Genetic Algorithm for Seismic Velocity Picking

Transcription

1 Proceedings of International Joint Conference on Neural Networks, Dallas, Texas, USA, August 4-9, 2013 Genetic Algorithm for Seismic Velocity Picking Kou-Yuan Huang, Kai-Ju Chen, and Jia-Rong Yang Abstract We adopt genetic algorithm (GA) for velocity picking in reflection seismic data. Conventional seismic velocity picking was to pick a series of peaks in a seismic semblance image (stacking energy) by geophysicists. However, it took human efforts and time. Here, we transfer the velocity picking to a combinatorial optimization problem. The local peaks in time-velocity seismic semblance image are ordered in a sequence with time first, then velocity. We define a fitness function including the total semblance of picked points, and constraints on the number of picked points, interval velocity, and velocity slope. GA can find an individual with the highest fitness value, and the picked points form the best polyline. We use simulation data and Nankai real seismic data in the experiments. We sequentially find the best parameter settings of GA. The picking result by GA is good and close to the human picking result. The result of velocity picking by GA is used for the normal move-out (NMO) correction and stacking. The stacking result shows that the signal is enhanced. This method can improve the seismic data processing and interpretation. V I. INTRODUCTION ELOCITY picking was to pick a series of peaks in the seismic semblance image (stacking energy) and was done by geophysical experts. However, velocity picking by human took much time. Some automatic methods had been done on seismic velocity picking. In 1974, Beitzel and Davis [1] used the graph theory method to do the velocity picking. Two predefined constraints, interval velocity and velocity slope, were used to delete the invalid edges, and the minimum spanning tree algorithm was applied to find a polyline from the valid edges. However, this method had to manually choose a skeleton as the final solution from the minimum spanning tree result. In 1992, Schmidt and Hadsell [2] applied the multilayer perceptron (MLP) in velocity picking. They trained two MLP models by predefined patterns representing the relation between time and semblance, and velocity and semblance. Then, they used another MLP model that was trained by the human picked polyline to validate the polyline which was determined by the previous two networks. In 1994, Fish and Kasuma [3] also used multilayer perceptron in velocity picking. In their study, they used the human picked peaks and the eight neighbors in a semblance image to form training patterns. The MLP was trained until it could This work was supported in part by the National Science Council, Taiwan, under NSC E and NSC E Kou-Yuan Huang is with the Department of Computer Science, National Chiao Tung University, Hsinchu, Taiwan. (Corresponding author kyhuang@cs.nctu.edu.tw) Kai-Ju Chen is with the Department of Computer Science, National Chiao Tung University, Hsinchu, Taiwan. ( chenkaiju@gmail.com) Jia-Rong Yang is with the Department of Computer Science, National Chiao Tung University, Hsinchu, Taiwan. ( yangcj@cs.nctu.edu.tw) approximate the expert picking result. Then, the MLP could be used to do velocity picking. The drawback was that the MLP needed the human picking result as the reference to validate the candidate picks. In 2002, Beveridge et al. [4] took velocity picking as a problem of choosing the best polyline from semblance peaks. They developed an energy function composed of inverse energy, average turning angle, and proximity to median. By using the steepest descent method to find the largest energy drop, the best polyline could be obtained. However, the interval velocity was not taken into consideration, and that might result in an ineligible solution. Here, we transfer the velocity picking problem to a combinatorial optimization problem. Then we use genetic algorithm (GA) to find the best solution. GA is a global optimization method proposed by Holland in 1975 [5]-[6]. It was ever used in correlating seismic horizons across fault [7] and Kirchhoff ray tracking modeling for seismic migration [8]. Based on the concept of survival in Darwin s evolution theory, GA use four operations: reproduction, crossover, mutation, and survivor selection to perform the evolution. These operations repeat for several generations, and