the number of data pairs in the separation interval h, and where ZX is the normal score transform (Eq. ) of either D or L.

In addition to the experimental semivariogram, the mean m(h) and the variance s2(h) were calculated as a function of h, m(h)=1N(h)∑i=1N(h)Zi and s2(h)=1N(h)∑i=1N(h)(Zi-m(h))2=1N2(h)∑i=1N(h)∑j>iN(h)(Zi-Zj)2, where N(h) is the number of observations for the separation interval h.

The experimental cross-semivariogram was estimated by expressing the two functions ZD(h) and ZL(h) as a sum of each other: W(h)=ZD(h)+ZL(h). This is possible because D and L were sampled in the same locations, and after the normal score transform (Eq. ) we know by definition that E[ZD]=0 and E[ZL]=0. In that case, the cross-semivariogram can be found by γ^ZDZL(h)=1/2[γ^W(h)-γ^ZD(h)-γ^ZL(h)], which is valid if ZD(h) and ZL(h) are stationary functions in space with finite variance. These properties are difficult to prove in practice, but suggests that if |γ^ZDZL(h)|≤γ^ZD(h)γ^ZL(h)1/2, then Eq. () is valid.

Anisotropy structures in the experimental data may be discovered by calculation of semivariogram and cross-semivariogram maps. The same equations (Eqs.  and ) are applied, but instead of the separation vector h the intrinsic values are calculated as a function of the north–south and east–west components (hx,hy) of the separation vector: γ^Z1Z2(hx,hy)=12N(hx,hy)∑i=1N(hx,hy)Z1(ui)-Z2(ui+(hx,hy))2, where Z1 and Z2 denote stochastic functions. If Z1=Z2 (i.e., the normal score transform of D or L), then Eq. () is the semivariogram map for ZD or ZL. If Z1=ZD and Z2=ZL, then Eq. () is equivalent to the cross-semivariogram map between ZD and ZL. The semivariogram (or cross-semivariogram) maps are similar to the experimental semivariogram function, but the semivariance is visualized in terms of a separation matrix instead of a separation vector. By calculating the semivariance in terms of a separation matrix, it is possible to reveal large-scale (systematic) directional dependencies – called anisotropy. If anisotropy in the observation material is evident, the next step is to calculate directionally dependent experimental semivariograms, where the direction of the searching sector is taken from the semivariogram map. The directionally dependent properties can be taken into account in the estimation procedure by using the directionally dependent searching directions derived from the semivariogram maps. An alternative is to transform the observation coordinates to an isotropic and orthogonal coordinate system .

The semivariogram model, which was fit to the experimental semivariogram had the following form: γ(h)=C0+C11-exp⁡βhaα, where C0, C1, a, and α were the fitting parameters. In geostatistical terms C0 is called the nugget (the variance at h→0), C0+C1 is the sill (the variance at h→∞), a is the range, and α is the exponential coefficient (1≤α≤2). The constant β determines the variance at h=a. In this case β=-ln⁡(20), which is equivalent to 95% of γ(∞). For that reason β is called the practical range in the literature.

The model parameters in Eq. () were obtained by minimizing the objective function Υ: Υ(h)=∑i|γ(hk)-γ^(hk)|,k=1,…,K, where K is the number of distance classes in the semivariogram. For the case study, the objective function was minimized by using the simulated annealing algorithm .

The kriging equations below are expressed in terms of the covariance function: C(h)=C0+C1-γ(h)=C1exp⁡βhaα, where the constant β=-ln⁡(20), and the parameters C0, C1, a, and α were found by minimizing Eq. ().

