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AN INNOvATIvE METhOD TO ATTENuATE gENETIC DRIFT IN gENETIC ALgORIThM OPTIMIZATIONS: APPLICATIONS TO ANALyTIC ObjECTIvE FuNCTIONS AND RESIDuAL STATICS CORRECTIONS. Pierini, M. Aleardi, A. MazzottiEarth Sciences Department, University of Pisa, Italy

Introduction. �he solution of geoph�sical non-linear inverse pro�lems presents several challenges mainl� related to convergence and computational cost. In addition, the performances of local optimization algorithms are strongl� dependent on the initial model definition. For this reason, glo�al optimization is often preferred in case of model spaces with complex topolog� (i.e. man� local minima, small gradient of the o�jective function in a neigh�ourhood of the glo�al minimum). �enetic Algorithms (�As) (Holland, 1975) are a class of glo�al optimization methods that have �een proven ver� effective in solving geoph�sical optimization pro�lems (�ajeva et al., 2016). In �A terminolog�, an individual, or chromosome, is a solution in the model space, whereas a population represents a set of individuals (i.e. an ensem�le of possi�le solutions). A ver� simple �A flow starts with the generation of a random population of individuals over which the fitness function (namel� the goodness of each solution) is evaluated. �he fitness value stochasticall� contri�utes to the selection of the �est individuals for the reproduction step in which a set of new solutions (offspring) is generated �� com�inations of parent individuals. �he offspring are mutated, the fitness is evaluated, then a new generation is created �� replacing some of the parent individuals with the generated offspring. �he algorithm iterates until convergence conditions are satisfied. B� assuming a population containing an infinite num�er of individuals, the convergence of the algorithm to the glo�al minimum is guaranteed �� the Holland theorem (Holland, 1975). For finite populations, the convergence of �As is not guaranteed: this characteristic is often called “genetic drift”. A more heuristic description of the genetic drift phenomenon can �e given in terms of population �ehaviour: after some generations, the chromosomes tend to converge in a convex neigh�ourhood of a minimum of the o�jective function (not necessar� the glo�al minimum), and thus the exploration of other promising portions of the model space is prevented. In the worst case, the population cannot escape from such convex neigh�ourhood, and a non-optimal solution is provided. �ver the last decades, man� strategies have �een proposed to attenuate the genetic drift effect (Eldos, 2008� Aleardi and Mazzotti, 2017). For example, a possi�le approach is the so called niched genetic algorithm (N�A), in which the initial random population is divided into multiple su�populations that are su�jected to separate selection and evolution processes.

In this work, we propose an innovative method to attenuate the genetic drift effect that we call “drift avoidance genetic algorithm” (DA�A). �he implemented method com�ines some principles of N�As and Monte-Carlo algorithm (MCA) with the aim to increase the exploration of the model space and to avoid premature convergence and/or entrapment into local minima.

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�he DA�A performances are first tested on two anal�tic o�jective functions often used to test optimization algorithms. �hen, the DA�A approach is applied to a well-known non-linear geoph�sical optimization pro�lem: residual statics correction. �he DA�A method is also compared with standard N�A and MCA implementations.

The DAGA method. With the aim to increase the exploration capa�ilities of �As, we first h��ridize N�A and MCA �� triggering catastrophic events over the evolving su�populations (Eldos, 2008). A catastrophe is a random phenomenon, centred around a su�population that, �asing on stochastic criteria, destro�s a certain num�er of chromosomes in the central su�population and, to a lesser extent, in the neigh�our su�populations. After a su�population has �een hit �� a catastrophic event, its num�er of individuals decreases. For this reason, the destro�ed chromosomes are progressivel� replaced over the following generations �� new individuals migrating from the other su�populations, and �� new randoml� generated solutions. �he centre of the catastrophe is stochasticall� selected �asing on the mean fitness values of the entire ensem�le of su�populations. �he intensit� of the catastrophic event (the num�er of chromosomes that will �e removed) is randoml� determined �asing on a gaussian distri�ution with a user-defined variance and an adaptive mean value, which decreases as the num�er of generation increases (i.e. as the minimum of the o�jective function is approached). Note that in case of a catastrophic event the �est chromosomes are alwa�s preserved to ensure that the optimization does not waste time re-discovering previousl� promising solutions.

