An Aquifer-Imitating Technique

An Aquifer-Imitating Technique Based on the Monte Carlo Method

Adam Szymanski

Numerics: Stochastic MethodsParallel Session (parallel33), 01.09.1998, 16:00 - 18:15Numerical Methods for Subsurface and Rock FlowAdvances in Hydro-Science and – Engineering Volume IIIProceedings of Abstracts and Papers (on CD-ROM) on the 3rd International Conference on Hydro-Science and –Engineering Cottbus/Berlin, Germany 1998

Abstract

Applications of the MIE reconstruction and simulation techniques (Minimum of the sequence of Informational Entropies) in such fields as;

· digital image processing (continuous-tone colour structures), and, · regional hydrology (a model for creating 3-D images of subsurface aquifer' zones),

are presented.

One considers deterministic and random MIE reconstruction methods. The random MIE reconstruction method is an extension of the deterministic one, and is based on the application of the Monte Carlo procedure. The basic idea of this technique is to represent a piecewise constant function by means of a set of the random functions, assuming that their number tends to infinity. The deterministic technique is simply a discrete interpolation of piecewise constant functions in 3-D Euclidean space, and consists of three submodels;

· a discretization model which describes a sampling procedure to replace a given structural image in continuous space with a discrete one,

· an interpolation model, where the minimum of the sequence of informational entropies was chosen as a decision making criterion, and,

· a space-scale model that characterizes space-scale properties of sampling data and supplies the discrete reconstruction structure.

The random MIE reconstruction method has been extented to the Monte Carlo technique for simulating complex geological structures in 3-D Euclidean space, where the unknown structure of the region studied is defined by means of the random translation field. The Markov chains theory, as a method for generating the correlated non-stationary random fields, is used as a model for the translation field.

The developed techniques are based on the application of two concepts; informational entropy, and variable continuous structuring elements. The structuring elements are the geometrical shapes, modelled as the closed balls in continuous Euclidean space, and are used

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for extracting the geometrical properties of sampling data. Both techniques are intrinsically discrete and allow parallel computations.

1. Introduction

The concept of the zonation is used in the field of regional hydrology as a method for describing the so-called regional- scale heterogeneity. In zonation an aquifer is divided into hydrologic facies or hydrostratigraphic units, using a geological and geophysical information such as; lithological logs, soil borings, cores, geological maps, cross-sections, geophysical logs, paleontological information, age data, and hydraulic and tracer test data. In addition, zonation is particularly useful for regional flow problems, where there is usually not enough data to define small-scale hydrostratigraphic units. Further, it is clear that above-mentioned information can also be applied to create the 3-D images of the subsurface aquifer' zones. Approaches for creating such images can be grouped into three categories; aquifer-imitating, process-imitating, and descriptive. Their classification can be found in [1].

Recently, Szymanski [2] proposed new aquifer-imitating method, called MIE reconstruction technique (Minimum of the sequence of Informational Entropies) which is based on the concepts of informational entropy and variable structuring elements. The structuring elements are geometrical shapes, modelled as the closed balls in continuous Euclidean space, which interact with the sampling points in order to extract useful geometrical characteristics. Their formal definition is presented in [2].

In [2] one considers deterministic and random MIE reconstruction techniques. The random MIE reconstruction method is an extension of the deterministic one, and is based on the application of the Monte Carlo procedure. The basic idea of this technique is to represent a piecewise constant function by means of a set of random functions, and specifying the greatest probability structure as an estimator. Its detailed description can be found in [3], where convergence properties are also discussed.

The deterministic technique is simply a discrete interpolation of the piecewise constant functions in 3-D Euclidean space, and consists of three models; a discretization model, an interpolation model, and a space-scale model (cf., [2]).

The random MIE reconstruction method has been extented in [3] to the Monte Carlo procedure for simulating complex geological structures in 3-D Euclidean space, where the unknown structure of the region studied is defined by means of the random translation field. The Markov chains theory, as a method for generating the correlated non-stationary random fields, is used as a model for the translation field.

The difference between the notions of the reconstruction and simulation can heuristically be described in the following way; having defined the appropriate set of the different sampling realizations of a given structure, we will speak about the reconstruction. Having defined, however, only one such a realization we will consider the simulation model. Indeed, it is clear that the quality of the input information is not comparable in both situations mentioned above. Some applications of the deterministic MIE reconstruction technique in the field of the regional



hydrogeology can be found in [4], where we used this technique as a preprocessing tool for the 3-D simulations of the regional groundwater flow in unconfined aquifers. In [4] we show the importance of incorporating information about aquifer heterogeneity, and spatial structure of geological formations into the regional studies of fluid flow in quaternary deposits, especially, when the flow have to be treated as unconfined. Other applications can also be found in [5], [6], [7]. For example, in [5] the comparison between the ordinary kriging and deterministic MIE reconstruction technique has been discussed. Applications of the random MIE reconstruction technique are presented in [3].

