Creating increasingly realistic groundwater models involves the
inclusion of additional geological and geophysical data in the
hydrostratigraphic modeling procedure. Using multiple-point statistics (MPS)
for stochastic hydrostratigraphic modeling provides a degree of flexibility
that allows the incorporation of elaborate datasets and provides a framework
for stochastic hydrostratigraphic modeling. This paper focuses on comparing
three MPS methods: snesim, DS and iqsim. The MPS methods are tested and
compared on a real-world hydrogeophysical survey from Kasted in Denmark,
which covers an area of 45 km

The presented MPS methods each have their own set of advantages and disadvantages. The DS method had average computation times of 6–7 h, which is large, compared to iqsim with average computation times of 10–12 min. However, iqsim generally did not properly constrain the near-surface part of the spatially dense soft data variable. The computation time of 2–3 h for snesim was in between DS and iqsim. The snesim implementation used here is part of the Stanford Geostatistical Modeling Software, or SGeMS. The snesim setup was not trivial, with numerous parameter settings, usage of multiple grids and a search-tree database. However, once the parameters had been set it yielded comparable results to the other methods. Both iqsim and DS are easy to script and run in parallel on a server, which is not the case for the snesim implementation in SGeMS.

Recent advances in groundwater modeling have shown the importance of accurate hydrogeologic models for management of increasingly sparse groundwater resources. Groundwater modeling predictions are sensitive to geologic heterogeneity (e.g., Freeze, 1975; Gelhar, 1984; Fogg et al., 1998; LaBolle and Fogg, 2001; Zheng and Gorelick, 2003; Feyen and Caers, 2006; Fleckenstein et al., 2006; Zhao and Illman, 2017). However, geological units include complexities not directly related to hydrofacies (Klingbeil et al., 1999). Instead, the concept of hydrostratigraphic units is used throughout this study, which effectively combines geological units and reduces the total number of units resulting in a closer relation to the hydrologic units. Improving the realism and quantification of uncertainty around hydrostratigraphic models is therefore an important step towards accurate groundwater modeling predictions. Hydrostratigraphic models are created using several approaches. A common approach is a manual co-interpretation of available geophysical, geological and/or hydrologic data. The geoscientist cognitively uses his/her refined knowledge of geological processes combined with the provided datasets to create a detailed cognitive geological model (e.g., Jørgensen et al., 2013; Royse, 2010). The cognitive geological model is then simplified to a hydrostratigraphic model. Even though the hydrostratigraphic model encapsulates the complexities related to geologic architecture, it does not reflect the hydrostratigraphic uncertainty. It is a so-called deterministic model, i.e., one version of the hydrostratigraphic subsurface. An alternative to cognitive modeling is stochastic modeling using geostatistical methods. The field of geostatistical modeling focuses on creating models depicting subsurface hydrogeology and/or reservoir properties. Geostatistics is currently applied in a number of geoscience fields, such as petrology (e.g., Okabe and Blunt, 2005), petroleum reservoir modeling (e.g., Journel and Zhang, 2006; Strebelle et al., 2002), hydrogeology (e.g., Huysmans and Dassargues, 2009) and hydrology (e.g., Michaelides and Chappell, 2009). Overall geostatistical methods provide a framework in which multiple equiprobable hydrostratigraphic models can be created in a semiautomated fashion. The individual stochastic models do not reflect the modeling uncertainty, but the model ensemble does. The multiple hydrostratigraphic models can be used as a set of input parameters for the groundwater model. By running the groundwater model several times with different hydrostratigraphic models, multiple predictions can be made, yielding an estimate of the prediction uncertainty. The ability to understand how the hydrostratigraphic uncertainty is related to the prediction uncertainty will help in understanding where to improve the hydrostratigraphic models in order to reduce the prediction uncertainty. This study will however not focus on groundwater modeling predictions, but on the presentation of a stochastic modeling framework for reconstructing subsurface hydrostratigraphic architecture.

Today state-of-the-art geostatistical tools are readily available to geoscientists. Traditional two-point statistics, or variogram-based methods, e.g., sisim (Journel, 1983) and sgsim (Deutsch and Journel, 1998), have been widely used in both research and in practice (e.g., Seifert and Jensen, 1999; Caers, 2000; Juang et al., 2004; Delbari et al., 2009). However, variogram-based techniques depend on two-point statistics for simulation of complex geological features. Depending on the complexity of the geological setting, such two-point statistical methods cannot recreate complex curvilinear geological features of the subsurface which are common in fluvial and glaciofluvial environments (e.g., Arpat and Caers, 2005; Hu and Chugunova, 2008; Journel and Zhang, 2006; Journel, 1993; Liu, 2006; Sánchez-Vila et al., 1996; Strebelle and Journel, 2001). An additional geostatistical modeling tool which should be mentioned is T-PROGS (Carle, 1999). T-PROGS is based on transition probabilities between categories and generates geostatistical realizations based on such constraints. In comparison with the indicator method, sisim, it allows for better integration of these transition probabilities and, hence, the spatial cross-correlations of soil/rock-type architecture into the groundwater models. However, T-PROGS also has difficulties in reconstructing curvilinear geological features. Kessler et al. (2013) made a detailed comparison between T-PROGS realizations and real-world cross sections in a gravel pit in Denmark. The result reveals a suboptimal pattern reproduction, in comparison to other simulation tools such as multiple-point statistics (MPS) (Mariethoz and Caers, 2014b). MPS is a recent alternative to classic two-point statistics. Here, additional multiple-point (MP) information from a training image (TI) is used to condition the simulations. The usage of MP information allows for reconstruction of more complex geological features, such as curvilinear features (Strebelle, 2002). A TI is any 2-D or 3-D image containing geometrical information relevant to the hydrostratigraphic model. The crux of the MPS approach is finding a relevant TI. Some examples of 2-D and 3-D TIs are categorical images of outcrops (2-D), categorical drawings of a geological system created by a geoscientist (2-D), and cognitive geological or hydrostratigraphic voxel models (3-D) (e.g., Høyer et al., 2015a). Today, MPS techniques are widely used in geoscientific research and studies, a few examples are Maharaja (2005), Meerschman et al. (2013) and Hermans et al. (2014). The MPS framework allows for conditioning of geological architecture/patterns, a stochastic framework and spatially constraining to both soft data and hard data (Arpat and Caers, 2005; Guardiano and Srivastava, 1993; Journel, 1993; Strebelle and Journel, 2001).