the individual with the highest fitness value is the solution. At first, we get the local peak points on semblance image by peak detection. Then, we choose several points with higher semblance values as candidate peak points. Next, we formulate a fitness function including the semblance value of picked points, and constraints on the number of picked points, interval velocity, and velocity slope. Then GA is adopted to find the best solution among the combination of candidate peak points. The best solution represents a polyline consisted of several high semblance points. Finally, the picking result is applied to do the normal move-out (NMO) correction and stacking to verify the result of velocity picking. II. SEISMIC VELOCITY PICKING BY GENETIC ALGORITHM A. Introduction to Seismic Data Acquisition We use Nankai real seismic data to describe the seismic data acquisition [9]. Fig. 1 shows the shot and one side receivers to get a one-shot seismogram. Then, we move the shot point and receivers at the same time and we can get the other seismograms. The spacing of each shot, as well as the spacing of each receiver, is m. Fig. 2 shows the seismogram of shot 1750 of Nankai data. It has 69 traces and 2,750 samples per trace with sampling interval seconds and total 11 seconds /13/$ IEEE 2722

2 Fig. 1. Shot and receivers at one-shot seismogram. Fig. 4. Shots and receivers geometry on a CMP gather. Fig. 2. Seismogram of shot 1750 of Nankai data. We collect those traces with the same reflection point to become a common mid-point (CMP) gather as shown in Fig. 3 and Fig. 4. Fig. 5 shows the CMP 933 gather of Nankai data. In order to find the correct velocities for NMO normal correction, we have to construct the semblance image of the CMP gather and perform velocity picking. In the generation of semblance image from CMP gather, each pixel in a seismic semblance image is the energy of all traces in the CMP gather at the certain time and certain velocity. The range of stacking velocity is from 1000 (m/s) to 7000 (m/s) and the velocity sampling interval is 25 (m/s). To obtain the energy of each pixel, we shift every 5 time samples with window length 10 time samples to calculate the energy of all traces within the window. Hence, the time interval of the semblance image is 0.02 sec ( ). Fig. 6 shows the semblance image of CMP 933. The result of velocity picking is used to find the correct velocities for the NMO and stacking. Fig. 7 shows the NMO correction on traces of CMP gather. The purpose of NMO correction is to correct the offset difference existing in the reflection signal to vertical reflection. Stacking is to enhance signal to noise ratio by stacking those corrected traces. Fig. 5. CMP 933 gather of Nankai data. Fig. 6. Semblance image of CMP 933 Fig. 7. NMO on traces of CMP gather. Fig. 3. Stack chart of selected traces of a CMP gather from seismograms. B. Introduction of Seismic Velocity Picking The energy of each pixel in a seismic semblance image is a normalized coherence measure of the traces between 0 and 1. Our purpose is to pick several peak points and get the best time-velocity polyline. The example of velocity picking by GA is shown below. Fig. 8 illustrates eight peaks in the semblance image. The local peaks in time-velocity seismic semblance image are 2723

3 ordered in a sequence with time first, then velocity. Eight peaks are ordered in A, B,..., and H. The points with earlier time link to the points with later time, and they become a polyline. We choose several peak points in the semblance image to estimate the velocities in different layers. For example, in Fig. 8, A C D G H is a possible solution for velocity picking. The value of picked point is 1, otherwise 0. We connect the picked point with value 1 as the picking result. Usually, the velocity in deeper layer is increased. Fig. 9 shows the proposed system of velocity picking by GA. In the semblance image, local peak points are selected after preprocessing. We define the fitness function and use GA to find the best solution where each element represents whether the local peak point is picked or not. After the selection of time-velocity pairs, we can apply it to the NMO correction and stacking. Fig. 8. Illustration of seismic velocity picking. Start Input a time-velocity semblance image Preprocessing: get candidate peaks by peak detection Define the fitness function of GA Find the best t-v pairs