Vol49,6,2006 1201 ANNALS OF GEOPHYSICS, VOL. 49, N. 6, December 2006 Key words Forced Neural Network – gravity anom- aly – modeling – synthetic model – Gulf of Mexico 1. Introduction A classical problem in gravity and magnetic exploration is the computation of theoretical anomalies caused by idealized models of known shapes. Many workers have published different methods for carrying out such computation, and textbooks on potential theory, e.g. Routh (1908), provided various formulas for these models. Ear- ly publications like Barton (1929) dealt with the computation of the gradients of the gravity field. Hubbert (1948) used line-integral approach for the computation of gravitational attraction of two-dimensional masses. Bhattacharyya (1964), Nagy (1966), and Plouff (1976) presented closed form of analytical solutions for prism shaped bodies. Talwani and Ewing (1960), Talwani (1965) used numerical integration techniques for the computation of the fields due to models of ar- bitrary shape by dividing them into polygonal prisms or laminas. Parker (1974) tried to find depth and density values using gravity data. Green (1975) studied an inverse solution of grav- ity profiles. Last and Kubik (1983) estimated un- derground density distribution with recursive in- verse solution techniques. Lines and Treitel (1984) applied a Singular Value Decomposition (SVD) approach for problems in evaluation of gravity and seismic projections. Mareschal (1985) used Fourier Transform for inverse solu- tion of gravity density distributions. Murty et al. (1990) focused on density differences of 2D and 3D gravity models. Murty and Rao (1993a,b) cal- culated inverse solution of gravity and magnetic A new approach for residual gravity anomaly profile interpretations: Forced Neural Network (FNN) Onur Osman (1), A. Muhittin Albora (2) and Osman Nuri Ucan (3) (1) Istanbul Commerce University, Eminonu, Istanbul, Turkey (2) Istanbul University, Engineering Faculty, Geophysical Department, Avcilar, Istanbul, Turkey (3) Istanbul University, Engineering Faculty, Electrical & Electronics Dept, Avcilar, Istanbul, Turkey Abstract This paper presents a new approach for interpretation of residual gravity anomaly profiles, assuming horizontal cylinders as source. The new method, called Forced Neural Network (FNN), is introduced to determine the un- derground structure parameters which cause the anomalies. New technologies are improved to detect the borders of geological bodies in a reliable way. In a first phase one neuron is used to model the system and a back prop- agation algorithm is applied to find the density difference. In a second phase, density differences are quantified and a mean square error is computed. This process is iterated until the mean square error is small enough. After obtaining reliable results in the case of synthetic data, to simulate real data, the real case of the Gulf of Mexico gravity anomaly map, which has the form of anticline structure, is examined. Gravity anomaly values from a cross section of this real case, result to be very close to those obtained with the proposed method. Mailing address: Dr. Onur Osman, Istanbul Commerce University, Ragip Gumuspala Cad. No. 84 Eminonu, 34378 Istanbul, Turkey; e-mail: oosman@iticu.edu.tr 1202 Onur Osman, A. Muhittin Albora and Osman Nuri Ucan anomalies of polygonal structures using Mar- quart algorithm. Murthy and Rao (1993b) pro- posed some methods in inverse solution of gravi- ty anomalies for circular, cylindrical, and vertical discs. Mosegaard and Tarantola (1995) applied Monte Carlo method. Tsokas and Hansen (1997) studied on crustal thickness with gravity anom- alies in Greece. Artificial neural networks are part of a much wider field called artificial intelligence, which can be defined as the study of mental facilities through the use of computational models (Char- niak and McDermott, 1985). They encompass computer algorithms that solve classification, parameter estimation, parameter prediction, pat- tern recognition, completion association, filter- ing, and optimization problems (Brown and Poulton, 1996). They have gained popularity in geophysics during the last decade because these tools can approximate any continuous function with an arbitrary precision (Van der Baan and Jutten, 2000). The location of the buried steel drums is estimated from magnetic dipole source using supervised artificial neural network (Salem et al., 2001). Neural networks are used to speed up the detection of ferro-metalic ob- jects (Selam and Ushijima, 2001). Depth and ra- dius of subsurface cavities are determined from microgravity data using back propagation neural networks (Eslam et al., 2001). Neural networks are studied to solve 1D and 2D resistivity in- verse problems (El-Qady and Ushijima, 2001). For 2D modeling CNN (Cellular Neural Net- works) is applied to the separation of regional/ residual potential sources in geophysics by Alb- ora et al. (2001a,b). Artificial neural networks can be divided into two main categories: unsupervised recurrent and supervised feed-forward networks. In the unsu- pervised recurrent type, the networks allow infor- mation to flow in both directions. These modals are called unsupervised because there is no teacher to set the input-output mapping relation during the learning phase. In the supervised be- cause through a set of correct input-output pairs, called the training set, the network learns the re- lation between the input-output pairs. In this paper, a new algorithm, denoted «Forced Neural Network (FNN)» is proposed. The aim of FNN is to estimate the physical pa- rameters of buried objects. It is first applied to synthetic examples and then real data. We have found satisfactory results for both cases. 