A direct inversion scheme for deep resistivity soundingdata using artificial neural networks
Jimmy Stephen∗, C Manoj and S B Singh
National Geophysical Research Institute, Hyderabad 500 007, India∗e-mail: [email protected]
Initialization of model parameters is crucial in the conventional 1D inversion of DC electricaldata, since a poor guess may result in undesired parameter estimations. In the present work, weinvestigate the performance of neural networks in the direct inversion of DC sounding data, withoutthe need of a priori information. We introduce a two-step network approach where the first networkidentifies the curve type, followed by the model parameter estimation using the second network.This approach provides the flexibility to accommodate all the characteristic sounding curve typeswith a wide range of resistivity and thickness. Here we realize a three layer feed-forward neuralnetwork with fast back propagation learning algorithms performing well. The basic data sets fortraining and testing were simulated on the basis of available deep resistivity sounding (DRS) datafrom the crystalline terrains of south India. The optimum network parameters and performancewere decided as a function of the testing error convergence with respect to the network trainingerror. On adequate training, the final weights simulate faithfully to recover resistivity and thicknesson new data. The small discrepancies noticed, however, are well within the resolvability of resistivitysounding curve interpretations.
1. Introduction
Solving a geoelectrical inverse problem consists ofusing a set of measurements to evaluate subsurfaceparameters like resistivity and thickness of differentlayers. A proper mathematical relation should bedeveloped between the observations and subsurfaceparameters. The ‘good fit’ between the measure-ments and unknown model parameters (resistivityand thickness) is related to an error function,normally a non-linear function, of the subsurfaceparameters. An optimal solution is derived by theiterative minimization based on local derivativesof the error function. Several attempts were madein the last three decades for 1D resistivity inver-sion (Kunetz and Rocroi 1970; Ghosh 1971; Zohdy1989; Meheni et al 1996). None of these attemptsguarantee a geologically plausible model since mostof these algorithms critically depend on the ini-tial parameters given to it. A Monte-Carlo method,
which uses randomly drawn models, is useful whena good starting point is not available (Rubinstein1981). Simulated annealing (Kirkpatrick 1983) andgenetic algorithms (Horne and MacBeth 1994) usesprevious model evaluations to drive securely in themodel parameter space to the most promising partof error surface. These methods are computation-ally expensive than the linearized techniques andtherefore are more useful only when a good start-ing model is not available.
In the present study we try to investigate theapplication of artificial neural networks (ANN)to interpret one-dimensional (1D) electrical deepresistivity sounding (DRS) data over a wide rangeof model parameters. The use of ANN is moti-vated by its resident intelligence that resemblesthe biological neural network having the capa-bility of perceptual interpretation, abstractionand learning (Lippman 1987). Recent applicationareas of ANN involves the pattern recognition,
Keywords. Artificial neural networks; back propagation; deep resistivity sounding; 1D inversion.
clustering/classification, image processing, contentaddressability, optimization and control systems.Unlike the conventional methods that incorporatea fixed algorithm to solve a particular problem ingeophysics, ANN performs an intelligent non-linearmapping between input and output data allow-ing the network to acquire important informationon the problem being solved. It is featured withits adaptive discrimination or ‘learning’ throughrepeated exposure to examples and the versa-tile generalization capability. The use of ANN inlocating subsurface targets from geophysical datahas been studied by Paulton et al (1992). Recentstudies show the efficacy of ANN in providingautomation for geophysical inversions (Spichak andPopova 2000) and noise discriminations (Manojand Nagarajan 2003). To the interest of this paper,Macias et al (2000) discusses the implementationof ANN for the otherwise complex problem of geo-physical parameter estimation with emphasis onvertical electrical sounding and seismic data. Theelectrical resistivity data inversion is also addressedby Qady and Ushijima (2001) and discusses theperformance of different ANN paradigms.