by GA Normal move-out correction Stacking Stop Fig. 9. Flowchart of velocity picking by genetic algorithm. C. Preprocessing For an input semblance image, we use a 3-by-3 window to identify the peak points. We compare the semblance value of a point with its eight neighbors. The point is considered as a peak point if it has the largest semblance value. We move the window from left to right and top to bottom to find the peak points. Then, we choose top Q points with higher semblance values as candidate points. The candidate points are arranged with the increasing order of time and velocity. And several points are selected from the Q candidate points as the result of velocity picking. We use a vector s = [x 1, x 2,, x Q ] T where x i is 1 if the i th point is picked and 0 otherwise, to represent a possible combination of Q candidate points. The points x i with value 1 will be linked as a polyline. D. Define the Fitness Function The fitness function for velocity picking by GA in s is expressed as follows. E s E s E s E s E s (1) p p npts npts where α p, α npts, α vi, and α vs are constants. The first term in the fitness function is the total semblance value of the picked points. E p( s i i vi s ) x p( x ) (2) where p(x i ) is the semblance value of the i th point. The second term is for the total number of picked points, K. ( s ) sum( s) K (3) E npts 2 where K is a preset constant and K Q. The third and fourth terms consider the constraints of interval velocity (vi n-1,n ) and velocity slope (vs n-1,n ) to remove the ineligible polyline as in Beitzel and Davis [1]. The interval velocity, which is related to Dix function, is used to restrict the calculated interval velocity [10]. They are defined as 2 2 t v t v t t vi (4) n1, n n n n1 n1 / n n1 n1, n ( vn vn 1)/( tn tn 1) vs (5) where t n-1 and t n are the two-way vertical travel time of layer n-1 and n respectively, and v n-1 and v n are the stacking velocity of layer n-1 and n respectively. Penalties of constrains for interval velocity (csvi) and velocity slope (csvs) are 0 max, VI min vin 1, n VI csvivin 1, n (6) 1, otherwise 0, VS min vsn 1, n VS max csvsvsn 1, n (7) 1, otherwise where VI min, VI max, VS min, VS max are the predefined values for constraints. Therefore, the total interval velocity penalty (E vi ) and the total velocity slope penalty (E vs ) in s are defined as E vi s csvivi n 1, n (8) s s csvs E vs vs n 1, (9) s n After the fitness function is defined, GA is adopted to find a solution that maximizes the fitness function. The algorithm of GA for seismic velocity picking is as follows. Algorithm: Genetic algorithm for seismic velocity picking. vi vs vs 2724

4 Input: A seismic stacking velocity semblance image and Q candidate peak points. Output: Vector s = [x 1, x 2,, x Q ] T representing the optimal picking result. Step 1: Initialization 1. Set the maximum number of generation N gen and the mutation rate σ between [0, 1]. 2. Set the number of individuals M and randomly generate M individuals. Each individual represents a vector s with length Q where each element x i in s is 0 or 1 with equal probability 0.5. Step 2: Fitness value calculation Calculate the fitness value of each individual by the fitness function E(s) in (1). Step 3: Reproduction 1. Randomly select any two individuals, s 1 and s 2, from these M individuals. 2. Compare their fitness values, E(s 1 ) and E(s 2 ). The individual with a higher fitness value is the parent. Repeat Step 3 M times to obtain M parents. Step 4: Crossover 1. Randomly pair every two parents obtained in Step 3. There are M/2 parent pairs. 2. Randomly choose two crossover points with equal probability for each pair of parents. 3. Exchange the data of each pair of parents between their crossover points and generate two children. After the exchange is completed, the corresponding M children are generated. Step 5: Mutation 1. Generate a random number r with uniform distribution over [0, 1] for each element of each child. 