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Geophysical interpretation — A pattern recognition system
Geoexploration, 11 (1973): 45- 50 © Elsevier Scientific Publishing Company, Amsterdam -. Printed in The Netherlands Short Communication GEOPHYSICAL
INTERPRETATION
-
A PATTERN
RECOGNITION
SYSTEM
SUBHASH C. GARDE and SAURABH K, VERMA National Geophysical Research Institute, l~vderabad (India) (Accepted for publication October 2, 1972)
ABSTRACT Garde, S.C. and Verma, S.K., 1973. Geophysical interpretation - a pattern recognition system. Geoexploration, 11:45-50. The general interpretation system is defined by a cyclic chain of deductions and inductions within the premises of the general pattern recognition system. The major problem of adducing the best-fitting model for the determination of target parameters through their field response can be achieved by designing an adaptive pattern recognition scheme. Such a scheme has been resolved into three major areas: Prognosis, Sophistication and Diagnosis. Sophistication has been considered as the key to simulate natural settings. The dependence of this sophistication on the type of physical parameter involved has been shown through statistical approach of Kernel approximation.
INTRODUCTION In recent years a great deal of effort has been devoted to research in automated systems for character recognition, weather forecasting, voice typewriting, medical diagnosis, data classification, target identification and finger print interpretation which appear at first glance to be quite unrelated. However, if We take into consideration the fact that in all these cases some type of abstract pattern is assumed as a basis for recognition, we realize that these various problems are really similar in nature. To all problems of this kind, a general term has been used, namely "Pattern Recognition". The basic problems in pattern recognition are measurement, feature extraction, training for discrimination and generalization. The physico-mathematical procedures o f unfolding the pages of earth's history and exploration of mineral resources treasured in the shallow depths o f the crust define the premises for various disciplines of geophysics. This is essentially a pattern recognition problem in which the received input data are the measurements o f different geophysical parameters and the output response o f the system is the delineation of physical processes, giving rise to different geologic and tectonic settings. Because o f the wide
46
S.C. GARDE AND S.K. VERMA
range of spatial dimensions of these settings, the problem of geophysical interpretatron is classified into those related to global scale, and local or regional scale. This is due, mainly, to the fact that suitability of the techniques in field observations on regional scale fails when the dimensions increase to that of the global scale. This failure may be attributed to the dynamic behaviour of the geological phenomenon which was assumed to be static on a regional scale, an assumption involving no error in adducing a picture of geology at depth. The domain of dynamic geophysics is comparable (in a reverse way) with the domain of quantum mechanics in physics when the laws of classical mechanics failed to explain the particle behaviour at the dimensions of the size of atoms. The principles of this dynamic geophysics are based on the concepts of: (1) ocean-floor spreading and continental drift; (2) transform faults; and (3) creation and consumption of lithosphere. These concepts are not accessible directly to analytical manipulation and they are often complex. This has given rise to vague suggestions which become hypotheses and the hypotheses are called theories. Improbable theories abound and are difficult to dispose of convincingly, if only because someone can find another, possibly irrelevant factor which has not previously been considered. However, the domain of static geophysics has passed through this stage of evolution and hence construction of stochastic models to extract the geologic and tectonic features through analytic manipulations has become a possibility. The construction of stochastic models involves the development of a cyclic chain of deductions and inductions. The purpose of the present note is to fit this chain in the geophysical interpretation system by defining its different components. FEATURE EXTRACTION •
The feature extraction can be considered to be a two-stage process of classification. This process attempts to characterize the input by a set of properties which allow efficient classification. The algorithm of a feature extraction system is illustrated in a simplified block diagram (Zobrist, 1971) in Fig. 1.
~ - i Feature extraction){ [ REPRESENTATION OF (Meosurables)
J,, FEATU,RES (Anomaly indication)
Classification (Predictables)
Fig. l. Information flow through a general feature extraction system with corresponding operations in the geophysical interpretation.