2. Forced Neural Network The artificial neural network is composed of many simple processing elements, which are massively interconnected and operate in paral- lel. The processing elements commonly known as neurons, receive the input from previous ele- ments and send the output to other elements through synaptic connections. These connec- tions have different weights. In order to find the effective values of inputs and outputs, these val- ues are multiplied by these weights. The main purpose of neural networks is to compute such weights giving the best output. To obtain the el- igible values for weights, back propagation method being the most popular learning algo- rithm for neural networks, is used in this study. 2.1. Back propagation algorithm The error signal at the output of neuron j at iteration n, is defined by (2.1) where neuron j is an output node, dj(n) is de- sired output and yj(n) is actual output of Neural Networks (NN). The instantaneous value of the error energy for neuron j can be defined as . Correspondingly, the instantaneous value E(n) of the total error energy is obtained by summing over all neurons in the output layer; only «visible» neurons are the ones for which error signals can be calculated directly. We may thus write, (2.2) where, the set C includes all the neurons in the output layer of the network (Haykin, 1999). Let N denote the total number of patterns (exam- ples) contained in the training set. The average squared error energy is obtained by summing ( ) / ( )E n e n1 2 j j C 2= ! / / ( )e n1 2 j 2 / ( )e n1 2 j 2 ( ) ( ) ( )e n d n y nj j j= − 1203 A new approach for residual gravity anomaly profile interpretations: Forced Neural Network (FNN) E(n) over all n and then normalizing with re- spect to set size N, as shown by, . (2.3) The instantaneous error energy E(n), and there- fore the average error energy Eav, is a function of all the free parameters (i.e. synaptic weights and bias levels) of the network. For a given training set, Eav represents the cost function as a measure of learning performance. The objective of the learning process is to adjust the free parameters of the network to minimize Eav. To do this mini- mization, we use an approximation similar in ra- tional to that used for the derivation of the Least Mean Square (LMS) algorithm. We consider a simple method of training in which the weights are updated on a pattern-by-pattern basis until one epoch, that is, one complete presentation of the entire training set has been dealt with (2.4) where δj(n) is the local gradient and η is learn- ing speed (Haykin, 1999). Local gradient points are required changes in synaptic weights and we obtain Back-Propagation (BP) formula for the local gradient δj(n) as ( ) ( ) ( )w n n y nij J iηδ∆ = / ( )E E nN1av n N 1 = = / (2.5) neuron j is hidden. Figure 1 shows the signal-flow graph repre- sentation of eq. (2.5), assuming that the output layer consists of mL neurons. The factor involved in the compu- tation of the local gradient δj(n) in eq. (2.5) de- pends solely on the activation function associat- ed with hidden neuron j. The remaining factor involved in this computation, namely the sum- mation over k, depends on two sets of terms. The first set of terms, δk(n), requires knowledge of the error signals ek(n), for all neurons that lie in the layer to the immediate right of hidden neuron j, and that are directly connected to neu- ron j which is shown in fig. 1. The second set of terms, wkj(n), consists of the synaptic weights associated with these connections. We may redefine the local gradient δj(n) for hidden neuron j as (2.6) (2.7) neuron j is hidden. ( ( ))v nj( ) ( ) y n E n j j2 2 ϕ= − l ( ) ( ) ( ) ( ) ( ) n y n E n n y n vj j j j 2 2 2 2 δ = − ( ( ))v nj jϕl ( ( ))v n( ) ( ) ( )n n w nj j j k kj k δ ϕ δ= l / Fig. 1. Signal flow graph of a part of the adjoint system pertaining to Back-Propagation of error signals. 1204 Onur Osman, A. Muhittin Albora and Osman Nuri Ucan The local field parameter vj(n) produced at the input of the activation function associated with neuron j is therefore (2.8) where m is the total number of inputs (excluding the bias) applied to neuron j (Haykin, 1999). The synaptic weight wj0 (corresponding to the fixed input y0 = +1) equals the bias bj applied to neuron j. Hence the function signal yj(n) appearing at the output of neuron j at iteration n is . (2.9) Next differentiating eq. (2.9) with respect to vj(n), we get (2.10) where the use of prime (the right-hand side) signifies differentiation with respect to the ar- gument Haykin (1999). 