2. If r mutation rate σ, then mutate the element by reversing the bit value. Otherwise, go to next element. Repeat Step 5 until all elements are handled for each child. Step 6: Fitness value calculation Calculate the fitness value in (1) for each child. Step 7: Survivor selection Choose top M individuals with higher fitness values from the M parents and M children as the individuals of next generation, and then repeat Step 3 to Step 7 until reach the preset number of generation N gen. Finally, we choose the individual with the highest fitness value as the solution of velocity picking. III. EXPERIMENTAL RESULTS A. Experimental Results on Simulation Data 1) Introduction to Simulation Data: We generate simulation data in Fig. 10. There are 65 candidate peak points in the semblance image. The semblance value of the point is either 0.8, 0.4 or, 0.1. The red dots in Fig. 10 represent the points with semblance value 0.8, the blue star symbols represent the points with semblance value 0.4, and the green square symbols represent the points with semblance value 0.1. The stacking velocity is from 1000 (m/s) to 7000 (m/s) and the velocity sampling interval is 25 (m/s). The sampling interval is 0.02 sec in total 6 seconds. We pick 20 out of the 65 candidate peak points to form a polyline as the human picking result shown in Fig. 10. For GA processing in the simulation data, we set that the vector length Q is 65 and K is 20. Then, we perform velocity picking by GA. Fig. 10. Simulation data and picking result by human. In the performance evaluation, we compare the picking polyline result of GA with that of human shown in Fig. 10. The average absolute difference of velocity (V diff ) is used for comparison and defined as N 1 V t V t (10) V diff GA human N t where N is the number of calculated time samples by interpolation, V GA (t) and V human (t) are the velocity picked by GA and human. Table I shows the constants in the fitness function. We set them with equal weighting. According to Table I, if two picked points possess the largest semblance value, their total semblance value is 1+1=2. However, if they also violate the two constraints, interval velocity and velocity slope, the total penalty is also 1+1=2. Table II shows the values of VI min, VI max, VS min, and VS max. According to Fig. 10 and Table II, the feasible regions of interval velocity and velocity slope are shown in Fig. 11 and Fig. 12, respectively. In Fig. 11, if the circled point links to the other points in the yellow region, it violates the interval velocity constraint. In contrast, if the circled point links to the other points in the white region, it does not violate the interval velocity constraint. Similarly, in Fig. 12, if the circled point links to the other points in the yellow region, it violates the velocity slope constraint. And if the circled point links to the other points in the white region, it does not violate the velocity slope constraint. 2725

5 TABLE I THE CONSTANTS IN THE FITNESS FUNCTION. α p α npts α vi α vs TABLE II THE VALUES OF VI MIN, VI MAX, VS MIN, VS MAX. VI min VI max VS min VS max we choose = 0.04 for getting the smallest mean value. TABLE III MEAN OF V DIFF UNDER DIFFERENT MUTATION RATE AFTER 300 Mutation rate Mean of V diff (m/s) To find the best parameter of generation number N gen, we fix the = 0.04 and M =50, then vary the N gen with 100, 150, 200, 250, and 300. Table IV shows the mean of V diff of parameter N gen. From the result, we choose N gen = 300 for getting the smallest mean value. TABLE IV MEAN OF V DIFF UNDER DIFFERENT GENERATION NUMBER N GEN AFTER 300 Generation number N gen Mean of V diff (m/s) Fig. 11. Feasible region of interval velocity at one point on the simulation data. To find the best parameter of individual number M, we fix the = 0.04 and N gen = 300, and vary the M with 10, 30, 50, 70, and 90. Table V shows the mean of V diff of parameter M after 300 experiments under different M. From the result, we choose M = 90 for getting the smallest mean value. Through these experiments, we sequentially find the best parameter setting for GA. Table VI shows the GA parameters for the further experiments. TABLE V MEAN OF V DIFF UNDER DIFFERENT INDIVIDUAL NUMBER M AFTER 300 Individual number M Mean of V diff (m/s) Fig. 12. Feasible region of velocity slope at one point on the simulation data. 2) Parameter Determination for Simulation Experiments: The parameters of GA are mutation rate, generation number N gen, and individual number M. We use a vector p = [, N gen, M] to present the parameter setting and sequentially find the best value of each parameter. We set the mutation rate to 0.02, 0.04, 0.06, 0.08, and 0.1; the generation number N gen to 100, 150, 200, 250, and 300; and the individual number M to 10, 30, 50, 70, and 90. The initial setting of p is the average of lower and upper bounds of