Though the earlier attempts to invert DC resis-tivity data were successful, the application on real-world data was limited, as they depend mainly onthe synthetic data with less parameter variationsin more specific curve types. Here we made anattempt to overcome some of these limitations bydesigning a network that can incorporate a widespectrum of model parameters. The inversion isdone in two steps,
• identifying the curve type and• the estimation of model parameters.
In both the cases we use feed-forward neural net-work (FNN) as a mapping function between inputsand outputs. Fast back propagation with momen-tum (FBPM) is used as the learning rule, which issufficient to design any kind of non-linear classifierand by far the most popular learning algorithm(Guyon 1991; Haykin 1994). Since an optimum sizeand efficiency of the training data set is impor-tant to render generalization capability to the net-work (Kung and Hwang 1988; Hush and Horne1993), the database is generated cautiously. Ade-quate cross-validation is done for ensuring the per-formance of network on new data examples. Thedetails of network and learning schemes are dis-cussed later.
2. ANN architecture
Figure 1 shows the architecture of the 3 layer FNNwe adopted in the present study. The structure ofthe processing element (PE), called neuron, which
has a non-linear activation function is also shown.The input layer receives single inputs at each node(1 to I). Second and third layers represent the hid-den layer and output layer respectively and havethe same activation function in all neurons. Herewe use a logistic (unipolar) sigmoid function as theactivation function for neurons. Sigmoidal functioncan produce outputs with reasonable discriminat-ing power (Bishop 1995) and its output functionsare differentiable, which is essential for the back-propagation of errors (Yegnanarayana 2001). ItsGaussian shaped derivatives help to stabilize thenetwork and compensate for over-correction of theweights (Caudill 1988). During FNN training, eachhidden and output neuron process inputs by multi-plying each input by its weights. The products aresummed and processed using the activation func-tion. The expression for logistic sigmoid functionand its derivative are given below.
f(x) =1
(1 − e−x), (1)
f ′(x) = f(x)[1 − f(x)]. (2)
The number of neurons in the hidden layervaries as per the requirement of optimum perfor-mance, which could be decided on trial and errorbasis. Initial weights are assigned at random in therange suitable for the activation function at neu-rons. The signal is fed forward through the net-work as shown in figure 1 and hence the namefeed-forward. The learning cycle begins with theupdating of the weights of output layer (W2kj)and utilized to adjust the input weights (W1ji) toobtain the desired output, known as a back prop-agation, or otherwise generalized delta rule (Wer-bos 1974; Werbos 1990; Rumelhart et al 1986).Back propagation is a gradient descent algorithm,as the weight updating is in the direction of neg-ative gradient of the defined error function. Theability to obtain strict convergence for a lengthyset of data speaks strongly of the pattern recog-nition capability of the neural network technique,but the true test of a model is its ability to repro-duce features that were not included in its trainingset. In other words, the performance of a networkdepends greatly on its generalization capabilities,rather than being a look-up table.
Using the back propagation algorithm, theweighting coefficients and biases are converged. Ina three layer FNN, the outer two layers repre-sent the input and output, while the inner onerepresents the hidden layer (figure 1). The num-ber of neurons in all these layers is decided onthe interest of the user. The hidden layer main-tains the connectivity between the input and theoutput through the activation functions assignedto its neurons. Here X1, X2, . . . Xi, . . . XI , form the
ANN inversion of deep resistivity sounding data 51
Figure 1. Neural network architecture; (a) structure of the 3 layer feed-forward ANN used for present study, (b) modelof the artificial neuron/processing element (PE).