Within the geostatistics framework the creation of hydrostratigraphic models requires the inclusion of data from multiple sources, often geophysical models (soft data), borehole data (hard data) and a TI. The different data sources each provide relevant information. Geophysical models provide information regarding the large-scale hydrostratigraphic architecture. Boreholes, on the other hand, provide detailed yet usually sparse information regarding hydrostratigraphic units. Each data source is a piece of the puzzle; combining the individual pieces improves the resulting hydrostratigraphic models. The inclusion of several types of data is, however, not trivial since information regarding their mutual relationships, e.g., the hydrostratigraphic–petrophysical relationship, is required. An important source of information which helps to combine the different sources of data is geologic knowledge. Geologic knowledge can be defined as information regarding geologic processes, geomorphologic patterns and structural geology. Incorporating geological knowledge into hydrostratigraphic models is often difficult and done ad hoc. Geologic information, as described above, complements the soft data and helps to create more realistic hydrostratigraphic models. However, within the MPS framework this type of information can be implemented via the TI.

This study focuses on comparing and testing three MPS methods on a real-world dataset from a groundwater survey in Kasted, Denmark. An important part of the dataset is the airborne geophysical survey, providing a set of resistivity models containing information regarding the large-scale hydrostratigraphic architecture of the area. The MPS tools are used to reconstruct an intricate system of interconnected buried valleys. The end result is an ensemble of hydrostratigraphic models. A 3-D hydrostratigraphic voxel model of the area is used as a TI, containing detailed MP information regarding the hydrostratigraphic features of the survey area. Information regarding the geological architecture and the relationship between hydrostratigraphy and petrophysical properties are contained in the TI. The hydrostratigraphic–petrophysical relationship is explicitly known since the hydrostratigraphic model spatially overlaps with the geophysical and borehole lithology logs. Spatially constraining the simulation to the soft data, consisting of the resistivity models, ensures that simulated geological patterns are placed concurrently to the real world. However, such geophysical soft data have several types of related uncertainty, e.g., spatial uncertainty related to incomplete datasets, resolution capabilities and signal-to-noise ratio decrease with depth. Incomplete geophysical datasets are a common problem and are typically reconstructed using geostatistics – often in a deterministic fashion. A common approach is to use variogram-based geostatistics, such as kriging interpolation, to reconstruct the incomplete resistivity grid (Isaaks and Srivastava, 1989). We have used the stochastic direct sampling (DS) grid reconstruction routine proposed by Mariethoz and Renard (2010). Here, the grid reconstruction uncertainty is reflected by multiple resistivity grids, yielding variable patterns in the multiple reconstructed grids. The reconstructed grids are then used in conjunction with the hydrostratigraphic TI to create a set of stochastic hydrostratigraphic realizations of the hydrostratigraphy of the modeled area.

In relation to the Danish groundwater mapping campaign (Thomsen et al., 2004), detailed geophysical datasets (Møller et al., 2009) and hydrostratigraphic models exist. A selection of the 3-D geologic and hydrostratigraphic voxel models is reported in the literature (e.g., Høyer et al., 2015a, b; and Jørgensen et al,. 2015). Additionally, the study by Høyer et al. (2017) presents a framework for making large-scale MPS models based on geological 3-D voxel models, as well as seismic and borehole data. In this study, we will show how MPS methods can be utilized to model a new survey area. An existing cognitive model from one area is used as a TI for simulating another survey area with similar geological characteristics.

To our knowledge, no vigorous studies comparing multiple MPS methods have been carried out on real-world hydrogeophysical datasets. By applying several measures to assess and compare the modeling results, the selected MPS tools are tried, tested and compared on real-world data. The MPS methods are tested in a pseudo-synthetic environment, where an actual 3-D hydrostratigraphic model of the Kasted survey area is used as a TI. This guarantees a controlled modeling environment in which the TI contains highly relevant hydrostratigraphic architecture. The main contributions of this study are (1) a practical real-world example of stochastic reconstruction of incomplete geophysical datasets; (2) comparison of three MPS methods for integrating geophysical data – snesim (Liu, 2006; Strébelle and Journel, 2000), direct sampling (DS) (Mariethoz et al., 2010) and image quilting (iqsim) (Hoffimann et al., 2017; Mahmud et al., 2014); (3) validation of the comparison results by (a) visual inspection, (b) a mathematical comparison method called the analysis of distance (ANODI) (Tan et al., 2014) and (c) comparison of the simulation results against the borehole lithology logs; and (4) to show the strengths and weaknesses of a stochastic hydrostratigraphic modeling framework, and (5) an example using the direct sampling method and showing how to use the cognitive hydrostratigraphic interpretation of one area to directly generate hydrostratigraphic models of new areas, using only the soft data from the new area.