each parameter where p = [0.06, 200, 50]. We change a parameter in its range at a time while other parameters are fixed. For each candidate of a parameter, we perform 300 experiments and calculate the V diff and the mean of V diff. According to the mean values of V diff, we choose the candidate with the smallest mean of V diff as the best parameter. Then, sequentially we find the best value for the next parameter. To find the best parameter of mutation rate, we fix the N gen = 200 and M =50, and vary the mutation rate with 0.02, 0.04, 0.06, 0.08, and 0.1. Table III shows the mean of V diff of parameter after 300 experiments. From the result, TABLE VI BEST PARAMETERS FOR GA. N gen M ) Results of Simulation Experiments: We use constants of fitness function in Table I, the ranges of interval velocity and velocity slope in Table II, and the best parameters for GA in Table VI as the configuration of experiments. We perform 3,000 experiments and calculate the mean and standard deviation of V diff. Table VII shows the best case, worst case, mean, and standard deviation of V diff of GA on the simulation data. In Table VII, the best case is zero and that means that our method can obtain the same result as human result shown in Fig. 10. Fig. 13 shows the histogram of V diff of 3000 experiments on the simulation data. As in Fig. 13, most of the V diff of experiments is zero. Fig. 14 shows the fitness value versus generation of one experiment on the simulation data. 2726

6 TABLE VII THE BEST CASE, WORST CASE, MEAN, AND STANDARD DEVIATION OF V DIFF USING GA FOR VELOCITY PICKING ON SIMULATION DATA. Best (m/s) Worst (m/s) Mean (m/s) Standard deviation Fig. 13. Histogram of V diff of 3000 experiments on the simulation data. Because the 15 CMP gathers are obtained from the same area, we use one of the 15 CMP gathers, CMP 1008, to determine the best parameters for Nankai experiments. For each candidate of a parameter, we perform 1000 experiments and calculate the V diff and the mean of V diff. According to the mean values of V diff, we choose the candidate with the smallest mean of V diff as the best parameter. Then, sequentially we find the best value for the next parameter. To find the best parameter of mutation rate, we fix the N gen = 200 and M =50, and vary the mutation rate with 0.02, 0.04, 0.06, 0.08, and 0.1. Table VIII shows the mean of V diff of parameter after 1000 experiments under different. From the result, we choose = 0.06 for getting the smallest mean value. TABLE VIII MEAN OF V DIFF UNDER DIFFERENT MUTATION RATE AFTER 1000 Mutation rate Mean of V diff (m/s) Fig. 14. Fitness value versus generation of one experiment on the simulation data. To find the best parameter of generation number N gen, we fix the = 0.06 and M =50, then vary the N gen with 100, 150, 200, 250, and 300. Table IX shows the mean of V diff of parameter N gen after 1000 experiments under different N gen. From the result, we choose N gen = 200 for getting the smallest mean value. B. Experimental Results on Nankai Data 1) Parameter Determination for Nankai Experiments: We use Nankai real seismic data for experiments on the 15 CMP seismic gathers of Nankai [9]. They are CMP 933, 958, 983, 1008, 1033, 1058, 1083, 1108, 1133, 1158, 1183, 1208, 1233, 1258, and Since there is a water layer whose two-way travel time is approximately 6 sec, we perform velocity pickings between 5.5 sec and 11 sec. For experiments, we compare the picking polyline result of GA with that of human [9] and calculate the V diff by equation (10). Fig. 15 shows the number of human picked points on each CMP gather [9]. The vector length Q is 50, i.e., the number of candidate peaks is 50. The number of picked points K by GA in each CMP gather is the same as the number of human picked points in each CMP gather shown in Fig. 15. K is from 2 to 5 points. We will pick K points from 50 candidate peaks in each semblance image by GA. TABLE IX MEAN OF V DIFF UNDER DIFFERENT GENERATION NUMBER N GEN AFTER 1000 Generation number N gen Mean of V diff (m/s) To find the best parameter of individual number M, we fix the = 0.06 and N gen = 200, and vary the M with 10, 30, 50, 70, and 90. Table X shows the mean of V diff of parameter M after 1000 experiments under different M. From the result, we choose M = 50 for getting the smallest mean value. Through these experiments, we sequentially find the best parameter setting for GA. Table XI lists the best parameters of GA in our experiment on Nankai real data. TABLE X MEAN OF V DIFF UNDER DIFFERENT INDIVIDUAL NUMBER M AFTER 1000 Individual number M Mean of V diff (m/s) TABLE XI BEST PARAMETERS FOR GA. N gen M Fig. 15. The number of human picked point on each CMP gather. 3) Results of Nankai Experiments: We use constants of fitness function in Table I, the ranges of interval velocity and velocity slope in Table II, and the best parameters for GA in 2727