For this project the kriging and co-kriging equations were implemented in , which makes it convenient to express the equations in terms of matrix notation. A thorough mathematical derivation of the equations can be found in . In matrix notation the estimation is expressed as Z^=ZobsΛ, where Z^ is the estimated variable in location u. If k=1,…,m variables are involved, then Z^ is a row vector with m entries (1×m matrix), Zobs contains the observations in a 1×m matrix, and Λ is an m×m matrix where the column vectors are the estimation weights. In this case, m=2, Eq. () is written as Z^D,Z^L=ZD(obs),ZL(obs)ΛDDΛDLΛLDΛLL. For the present case study, the observations ZD(obs) and ZL(obs) were available in the same locations ui,i=1,…,n. The weights Λ are found by solving the kriging equations : X=C-1C0, where C-1 denotes the inverse of the matrix C, which in this case reads as C=CDD(h)CDL(h)IDDTIDLTCLD(h)CLL(h)ILDTILLTIDDIDL00ILDILL00, where Ckk(h) is the covariance model (Eq. ) and where k=D,L. IDD=ILL are row vectors of ones (1×n), and IDL=ILD are row vectors of either ones or zeros depending on whether all weights should sum up to one or not (IT is the transposed of I).

The matrix C0 denotes the covariance between the point of estimation (u) and the observations: C0=C0DD(h)C0DL(h)C0LD(h)C0LL(h)10*0*1, where C0kk(h) is given in Eq. () and where k=D,L. The symbol 0* indicates that the entry might be one or zero, depending on the Lagrange condition that all weights should sum up to one or only the weights for the single variable estimation problem. Again, zero is the default value. The estimation weights Λkk(h) and the Lagrange multipliers μkk(h) (k=D,L) are contained in the X matrix: X=ΛDDΛDLΛLDΛLLμDDμDLμLDμLL.

The estimation variance σK2 can then be written as σK2(u)=Var[Z]-XTC0, where the total variance is Var[Z]=Var[ZD] for ordinary kriging and Var[Z]=Var[ZDZL] for co-kriging. Hence, the total variance is equivalent to the sum of the diagonal entries in CDD(h) and CLL(h)), where C0 and X are given in Eqs. () and ().

The quality of the estimation method depends on the absolute error AE, which is the difference between the observed value and the estimated value: AE(ui)=|Zobs(ui)-Z^(ui)|. Average values for all estimates are given by the mean absolute error MAE, MAE=1n∑inAE(ui), and the standard deviation of the absolute error SAE: SAE=1n-1∑inAE(ui)-MAE21/2, where n is the number of cross-validated observations.

In addition, it is necessary to quantify the precision of the estimates. Two concerns are taken into account in this study: first, if the estimate is within a given confidence interval (PR): PR(ui)=AE(ui)-ωσK, where ω depends on the level of confidence. The accuracy AC of the estimates is then given by AC(ui)=1ifPR(ui)≤0elseAC(ui)=0, and the accuracy is given as a fraction of the total number of observations FAC: FAC=1n∑inAC(ui), where n is the number of cross-validated observations.

If two methods have the same level of accuracy, then the method that gives the best precision should be preferred. Precision can be taken into account by scaling the absolute error by the estimation uncertainty: ξ(ui)=AE(ui)/σK(ui), and the scaled precision SP is written as SP(ui)=ξ(ui)AC(ui), and with the mean scaled precision MSP, expressed as MSP=1n∑inSP(ui), and the standard deviation of the mean scaled precision SSP: SSP=1n-1∑inSP(ui)-MSP21/2, where n is the number of cross-validated observations.

The Granada boreholes used in the current study were clustered in urban areas (Fig. ). To minimize the impact of this uneven spatial sampling, declustering weights were calculated according to the procedure described in Sect . The window sizes (w=wx=wy) applied to calculate the declustering weights were w=[500;1000;2000;4000] m. Average declustering coefficients were calculated by moving the grid in seven steps p=7, with an offset δ=100 m (Eq. ).