52 Jimmy Stephen et al
input vector of examples submitting to the net-work. At the input layer, these inputs are acti-vated by the activation function of the layer andoutputs the activation vector of input layer givenby x = x1, x2, . . . xi, . . . xI . These vector elementsare generally known as the input neurons. Each ofthese input neurons make connections with all theJ hidden neurons by a connection weight W1ji andeach hidden neuron holds a bias to the input vec-tor. These vector units get activated at the hiddenlayer and passes the activation vector of hiddenlayer y = y1, y2, . . . yj, . . . yJ , to the right hand side.Similar to the input layer, all the components ofthis activation vector also make connections withall the K nodes of output layer by a connectionweight W2kj. These output layer nodes also estab-lish a bias with the hidden layer. The activationfunction of the output layer transforms the input tothis layer to the activation vector of output layer,viz., z = z1, z2, . . . zk, . . . zK . The number of nodesor neurons in the outer layers are decided on thebasis of our problem definition, whereas the hid-den neurons are fixed on trial and error methodsto achieve the optimum network performance. Theincrease and decrease in the learning rate enablesthe network to learn and converge fast. As thetraining progresses the learning rate is adjustedwith respect to the weight adjustments. The detailsof network components are discussed later in thispaper, separately for both the networks.
3. FNN training
At any stage of the training, say at a step m, let(x, d) be the current sample of the function map-ping the input space to the output space RI → RK .Then the mean squared error function at ‘m’ isgiven as:
E(m) =1K
K∑k=1
(d(m)k − z
(m)k )2 (3)
where, dk is the desired output and zk is the net-work output given by
zk = f
(J∑
j=1
W2kj · Yj
)(4)
where, ‘f ’ is the sigmoidal function defined byequation (1). The subscripts i, j and k used in allthese equations represent the inputs, hidden neu-rons and outputs respectively and can be easilyrealized from figure 1.
Here the error function conceived in the n-dimensional hyperspace can be deemed as a bowl,whose bottom-most point indicates the optimum
set of weights (Lippmann 1987). Weights are to bechanged in the direction of negative gradient of thiserror function. Since the input vector ‘x’ is givenat the input layer and the desired output d is avail-able only at the output layer, the error between thedesired output vector and the actual output vector‘z’ is available only at the output layer. Using thiserror, it is necessary to adjust the weights (W1ji)from the input units to the hidden units and theweights (W2kj) from the hidden units to the out-put units. It is done in two steps, initially the out-put layer weights are updated and subsequently theprevious (input) layer weights are updated. Themathematical treatment for these two-step weightadjustments is given below. A detailed derivationand weight adjustment procedure can be obtainedfrom Lippmann (1987); Yegnanarayana (2001) andHaykin (1994).
3.1 Updating of layer weights
In a multi-layer FNN, the output layer is firstadjusted for the weight variations. The gradientdescent along the error surface to determine theincrement in the weight connecting the units k andj at current iteration step ‘m’ is given by,
∆W2(m)kj = −η
∂E(m)
∂W2(m)kj
, (5)
where η > 0 is the learning rate, which may alsovary for each presentation of the training pair. Inthe above equation, substituting for ‘E’ from equa-tion (3) in the above equation, we have,
∆W2(m)kj = ηδm
k ymj , (6)
where, yj is the hidden layer output and δk is thederivative of output zk as per the equation (2),given by,
δmk = zm
k (1 − zmk ). (7)
The weight update is now given by,
W2(m+1)kj = W2(m)
kj + ∆W2(m)kj
= W2(m)kj + ηδ
o(m)k y
(m)j
for j = 1, 2, 3, . . . J. (8)
The above equation is repeated for k = 1, 2,3, . . . K, thus completing update of all (k × j)weights of the output layer. Similarly for the inputlayer updating,
∆W1(m)ji = ηδ
(m)j x
(m)i . (9)
ANN inversion of deep resistivity sounding data 53
The input weight update is given by,
W1(m+1)ji = W1(m)
ji + ∆W1(m)ji
= W1(m)ji + ηδ
(m)j x
(m)i
for i = 1, 2, 3, . . . I. (10)
The above equation is repeated for j = 1, 2, 3, . . . J ,thus completing update of all (j × i) weights of theinput layer. To stabilize the convergence, a momen-tum gain (α) is added to the weight change givenby,
∆W1(m)ji = α∆W1(m−1)
ji + ηδ(m)j x
(m)i , (11)
where α(0 ≤ α < 1) is the momentum constantand m represents the current iteration index. Themomentum constant helps to stabilize the oscilla-tions during the learning process, accelerates thedescent to the minimum of the error surface andreduces the effects of local minima of the errorsurface (Plaut et al 1986; Rumelhart et al 1986;Fahlman 1989). Momentum gain can also speed uptraining in very flat regions of the error surface andsuppresses the weight oscillations in steep valleysor ravines (Schiffman et al 1992). All the weightadjustments illustrated above represents one itera-tion for a given example. Likewise, adjustments ofweights are done for other examples in the train-ing database. Presentation of the entire database(all training examples) and subsequent adjustmentof weight constitute one epoch. The training canbe stopped on the convergence of weighting coeffi-cients at which the mean square error at the out-put falls below a desired error goal or at the com-pletion of a certain number of iterations. Moredetails on the convergence of weighting coefficientsare discussed by Luo and Unbehauen (1997). Indeciding the optimum training, the testing perfor-mance on new data sets is also important. Becausea monotonically observed convergence in trainingmay not hold true with the case of testing sampleswhere the generalization capability of network isan important factor. The details are discussed inthe later portions.
