Optimal time-delay spiking deconvolution and its application in the physical model measurement

Transcription

1 Optimal time-delay spiking deconvolution and its application in the physical model measurement Zhengsheng Yao, Gary F. Margrave and Eric V. Gallant ABSRAC Spike deconvolution based on iener filter theory usually has the desired output as a zero-lag spike, hich orks ell for avelets of minimum phase. In this paper, the algorithm that extends the standard spiking deconvolution to mixed-phase avelet is presented. he optimal output spike has a time delay, determined from the projection matrix, and forms an optimal time-delay spiking deconvolution. he application of this algorithm to physical model measurements shos that this technique is efficient for the case hen the avelet is ell determined. INRODUCION In seismic data processing, spiking deconvolution is routinely applied to compress the effective source avelet contained in the seismic traces to improve temporal resolution. For the case of a normal incidence plane ave (and neglecting multiples), a seismic trace, the seismic recorded seismogram x( is a convolution beteen the primary reflectivity r( and the seismic avelet (, i.e. x( = r( ( (1) here " " denotes convolution (Yilmaz, 1987). In equation (1) noise is not considered. he goal of the spiking deconvolution process is to recover r( by removing the effects of ( from x(. If a filter operator f( ere defined such that convolution of f( ith the knon seismogram x( produces an estimate r ˆ(, then By substituting equation () into equation (1), e have therefore, in order to have a best estimate of r(, e need r ˆ( = x( * f ( () r ˆ( = r( * ( * f ( (3) δ ( t ) = ( * f ( ) (4) t here δ ( represents the zero time delay delta function, i.e. δ (=1 hen t= and δ(= hen t. hen the avelet ( is ell estimated, the problem of deconvolution defined in equation () becomes that of solving equation (4) to obtain the inverse filter operator f(. In real applications, both ( and f( can be represented as discrete forms ith discrete samples l and n, respectively, and equation (4) can be ritten in a matrix form (e.g. Hatton et al., 1988)

2 Yao, Margrave, and Gallant here F = Y (5) = m 1 m n and F = (f 1, f,...f n-1 ), Y = ) ( y, y1,..., y m 1 here m=n+l. Generally, equation (5) does not have an exact solution because m is alays larger than n and ( is bandlimited. he inverse filter, F, hich satisfies equation (5) in the least-squares sense, is ritten in the form of F = ( 1 ) is called a iener filter. he operator F corresponding to a zero time-delay spike delta function forms the standard spiking deconvolution operator, hich is routinely applied to seismic data processing. It is ell knon that standard spiking deconvolution orks ell hen the avelet is minimum phase (e.g. Yilmaz, 1987). he physical explanation of this design is that the distribution of energy of minimum phase avelet is concentrated at the beginning. herefore, the standard spiking deconvolution may not ork optimally if the avelet is not a minimum phase. For example, if the avelet is a maximum phase, the desired output should be a spike ith a maximum phase delay, i.e. (e.g. Karl, 1989) Y = δ ) Y max ( = (,,...,, y m 1 As increasing knoledge of the mixed delay nature of seismic avelets is obtained, more ork is done on the spiking of mixed delay avelets (e.g. Ziolkoski, 197; Eisner & Hampson, 199). In this paper, e present a simpler algorithm for the case that the mixed phase avelet is knon. his method can be combined ith methods of seismic avelet estimation technique for a complete deconvolution. he example of spiking deconvolution of the physical model data illuminates the efficiency of our algorithm.. (6) (7)

3 OPIMAL IME DELAY SPIKE DECONVOLUION Equation (6), hich is derived by a least squares method, often gives an impression that the inverse filter operator obtained from this equation is alays the best because object function F Y is minimum. Hoever, this impression is sometime misleading. aking a thought experiment as an example: assume e have a maximum phase avelet and solve equation (6) for the desired outputs as the zeroand maximum-lag spikes, respectively. hen, comparing the values of F Y for these to different cases, e ill find that even though in each case the object function is minimum, the values are different. herefore, as Ford (1978) noticed, in spiking deconvolution, the object function depends on the different delays of the desired output spike. he optimal time delay spike deconvolution is the one here the object function is the smallest among the all the possible time delays (Robinson and reitel, 198). Substituting equation (6) into (5), the predicted output ith F is given by Defining Yˆ = ( 1 ) Y. (8) P = ( as a projection matrix, this matrix shos ho the designed output Y can be mapped into the predicted output. Since the designed output of spiking deconvolution is a spike ith a time delay, the predicted output is actually one of the column vectors in the projection matrix that corresponds to the specific time delay. For example, if the time delay is the j-th sample in Y, then is the j-th column in the projection matrix. herefore, the matrix P gives total information for all possible effects of the deconvolution. Ideally, if the matrix P is an identity matrix, then Yˆ is the same as Y. Hoever, P is often not the identity. herefore, e need to find the one column vector that is most close to a spike, hich is equivalent to finding the one that makes the objective function minimal. he optimal time delay can be obtained by visual inspection of the projection matrix. It can be also performed by minimizing the folloing function φ( ) 1 Φ j [ p i, j / max( pi, j )], i j± q i = (9), j = 1,,... n (1) here q is the resolution idth. If Yˆ is a spike at the j time sample delay, then σ j is zero. APPLICAION IN HE PHYSICAL MODEL MEASUREMEN he seismic measurement as carried out on the physical made of acrylics ith its volume of (length, idth, heigh mm 3 (figure 1). he velocity of