The skewness given by the ratio of the median to the mean for the different declustering windows w shows that maximum skewness appears for w=500 m (Table ). For w=1000 m, however, the skewness was more similar to the original (raw) observations; thus, for the cross-validation analysis the declustering weights were calculated with w=1000 m. The declustering coefficients show that about 13 % of the boreholes had 10 or more boreholes located within a neighborhood of 5 km. More than 50 % of the boreholes had two or more boreholes within a search radius of 5 km, and about 23 % had no other wells within a 5 km neighborhood. The normal score transform (Eq. ) yields by definition a normal pdf of the variables involved. The transform relies, however, on the experimental data, which means that sampling of extreme values has an impact on the results. The dataset used for calculations (N=19682 samples) had a minimum observed D=0.05 m and a maximum D=229 m (Fig. ). Some of the extreme high values may represent outliers or recording errors; thus, for the cross-validation, study boreholes with recorded sediment thicknesses of more than 100 m were not included in the calculations. The scatter plot of the raw observations shows the censored character of the data with a high frequency of recordings at even numbers (0.10; 0.20; 0.30 m; etc.). This is very clear from 0.1 to 1 m, and to some degree from 1 to 10 m (Fig. a). After declustering, the censored character was less obvious (Fig. b). The semivariogram analyses and the kriging procedures were employed on the normal score data (ZD and ZL). After kriging, the estimation results were transformed back by the inverse normal score transform Eq. () and divided by the declustering coefficients.

Semivariogram maps (Eq. ) of depth to bedrock ZD and horizontal distance to outcrop ZL were calculated to detect large-scale anisotropy in the data material. Anisotropy might be identified in Fig.  for the range (correlation length) of ZD. The range varies apparently as a function of direction with the slowest decay in the north-westerly direction (N35W–N45W) and with a somewhat faster decay in the south-easterly direction. The number of observation pairs had, however, a similar structure, which indicates that the apparent anisotropy might be an artifact due to the clustering of the observations. This presumption was tested by calculating artificial semivariogram maps based on the same borehole locations but where the observations were substituted by a random number. The artificial semivariogram maps revealed similar structures that can be seen in Fig. . Hence, the presumption of an artifact due to clustering cannot be ruled out. For this reason no directional experimental semivariograms were calculated as part of this case study.

The results of the semivariogram analysis confirm the existence of a correlation structure in the data (Fig. ) that might be capitalized when estimating D(u). The model parameters given in Fig.  were obtained by minimizing the objective function (Eq. ) by the simulated annealing algorithm . First, all parameters [C0,C1,a,α] were optimized; and then, secondly, C0 was fixed and the remaining parameters [C1;a;α] were simulated. This automatic procedure gave the model parameters shown in Fig. . The minimum of the objective function is not well defined everywhere and different combinations of model parameters gave almost similar results. The model parameters in Table  were evaluated in the cross-validation procedure below. The automatic calibration procedure gave an optimal correlation length of about a=10 km for depth to bedrock ZD(h) (Fig. a). The most prominent feature, however, is the large nugget value C0, which in this case is about 50 % of the total variance: C0+C1. The experimental semivariogram for the horizontal distance to outcrop ZL(h) had a minor nugget value compared to the total variance (Fig. b). At the same time the correlation length (a=5.9 km) was somewhat shorter compared to ZD(h). The experimental cross-semivariogram between ZD(h) and ZL(h) was calculated according to Eqs. () and (). The nugget value was about 10 % of the total variance in this case, with a correlation length of a=2.7 km. Finally, the cross-semivariogram was tested according to Eq. (), but none of the parameter combinations in Table  violated the criterion.

The purpose of the cross-validation was to evaluate the impact of using horizontal distance to outcrop as an additional variable for estimation of sediment thickness above the bedrock. In this case, the cross-validation was performed by leaving one observation out. At the point where the observed value was left out, ordinary kriging (OK) and co-kriging (CK) were performed by using the global model parameters given in Table . The differences between the estimation results and the observations left out were used to quantify the quality of the estimation procedure. Three criteria were used to distinguish the two estimation procedures: the mean absolute error (Eqs.  and ); the accuracy of the estimation results (Eqs.  and ); and the precision of the estimation results (Eqs.  and ).