In case of genetic drift, the genetic diversit� is lost, or in other words, the entire population contains several copies of the same individual. In this case a considera�le amount of time is wasted in performing forward modelling for ver� similar solutions. �o tackle this issue, we include in our algorithm an additional strateg� that s�stematicall� performs random replacements in each generation. Let t �e a user-defined parameter� the algorithm identifies t couples of similar individuals for each su�population and one of the individuals for each couple is replaced with a randoml� generated solution. �imilarl� to the catastrophic event, we again preserve the �est individuals. �hanks to this operation, the genetic diversit� within each population is preserved and the exploration of the model space is maximized.

�he previous considerations make it clear that the DA�A approach is mainl� aimed at increasing the exploration capa�ilit� of a standard niched genetic algorithm and at efficientl� exploring the most promising portions of the model space. ��viousl� in appl�ing the DA�A method a good compromise must �e found �etween exploration and exploitation of the algorithm. In practical applications, this translates in finding (usuall� �� a trial and error procedure) an optimal set of user-defined parameters for the pro�lem at hand. Indeed, a too strong exploration will heavil� increase the computational time and slow down the convergence, while a too strong exploitation will result in premature convergence toward su�-optimal solutions. As a final remark, note that the inclusion of several Monte Carlo principles into the genetic algorithm optimization kernel, results in an increased computational effort. For this reason, an accurate code optimization is crucial to ensure the applica�ilit� of the method.

Analytic test functions. �he performances of the DA�A method are first evaluated on the anal�tic Rastrigin and �chwefel functions. �he DA�A method is also compared with standard implementations of N�A and MCA. In particular, for each method and for each o�jective function, we perform a set of 50 tests, and for each generation we progressivel� count the num�er of tests that attain convergence. �he convergence criterion is �ased on a Euclidean measure of distance from the glo�al minimum (�ajeva et al., 2017). �o allow for a meaningful comparison, the total num�er of evaluated models is maintained constant for each considered method.

�he Rastrigin function is a linear com�ination of a para�oloid and a harmonic function: (1)

where n is the dimension of the model space. Fig.1a shows a 2-D Rastrigin function computed within [-5, 5] n. �he glo�al minimum is located at [0, …, 0] and is surrounded �� regularl�

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distri�uted local minima. In this case the algorithms are tested on a 4-dimensional Rastrigin function, and considering (for DA�A and N�A) a population of 26 individuals divided into 2 su�populations and a maximum num�er of 250 generations. �he results (Fig. 1�) are represented as incremental curves showing, for each generation, the cumulative sum of tests that attained convergence. In this case we o�serve that, as expected, the MCA is not a�le to efficientl� explore the model space and converges in onl� 7 out of 50 tests, whereas N�A and DA�A show ver� similar performances. In particular, this example confirms that �As are ver� efficient in finding the glo�al minimum in case of regularl� distri�uted minima (�ajeva et al., 2017). For this example, considering a serial Matla� code running on an intel i3@1.70�Hz, the average convergence time for ��A and DA�A are 0.21 s and 0.49 s, respectivel�.

�ajeva et al. (2017) demonstrated that �As severel� suffer in case of o�jective functions with irregularl� distri�uted minima. For this reason, we now compare the different algorithms on the �chwefel function:

(2)

Fig. 1 - a) 2-D plot of the Rastrigin function. �) Comparison of the performances of the different algorithms. Note the similar performances of N�A and DA�A.

Fig. 2 - a) 2-D plot of the �chwefel function. �) Comparison of the performances of the different algorithms. Note that the N�A is strongl� affected �� genetic drift, whereas the DA�A algorithm converges for all the tests.

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Fig. 2a shows the 2-D �chwefel function computed within [500, 500]n. Note that this function is characterized �� a much more complex topolog� than the Rastrigin function. In particular, in the �chwefel function the local minima are more irregularl� distri�uted, and important local minima are distant from the non-centred glo�al minimum (located at [420.9687, …,420.9687]), or are even located at the opposite edge of the model space. In the following tests, we consider a 4-D space, a population of 100 individuals divided in 4 su�populations, and a maximum num�er of generations equal to 800. �he final results are represented in Fig. 2�. Note that none of the MCA tests satisfies the convergence criterion, and that for the N�A, onl� 22 out of 50 tests converge. Differentl�, the implemented DA�A approach converges for all the tests. From the one hand, these results confirm that the �A approach is strongl� affected �� genetic drift in case of irregularl� distri�uted minima. From the other hand, Fig. 2� proves that the DA�A approach effectivel� attenuates the genetic drift and is a�le to find the glo�al minimum even in case of o�jective functions with ver� complex topolog�. For this test, the average convergence time for ��A and DA�A is 1.76 s and 3.86 s, respectivel�.