The Kasted survey area is located in Denmark, in the eastern part of
Jutland, close to the city of Aarhus (Fig. 1a).
The 45 km

An overview map of the Kasted survey area.

The dataset used in this study consists of a dense airborne geophysical SkyTEM survey, near-surface boreholes from the Danish borehole database, and a cognitive geologic model created by an experienced geoscientist. In the following we will summarize the key features of these datasets.

The SkyTEM system (Sørensen and Auken, 2004) is a helicopter transient electromagnetic system allowing for rapid collection of large geophysical datasets, with high spatial density. The Kasted SkyTEM survey contains 333 line km with a line spacing of roughly 100 m (Fig. 1b). The SkyTEM data are inverted and modeled according to the scheme described by Viezzoli et al. (2008), with the end result being a collection of spatially constrained inversion models. In Denmark it is standard protocol to calibrate the SkyTEM system at an official calibration site, as described by Foged et al. (2013), ensuring data of high-quality and reproducible results. Therefore, the resistivity values from a calibrated SkyTEM survey are comparable to other calibrated SkyTEM surveys. The SkyTEM system is sensitive towards large-scale conductive trends in the subsurface, especially when a significant contrast between a conductive and a resistive feature exists. In the eastern part of Jutland it is common that the lower confining boundaries of the buried valleys are well resolved since these buried valleys are often quite resistive and are eroded into conductive hemipelagic Paleogene clays.

The Danish borehole database, JUPITER (Hansen and Pjetursson, 2011), contains about 280 000 shallow boreholes which have been drilled for a variety of purposes, mainly in relation to drinking water and raw materials exploration, but also in relation to research and geotechnical studies. The JUPITER database contains information on location, drilling method, lithology, geologic age, filter position and water chemistry.

The cognitive geologic model was created using all available data, including the 333 line km of SkyTEM data, information from 435 boreholes and prior geological knowledge of the area. The model was created using the cognitive modeling scheme, which is introduced by Jørgensen et al. (2013). The geological model is described in great detail by Høyer et al. (2015a). The geologic model is detailed and contains a set of 21 interconnected buried valleys. The final 3-D voxel model contains 42 unique geological units, which are simplified into three overall hydrostratigraphic units in this study. The three hydrostratigraphic units are chosen for the purpose of covering the overall hydrogeological features of the groundwater modeling area. The cognitive hydrostratigraphic model will act as the TI as well as a baseline for assessing the performance of the three MPS methods, and the stochastic modeling results will be compared against the cognitive model.

MPS provides a degree of flexibility, which assists the modeler in creating geologically realistic hydrostratigraphic models. The idea is to create a suite of hydrostratigraphic models, which span a realistic subset of possible model architectures, as opposed to a deterministic model, which spans a single possible model architecture. The term realistic refers to models, which comply with the underlying datasets mentioned above, i.e., borehole lithological logs, geophysical resistivity models and the cognitive geological model. The underlying datasets have associated uncertainties describing ranges of possible models. The suite of equiprobable hydrostratigraphic models can be used as input to a groundwater model, making it straightforward to test the sensitivity of specific groundwater model predictions.

MPS methods use a training image to condition a model simulation to a prior geological conceptualization. As opposed to two-point statistics, the joint variabilities of multiple points are assessed at the same time during simulation. The MP joint variabilities cannot be inferred from sparse data and are therefore taken from a relevant exhaustive TI. The justification that a given TI can be used to infer the joint variability of MPs heavily lies on the choice of a relevant TI. A TI should always contain geologically realistic and relevant information (Journel and Zhang, 2006). Finding and/or creating a realistic TI is thus important to the MPS methodology. A TI is essentially any categorical or continuous image. which contains the geological conceptualization of the target variable (Mariethoz and Caers, 2014a). It is not a subsurface model itself, but a quantitative conceptual depiction of it. The user chooses the TI based on his/her prior understanding of the local hydrogeological system. The TI does not need to carry any locally accurate information; i.e., it does not need to contain the actual geographical positions of the hydrostratigraphic architecture, just the general patterns. It needs to reflect a prior geological or structural concept (Strebelle and Journel, 2001).

The MPS methods chosen in this study have been selected to reflect recent advances in MPS methods. The MPS methods in this study include the single normal equation simulation (snesim) (Strébelle and Journel, 2000) implemented in the Stanford Geostatistical Modeling Software (SGeMS), direct sampling simulation (DS) (Mariethoz et al., 2010) implemented in the DeeSse software package (Straubhaar, 2011) and image quilting simulation (iqsim) (Hoffimann et al., 2017) implemented in ImageQuilting.jl.

The snesim method is a traditional MPS method. It fits into the so-called
“probability framework” where geophysical models (not data) are considered soft
information, and as such needs to be converted into probabilities. Suppose
we have a categorical random variable

In snesim, the TI is stored in a dynamic data structure called a search tree. The search tree is a database and can be seen as a condensed summary of the full TI. It contains the spatial information to which the simulation is conditioned; for more detail see Strebelle (2002). To avoid repetitive scanning of the TI, which is computationally expensive, the TI is stored in a search-tree database ahead of the simulation (Roberts, 1998). This is done once. TI patterns can then be retrieved from the database without scanning the entire TI. Depending on the amount of detail stored in the search tree this can be quite CPU intensive, since the entire search tree is stored in memory, and therefore there is an upper limit to the size of the search-tree pattern database. However, advances in computers have increased the upper limit for available CPU.

