Electronic Geosciences

ISSN 1436-2511

Electronic Geosciences (2000) 5:1

A New Approach to Gravity Anomaly Separation:

Differential Markov Random Field (DMRF)

O. N. Uan1, B. Sen2, M. A. Albora3, and A. zmen1

1 Istanbul University, Engineering Faculty, Electrical & Electronics Department, 34850, Avcilar,Istanbul, Turkey2 TUBITAK Marmara Research Center, P.O:21 41470 Gebze, Kocaeli, Turkey3 Istanbul University, Engineering Faculty, Geophysical Department, 34850, Avcilar, Istanbul,Turkey

Correspondence to: O. N. Uan (email: uosman@istanbul.edu.tr)

B. Sen (email: bulents@mam.gov.tr))M. A. Albora (email: muhittin@istanbul.edu.tr A. zmen (email: aozmen@istanbul.edu.tr)

Received: 16 June 1999 / Revision: 16 November 1999 / Accepted: 4 February 2000

Abstract. In this paper the authors introduce a novel approach to stochastic image processing,denoted as Differential Markov Random Field (DMRF), which has been applied to gravityanomaly separation problems. The advantages of the method are that it introduces only littledistortion into the shape of the original image and that it is not affected significantly by factorssuch as the overlap power spectra of regional and residual fields. Testing of the proposed methodusing synthetic examples gave excellent results.

Key words: Differential Markov random field (DMRF) Gravity anomaly Residual andregional anomalies

Introduction Gravity anomaly Maps - Statement of Problems Differential Markov Random Field (DMRF) Approach

Basic Definitions of Gibbs Distributions Differential Markov Random Field Model (DMRF) for Gravity Anomaly Map

Examples of the DMRF Approach Discussion Appendix A References

Introduction

One of the main purposes of geophysical mapping is the identification of features that can berelated to the unknown geology of a region. In classical approaches to gravity anomaly of theoriginal images, correlation of neighbouring pixels and stochastic property of the imagesexamined are not considered together.

The introduction of stochastic models into image analysis has led to the development of manypractical algorithms such as the Markov Random Field, MRF algorithm (Geman and Geman,1984; Derin and Elliot, 1987). The MRF approach is a stochastic relaxation algorithm whichgenerates a sequence of images that converges in an appropriate sense to the maximum aposteriori (MAP) estimate. This sequence evolves by local changes in pixel grey levels and inlocations and orientations of boundary elements. The objective of modelling in image analysis isto capture the intrinsic character of images in a few parameters and thus to gain an understandingof the nature of the phenomena generating the images. Image models are also useful inquantitatively specifying natural constraints and general assumptions about the physical world andthe imaging process. The success of MRF algorithms is largely due to the incorporation ofneighbourhood relations and stochastic modelling of the 2-D images.

This paper presents a new unsupervised algorithm is which is based on recursive Bayes smoothingof gravity anomaly maps modelled by Markov random fields. The Bayes method yields the aposteriori distribution of the map at each pixel, given the total noisy map, in a recursive manner.In our algorithm, both the absolute difference between neighbouring pixels and the stochasticproperties of the gravity anomaly map have been evaluated in real time and a new stochasticapproach known as the "Differential Markov Random Field" (DMRF) has been proposed. Sincethe neighbourhood relations and the statistical properties of the original data are evaluated andoptimised simultaneously, our approach gives satisfactory results. The DMRF method does notrequire a priori knowledge about the gravity anomaly map. Thus it can be applied to real datawithout prior information.

This paper is organised as follows. Section II is entitled Gravity Anomaly Maps Statement ofProblem. In Section III, the Differential Markov Random Field Approach is explained. In SectionIV, the proposed method is tested using a synthetic example and excellent results have beenobserved.

Gravity anomaly Maps - Statement of Problem

Geophysical maps usually contain a number of features (anomalies, structures, etc.) superposed onone another. An interpretation of such maps aims to extract as much useful information as possiblefrom the data. Since one type of anomaly often masks another, the need arises to separate thevarious features from one another.