Both above-mentioned techniques are intrinsically discrete and allow parallel computations. In terms of the MIE reconstruction algorithm, each 2-D picture or 3-D structure can formally be treated as a piecewise constant function. Thus, the random MIE reconstruction technique can also be applied in the field of digital image processing. Indeed, in computer graphic, an image is defined by a function which associates a colour to each pixel. Usually, a colour is described by a single discrete value (grayscale image or colourmap image) or by a triplet (a, b, c) where a, b, and c are discrete values (RGB image, HLS image, etc.).

The present paper is organized as follows; in section 2 we show the application of the MIE random reconstruction technique to the processing 2-D images. As an example the colour photography, which is represented by means of the so-called three-spectral-system RGB (Red, Green, Blue) is reconstructed. Section 3 includes the application of the deterministic MIE reconstruction technique to the reconstruction of the complex 3-D quaternary sequences. The application of the MIE simulation technique to 3-D case is presented in section 4. Finally, in section 5 we discuss the result obtained.

2. Reconstruction of the colour images

To illustrate one of the possible applications of the random MIE reconstruction technique we consider 2-D colour photography shown in fig. 1.

fig. 1: Original photography used as an example for the reconstruction

Since presented image is characterized by a continuous-tone colourtable, its reconstruction is a very difficult task. This image was given in a form of a postscript file. Thus, we carried out the conversion to the discrete structure (70797 pixels) using the graphic user interface of the MIE-SYSTEM which



is based on the X11-Window R5/Motif R1.2 software (see APPENDIX A). The RGB-colourmap of the converted image and its colour-frequency spectrum are shown in figs 2, and 3, respectively. Unfortunately, some of the frequencies are not visible due to the vertical scale that has been chosen.

fig. 2: RGB-colourmap of the original image shown in fig.1 (43 RGB colours)

fig. 3: Colour-frequency spectrum the original image shown in fig.1

Formally, the RGB-colourtable of the MIE graphic user interface includes 256 different colours (cf., APPENDIX A). However, after conversion the original image is only characterized by 43 different RGB colours. According to the homogenization procedure presented in [2], its RGB-colourmap has been mapped onto the subset of natural numbers, giving 43 different attributes for the random reconstruction. Further, using the random MIE reconstruction technique where the number of sampling points was equal to 1000, and the number of generated realization was taken to be equal to 10, 30, 50, we obtained the reconstructed images which are presented in figs 4, 5, and 6, respectively. Note that a basic discretization element, within both considered here MIE techniques, is simply a point in 3-D Euclidean space. The structural diagram of the random MIE reconstruction technique is described in [3], where the notion of convergence is also defined.



fig.4 fig.5 fig.6

fig. 4: Reconstructed image (1000 sampling points, 10 random realizations) fig. 5: Reconstructed image (1000 sampling points, 30 random realizations) fig. 6: Reconstructed image (1000 sampling points, 50 random realizations)

Additionally, figs 7 and 8 show the RGB-colourmap and the colour-frequency spectrum for the reconstruction shown in fig. 4 (10 random realizations).

fig. 7: RGB-colourmap of the reconstructed image shown in fig.4 (29 RGB colours)

fig. 8: Colour-frequency spectrum of the reconstructed image shown in fig.4



Note that the RGB-colour 254 was used to describe the attribute where the space-scale model was not able to make the interpolation decision (the so-called attribute 0, cf., [2]). The points characterized by the RGB-colour 254 should be interpreted as a noise, induced due to the lack of appropriate information for the interpolation or space-scale models. Its graphical representation, for the reconstruction presented in fig. 4, is shown in fig. 8a.

fig. 8a: 2-D representation of the noise for the reconstruction shown in fig.4

This problem is related to the scale-space properties of the sampling data, and is too technical to be considered here. For the formal explanation the reader is referred to [2]. The above-mentioned problem is directly related to the convergence characteristics of the random MIE reconstruction technique and is discussed in [3].