Another caveat of snesim is the usage of multiple grids (Tran, 1994). Due to limitations in relation to the search neighborhood, the simulation of structures on all scales requires the usage of multiple grids. The simulation is carried out on a series of multiple simulation grids with varying density, ensuring pattern reproduction at all scales. The search-tree formulation and multiple grid approach add to the overall complexity of parameterization in snesim, but at the same time ensure stable and reliable MPS modeling results. The increased number of user-defined parameters makes it less intuitive, since it is relatively difficult to determine the optimal parameter values for a given dataset.

The direct sampling simulation (DS) method consists, for the simulation of each cell, in randomly scanning the TI until a pattern similar to the pattern centered at the simulated cell is found and then in copying the value in the center of the pattern from the TI to the simulation grid. Consequently, contrary to snesim, no probability is explicitly computed to draw a value at a simulation grid cell. In this paper, we use the DeeSse implementation of DS, presented by Straubhaar (2011). This bypasses the necessity of saving spatial patterns in a search-tree database; instead, spatial patterns are conditioned by directly scanning the TI.

One issue which needs to be solved is how to constrain a soft data variable.
In DS, this is accomplished by introducing an auxiliary variable. The
auxiliary variable is roughly a translation of the TI into a soft data
variable. Suppose a forward operator, denoted by

DS is more flexible than traditional MPS methods, such as snesim. As no search-tree database is required, the multiple grid formulation used in snesim is not required in DS, which effectively reduces the number of parameters and makes the parametrization relatively simple. Furthermore, one can simulate continuous variables and/or discrete variables with no limitation to the maximum number of categories (e.g., hydrostratigraphic units). In our case, any number of geophysical datasets collocated or not can be included as long as a corresponding auxiliary variable is added to the multivariate TI. However, it can be a cumbersome process generating the auxiliary variable. Furthermore, it is even possible to use probability grids in place of the actual soft data variable, as in snesim, if desired (Mariethoz et al., 2015). Depending on the setup and dataset, DS can be computationally as fast as snesim. Moreover, the DS implementation used in this work is amenable to scripting yielding the possibility of improving computation times on computer clusters or servers, if available.

The image quilting simulation (iqsim) method has been borrowed from the
computer vision literature (Efros and Freeman, 2001). The algorithm is
originally designed to synthesize and/or replicate patterns from 2-D images
but has since been modified to accommodate conditioning data and 3-D
geoscience problems (Mahmud et al., 2014). The concept of the iqsim method
is straightforward. In essence, iqsim cuts the TI into user-defined patches
or blocks and then reassembles the patches to create a simulation. The
difficult part is how to reassemble the patches to create meaningful and
seamless realization results, which can be constrained to a soft data
variable. These difficulties have been solved (for more detail see
e.g., Hoffimann et al,. 2017)

Software is available at

A common problem in hydrogeophysics is that datasets, albeit spatially dense, do not cover the entire modeling grid. In electromagnetic methods human infrastructure causes electromagnetic interference with the signal. Such noisy soundings, referred to as coupled soundings, are removed during processing, as presented by Auken et al. (2009), resulting in an incomplete dataset with gaps scattered throughout the survey area (Fig. 1b). Several approaches to manage with incomplete datasets exist. One approach is to leave the incomplete dataset as is, meaning gaps are reconstructed during simulation of the hydrostratigraphic model without spatially constraining the simulation gaps. The gaps are filled out solely by conditioning to the TI. Alternatively, dataset gaps can be filled prior to simulation, which is primarily done if the dataset has a high spatial density and/or the underlying random variables describing the data are not assumed to be especially complicated. The soft data utilized for constraining in this study are SkyTEM models. The raw SkyTEM data undergo processing and inversion (Auken et al., 2009), resulting in a series of spatially constrained 1-D resistivity models at the sounding locations (Viezzoli et al., 2008) (Fig. 1b). The SkyTEM resistivity models are then assigned to the nearest sampling grid cells by simple kriging with a 50 m search radius. The end result is a spatially dense incomplete 3-D resistivity grid (Fig. 2a). The high spatial density makes it possible to reconstruct the dataset using geostatistical tools, such as pixel-based kriging techniques, a so-called two-point statistical tool, for reconstructing incomplete datasets (Goovaerts, 1997). Another approach for reconstruction of incomplete datasets is the method using DS presented by Mariethoz and Renard (2010). Since the density of the data points is sufficiently large, the resistivity grid itself can be used as both a TI and soft data variable to stochastically simulate the missing values in the resistivity grid, i.e., the gaps in Fig. 2a. The MPS dataset reconstruction approach (Fig. 2c and e) is advantageous over the variogram-based kriging estimation (Fig. 2b and d) since it only requires setting up a few parameters. Furthermore, the DS approach uses MP information to condition the reconstruction of the dataset. Here, it is important to note that the kriging method is an estimation method, while the DS approach is a simulation method. An estimation method estimates a “best” value, while a simulation method makes a stochastic ensemble of equiprobable guesses. The end result of the DS reconstruction approach is an ensemble of stochastic resistivity grids, of which one realization is compared against a corresponding kriging reconstructed grid in Fig. 2d and e. The close-ups in Fig. 2b and c reveal some key differences in the reconstruction of gaps using kriging and DS. The resistive peak fringing the border of the gap in the westernmost resistive buried valley is smeared into the gap in the kriging reconstructed grid (see close-up in Fig. 2b). However, the single DS reconstruction presented here does not smear the resistive peak into the gap (see close-up in Fig. 2c). The usage of MP information in DS allows the possibility that the resistive peak is not part of the gap.