Here we discuss the approach introduced in thispaper to use FNN based technique for direct para-meter estimation. We propose two networks in thesame architecture discussed above for inverting themeasured DRS apparent resistivity values. Thistwo-stage network scheme is shown in figure 2. Theapparent resistivity data sampled at 30 predeter-mined AB/2 values are fed into the first networkwhere it identifies the curve type. Then it is passedinto the second network, which outputs the modelparameters. A total of 20 curve types compris-ing most of the commonly encountered 3, 4 and 5layer resistivity cases were included in the study.
In each case the ‘loading’ problem, the determina-tion of weights from the training examples (Judd1990; Blum and Rivest 1992), is better solved andweights are saved for further simulation with newexamples.
4. The database
The efficacy of any network application dependsmainly on the database used for training and test-ing the network. The training data set should con-tain all the possible curve types with an optimumnumber of data points depending on the problemto be addressed. In this work we use both fielddata and synthetically generated data to train theneural network. A wide range of resistivity andthickness were included in the data to representranges observed in crystalline terrains as such overthe Southern Granulite Terrain (SGT) of India. Itexposes the exhumed lower crust characterized byhigh resistive granulite and gneissic complexes withmany deep-seated faults (Grady 1971). The regionalso provides the window for the oldest formingcrusts of the world with wide spectrum of meta-morphic events comprising a number of shear zonesseparating diversified crustal blocks (Drury et al1984). Deep Resistivity Sounding (DRS) measure-ments were carried out along the corridors of amajor N-S geo-transect extending from Kuppam toPalani as part of an integrated geophysical program(Singh et al 2000; Reddy et al 2001; Singh et al2003). A simplified geological map of this region isshown with the DRS locations in figure 3. Schlum-berger electrode configuration with an electrodespread (AB) of 10 km was used to probe the deeperresistive structures. The model parameter rangesfor generating the synthetic data in the presentANN scheme were selected on the basis of theseDRS measurements. Figure 4 shows the layer-wiseparameter ranges for three orders of curve typesused in the present study.
A total of 20 DRS curve types belonging to 3, 4and 5 layered resistivity structures (shown in fig-ure 4), frequently encountered along the crystallineterrain in the scope of the present study, were con-sidered. To represent each type, 150 examples wereused among which two-thirds were used for train-ing and one-third was used for testing the network.The examples are selected in such a way that, therange of layer parameters falls within the specifiedrange and covers all the possible model parametercombinations in that particular curve type, whichis essential to obtain the generalization capabilityfor the trained network. Apparent resistivities at30 successive AB/2 positions, 1.5 m to 5000 m, wereused as input. The large range of apparent resistiv-ity necessitated a logarithmic scale to represent
54 Jimmy Stephen et al
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ANN inversion of deep resistivity sounding data 55
Figure 3. The distribution of DRS data points (solid circles) along with the geological setting of the granulite terrain ofSouth India. The encircled symbols show the four test data used to analyze the network performance.