To show the influence of the number of sampling points on the reconstruction results, we carried out the random reconstruction taking their number equal to 10000. figs 9, 10, and 11 show these reconstructions using 5, 10, and 20 random realizations, respectively.

fig.9 fig.10 fig.11

fig. 9: Reconstructed image (10000 sampling points, 5 random realizations) fig. 10: Reconstructed image (10000 sampling points, 10 random realizations) fig. 11: Reconstructed image (10000 sampling points, 20 random realizations)

In addition, figs 12 and 13 illustrate the RGB-colourmap (36 RGB colours) and the colour- frequency spectrum for the reconstruction shown in fig. 11.



fig. 12: RGB-colourmap of the reconstructed image shown in fig. 11 (36 RGB colours)

fig. 13: Colour-frequency spectrum of the reconstructed image shown in fig. 11

Note that the frequency of the so-called noise-colour 254 decreases implying better reconstruction results. Indeed, the 2-D representation of the noise, for the reconstructed image shown in fig. 11, is illustrated in fig. 13a.

fig. 13a: 2-D representation of the noise for the reconstruction shown in fig. 11



Further, fig. 14 visualises the similarity between the original image shown in fig. 1, and its reconstructions presented in figs 9, 10, and 11. To describe the similarity of the reconstructed structures the concept of the similarity function (similarity measure) has been introduced. Its detailed description is presented in [2], and [3] where the set of the reconstructed structures is considered as a discrete metric space. For example, the similarity between the original image and its reconstruction shown in fig. 11, is approximately equal to 50% (20 random realizations). This means that to obtain better reconstruction results the greater number of the random realizations should be used(cf., [3]).

fig. 14: Similarity relations between reconstructed images and the original one

3. Reconstruction of the 3-D geological structures

To illustrate the application of the deterministic MIE reconstruction technique in the field of quaternary geology we consider the 3-D geological structure presented in fig. 15.

fig. 15: 3-D view of the original geological structure

Its geometry is as follows; the length/width/thickness are equal to 1000/1000/100 [m], respectively. In addition, one has assumed that the geometrical location of the quaternary base is constant and is equal to 0 [m]. Five different geological formations mimic horizontal stratification of quaternary sequences. The above-shown structure has been generated using the so-called "geological



generator" which is based on the work by Mengeling [8]. In this section we follow the concepts presented in [8], and carry out the reconstruction using the special net-discretization model suggested by Mengeling. To construct such a discretization model we used the notion of the pseudo-borehole which was introduced in [3]. fig. 16 shows the horizontal projection of the net-discretization model suggested.

fig. 16: Horizontal projection of the 3-D net-discretization model Using 25 boreholes we have constructed the appropriate cross-sections between the given boreholes, similarly as was presented in [8]. Of course, the given boreholes were treated as the so-called "hard" data. The heuristical definition of the notion of "hard" data is presented in [3], and will be not repeated here. In such a way one obtains as a discretization model a 3-D net which consists of 100971 discrete sampling points. The 3-D view of a part of above-mentioned discretization net is shown in fig. 17.

fig. 17: 3-D view of a part of the net-discretization model



Appling such a discretization model we may carry out the 3-D reconstruction. However, for simplicity of graphical presentation, the horizontal cross-section 50 [m] over the quaternary base was only chosen for the reconstruction. For comparative reasons, fig.18 shows this cross-section of the original geological structure (cf., fig. 15). The reconstructed cross-section is presented in fig. 19.

fig. 18 fig.19

fig. 18: Horizontal cross-section of the original structure (50 [m] over the quaternary base , see fig. 15) fig. 19: Reconstructed cross-section using the 3-D net-discretization model (50 [m] over the quaternary base) The example presented above shows that the net-discretization models developed manually by geologists can be effectively extented to 3-D models using, for instance, the deterministic MIE reconstruction approach. Of course, manual generation of a 3-D net of cross-section, as was presented in [8], is a time-consuming procedure, and should not be treated as a standard methodology in geological practice [H. Mengeling, personal communication 1996].

4. Simulation of the 3-D geological structures Above applied discretization model allows for the use of the deterministic MIE reconstruction technique. In fact, the comparison of the original and reconstructed cross-sections (see figs 18 and 19) shows that the sampling used is fine enough to guarantee for the reasonable reconstruction results. Let us now consider the discretization model shown in fig. 20.

fig. 20: 3-D view of the discretization model



Note that we only have 25 boreholes at our disposal as "hard" data. In this case it seems to be reasonable to apply the MIE simulation model presented in [3]. This technique is an extension of the random MIE reconstruction model and is based on the notion of the random translation field. This field is used to introduce the "soft" information such as; regional geological analyses or conceptual depositional models. This is necessary in a case of sparse "hard" data. The Markov chains theory is applied as a model for the translation field. The choice of the parameters for generating the translation field should be based on the practical experience of geologists, and is considered as a part of the so-called modelling procedure. Its detailed description can be found in [3]. We restrict our attention here to the presentation of an numerical example. fig. 21 shows simulated horizontal cross- section 50 [m] over the quaternary base. To generate this cross- section 25 pseudo-boreholes, and 1000 pseudo-boreholes have been used as "hard" and "soft" data, respectively. Another view of the simulated cross-section illustrates fig. 22.

fig.21 fig.22

fig. 21: Simulated horizontal cross-section (50 [m] over the quaternary base) fig. 22: Simulated horizontal cross-section (another 3-D view)

Moreover, using the MIE simulation technique we are able to generate multiple geological maps or 3-D structures of an aquifer. In this context, zonation is subjective and dependent on the quality of "hard" data and experience of hydrologist developing the aquifer image. Further, it is clear that the MIE simulation technique allows for examining the effects of uncertainty associated with number of zones and their geometry, the number of geologic units, and the assumptions of the homogeneity within hydrostratigraphic units. The diagram of the MIE simulation technique that shows the hierarchical structure of the algorithm proposed is described in [3].