The uncertainty related to the stochastic resistivity grids is different from the kriging resistivity grid uncertainty. The standard deviation (SD) related to the kriging reconstructed grid is closely related to the distance to the nearest data point (Fig. 2f), whereas the uncertainty on the stochastic resistivity grids reveals values much more correlated to the patterns of the geophysical information.

Comparison of the deterministic kriging and stochastic DS
resistivity grid reconstruction and their corresponding standard deviation.
The presented horizontal slice is centered on 20 m b.s.l.

It is important to note that the resistivity parameter uncertainty has
neither been included in the kriging nor the DS reconstruction, enabling the
comparison of the SD maps. As an example, a gap present in the homogeneous
conductive units with resistivity values between

In summary, the uncertainty of the DS reconstruction provides additional information regarding the reconstructed resistivity patterns over, for instance, a kriging approach. Also, the MPS reconstruction of the incomplete dataset is less smooth, easier to parameterize, and stochastic, and the uncertainty is related to pattern reconstruction and not the distance to the nearest data point.

The MPS grid reconstruction procedure is used to generate an ensemble of
resistivity grids without gaps (Mariethoz and Renard, 2010). The
reconstructed resistivity grids are used as soft data for constraining the
simulation of the hydrostratigraphic models, with the cognitive 3-D
hydrostratigraphic model used as a TI. The full cognitive geological model
contains a total of 42 different geological units (Høyer et al., 2015a),
which have been grouped together to form three key hydrostratigraphic
categories. The three categories are described as follows.

Sand and gravel: Miocene sand, Quaternary meltwater sand and sand till, within and above the Quaternary buried valleys.

Glacial clay: Quaternary clay till and meltwater clay within and above the buried valleys.

Hemipelagic clay: hemipelagic, fine-grained Paleogene and Oligocene clays.

The overall workflow can be seen in Fig. 3. In
detail, the steps are described as follows.

The SkyTEM resistivity grids are reconstructed using the methodology of Mariethoz and Renard (2010) as described in Sect. 3.2 “Reconstructing incomplete dense geophysical datasets”.

The ensemble of reconstructed SkyTEM resistivity grids is used as soft data
for constraining the three MPS methods.

A reconstructed resistivity grid and the TI are used in the snesim
framework.

Using histograms created using the resistivity atlas approach presented by Barfod et al. (2016) (Fig. 4c and d) a single reconstructed resistivity grid is translated into a set of probability maps (Fig. 5).

The TI is used for conditioning in conjunction with the probability maps, which are used for spatially constraining the snesim simulations using the tau model (Journel, 2002). The end result is a realization of a hydrostratigraphic model.

A reconstructed resistivity grid is selected and used in combination with
the TI for running DS.

The soft data variable (the resistivity grid) is used for both constraining and as the auxiliary variable. The soft data grid is directly available as an auxiliary variable since it geographically overlaps with the categorical TI variable. The combination of the cognitive hydrostratigraphic model and auxiliary variable creates a bivariate TI.

The bivariate TI is used together with the soft data grid to simulate a realization of the hydrostratigraphic model.

A reconstructed resistivity grid is used together with the TI for running
iqsim.

As with DS, the soft data grid is used as an auxiliary variable and for spatially constraining the simulations. The TI and auxiliary variable are combined into a bivariate TI.

The bivariate TI is used to create a simulation of the hydrostratigraphic model.

Workflow diagram showing the stochastic modeling procedure for a single realization. Each simulation is run with snesim, DS and iqsim.

Spatially constraining the simulations to the soft data requires information
regarding the relationship between hydrostratigraphic units and, in this
case, resistivity values. In DS and iqsim the information is contained in
the bivariate TI, which in this case consists of a categorical and a
continuous auxiliary variable. As discussed in Sect. 3.1.2 the setup used in
this paper avoids using the

The hydrostratigraphic–resistivity relation shown as a series of
histograms.

In the snesim framework, constraining to the soft data requires a translation of
the soft resistivity data into a set of probability maps, one for each of
the hydrostratigraphic units. This is achieved by using prior information
regarding the hydrostratigraphic–resistivity relationship. Often this
information is difficult to obtain, unless a large number of boreholes are
available. If boreholes are readily available the resistivity atlas
framework (Barfod et al., 2016) can be utilized. The raw resistivity atlas
histograms are seen in Fig. 4c. Due to the
general coarse nature of the histograms the mean and interquartile range
from the coarse histograms (Fig. 4c) were
computed and used to create a set of smooth histograms with identical
summary statistics (Fig. 4d). By comparison the
resistivity atlas histograms are quite similar to the kriging grid
histograms (Fig. 4b). However, the separation
between the (i) sand and gravel and (ii) glacial clay histograms is even larger in the resistivity atlas
histograms. The respective median values are 59 and 34

The summary statistics table for the histograms in
Fig. 4. The first section, named DS, shows
summary statistics for the three histograms seen in
Fig. 4a. The second section, named kriging, shows
the summary statistics for the histograms in Fig. 4b. The last section, labeled resistivity atlas, shows the summary
statistics for the resistivity atlas histograms
in Fig. 4c and d. All the values presented in the
table are resistivities (