Traditionally, gravity maps are subjected to operations approximating certain functions such assecond derivative and downward continuation. The gravity expression valid for a geometricalbody having depth and amplitude coefficient m, is given as (Pick et al., 1973)

g = mW (1)

where W is a known analytical function in depth that has positional coordinates. Gravity dataobserved in geophysical surveys are the sum of gravity fields produced by all undergroundsources. The targets of specific surveys are often small-scale structures buried at shallow depths,and these targets are embedded in a regional field that arises from residual sources which areusually larger or deeper than the targets or are located farther away. Correct estimation andremoval of the regional field from initial field observations yields the residual field produced bythe target sources. Interpretation and numerical modelling are carried out on the residual field data,and the reliability of the interpretation depends to a great extent upon the success of theregional-residual separation.

In the literature some classical methods are proposed for the separation of gravity maps. Thesimplest is the graphical method in which a regional trend is drawn manually for profile data.Determination of the trend is based upon interpreters understanding of the geology and relatedfield distribution This is a subjective approach and also becomes increasingly difficult with large2-D data sets. In the second approach, the regional field is estimated by least-squares minimisationof the observed field (Abdelrahman et al., 1991). This reduces subjectivity, but still requiresspecification of the order of the polynomial and selection of the data points to be fitted. The thirdapproach applies a digital filter such as Wiener filtering to the observed field (Pawlowski et al.,1990). On a regional scale, gravity maps are the most useful tools presently available, althoughother techniques such as conductivity mapping (Palacky, 1986) or remote sensing (Watson 1985are very helpful in locating lithological boundaries. Their interpretation, which makes extensiveuse of enhanced maps of gravity data, often involves initial steps to eliminate or attenuateunwanted field components in order to isolate the desired anomaly (e.g., residual-regionalseparations). These initial filtering operations include the radial weights methods (Griffin, 1989least squares minimisation (Abdelrahman et al., 1991), the Fast Fourier Transform methods(Bhattacharyya, 1976) and recursion filters (Vaclac et al., 1992), and rational approximationtechniques (Agarwal and Lal, 1971).

Gravity anomaly separation can also be affected by such wavelength filtering when the gravityresponse from the geological feature of interest (the signal) dominates one region (or spectralband) of the observed gravity fields power spectrum. Pawlowski et al. (1990) have investigated agravity anomaly separation method based on frequency-domain Wiener filtering. Hsu et al. (1996)have presented a method for deriving geological boundaries from potential-field anomalies. Li andDougles (1998) achieved separation by inverting the observed data from a large area to construct aregional suspectibility distribution.

Since classical approaches need either prior information on data or offer simple solutions thatcannot be applied to real problems, a new unsupervised stochastic model has now been proposedand denoted as "Differential Markov Random Field (DMRF)". In this, our new approach, both theabsolute difference of neighbouring pixels and the stochastic properties of the gravity anomalymap are evaluated in real time. In the DMRF model, the neighbourhood relation and the statisticalproperties of the original data are optimised simultaneously, resulting in satisfactory performance.

Differential Markov Random Field (DMRF) Aproach

Stochastic models based on the Markov Random Field (MRF) approach in 2-D data analysis haveled to the development of many practical algorithms (Dubes et al., 1989, Geman and Geman,1984; Derin and Elliot, 1987) that would not have been realised with ad-hoc processing. Arandom field is a joint distribution imposed on a set of random variables representing objects ofinterest, such as pixel intensities, that imposes the statistical dependence in a spatially meaningfulway.

The objective of modelling is to capture the intrinsic character of data in a few parameters so as togain an understanding of the nature of the phenomena generating the data. The literature of 2-Ddata analysis has experienced a resurgence of the use of stochastic models to represent image dataand to express prior, generic knowledge.

In recent years, there has been increasing interest in use of statistical techniques for modelling andprocessing data. The reports by Dubes et al., 1989, Geman and Geman, 1984; Derin and Elliot,1987, all make use of the Gibbs distribution (GD) for characterising MRF.

This paper presents a new approach to the application of GD to problems in gravity anomalymaps. We therefore give the basic definitions of GD and a particular class of GD used in gravityseparation. Gravity anomaly separation can also be effected by such wavelength filtering when thegravity response from the geological feature of interest (the signal) dominates one region (orspectral band) of the observed gravity fields power spectrum.

Basic Definitions of Gibbs Distributions

The gravity anomaly map we are investigating is assumed to be a finite N1 N2 rectangular latticeof points (pixels) defined as . A collection of subsets of Ldescribed as,

(2)

is a neighbourhood system on L if and only if the neighbourhood of pixel (i,j) is such that

if then for any .