2.2. Forced Neural Network for gravity anomaly This method could be used in modeling ar- bitrary subsurface body geometry and density contrasts. We begin with a horizontal cylindri- cal structure, whose gravity anomaly function is shown below, (2.11) ∆ρ is density difference, H and X are the depth and the total length of the cross section respec- tively, i and j are the levels of the depth and the distance of the cylinder from the starting point, and finally xref is the concerned distance point where the anomaly value is observed. We use as an input of the neuron, which is shown in fig. 2, and there should be (H × X) inputs and these inputs are constant for every A(xref). In fig. 2, ϕ(.) is an ac- tivation function. We use partially linear activa- ( ( )K i i j xref 2 2$ + −7 A ( ) ( ( ) ) A x K i j x i ,ref ref i j j X i H 0 1 1 2 2$ $ρ∆= + − = − = // ( ( ))v n ( ) ( ) v n y n j j j j2 2 ϕ= l ( ) ( ( ))y n v nj j jϕ= ( ) ( ) ( )v n w n y nj ij i m i 0 = = / tion function (Haykin, 1999), which gives lin- ear output values between zero and ∆ρ depend- ing on its input. The neuron can be modeled as below: In the method, weights of the neuron are as- signed as ∆ρi,j for each pixel and linear function is assumed as an activation function. After us- ing the back propagation, ∆ρi,j are updated and the output of the neuron gives the gravity anom- aly. Although the density differences are found, the results of this system are not sufficient be- cause of non-uniqueness and horizontal loca- tions that are constrained. Therefore, the value of ∆ρ is set to zero if its value is very close to the zero according to the density difference which is obtained form geological features of the region. Otherwise, the value of ∆ρ is set to the density difference of the geological region after back propagation. Forced neural network means that after suf- ficient epoch is applied, fixed values are as- signed to the output of the neuron according to the density difference ∆ρ, and this process is continued until the mean square error of the output, A(xref) which is shown in fig. 2, be- comes sufficiently small. 3. Performance of the algorithm in synthetic data Our synthetic data are obtained from a cylin- drical structure of having a depth of 1 m and a ra- Fig. 2. Forced Neural Network (FNN) design for gravity anomaly. 1205 A new approach for residual gravity anomaly profile interpretations: Forced Neural Network (FNN) dius of 2 m for ∆ρ= 1 mGal as shown in fig. 3. The anomalies of this model are considered as the input data provided to the FNN. In synthetic examples, every learning cycle is comprised of 350 epochs, and two-level quantization (∆ρ or zero) is applied after every 10 learning cycles, which is found to be optimum through experi- ments. The estimated geological structure ob- tained via FNN application results in an anomaly profile (dashed line) that is similar to the ob- served anomaly (solid line), as shown in fig. 3. For a second synthetic model, we choose T- type prismatic structure with ∆ρ = 1mGal. We use the Talwani and Ewing (1960) 2D method. The estimated geological structure obtained via FNN application results in an anomaly profile (dashed line) that is similar to the observed anomaly (solid line), as shown in fig. 4. In both examples, satisfactory results are obtained. 4. Example of application on real data As an example for application of real data in FNN, we use the Bouguer anomaly reported by Nettleton (1943), whose reproduction is shown in fig. 5. The anomaly was recorded in the Gulf of Mexico about 241 km away from Galveston and at a small distance inside the edge of the continental shelf. The importance of basement architecture to the hydrocarbon exploration in the Gulf of Mexico Basin has been debated on for years. Alexander (1999) studied on tectonic and stratigraphic in Gulf Basin. The origin of the topographic feature was not established until the gravity survey indicat- ed a large closed minimum coincident with the contours of the elevated mound that could be accounted for only by the assumption of a salt dome. The survey was not extensive enough to Fig. 3. Results of FNN for synthetic horizontal cylinder. Fig. 4. Performance of FNN for synthetic T-type prismatic structure. 3 4 1206 Onur Osman, A. Muhittin Albora and Osman Nuri Ucan define the gravity anomaly, but judicious ex- trapolation indicated the maximum negative anomaly to be about 9 mGal. The gravity anom- aly map given in fig. 5 is obtained from Dobrin and Savit (1988). Figure 6 is composed from the AB cross section of this map and demon- strates Nettleton’s interpretation of the salt structure giving rise to the anomaly. The solid line shows the observed anomaly and dotted line shows the anomaly, which is derived from FNN. The results of the proposed method are very close to the observed one. 5. Conclusions The Forced Neural Networks (FNN) pre- sented in this paper shows that the gravity field at any point due to a solid body with uniform volume density can be computed as the field due to a fictitious distribution of surface mass-densi- ty on the same body. First of all, we applied the FNN technique to two synthetic data. These tests provide successful results in fitting the cal- culated to observed data. As a real data applica- tion, a salt dome gravity anomaly map taken Fig. 5. Gravity map observed over inferred salt dome causing anomaly in water-bottom tomography in Gulf of Mexico (contour interval is 5 mGal) (modified form Dobrin and Savit, 1988). 1207 A new approach for residual gravity anomaly profile interpretations: Forced Neural Network (FNN) from the NW part of the Gulf of Mexico is con- sidered. This anomaly shows a negative closure from 1060 mGal to 990 mGal. The reason for this negative closure is mostly because of the geological properties of the salt dome. The den- sity contrast in salt dome of the Gulf is lower than those in the surrounding rock formations. The anomaly of AB cross-section is modeled us- ing FNN and the anomaly of this model is very close to the observed one. To make a compari- son between the methods of Nettleton and FNN, we can see that by using FNN the model better fits the observed anomaly. 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