7 Table XI as the configuration for the Nankai experiments. For each CMP gather, we do 10,000 experiments repeatedly and calculate the V diff of each experiment. Fig. 16 shows the mean and standard deviation of V diff on each CMP gather. The triangles are the mean values. And error bars are the standard deviations. (a) Fig. 16. Mean and standard deviation of V diff on each CMP gather. The picking result by GA with the smallest mean of difference V diff among 15 CMP gathers is CMP We choose the best result for demonstration. Fig. 17 shows the fitness value versus generation of one experiment on CMP Fig. 18 (a) shows the CMP gather Fig. 18 (b) is the plot of picking result, where the pink color in semblance image represents the high semblance region. The black squares are the GA picking result and the red dots are the human picking result. Fig. 18 (c) shows the result of NMO correction and Fig. 18 (d) is the stacked trace. In Fig. 16, the mean of V diff on CMP 1008 is m/s. Meanwhile, in Fig. 18 (b), the human picking result is close to 1500 m/s. Therefore, the error ratio is 14.62/1500 = It is close to human picking result. (b) (c) Fig. 17. Fitness value versus generation of one experiment on CMP 1008 by GA. (d) Fig. 18. GA picking on the CMP (a) CMP1008 gather, (b) picking result, (c) result of NMO correction, and (d) the stacked trace. IV. CONCLUSIONS The velocity picking problem is transferred to a series of 2728

8 binary representation. GA is a global optimization method. We adopt GA for velocity picking and define a fitness function that includes the total semblance value and constraints for the number of picked points, interval velocity, and velocity slope. The defined fitness function provides the optimal result with a high total semblance value and satisfies the velocity constraints. We have experiments on the simulation data and Nankai seismic gather data. By GA, for the simulation data, we pick 20 points from 65 candidate peaks, and for Nankai data, we pick 2~5 points from 50 candidate peaks in each semblance image. We sequentially find the best parameter settings for the two experiments. The picking result is good and close to the human picking results even on the case with many picking points. For Nankai data, the velocity picking is further used for the NMO correction and stacking. The stacking result shows that the signal is enhanced. This method can improve the seismic data processing and interpretation. REFERENCES [1] J. E. Beitzel and J. M. Davis, "A computer oriented velocity analysis interpretation technique," Geophysics, vol. 39, pp , [2] J. Schmidt and F. A. Hadsell, "Neural network stacking velocity picking," SEG Technical Program Expanded Abstracts, vol. 11, pp , [3] B. C. Fish and T. Kusuma, "A neural network approach to automate velocity picking," SEG Technical Program Expanded Abstracts, vol. 13. pp , [4] J. R. Beveridge, C. Ross, D. Whitely, and B. Fish, "Augmented geophysical data interpretation through automated velocity picking in semblance velocity images," Machine Vision and Applications, vol. 13, pp , [5] J. H. Holland, Adaptation in natural and artificial systems, University of Michigan Press, [6] J. H. Holland, "Genetic algorithms," Scientific American, vol. 267, pp , [7] M. Aurnhammer and K. D. Tonnies, "A genetic algorithm for automated horizon correlation across faults in seismic images," Evolutionary Computation, IEEE Transactions, vol. 9, pp , [8] H. Liangjie, L. Shiliang, and R. P. Bording, "A computational engine for petroleum applications using Genetic Algorithms," Canadian Conference on Electrical and Computer Engineering (CCECE), pp , [9] D. Forel, T. Benz, and W. D. Pennington, Seismic data processing with Seismic Un*x: a 2D seismic data processing primer, Society of Exploration Geophysicists, pp pp , [10] Ö. Yilmaz and S. M. Doherty, Seismic data analysis: processing, inversion, and interpretation of seismic data: Society of Exploration Geophysicists,