The hemipelagic clays have unique properties. They are aquitards with low hydraulic
conductivity and often used as a hydraulically confining no-flow boundary at
the bottom of a groundwater model in parts of Denmark. When hemipelagic clay is encountered
during drilling, the drilling is halted and generally hemipelagic clay is sparse in Danish
borehole lithology logs. For this reason the resistivity atlas based on
transient electromagnetic data does not provide a lot of information on
hemipelagic clays. However, the hemipelagic clays are regionally extensive and homogeneous.
From wireline resistivity logs in eastern Jutland they are found to be
conductive, with median resistivities ranging between 4–7

The model setup is different for the three MPS methods. When running DS and
iqsim the hydrostratigraphic–resistivity relationship is explicitly given
due to the geographically overlapping resistivity grid and
hydrostratigraphic TI. Normally the auxiliary variable has to be created for
the given TI using the

The SkyTEM soft data grids are translated into three sets of
probability grids, one for each lithological category to be simulated;

Comparing 153 3-D models each with 1 187 823 grid cells is not trivial. Visual comparison is used mainly to check if the results are geologically realistic, but a detailed visual comparison would be time consuming and subjective. Therefore, a set of tools are used to compare how similar the simulation results are to each other and how different they are from the TI.

In this study, a distance measure is used as a measure of similarity between
3-D model simulations. The chosen distance measure is the modified Hausdorff
distance (

Dubuisson and Jain (1994) found that the

A 2-D example is presented to illustrate the overall

A 2-D example of the binary categorization of the
hydrostratigraphic models and example of the Roberts cross operator for
edge tracing.

From here on, we leave the 2-D example and consider the entire 3-D model. In
this study, the

The initial step is to create a matrix containing all

The other evaluation measure, which can be calculated from

It is also possible to evaluate the

In reservoir modeling, boreholes are considered to be hard information, due
to their overall high quality. However, in many surveys related to
groundwater modeling, boreholes cannot be considered as reliable hard data
due to variable quality – such as seen in Barfod et al. (2016) and He et
al. (2014), where boreholes were divided into quality groups. Therefore, the simulations
are run without constraining against boreholes, and then the realizations
are compared against the boreholes as an independent measure of geological
realism. A method for comparing similarity between the simulated
hydrostratigraphic models and the boreholes was developed. The method does
not use the

The end result is three arrays, one for each hydrostratigraphic unit, each containing one average distance per realization for the given MPS method. The distance arrays for each individual MPS method can then be compared to the distance arrays of the other MPS methods.

An example of how a single lithology log is categorized and sorted for the purpose of calculating the borehole distance. The first step is to translate the raw lithology log into a hydrostratigraphic log, which is achieved by categorizing the multiple lithological categories into a subset of three hydrostratigraphic categories corresponding to the target categories we wish to model. Note that some categories do not fit into the overall hydrostratigraphic categories and are therefore not translated, e.g., the meltwater sand category in this example. The final step is then to assign the hydrostratigraphic logs to the regularized sampling grid and create one binary log for each of the three target modeling categories. This is done by simply asking whether or not the given hydrostratigraphic category is present (true) or not (false) for the given sampling grid interval.

The hydrostratigraphic simulation results include 153 3-D hydrostratigraphic realizations, each containing 1 187 823 grid cells. The models can be subdivided into 51 snesim realizations, 51 DS realizations and 51 iqsim realizations. A visual presentation of the hydrostratigraphic model or TI as well as two realizations for each of the three different MPS methods is seen in Fig. 8. The cognitive hydrostratigraphic model (Fig. 8a) shows clear-cut and smooth buried valley architecture with almost no short-scale variability. Comparing the cognitive hydrostratigraphic model to the stochastic MPS hydrostratigraphic models reveals the more erratic nature of both snesim and DS; i.e., both MPS methods yield models containing short-scale variability (Fig. 8b and c).

Overall snesim (Fig. 8b) and DS (Fig. 8c) realizations are similar in nature. In the example provided, Fig. 8c, the west–northwest- to east–southeast-trending glacial clay valley (see box in Fig. 8a) is uninterrupted in one realization, but intersected by hemipelagic clay in the other realization. In 47 of the 51 snesim realizations, the glacial clay valley is uninterrupted; in the remaining 4 realizations the valley is intersected by hemipelagic clay. The presented soft data grid in Fig. 5d shows a small probability of approximately 10 % for hemipelagic clay at the position of the valley gap. The 4 realizations which yielded an interrupted glacial clay valley amount to 8 % of the 51 realizations, which is close to the probability found in the probability grids. The DS realizations shows valley architecture with less resemblance to the soft data, i.e., the valleys are not conditioned in accordance to the soft data grids. In 11 of the 51 simulation results the valley is intersected by hemipelagic clay, amounting to 22 % of the 51 realizations.

The iqsim results are the most similar to the cognitive hydrostratigraphic model with regards to short-scale variability, which is generally nonexistent. Generally, realizations will reflect the TI, and short-scale variability is only introduced if present in the TI. This is due to the nature of iqsim, which is not a pixel-based algorithm, like snesim and DS. Instead, iqsim cuts the TI into patches and then reassembles the patches, which means that noise patterns which are smaller than the patch size cannot be fabricated, unless actually present in the TI. The iqsim realizations show smooth and clear-cut valley architecture. The main issue with the iqsim realizations is that artifacts are introduced near the surface of the model, which is evident if the vertical iqsim cross sections (Fig. 8d) are compared to the remaining vertical cross sections of the TI, snesim and DS (Fig. 8a–c). This is neither reflected in the resistivity grid (Fig. 5) nor in the TI (Fig. 8a). Close to terrain hydrostratigraphic layers consist of either glacial clays or sand and gravel, and conductive hemipelagic clays are not evident. Since the soft data does not support the presence of the hemipelagic clays in the upper part of the hydrostratigraphic model, the soft data can be concluded to be improperly constrained with this specific setup. Another observation is that in 43 out of the 51 realizations, amounting to 84 %, the referenced glacial clay valley is intersected by hemipelagic clay.