A hierarchically ordered sequence of neighbourhood systems commonly used in modelling are is , consisting of the closest four neighbours of each pixel; this is known as the

nearest-neighbour model (Derin and Elliot, 1987). is such that consists of eight

pixels neighbouring (i,j). The neighbourhood structures are given in Figures 1 and 2. Theneighbourhood system is referred to the mth order neighbourhood system.

Fig. 1 Hierarchically arranged neighbourhoodsystem.

The cliques related with a lattice-neighbourhood pair , denoted by c, is a subset of L suchthat

c consists of a single pixel, or for and implies that .

The collection of all cliques of is denoted by C = C . The types of cliques associatedwith and are shown in Figure 2.

Fig. 2Neighbourhoodsystems and

and their

associatedclique types.

Let be a neighbourhood system defined over the finite lattice L. A random field

defined on L has Gibbs Distribution (GD) or equivalently is a Gibbs Random Field

(GRF) with respect to if and only if its joint distribution is of the form,

(3)

where

, defined as energy function

potential associated with clique c.

:partition function is simply a normalising constant.

The joint distribution expression in (3) has the following physical interpretation: the smaller theenergy function U(x), i.e. the energy of realisation of x, the more likely that realisation is [i.e.,larger P(X=x)]. The GD is basically an exponential distribution. On appropriate choice of theclique potential , a wide variety of distributions both for discrete and continuous random

fields can be formulated as GD. Besag, 1974, proved that there is one-to-one correspondencebetween Markov Random Field (MRF) and Gibbs Random Field (GRF). Thus any random fieldcan be considered as a MRF, and consequently as a GRF, with respect to a large enoughneighbourhood system.

Differential Markov Random Field Model (DMRF) for Gravity Anomaly Map

In this paper, we describe a gravity anomaly map y={y ij} as an N 1 N 2 matrix of observations.

It is assumed that this matrix y is a realisation of a random field Y = {Y ij} which is the sum of

gravity fields produced by all underground sources. In our case, the targets for specific surveys areoften small-scale structures buried at shallow depths, and the scene including these targets isdefined as a residual random field X = {Xij}. The random field X is a discrete valued random field,where takes M quantisation level values as defined by .

Correct estimation and removal of the regional field from the initial field observations yields theresidual field produced by the target sources. Interpretation and numerical modelling are carriedout on the residual field data, and the reliability of the interpretation depends to a great extentupon the success of the regional-residual separation. In other words, given a gravity anomaly maprealisation y, it is desired to determine the residual scene x that gives rise to y. The scenerealisation of x, of course, is not observed and cannot be obtained deterministically from y. So theproblem is to obtain an estimate x * = X *(y) of the scene X, based on a realisation y. Having set upthe problem statistically, a maximum a posteriori (MAP) estimation is chosen as statisticalcriterion (Derin and Elliot, 1987). So the objective now is to have an estimation rule, i.e., analgorithm which will yield x * that maximises the a posteriori distribution for agiven y. Applying Bayes rule, the a posteriori distribution can be written as,

= (4)

Since the probability P (Y = y) does not effect maximisation, the logarithmic form of Eq. (4) canbe written as,

ln = ln P(X = x) + ln (5)

To maximise Eq. (5) for gravity anomaly maps, we proposed use of the Differential MarkovRandom Field (DMRF) approach since residual information is present in gravity anomalies at theboundaries of regions of the original data. It is also clear that, traditionally, magnetic and gravitymaps are subjected to operations approximating certain functions such as second derivative anddownward continuation.

The gravity anomaly map we investigate is assumed to be a finite N1 N2 rectangular lattice ofpoints (pixels) defined as {y ij} are passed through a Differential Markov Random Field (DMRF)

precoder resulting {s ij} parameters as,

(6)

where M is the number quantisation levels obtained for a residual map. the neighbourhood of

pixel (i,j) as given in Eq. (2).

Then the two components of the joint log-likelihood in Eq. (5) can be expressed for DMRF as,

ln (7)

ln (8)

where S m = {(i,j) }. Z is defined in Eq. (3), qm is the transient quantisation

level of a residual map during optimisation and is the potential associated with clique c. The

joint log-likelihood ln P(X = x,Y = y) (Eq. 5) is the sum of Eq. (7) and (8) and is to be optimised.Then we need to compute the second component of Eq. (7).