In general, both OK and CK overestimate minor depths to bedrock and underestimate large depths (Fig. ). The most important estimation criterion is usually considered to be the mean absolute error (Eq. ). With the model parameters tested in Table  there are only minor differences in the mean absolute error (Eq. ) between the OK and CK estimates (Table ). The CK estimates have slightly lower mean absolute errors than the OK estimates unless the nugget value (C0) for the cross-covariance between D and L approaches half of the total variance: C0+C1 (Table ).

In cases with a minor difference in the mean absolute error, the estimation results might be ranked according to criteria for estimation accuracy (Eq. ) and precision (Eq. ). For the present case study, the definitions of accuracy and precision were both related to the estimation variance (Eq. ), and in this respect, CK was superior compared to OK (Fig. ).

In Fig.  scaled precision (Eq. ) is sorted and given as a function of cumulative accuracy SAC: SAC=ΣijAC(ui)/nmax⁡,j=1,…,nmax⁡, where nmax⁡ is the number of estimates where AC=1.

As long as the absolute estimation errors (Eq. ) are similar, OK yields a higher accuracy than CK because CK has a lower estimation variance. This result follows directly from the definitions in Eqs. () and (). With ω=1 in Eq. (), the OK estimates gave an accuracy from 60 to 65 %, while CK had an accuracy of 50–60 %. At the same time CK yields an overall higher precision than OK because of lower estimation variances (Fig. ).

A final result that deserves some attention is the location of estimates that did or did not fulfill the accuracy criteria. This is illustrated for mainland Norway and the Oslo area in Fig. . Three categories were visualized: (i) locations with low accuracy (AC=0, Eq. ); (ii) locations with good accuracy (AC=1, Eq. ) obtained either by OK or CK; and (iii) locations with good accuracy (AC=1, Eq. ) obtained only by the CK method. For all cases ω=1 (Eq. ).

For the current case study, D(u) was derived from the Granada open-access database . Public databases are prone to preferential sampling. In this context, preferential sampling implies two specific challenges that need to be discussed, namely clustering and bias. Clustering is due to the fact that wells and boreholes are located where people need them; thus, the spatial frequency of boreholes mirrors the population density (Fig. ). Clustering of observations has an impact on statistical inference regarding statistical moments and semivariograms. Different approaches have been suggested to control the clustering effects. suggested removal of wells randomly in areas with high density of observations, and then recalculation of the experimental semivariograms based on the remaining observations. The experimental semivariograms, however, turned out to be sensitive to the size of the searching window where clustered observations were removed. Thus, this method was disregarded in the current case study because the algorithm did not yield robust results.

suggested controlling clustering effects in the semivariogram by calculating weights that were inversely proportional to the Thiessen polygons for each observation. This method provides a set of weights that are mathematically sound, but it is relatively expensive with respect to computer resources, especially if the number of observations is large. Instead of Thiessen polygons a less computer demanding algorithm was employed, namely the moving grid method . By this method the declustering weights were inversely proportional to the average number of observations within the moving window (cf. Sect. ). The declustering weights depend on the size of the window (Table ). In general it is recommended to use the window size w that maximizes the skewness of the pdf(s), which in this case was w=500 m. However, w=1000 m gave a skewness for D(u) that was closer to the original data; thus, the semivariograms were based on a declustering window w=1000 m. The mean value from raw (not declustered) data was 5.5 m, but the declustered mean was reduced to 3.4 and 2.8 m for w=500 and w=1000 m, respectively (Table ).

The problem of biased recordings of D(u) in the database is more difficult to assess. There are good reasons to expect that bias exists and that minor sediment thicknesses are overrepresented in the database. One indication is that mean and standard deviation are highest at minor separation distances, which indicates that willingness to continue drilling is less if D(u) is large and if there are no other wells in the close neighborhood (Fig. ).