The hydrostratigraphic MPS realizations are presented as
horizontal slices centered on 20 m b.s.l. and vertical cross sections
intersecting at UTMY 6 230 150 m.

An advantage of the iqsim implementation used
(Hoffimann et al., 2017) is the favorable
computation time. On an Intel®HD Graphics Skylake ULT GT2 GPU
of a Dell XPS 13 laptop, iqsim runs with an average simulation time of 10–12 min per realization with the attempted setup. On a different laptop running
a 64 bit Windows system, with 8 GB RAM, an SSD hard disk, with an Intel®Core
i7-3520 M CPU at 2.9 GHz, the computation times for snesim were on average
between

A table presenting the average computation times per realization
for each of the three MPS methods and the approximated computation times
needed for running 51 realizations with the setup used in this study.

The

Summary of the modified Hausdorff distance (

The

The

The final comparison of the MPS methods regards the average Euclidean distance between the simulation results and the regularized binary hydrostratigraphic logs. The sorted average distances between each individual simulation and the boreholes are seen in Fig. 10.

The borehole distance results are presented for each of the three
MPS methods: snesim, DS and iqsim.

The average distances between the simulated hydrostratigraphic models and the boreholes are presented according to the three key hydrostratigraphic units. The average distance between sand and gravel units in the hydrostratigraphic realizations and sand and gravel units in the boreholes seems to be the largest for the modeling results of all three MPS methods (Fig. 10); i.e., the red curve is always on top. The average values of the individual curves in Fig. 10 are computed and presented in Table 4. An overall borehole distance average for each of the three MPS methods is computed as the average of each row in Table 4. The sand and gravel average in Table 4 reflects the large distances between resistive sand and gravel units in the realizations and the hydrostratigraphic logs. By comparing the individual frames of Fig. 10 it is seen that the average values for the hydrostratigraphic models created using iqsim have a higher average distance. The iqsim average for sand and gravel is centered on 5.8 m, while for snesim and DS it is centered on 3.8 and 4.9 m, respectively. The iqsim average distance to glacial clay is centered on a relatively large value of 3.5 m, as opposed to 2.1 and 2.8 m for snesim and DS, respectively. The hemipelagic clay units show a different pattern where iqsim has the lowest average distance of 0.2 m, while the snesim and DS distances are 1.6 and 0.8 m, respectively. The snesim method has the smallest borehole distance row average of 2.5 m, while DS and iqsim have row averages of 2.8 and 3.2 m, respectively.

The borehole distance results are summarized in this table. The borehole distances are the 3-D Euclidean distances calculated using the concept presented in Sect. 3.6. The presented distance values are the averages of the curves shown in Fig. 10, one average for each of the individual hydrostratigraphic units for each of the presented methods: snesim, DS and iqsim realizations. The last column shows the average distances for each of the three MPS methods.

In areas of groundwater interest, the initial step is to collect different
types of data relevant to the hydrogeological properties of the subsurface.
Among these data are dense geophysical datasets, e.g., SkyTEM, which can be
collected quickly and usually cover a significant part of the survey area.
The different datasets are processed and modeled and used in conjunction
with the borehole lithology logs to create a single geological and/or
hydrostratigraphic model. This model is only one version of the subsurface,
encasing only part of the complexity related to the given hydrological
system. We present an example of stochastic simulation of hydrostratigraphic
models. The result consists of multiple hydrostratigraphic realizations,
covering a larger span of possible models. Using the cognitive
hydrostratigraphic model from area

An overview of the setup for simulating new survey areas and the
hydrostratigraphic modeling results using the Kasted dataset. The presented
horizontal slices are centered on 20 m b.s.l., and the vertical cross
section intersects at UTMY 6 230 150 m.

The example presented in this study is synthesized from the Kasted dataset. The dataset is divided in two along the UTMX coordinate 569 025 m (Fig. 11a). The left half of the cognitive hydrostratigraphic model is then used as a TI to simulate the right half of the model. The reconstructed resistivity grid is also cut in half (Fig. 11b). The left half of the resistivity grid is used as an auxiliary variable describing the hydrostratigraphic–resistivity relationship, as seen in Fig. 4a, while the right half is used for spatially constraining the simulation. In this example 10 stochastic hydrostratigraphic realizations are created using DS. The DS method was selected since it is both easy to parameterize and to run in parallel on a computer cluster. A single hydrostratigraphic realization is seen in Fig. 11c, while the mode of the hydrostratigraphic model ensemble is seen in Fig. 11d. Using the same splitting of data and TI from the Kasted survey area, simulations using iqsim are presented by Hoffimann et al. (2017).