Using an optimisation approach similar to that of Derin and Elliot, 1987, we present theformulation in terms of a second order neighbourhood system , although its extension to any

order is possible. Consider a site (i,j) and its neighbourhood in the residual map of X. Let qmbe the transient quantisation level of a residual map during optimisation at (i,j) pixel and t represent the vector of neighbouring values of qm

at (i,j).

t = [u 1, u 2, u 3, u 4, v 1, v 2, v 3, v 4]T (9)

where the location of u is and v i s with respect to qm

are shown in Figure 3.

Fig. 3 qm and of residual gravity map.

We define indicator functions,

(10)

is the approximation defined as,

(11)

where is a small value given with regard to optimisation tolerance. Another indicator

defined as,

(12)

Using these indicators we can express the potential function (7) of all cliques that contain (i,j), the

site of . I.e.

(13)

where is the parameter vector. Thus both Eq. (7) and Eq. (8) are defined as a function of ,

resulting in a common optimisation which greatly improves the DMRF performance.

is defined as,

(14)

Then we can rewrite Eq. (13) as,

(15)

where,

Suppose is the joint distribution of random variables on 33 block centered at (i,j) and

P(t) is the joint distributions on only. Then the conditional probability, using Bayes rule can

be written as,

(17)

where

(18)

On rearranging Eq. (17) we obtain

(19)

Considering only the left-hand side of Eq. (19), for two distinct values of values

we have,

(20)

Taking the natural logarithm of Eq. (20) and replacing Eq. (15), we obtain,

(21)

In Eq. (21), the vector is determined easily for any j, k, t while is the

unknown parameter to be estimated. The question that remains to be answered is how to determine values. We accomplish this using histogram techniques.

As a result, we have explained how to extract residual gravity map using a Differential MarkovRandom Field approach. In the next section examples will be given to support the results obtainedhere.

Examples of the DMRF Approach

We have tested our proposed method for some examples by using synthetic data. In Appendix Athe necessary program is attached (Matlab Version 5.0). A histogram technique is used andquantisation is carried out by observing the frequency of images. The number of quantisationblocks is selectable. The purpose of this process is to segment the image into a fewer number ofblocks so as to emphasise the object, but the disadvantage of this operation is reducing sensitivity.In this program, the estimated parameter vector , given in Eq. (14), is taken as an average ofmany subgroups of the examples such that the considered parameter vector gives the bestinformation about the stochastic properties of the gravity anomaly map. In our program, theneighbourhood relation and statistical properties of the original data are optimised simultaneously,resulting in satisfactory performance.

All the units used in examples are normalised values. In the first example given as Figure 4, thereare five spherical structures with different densities, depths, and radii. They are chosen to form aBouguer anomaly map as shown in Table 1. As DMRF output, essentially perfect results areobtained and only the spheres closer to the ground with real parameter values are extracted(Figures 5 and 6). Thus a residual anomaly map has been extracted perfectly giving the desiredsolution of our problem in 2-D and 3-D dimensions. In a second example (Figure 7), fourspherical structures are used. For increasing regional effects on the Bouguer anomaly map, thelarge sphere with greatest radius is chosen deeper than the others (Figures 8 and 9). To increasethe residual effect, the other small spheres are placed closer to the ground as shown in Table 2.

Fig. 4 Example of Bouguer anomaly map of fivespheres with different depth, radius and gravity.

Table 1 Parameters of Bouguer anomaly map of Example 1.

ParametersSphere 1Sphere 2Sphere 3Sphere 4Sphere 5

h 30 2 5 3 4

r 20 1.5 3 2 3

r (gr/cm3) 1 1.2 1 1.5 1

Fig. 5 Residual anomaly of Figure 4 using DMRFapproach.

Fig. 6 3D representation of Figure 5.

Fig. 7 Example of Bouguer anomaly map of fourspheres with different depth, radius and gravity.

Fig. 8 Residual anomaly of Figure 4 using DMRFapproach.

Fig. 9 3D representation of Figure 8.

Table 2 Parameters of Bouguer anomaly map of Example 2.

Parameters Sphere 1 Sphere 2 Sphere 3 Sphere 4

h 100 6 5 5

r 30 4 4 3

r (gr/cm3) 1.8 1.2 1 1.3

Discussion

In this paper, a new stochastic image processing model has been proposed and denoted as theDifferential Markov Random Field (DMRF). We have applied our new approach to gravityanomaly separation problems. It is shown that the proposed method is one of the best approachesavailable in the literature.

Acknowledgement

This work was supported by the Research Institute of Istanbul University. The project number:1247/050599

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