Biased observations are a common problem for datasets sampled in open large-scale environments. The impact of bias may be controlled if there exists independent information on processes related to the variable of interest. did a case study based on biased observations of arsenic concentration in groundwater. They used geological maps and utilized knowledge of arsenic concentration in specific geological units to control the bias. reported biased recordings of precipitation from a meteorological gauge station. In this case the bias was due to turbulence in the wind field around the gauge equipment. They recorded wind speed and temperature together with precipitation and other meteorological variables, and derived functions for bias correction by application of Bayesian statistics. A similar token was applied in the current study. Here, horizontal distance to outcrop L(u) was evaluated as secondary information to control the impact of biased observations of sediment thickness D(u).

The cross-validation exercise presented here cannot verify a general relation between D(u) and L(u), but the results show that the estimation uncertainty was reduced by using L(u) as a secondary function. Non-biased relations between D(u) and L(u) ought to be investigated by further research, for example by utilizing datasets from geotechnical probe drillings. Results from such studies would increase the value of the Granada database and other similar databases.

The cross-validation analysis indicates low estimation accuracy in urban areas. One reason for this result might be anthropogenic reallocation of unconsolidated matter, which includes removal of sediments in some places and deposition of unconsolidated matter in others. Similar problems might also be valid for identification of horizontal distance to outcrop. For further studies such locations might be disregarded or given less weights. One option is to allocate a quality tag to the D(u) recordings in the same manner as was done for recordings of geographical coordinates.

Both OK and CK overestimated small D(u) and underestimated large D(u) (Fig. ). Such results are typical for Gaussian estimation methods applied on observations with positively skewed pdfs. Other case studies report similar results , but it should be noticed that the double logarithmic scale exaggerates the deviations especially for minor depths.

The observations of D(u) had a high fraction of small-scale noise (C0 in Fig. ) relative to the total variance: C0+C1 (Fig. ). Efforts should be taken to control C0. One abatement measure might be achieved by attaching a quality assurance tag to D(u). In this way low-quality recordings could receive less weights or be filtered out. These kinds of measures would increase the quality of the Granada database.

Despite these uncertainties the cross-validation shows that the accuracy is higher than 60 % for the model parameters with highest scores (Table ). For this case study, the estimation accuracy was set equal to one if the absolute estimation error was less than one standard deviation of the estimation uncertainty and zero for all others (Eqs.  and ). By this definition, the accuracy increases by increasing estimation variance, which means that accuracy should be evaluated together with the estimation variance (Eq.  and Fig. ). For stochastic simulation the precision of the estimates is of primary interest. In such cases, the probability of extreme realizations may also be quantified. For such applications, the precision is more important than the accuracy of the estimation method. The cross-validation results show that the precision in general is higher if the horizontal distance to the outcrop L(u) was included (Fig. ). Because precision increases as a function of decreasing estimation variance, the cross-validations show that L(u) should be included despite the uncertainties in the experimental data.

These results indicate that more advanced estimation procedures should be considered. In this case study, the total estimation domain (mainland Norway) was considered as homogeneous with respect to variance and correlation length. Methods that take local model parameters and local anisotropy into account may reduce the absolute estimation error but not necessarily the estimation variance. The same is true with respect to estimation methods that are more robust with respect to estimation of extreme realizations. For estimation of most likely minimum and maximum thickness of sediments within a given estimation area, stochastic simulations are recommended.

After initiation of this case study, the number of recorded boreholes, wells, and probe drillings in the Granada database has increased . The new recordings might be used as an independent dataset for cross-validation purposes. One interesting candidate for further work is the approach suggested by . They approximate the estimation problem to stochastic partial differential equations. In this method non-stationarity of statistical moments are taken into account, but at the same time less computer resources are spent on matrix inversions which is a challenge for applications with a large number of observations .

Finally, it is appropriate to recall that the primary purpose of the Granada database is not the recording of sediment thickness D(u) alone, but to provide information on groundwater resources in general . Hence, in this context, the present study is a call to explore public data to obtain important estimates for science and society.