The simulation results show that one hydrostratigraphic realization represents the overall architecture of the resistivity grid (compare Fig. 11c and b). Comparing the single hydrostratigraphic realization (Fig. 11c) to the original cognitive model (Fig. 11a) reveals that one realization largely reflects the variability in the soft data grid. The mode of the model ensemble on the other hand (Fig. 11d) has a closer resemblance to the cognitive hydrostratigraphic model (compare Fig. 11a and d). This means that the individual realizations do on average resemble the original cognitive model. The end goal is not to create a set of hydrostratigraphic models which match the cognitive hydrostratigraphic model. The goal is to create a suite of realistic hydrostratigraphic models. Generally, short-scale variability is introduced in both the single hydrostratigraphic realization and in the ensemble mode model, but is generally not present in either the TI or the resistivity grid.

The snesim setup is different from the DS and iqsim setups. The snesim setup
differs in the usage of the probability framework and in the choice of the
implicit resistivity atlas histograms (Barfod et al., 2016). The implicit
histograms (Fig. 4d) are used to directly
translate the resistivity grids into probability grids. This illustrates the
utility of the resistivity atlas framework in relation to geostatistical
modeling. The DS and iqsim frameworks would normally, in real-world cases,
require the usage of a

The snesim and DS realizations portray some differences, which are related to
the choice of the implicit resistivity atlas histograms for translating the
resistivity grid. This can help us understand some of the basic differences
in the information provided by the implicit resistivity atlas histograms and
the explicit auxiliary variable. In DS probable hydrostratigraphic units are
not conditioned properly. An example of this is the aforementioned
west–northwest- to east–southeast-trending glacial clay valley (see
Fig. 8a), which is uninterrupted in 78 % of the DS realizations. The same
valley is clearly represented in the resistivity grid (Fig. 5a) and in the
cognitive model (Fig. 8a). However, the explicit auxiliary variable
histograms show increased overlapping resistivity values for the glacial
clay and sand and gravel histograms (Fig. 4a, d). The auxiliary variable
histograms (Fig. 4a) reveal approximately equal probability of glacial clay
and sand and gravel resistivities lying close to 40–45

The

In conclusion, snesim and DS yield similar realizations, portrayed by the
relatively small

Short-scale variability is present in the snesim and DS realizations. This can be seen as an artifact introduced by the algorithms themselves and does not reflect the underlying datasets, i.e., the soft data or TI. As Linde et al. (2015) discuss, fine-scale patterns are present in the real-world hydrostratigraphic subsurface but are only slightly resolved in geophysical models. Two of the three presented stochastic MPS methods introduce fine-scale variations in the form of short-scale variability to the overall hydrostratigraphic architecture, with the overall architecture resembling the underlying datasets. This adds complexity to the realizations and the resulting equiprobable hydrostratigraphic models span a larger subset of possible models. The question, however, is whether this short-scale variability is similar to the real-world short-scale variability missing from our geophysical data, which is difficult to answer. The importance of short-scale variability also depends on the type of prediction for which the hydrostratigraphic model is to be used.

An important difference in the iqsim realizations, compared to snesim and DS,
is the lack of fine-scale variability and the resulting valley architecture.
The

In relation to the new survey example, it is worth mentioning a caveat. When cutting the TI in half the 3-D objects are reduced in size and some of the 3-D objects are entirely removed. Generally the TI should contain the objects which are to be conditioned during simulation (e.g., Emery and Lantuéjoul, 2014; Journel and Zhang, 2006; Strebelle, 2002). If the 3-D objects are not fully represented in the TI, it cannot be guaranteed that they will be reproduced in the resulting realizations (Emery and Lantuéjoul, 2014). It is therefore important to state that the example simply just exemplifies an important application of MPS in relation to dense geophysical datasets but is not a valid practical application.

The three MPS methods – snesim, DS and iqsim – are used for stochastic hydrostratigraphic modeling. The modeling results are compared in an elaborate framework of comparing the modeling results visually, mathematically and against boreholes. Each individual MPS method has its own set of advantages and disadvantages which are covered in this study. Overall the DS method had the highest computation times. An average DS realization takes 6–7 h, while for snesim it takes 2–3 h and for iqsim 10–12 min. We emphasize that these times are for a specific setup and that they will likely change for different configurations. Both the snesim and DS methods yield realizations with sufficient soft data conditioning, as reflected in the modeling results in Fig. 8a–c. The iqsim realizations showed erratic results in regards to the overall valley architecture, compare Fig. 8a, d, which was due to insufficient soft data conditioning.

The presented example for modeling new survey areas uses a cognitive hydrostratigraphic model from one area as a TI to simulate the new area without a preexisting cognitive model. The requirements are two-fold: (1) the geological settings of the two areas need to be similar and (2) the statistical hydrostratigraphic–petrophysical relationship needs to be stationary between the two areas. The presented example shows a case where the two requirements are true, and the set of stochastic models is consistent with the cognitive geological model.

Finally, the importance of the underlying resistivity–hydrostratigraphic relationship has been shown. The relationship contains information on the translation of the continuous soft data variable into subsurface hydrostratigraphic units and is indirectly used for soft data conditioning. The MPS modeling results are therefore sensitive towards the resistivity–hydrostratigraphic relationship, and the more information acquired regarding the relationship, the better the realizations.

The Kasted data are publicly available and can be downloaded by using the interactive maps
found on

The authors declare that they have no conflict of interest.

We would like to thank the two anonymous referees for their contributions towards improving the research enclosed in this paper. This study is supported by HyGEM, integrating geophysics, geology and hydrology for improved groundwater and environmental management, project no. 11-116763. The funding for HyGEM is provided by the Danish Council for Strategic Research. Edited by: Monica Riva Reviewed by: two anonymous referees