MEASURABLES,
The first operation in the feature extraction process involves the measurement of a convenient physical parameter. Very precise instruments have been and are being designed
GEOPHYSICALINTERPRETATIONSYSTEM
47
to measure changes in a suitable physical parameter of a potential field which are directly proportional to the distribution of a physical property of the geologic target under investigation. The potential fields may be static or dynamic depending upon whether source varies with time or remains constant. The time-invariant fields are the natural fields caused due to the presence of a geologic inhomogeneity whose manifestations are represented in variations of acceleration due to gravity (the gravity methods), the earth's magnetic field (magnetic methods), and the selfpotential field (S.P.-method), etc. Each of these measurements corresponds to the distribution of density, susceptibility and spontaneous polarization factor etc. in the pertinent medium. The time-variant fields involve the stimulation of the causative body by a temporal source and the measurement of the resulting response. This stimulation can be natural (magneto-telluric, audio-frequency magnetic fields (afmag), and radioactive, etc.) or artificial (resistivity, electromagnetic, induced polarisation, and seismic, etc.). ANOMALY INDICATION Whether it is variant or invariant with time a field when measured through a suitable parameter will give rise to geophysical anomalies which can be broadly divided into two classes (Parasnis, 1966): (1) symmetric, and (2) anti-symmetric, depending upon whether it does or does not possess two identically shaped halves. The operation of the second block is to identify the concerned geophysical anomaly from the observed data and attribute it to one of the above classes. EREDICTABLES This classification on a purely qualitative basis indicates the geometric disposition of the causative body. However, the determination of its exact geometric and physical parameters is impossible, both in practice and theory, because of the inherent ambiguity of the inverse problem in potential theory. These physical and geometrical parameters thus are the predictables in the feature extraction system (Fig. 1). This inherent ambiguity exists mainly because the predictables outnumber the measurables by an order of infinity (Roy, 1962; Paul et al., 1966). THEORY The imbalance between the measurables and the predictables may apparently lead us to the conclusion that the geophysical interpretation is an exercise in futility. However, the predictables can be brought close to the reality by performing a retrograde precession to build a stochastic model of the system. The a priori requirement for this stochastic model is: (1) the characterization of statistical features determined from the observed samples; and (2) the environmental premises to define a heuristic model where response
48
S.C. GARDE AND S.K. VERMA
is approximated to the statistical pattern classes. Let fix) be a statistical feature function which is to be extracted from a set of observed data (x l,x 2,x 3..... x k), and-((x) be the anomaly due to the heuristic model. The general expression for the message constituted by the anomaly pattern is given by Ward and Rogers (1967):
( 1)
f(x) = (IF) (S~O (ePF) ( O ~
where f(x) = anomaly, IF = inducing field, SF -- size factor, PPF = physical property factor, and GF = geometric factor. It can be shown that, in general, an approximation of f(x) to -f(x) is obtained by (Tou, 1968):
f(x)
¥(x) = K (x,y) f(y) dy
(2)
where Kn(x,y ) is the known Kernel function corresponding to the given source and target conditions. By the strong law of large numbers as applied to the observed data f(x) can be estimated from the K independent samples by; k
!
:
k ;:i
(3)
Kn(X,x
For each x, -fk(x) converges to -{(x) with probability 1 as k ~ oo, and f(x) converges to fix) uniformly as n ~ oo. The quality of approximation given in eq. 2 depends on the nature of the Kernel Kn(x,y ). Once an appropriate Kernel has been selected, good approximation to f(x) can be determined from eq. 3. This approximation amounts to adaptive matching and contextual analysis of the heuristic model with the environmental premises. Thus, a complete system of general geophysical interpretation can be expressed by the simplified block diagram in Fig. 2.