Residual statics correction. �he residual statics correction is a highl� non-linear optimization pro�lem that is often solved �� appl�ing glo�al optimization methods (Rothman, 1985). In the following test, we compare the N�A and DA�A methods on CMP-consistent residual statics correction performed on a s�nthetic CMP gather. �ajeva et al. (2017) showed that this geoph�sical optimization pro�lem is characterized �� an o�jective function that

shows some similarities to �oth the Rastrigin and �chwefel functions. For this reason, the �A encounters some pro�lems in finding the glo�al minimum in case of high-dimensional model spaces. In this case we use actual well log information and a convolutional forward modelling to generate the reference CMP gather (without residual statics� Fig. 3a). �o simulate residual statics in the data, we appl� to each trace in the reference CMP random time shifts uniforml� distri�uted over the range -15/+15 ms (Fig. 3�). In the su�sequent optimization process we allow time shifts within the range -25/+25 ms. In this case the model space dimensionalit� is equal to 40 (i.e. we consider 40 seismic traces), while the total num�er of individual is 200 divided into two su�populations. Figs. 3c and 3d show the inversions results for the N�A and DA�A approaches, respectivel�. We note that the DA�A method returns a final corrected CMP gather (Fig. 3c) ver� similar to the reference one, whereas the final CMP �ielded �� the N�A approach shows man� misalignments of the reflections and c�cle-skipped traces (Fig. 3d). In Fig. 3e, we also o�serve that the

Fig. 3 - a) �he s�nthetic reference CMP gather, �) the trace-shifted CMP, c) the final DA�A CMP, d) the final N�A CMP. e) Energ� curves representing the evolution of the energ� of the stack trace over iterations.

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energ� of the stack trace associated to the DA�A CMP gather is ver� close to the energ� of the stack trace pertaining to the reference CMP. Note that differentl� from the N�A CMP, the final DA�A CMP could �e used as a valid starting model for an� local optimization method for a further refinement of the residual statics estimation. Due to the limited num�er of model evaluations the DA�A resulted to �e onl� 1.11 times slower than N�A.

Conclusions. We presented an innovative strateg� to attenuate the genetic drift and to increase the exploration of the model space in a �A optimization. In the proposed DA�A approach, a standard N�A is h��ridized with principles coming from MCA. �he test on anal�tic o�jective functions and on residual statics computation, demonstrated that, differentl� from the N�A, the implemented DA�A approach does not suffer the genetic drift effect. In particular, our tests confirmed that the DA�A approach grants the convergence in case of o�jective functions with ver� complex topolog�, where the standard N�A and MCA fail to converge. Differentl�, in cases of simpler topologies the N�A and the DA�A algorithms achieved ver� similar performances. �he implemented DA�A method is more computational demanding than the standard N�A and for this reason an accurate code optimization has �een performed. After additional code optimization (i.e. parallel implementation), the next step of our research is to appl� the DA�A method to 2D acoustic full-waveform inversion that is a highl� non-linear geoph�sical optimization pro�lem with expensive forward modelling.ReferencesAleardi M., and Mazzotti, A. (2017). 1D elastic full-waveform inversion and uncertainty estimation by means of a

hybrid genetic algorithm–Gibbs sampler approach. �eoph�sical Prospecting, 65(1), 64-85.Eldos �. (2008). Distributed genetic algorithms: a scheme for genetic drift avoidance, Journal of electrical engineering,

59(1), 45.Holland J. (1975). Adaptation in Natural and Artificial Systems, �he MI� Press, Cam�ridge.Rothman D.H. (1985). Nonlinear inversion, statistical mechanics, and residual statics estimation. �eoph�sics,�eoph�sics,

50(12), 2784-2796.�ajeva A., Aleardi M., �tucchi E., Bienati N., Mazzotti A. (2016). Estimation of acoustic macro models using a

genetic full-waveform inversion: Application to the Marmousi model. �eoph�sics, 81(4), R173-R184.�eoph�sics, 81(4), R173-R184.�ajeva A., Aleardi M., �aluzzi B., �tucchi E., �padavecchia E., Mazzotti A. (2017). Comparing the performances of

four stochastic optimisation methods using analytic objective functions, 1D elastic full-waveform inversion, and residual static computation. �eoph�sical Prospecting. In print. doi:10.1111/1365-2478.12532.