the apparent resistivities. The entire database isthen remapped between zero and one. This nor-malization of inputs is essential as 0 and 1 are thelimiting values of the sigmoidal transfer function of
neurons. There are two sets of output for a giveninput data, viz., curve identity and model parame-ters. The curve identity output vector is all zeroesbut only one non-zero value, location of which tells
56 Jimmy Stephen et al
Figure 4. Range of model parameters, resistivity and thickness, used to make the database for network application areshown on the layered basis. (a) 3 layer models, (b) 4 layer models and (c) 5 layer models. Here, R and T represent theresistivity and the thickness of the individual layers.
the curve type. The model parameters are the resis-tivity and thickness itself. These model parameterswere derived by careful modeling of the observedresistivity curve. Figure 5 shows the input and out-put patterns used in the present network, where
the symbols represent the neurons. The number ofinput neurons (i.e., the AB/2 spaces) was chosento be 30 in all cases. But number of output neu-rons varies, with 20 for curve identification (i.e., thenumber of curve types used in the present study)
ANN inversion of deep resistivity sounding data 57
Figure 5. The normalized input (a) and output patterns (b & c) used for training the networks. (b) shows the binaryoutput pattern used in the curve identification network and (c) shows the output pattern used in the inversion network.
Figure 6. Problems due to the lack of training (a) and (b) and excess training (c) and (d).
and 5, 7 and 9 for 3 layer, 4 layer and 5 layer casesrespectively (i.e., the total number of model para-meters).
5. Inversion scheme procedures
5.1 First step: curve identification network
Curve identification is the first step for the ANNbased inversion scheme. Training data exampleconsists of apparent resistivities at 30 AB/2 and its
ANN inversion of deep resistivity sounding data 59
Figure 7. The performance of curve identifying network, showing the training error convergence and the error per cent ontest samples.
Figure 8. Examples of correct curve identification on training the network for the optimum number of epochs. The networkperformance is good, even if the network output is not the desired binary values of zeroes and one as seen in (b).
curve type. The database for training and testingconsists of 900 examples selected from the originaldatabase, discussed in the earlier section, with 45samples from each type giving true representation.The different network parameters like numberof hidden layers and neurons, learning rate andmomentum ratio are optimized in a trial and errorbasis. No appreciable change in network perfor-
mance (a function of error on training, testing andnumber of epochs taken for convergence) was foundby using more than one hidden layer. 30 hiddenneurons were found to be sufficient for the curveidentification. Table 1 shows the network parame-ters used in the present study. Maximum epoch isset on the basis of training convergence as well asthe testing performance.
Experiments show that both under-training andover-training will result in the poor performance ofnetwork. Choosing appropriate number of epochsis a tricky business. Though training error is anindicator, it seldom helps as the network perfor-mance on novel dataset (which was not used intraining) depends on another property of neuralnetwork, the ‘generalization ability’. A satisfactoryconvergence of the network on training data neednot imply a good performance with the testingdata. Some of the illustrations of the problems gen-erally risen due to under-training and over-trainingis shown in figure 6. An under-trained network
Figure 9. Nature of variations in the learning rate while training the inversion networks using fast back propagationscheme. Examples are shown for A (a), KQ (b), HAK (c) and QHA (d) curve type training networks.