[ PROGNOSS I ~ - - ~ $OPHS I TC I~ t.TO IN i------~ DIAGNOSIS i
I
I
Fig. 2. Simplified block diagram of general geophysical interpretation system. GENERAL INTERPRETATION SYSTEM
The general geophysical interpretation involves a system comprising a cyclic chain of prognosis and diagnosis. Based on the spatial distribution of the effect (anomaly pattern) due to a particular physical quantity (e.g., density, susceptibility, conductivity, etc.) a number of predictions (prognoses) about the causative body (target) can be made. The
GEOPHYSICALINTERPRETATIONSYSTEM
49
first'~tep in this translation involves the selection of a heuristic model representing the target. Improvements in this model are made through the contextual analysis conditioned by the geologic features of environment in general and the target in particular (geologic filters). Consequently a working model is evolved whose response is synthesized and matched with the observed anomaly pattern. The predicted model is improved and again tested through a chain of inductions and deductions until one of the prognoses is satisfactorily supported by the diagnosis, The parameters of this finally obtained model, offering the best fit, are taken as the representatives of the target parameters. A block diagram of such an interpretation system is shown in Fig. 3. The most significant part of this system is the sophistication which consists of the heuristic model, contextual analysis, synthesis and the adaptive matching. Evolutionary stages of the interpretation technique in a particular geophysical method can be precisely defined by the sophistication achieved up to the time
t INFORMATION I 1
PROG-~TAPROCESSING] NOSIS
I
/
i
)
\\
) Fig, 3. Cyclic chain of deductions and inductions through sophistication depicting the geophysicalinterpretation a pattern recognition system. -
50
s.c. GARDE AND S.K. VERMA
of assessment. The sophistication is governed by the following factors: (I) controllability of the source; (2) degree of idealization attainable for the target; (3) availability of the effective analytic techniques; (4) accountability for the pertinent environmental conditions; and (5) suitability of the source model system for the laboratory experiment. It is apparent from above that the suitability of any single geophysical method for obtaining the information about a causative body is directly proportional to the number of factors the method satisfies. CONCLUDINGREMARKS In the present note we have made a comparative assessment of the general geophysical interpretation system with that of the pattern recognition system. This has revealed that the sophistication of any interpretation technique corresponds to the feature extraction in pattern recognition thus making it susceptible to statistical approach. Kernel approximation to determine the best-fitting model is outlined. ACKNOWLEDGEMENT Thanks are due to Dr. J.G. Negi for encouragement and useful discussions, and to the Director of the National Geophysical Research Institute, Hyderabad, India, for according permission to publish this work. REFERENCES Parasnis, D.S., 1966. Mining Geophysics. Elsevier, Amsterdam, 356 pp. Paul, M.K., Datta S. and Banerjee, B., 1966. Direct interpretation of two-dimensional structural faults from gravity data. Geophysics, 31 (5): 940-948. Roy, A., 1962. Ambiguity in geophysical interpretation, Geophysics, 27 (1):190-99. Tou, J.T., 1968. Feature extraction in pattern recognition. Pattern Recognition, 1 (1); 3-12. Ward, S.H. and Rogers, G.R., 1967. Introduction. In: D.A. Hansen et al. (Editors), Mining Geophysics, 2. Society of Exploration Geophysicists, Tulsa, Okla., pp. 3-8. Zobrist, A.L., 1971. The organization of extracted features for pattern recognition. Pattern Recotmition, 3 (1): 23-30.
INTERPRETATION
-
A PATTERN
RECOGNITION
SYSTEM
SUBHASH C. GARDE and SAURABH K, VERMA National Geophysical Research Institute, l~vderabad (India) (Accepted for publication October 2, 1972)
ABSTRACT Garde, S.C. and Verma, S.K., 1973. Geophysical interpretation - a pattern recognition system. Geoexploration, 11:45-50. The general interpretation system is defined by a cyclic chain of deductions and inductions within the premises of the general pattern recognition system. The major problem of adducing the best-fitting model for the determination of target parameters through their field response can be achieved by designing an adaptive pattern recognition scheme. Such a scheme has been resolved into three major areas: Prognosis, Sophistication and Diagnosis. Sophistication has been considered as the key to simulate natural settings. The dependence of this sophistication on the type of physical parameter involved has been shown through statistical approach of Kernel approximation.