5. Summary and Conclusion

In papers [2], and [3] using the framework of the set-theoretical operations and the concept of informational entropy the random MIE reconstruction/simulation techniques, as the models for reconstructing/simulating continuous geological structures, were developed.

In the present paper we show the applications of these models in the fields of digital image processing and quaternary geology. Other applications, for instance, in the field of the regional hydrology are also mentioned.



We developed the above-mentioned techniques for better characterization of subsurface heterogeneity. Using such models one obtains better predictive capability of distributed parameters groundwater flow and transport models. Further, the models proposed here can also be used for prediction of uncertainty in regional hydrogeological studies or for designing the production and injection well systems in petroleum reservoirs.

For numerical computations presented here we used single workstation running the UNIX operating system. It would be of interest to implement both models mentioned applying parallel computers or distributed systems. In such a case one will have the opportunity to use the proposed algorithms in an efficient way.

Acknowledgements

The author is greatful to ARG for funding this work, and is much indebted to Peter Szymanski for the technical support during the implementation of the graphic user interface presented.

References

[1] Koltermann, E., CH. and S. M. GoerlicHeterogeneity in sedimentary deposits: A review of structure-imitating, process-imitatingand descriptive approaches, Water Resources Research,vol. 32, No. 9, pp 2617-2658,September 1996

[2] Szymanski, A.Reconstruction of geological structures for the simulation of regional transport processes,in A. Mueller (ed.): Hydroinformatics'96, vol.2, pp 565-571, A.A Balkema, Rotterdam, 1996

[3] Szymanski, A.Reconstruction and simulation of geological structures,(submitted to: MATHEMATISCHE GEOLOGIE, vol. 5, H. Thiergeartner (ed.), CPress VerlagDresden, 1998)

[4] Szymanski, A. and M. SchoenigerAnwendung von Grundwassermodellen feur die 3-D Simulationen der Stroemungsverhealtnisse im SFB-Untersuchungsgebiet Nienwholde, Sonderforschungsbereich 179, Abschlussbericht, Band 1, Inst. Geogr. Geooek. Tech. Univ. Braunschweig, 1996

[5] Szymanski, A. and M. SchoenigerEntwicklung und Anwendung von Modellen zur numerischen Simulation der Stroemungs undTransportvorgeange im paleaozoischen Festgestein (Grosses Mollentall/Oberharz),unveroeff. Abschlussbericht (gemeinsames Forschungsprojekt mit den Harzwasserwerken desLandes Niedersachsen, Harzbeache II), 1996

[6] Schoeniger, M. and A. SzymanskiDie Interpolationsmethode MIE zur Erstellung von reaumlichen Modellen des geologischen



Untergrunds, Arb.-H. Geologie, H. 1., 53-59, NLfB, Hannover, 1996

[7] Schoeniger, M.Practical applications of the geological reconstruction model,in A. Mueller (ed.): Hydroinformatics'96, vol.2, pp 565-571, A.A Balkema, Rotterdam, 1996

[8] Mengeling, H.Geologische Karte von Niedersachsen 1:25000 (Erleauterungen zu Blatt Nr. 3515 Hunteburg), NLfB, Hannover, 1994

APPENDIX A:

Graphic user interface of the MIE-SYSTEM

The above described MIE-SYSTEM is equiped with a graphic user interface developed under X-Window and Motif software, which allows not only the appropriate presentation of the results obtained, but is also useful for converting different continuous 2-D images to the discrete structures (a scan option). Such an option is particularly convenient in hydrogeological applications. Indeed, having at the disposal the old maps which are not in a digital form, applying the scanner device, one obtains the digital image of the document as a gif-image, PS-image, JPEG-image, XWD-image, etc., which can be easy converted to the discrete structure. If the high precision is not required above-proposed technique is a very comfortable solution. fig. 23 shows the graphic users interface of the MIE-SYSTEM. In addition, the colourtable used for the conversion procedure described in section 2 is also presented.

fig. 23: Graphic user interface of the MIE-SYSTEM



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An Aquifer-Imitating Technique