4 Yao, Margrave, and Gallant the material is 75 m/s for acoustic aves. Grooves ere cut at the bottom ith the idths increasing from 1 mm to 1 mm. he height of the grooves is 5mm. Nearly zero offset seismic data ere taken at the middle of the top along the line that is perpendicular to the grooves. Figure 1: Physical model for seismic experiments. Figure a is the image of the measured data set. It shos that the main signals are the aves reflected from the bottom of the model and the tops of the groves. he traces from 4 to 8 are plotted in figure b. a b ime (ms) ime (ms) Distance (mm) race numbers Figure. he image of the measured data set; the plots of traces from 4 to 8 of the measured data. he source avelet as estimated from reflections at the bottom on the left is shon in figure 3.

5 Figure 3. he estimated avelet from reflections. he image of project matrix is shon in figure 4a, hich gives all the information of the possible effects of different time lag spike deconvolutions. Figure 4b is the plots of the first columns of the matrix as the traces display. From the figure e can see that the standard spiking deconvolution does not ork very ell in this particular case. Ro index ime delay (in samples) ime (ms) ime (in samples) Column index Figure 4.. he image of the projection matrix and first columns plotted as the seismic traces. Figure 5 is the plot of the objective function Φ, hich shos that the optimal time delay is about 4 samples time delay. he results of the avelet deconvolution ith zero- and 4- sample delays are shon in figure 6 indicating that the 4 samples time delay operator produces more spiking result. he zero- and 4 time samples delay spiking deconvolution operators are applied to the measured data set, respectively and the results are shon images in figure 7 and their parts of the traces are in figure 8.

6 Yao, Margrave, and Gallant Column index Figure 5. he plot of Φ versus the lag of the desired output spike. ime (in samples) ime (in samples) Figure 6. he results of the avelet deconvolution ith zero lag and 4 time sample delays spiking operators. ime (ms) Distance (mm) Distance (mm) Figure 7. he images of the measured data after deconvolution ith zero- and 4 time samples time delay spiking deconvolution operators.

7 ime (ms) ime (ms) race number race number Figure 8. he traces plots (from 4 to 8) of the measured data after spiking deconvolution ith zero- and 4 time samples time delay spiking deconvolution operators. DISCUSSION AND CONCLUSIONS he projection matrix provides complete information about the possible effects of different time lag spiking deconvolutions. he time delay determined via the project ion matrix leads to an optimal time-delay spiking deconvolution. he application of this technique to a physical model measurement shos that, if source avelet is ell knon, this technique is effective. Based on the same idea, finding the smallest value of the function Φ(, Ford (1978) developed an algorithm using z-transform and Robinson and reitel (198) developed an algorithm by comparing the results of all possible time-day spiking deconvolutions. Hoever, ours is much simpler than these algorithms. Because the process of the calculation in our algorithm is exactly the same as iener filter, our algorithm should have all of the advantages of a iener filter. It should mentioned that the inverse of ( ) may not exist and therefore, the general inverse may need to be applied. ACKNOLEDGEMEN e thank the sponsors of the CREES Project for their financial support and Larry R. Lines for discussions. REFERENCES Ford,.., 1978, Optimal mixed delay spiking filters: Geophysics, 43, Karl, J.H., 1989, An introduction to digital signal processing: Academic Press, INC. Robinson, E.A. and reitel, S., 198, Geophysical signal analysis: Prentice-Hall, Engleood Cliffs, N.J. alden, A.., 1985, Non-Gaussian reflectivity entropy and deconvolution: Geoexpl., 16, 1-35.Eisner, E. and Hampson, G., 199. Deconvolution into minimum and maximum phase components. Geophysics, 55, Yilmaz, O., 1987, Seismic data processing, Society of Exploration Geophysics. Ziolkoski, A., 197, A method for calculating the output pressure aveform from an air gun: Geophys. J. Roy. Astr. Soc., 1,