INTRODUCTION In recent years a great deal of effort has been devoted to research in automated systems for character recognition, weather forecasting, voice typewriting, medical diagnosis, data classification, target identification and finger print interpretation which appear at first glance to be quite unrelated. However, if We take into consideration the fact that in all these cases some type of abstract pattern is assumed as a basis for recognition, we realize that these various problems are really similar in nature. To all problems of this kind, a general term has been used, namely "Pattern Recognition". The basic problems in pattern recognition are measurement, feature extraction, training for discrimination and generalization. The physico-mathematical procedures o f unfolding the pages of earth's history and exploration of mineral resources treasured in the shallow depths o f the crust define the premises for various disciplines of geophysics. This is essentially a pattern recognition problem in which the received input data are the measurements o f different geophysical parameters and the output response o f the system is the delineation of physical processes, giving rise to different geologic and tectonic settings. Because o f the wide
46
S.C. GARDE AND S.K. VERMA
range of spatial dimensions of these settings, the problem of geophysical interpretatron is classified into those related to global scale, and local or regional scale. This is due, mainly, to the fact that suitability of the techniques in field observations on regional scale fails when the dimensions increase to that of the global scale. This failure may be attributed to the dynamic behaviour of the geological phenomenon which was assumed to be static on a regional scale, an assumption involving no error in adducing a picture of geology at depth. The domain of dynamic geophysics is comparable (in a reverse way) with the domain of quantum mechanics in physics when the laws of classical mechanics failed to explain the particle behaviour at the dimensions of the size of atoms. The principles of this dynamic geophysics are based on the concepts of: (1) ocean-floor spreading and continental drift; (2) transform faults; and (3) creation and consumption of lithosphere. These concepts are not accessible directly to analytical manipulation and they are often complex. This has given rise to vague suggestions which become hypotheses and the hypotheses are called theories. Improbable theories abound and are difficult to dispose of convincingly, if only because someone can find another, possibly irrelevant factor which has not previously been considered. However, the domain of static geophysics has passed through this stage of evolution and hence construction of stochastic models to extract the geologic and tectonic features through analytic manipulations has become a possibility. The construction of stochastic models involves the development of a cyclic chain of deductions and inductions. The purpose of the present note is to fit this chain in the geophysical interpretation system by defining its different components. FEATURE EXTRACTION •
The feature extraction can be considered to be a two-stage process of classification. This process attempts to characterize the input by a set of properties which allow efficient classification. The algorithm of a feature extraction system is illustrated in a simplified block diagram (Zobrist, 1971) in Fig. 1.
~ - i Feature extraction){ [ REPRESENTATION OF (Meosurables)
J,, FEATU,RES (Anomaly indication)
Classification (Predictables)
Fig. l. Information flow through a general feature extraction system with corresponding operations in the geophysical interpretation.
MEASURABLES,
The first operation in the feature extraction process involves the measurement of a convenient physical parameter. Very precise instruments have been and are being designed
GEOPHYSICALINTERPRETATIONSYSTEM
47
to measure changes in a suitable physical parameter of a potential field which are directly proportional to the distribution of a physical property of the geologic target under investigation. The potential fields may be static or dynamic depending upon whether source varies with time or remains constant. The time-invariant fields are the natural fields caused due to the presence of a geologic inhomogeneity whose manifestations are represented in variations of acceleration due to gravity (the gravity methods), the earth's magnetic field (magnetic methods), and the selfpotential field (S.P.-method), etc. Each of these measurements corresponds to the distribution of density, susceptibility and spontaneous polarization factor etc. in the pertinent medium. The time-variant fields involve the stimulation of the causative body by a temporal source and the measurement of the resulting response. This stimulation can be natural (magneto-telluric, audio-frequency magnetic fields (afmag), and radioactive, etc.) or artificial (resistivity, electromagnetic, induced polarisation, and seismic, etc.). ANOMALY INDICATION Whether it is variant or invariant with time a field when measured through a suitable parameter will give rise to geophysical anomalies which can be broadly divided into two classes (Parasnis, 1966): (1) symmetric, and (2) anti-symmetric, depending upon whether it does or does not possess two identically shaped halves. The operation of the second block is to identify the concerned geophysical anomaly from the observed data and attribute it to one of the above classes. EREDICTABLES This classification on a purely qualitative basis indicates the geometric disposition of the causative body. However, the determination of its exact geometric and physical parameters is impossible, both in practice and theory, because of the inherent ambiguity of the inverse problem in potential theory. These physical and geometrical parameters thus are the predictables in the feature extraction system (Fig. 1). This inherent ambiguity exists mainly because the predictables outnumber the measurables by an order of infinity (Roy, 1962; Paul et al., 1966). THEORY The imbalance between the measurables and the predictables may apparently lead us to the conclusion that the geophysical interpretation is an exercise in futility. However, the predictables can be brought close to the reality by performing a retrograde precession to build a stochastic model of the system. The a priori requirement for this stochastic model is: (1) the characterization of statistical features determined from the observed samples; and (2) the environmental premises to define a heuristic model where response
48
S.C. GARDE AND S.K. VERMA
is approximated to the statistical pattern classes. Let fix) be a statistical feature function which is to be extracted from a set of observed data (x l,x 2,x 3..... x k), and-((x) be the anomaly due to the heuristic model. The general expression for the message constituted by the anomaly pattern is given by Ward and Rogers (1967):
( 1)
f(x) = (IF) (S~O (ePF) ( O ~
where f(x) = anomaly, IF = inducing field, SF -- size factor, PPF = physical property factor, and GF = geometric factor. It can be shown that, in general, an approximation of f(x) to -f(x) is obtained by (Tou, 1968):
f(x)
¥(x) = K (x,y) f(y) dy
(2)
where Kn(x,y ) is the known Kernel function corresponding to the given source and target conditions. By the strong law of large numbers as applied to the observed data f(x) can be estimated from the K independent samples by; k
!