may fail completely on novel data. As illustratedin figure 6(a), the desired output is HK (7), butthe network output classifies it more strongly asthe HAK (13) type and also gives some values forHKQ (15) and HK types. In another case shown infigure 6(b), the same input for a KHK (17) type isalso classified into KQ (9), both giving the binaryvalue. Both the above problems can be well solvedby allowing the iteration to go few more times.But it is very important to understand the levelup to which one can train a particular networkwith the given database. Because the networkafter considerable epochs of training could pro-duce good results on training database, but fail onnovel data set. Once we cross this optimum level,the network performance seems to be distortedas seen in figures 6(c) and (d). As the network isover-trained, the weights will try to adjust to theminor details of the training data set itself. It canbe compared to the ‘over-fitting’ of a polynomialto a given set of noisy data. This unnecessaryfitting will reduce networks performance on newdata sets. In both the above cases it is noticeablethat the network outputs more or less binary val-ues but with wrong results. Here the KHK typeis clearly outputted as a KQ type (c), where asearlier the network outputted the value 1 for boththe types. Experiments show that at some opti-mum level of training, the same network performsmore faithfully with a comparatively less error
ANN inversion of deep resistivity sounding data 61
Figure 10. Diagram showing the network training error convergence (thin line) and testing performance (thick line) fordifferent curve types. Figures (a & b) show the difference in convergence for the same network (KH type) due to differentweight initializations. (c) shows the case of a progressive convergence (HK type) and (d) shows error on training with KHKcurve type.
percentage. In figure 7, the sum squared error(SSE) on training and testing database is plottedagainst the training epoch itself. It is noticeablethat the training error is continuously decreasing,but the testing error initially decreases; reaches aminimum at 20,000 epochs and thereafter increasesagain. At the optimum chosen level of 20,000epochs, the network failure is only 2% (i.e., 6 outof 300). Figure 8 shows the network performanceon optimal training.
5.2 Second step: inversion network
The inversion of DRS data or the parameter esti-mation is worked out in the second FNN work-ing on the same architecture discussed earlier. Thisnetwork differs only in the design of network para-meters as well as the database. Apparent resistiv-ity values from the existing database were used asinput, whereas the output consists of its derivedmodel. Here the entire 20 curve types discussedearlier were used. A total of 3000 examples were
generated with each type representing 100 and 50examples for training and testing respectively, inthe discussed range (figure 4). The network para-meters such as learning rate, number of hiddenneurons, momentum constant, etc. are optimizedas we discussed earlier. A summary of the networkparameters used for all the curve types are givenin table 2.
A fast network convergence was obtained byvarying the learning rate than keeping it constantthroughout the network convergence. For exam-ple, in the case of a flat error plane, we can makefast learning by decreasing the learning rate. Theinitial learning rate and the increase and decreasein learning rate optimized for the fast networkconvergence are given in tables 1 and 2. The learn-ing rate adjustments with respect to the weightadjustments during training are shown in figure 9,for selective network types (here the network typeindicates the network for a particular curve type).It is evident that in case of less complex prob-lems with fewer output units, the learning rate
62 Jimmy Stephen et al
Figure 11. Bar chart showing the training (white bars) and testing (black bars) performance of networks for selective curvetypes. The training and testing errors (SSE) for the entire samples are shown for each model parameter.
variation is within a small range (figure 9a). Thetraining characteristics for some of the selectednetworks are shown in figure 10, where both thetraining and testing errors are plotted against theepochs. Figure 10 (a) and (b) demonstrates theimportance of a proper weight initialization tostart with the training procedure, though theredoesn’t exist any mechanism for a correct initial-ization other than a trial and error. It illustratesthe performance of a network for two differentsets of initial weights and biases. Figure 10(c)shows the case of a progressive error convergencein both training and testing, observed for a typical3 layer resistive structure. Error convergence onthese networks is fast because training and testingerror falls with larger epochs. The last case shown(figure 10d) is more complex since the curve typeis two orders high. Figure 10(d) depicts the mostdifficult case (KHK type) out of the 10 networksdesigned for 5 layer resistivity structures. Thesolid arrow in these figures show the optimumlevel of training we selected in each network. Theerrors shown are the total sum-squared errors forthe entire training (100 samples) and testing (50samples) data. Though we prefer the optimumnetwork training with reference to the test per-
formance, care has also been given to attain aminimum training SSE of 0.5 (i.e., 0.005 SSE persample for entire model parameters) on trainingdata.