:
k ;:i
(3)
Kn(X,x
For each x, -fk(x) converges to -{(x) with probability 1 as k ~ oo, and f(x) converges to fix) uniformly as n ~ oo. The quality of approximation given in eq. 2 depends on the nature of the Kernel Kn(x,y ). Once an appropriate Kernel has been selected, good approximation to f(x) can be determined from eq. 3. This approximation amounts to adaptive matching and contextual analysis of the heuristic model with the environmental premises. Thus, a complete system of general geophysical interpretation can be expressed by the simplified block diagram in Fig. 2.
[ PROGNOSS I ~ - - ~ $OPHS I TC I~ t.TO IN i------~ DIAGNOSIS i
I
I
Fig. 2. Simplified block diagram of general geophysical interpretation system. GENERAL INTERPRETATION SYSTEM
The general geophysical interpretation involves a system comprising a cyclic chain of prognosis and diagnosis. Based on the spatial distribution of the effect (anomaly pattern) due to a particular physical quantity (e.g., density, susceptibility, conductivity, etc.) a number of predictions (prognoses) about the causative body (target) can be made. The
GEOPHYSICALINTERPRETATIONSYSTEM
49
first'~tep in this translation involves the selection of a heuristic model representing the target. Improvements in this model are made through the contextual analysis conditioned by the geologic features of environment in general and the target in particular (geologic filters). Consequently a working model is evolved whose response is synthesized and matched with the observed anomaly pattern. The predicted model is improved and again tested through a chain of inductions and deductions until one of the prognoses is satisfactorily supported by the diagnosis, The parameters of this finally obtained model, offering the best fit, are taken as the representatives of the target parameters. A block diagram of such an interpretation system is shown in Fig. 3. The most significant part of this system is the sophistication which consists of the heuristic model, contextual analysis, synthesis and the adaptive matching. Evolutionary stages of the interpretation technique in a particular geophysical method can be precisely defined by the sophistication achieved up to the time
t INFORMATION I 1
PROG-~TAPROCESSING] NOSIS
I
/
i
)
\\
) Fig, 3. Cyclic chain of deductions and inductions through sophistication depicting the geophysicalinterpretation a pattern recognition system. -
50
s.c. GARDE AND S.K. VERMA
of assessment. The sophistication is governed by the following factors: (I) controllability of the source; (2) degree of idealization attainable for the target; (3) availability of the effective analytic techniques; (4) accountability for the pertinent environmental conditions; and (5) suitability of the source model system for the laboratory experiment. It is apparent from above that the suitability of any single geophysical method for obtaining the information about a causative body is directly proportional to the number of factors the method satisfies. CONCLUDINGREMARKS In the present note we have made a comparative assessment of the general geophysical interpretation system with that of the pattern recognition system. This has revealed that the sophistication of any interpretation technique corresponds to the feature extraction in pattern recognition thus making it susceptible to statistical approach. Kernel approximation to determine the best-fitting model is outlined. ACKNOWLEDGEMENT Thanks are due to Dr. J.G. Negi for encouragement and useful discussions, and to the Director of the National Geophysical Research Institute, Hyderabad, India, for according permission to publish this work. REFERENCES Parasnis, D.S., 1966. Mining Geophysics. Elsevier, Amsterdam, 356 pp. Paul, M.K., Datta S. and Banerjee, B., 1966. Direct interpretation of two-dimensional structural faults from gravity data. Geophysics, 31 (5): 940-948. Roy, A., 1962. Ambiguity in geophysical interpretation, Geophysics, 27 (1):190-99. Tou, J.T., 1968. Feature extraction in pattern recognition. Pattern Recognition, 1 (1); 3-12. Ward, S.H. and Rogers, G.R., 1967. Introduction. In: D.A. Hansen et al. (Editors), Mining Geophysics, 2. Society of Exploration Geophysicists, Tulsa, Okla., pp. 3-8. Zobrist, A.L., 1971. The organization of extracted features for pattern recognition. Pattern Recotmition, 3 (1): 23-30.