The network performance for new data set isanalyzed. We try to see the error statistics for theresistivity and thickness for all layers separately,to look into the performance on each parameterof all the curve types. Thereby the adaptability ofthe network is tested. Figure 11 shows the train-ing (white bars) and testing (black bars) errors forfew selected curve types. Here, the sum-squarederror of each model parameter calculated over theentire samples used for training and testing areshown. The detailed list of network errors is givenin table 3. Good results are observed with the 3layer type curves such as K and Q with very smallnetwork errors. At the same time, the networkerrors rise with higher order of layers as obtainedin the case of KHK curve type (see table 3). In allthe cases it is observed that the testing error onnew examples will be much greater than the train-ing errors as visible in the bar chart (figure 11).In fact, these errors for model parameters are wellwithin an allowable limit of a conventional inver-sion scheme.
ANN inversion of deep resistivity sounding data 63
Table 3. Error statistics for training (TR) and testing (TS).
The performance of trained network is tested onfour new DRS data over the SGT. The locationsof these data are also shown in figure 3. These fourcurves are selected from different geologic provinces
to represent the entire terrain. The complex resis-tive structures in the crystalline rock terrains likeSGT are very well reflected in the observed appar-ent resistivity curves over SGT. The DRS curvesand model parameters derived using the presentANN scheme as well as a conventional inversion
64 Jimmy Stephen et al
Figure 12. The ANN results for different field DRS data collected over the crystalline terrain of south India. The modelobtained by conventional inversion scheme (dashed line) and the ANN application (solid line) are shown for A, KH, AKHand KHK in (a), (b), (c) and (d) respectively.
scheme are shown in figure 12. The inverted mod-els show resistive upper crust with resistivitiesof the order of few thousands of Ohm-m as seenin the deeper parts with intermediate low resis-tive zones. Wherein, the DRS-07 near Mettur,marked by geologic boundaries, shows low resis-tivities even to the higher depths (figure 12d).The model parameters are well within the resolv-ability limits of resistivity technique and showa comparable match with the results of a con-ventional method as seen in the figure. The cur-rently proposed ANN based inversion (shown in
solid line), resulted in similar models well withinabout 10% error limit.
7. Conclusions
The conventional algorithm was constrained bythe requirement of an initial model, where a goodinitialization results with good results. Whereas,the ANN based inversion resulted in a geologicallyplausible resistivity model, without any a prioriinformation, making use of the internal constraints
ANN inversion of deep resistivity sounding data 65
stored as weights. These weights themselves areinherited from the training.
In this paper we have shown the adaptabilityof ANN in direct parameter estimation from DRSdata on a crystalline terrain involving a variety ofcurve types with a wide range of model parame-ter values. Though the problem gets complex withincrease in layers, it is found that a proper networkdesign can solve it, provided a good representativedatabase for the training. The inherent problemsof ambiguities, especially the equivalence problemin the case of 1D electrical sounding methods seemto be affecting the ANN inversion to some extent.We restrict to carefully selected 100 examples fortraining each network. The present approach oftwo-step network parameter estimation performseffectively and once trained, the model parametersare calculated fast on simulating the network withsaved weights and biases on new examples. Thenetwork also achieved the generalization capability,but within the limits of the presented model para-meter ranges representing a typical crystalline ter-rain. With due representation in a wider range orwith other selective terrain ranges, the scope of thetechnique is commendable. The technique can alsobe developed for assisting with first hand informa-tion for the model initialization in other conven-tional inversion schemes. This will help in a greaterextent for on-site interpretation of DRS data.
Acknowledgements
The authors are thankful to Dr. V P Dimri,Director, NGRI, for granting permission to pub-lish this work. They also thank the reviewers fortheir constructive suggestions, which helped tofurther improve the paper. Mr. Jimmy Stephen,greatly acknowledges the Council of Scientific andIndustrial Research (CSIR) for its Senior ResearchFellowship. Thanks are also due to all field partic-ipants of the DRS Group.
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