P125 Method for Calibrating Seismic Imaging Velocities

Transcription

1 P125 Method for Calibrating Seismic Imaging Velocities P.C. Docherty* (Fairfield Industries Inc.) SUMMARY Anisotropy can cause large depth errors in images obtained with isotropic velocity analysis and depth migration. Typically, the errors become apparent only where the seismic image intersects a well location, that is, where true depths are measurable. Sparsity of well information can lead to ambiguity - What criterion should be used to adjust the image at locations far from the wells? and risk - Is the prospect real? Here, the calibration problem is posed in terms of a three-dimensional mis-tie function, that is, in terms of the difference in depth between uncalibrated and calibrated images. By means of regularization, a mis-tie volume is obtained that produces an acceptable fit at the wells but does not introduce unnecessary structure into the calibrated image. Differentiation of mis-tie with respect to time yields a calibration velocity.

2 Introduction Isotropic velocity analysis followed by isotropic depth migration can result in large errors in seismic depth images where the earth is anisotropic. In the Gulf of Mexico, for example, it is not unusual to observe mis-ties of 1,000 ft, or more, at depths below 10,000 ft. As a consequence, isotropic images must frequently be adjusted to tie more accurate data at wells. A common approach uses the isotropic velocities to convert the image to time, followed by conversion back to depth with a calibration velocity. The problem is how best to estimate the calibration velocity; in particular, how to deduce it far from the nearest well. Robein (2003) describes many of the techniques in use today, while a recent paper by Shumaker et al. (2007) addresses the issue where salt is present. Because of the sparsity of well locations, calibrations are inevitably ambiguous. A clear pitfall to avoid is the creation of a prospect that is not in the isotropic image and is not supported by the wells. Given the isotropic velocities, I pose the problem in terms of a three-dimensional mis-tie function, that is, in terms of the depth difference between isotropic and calibrated images. Data for the problem are vertical traveltimes and corresponding depths, either from check shots or from interpreted formation tops. By regularizing mis-tie directly, I seek the most featureless update to the isotropic image that fits the data at the wells. The fit is not exact but takes into account uncertainty in the data. The method produces a mis-tie volume which can be used to adjust the isotropic image. Equivalently, the time derivative of mis-tie yields a calibration velocity to be applied as described above. For the case of polar anisotropy, the calibration velocity can be viewed as an estimate of the velocity of vertically propagating waves in the earth. As such, it is a useful starting point for further estimation of anisotropic earth parameters. Definition of mis-tie Let V 0 (x,y,t) be the (unknown) vertical wave speed in the earth and V iso (x,y,t) the wave speed obtained from isotropic velocity analysis. Starting at surface location (x,y,0), propagation along a vertical path for time t gives t z(x,y,t) V 0 (x,y, )d (1) 0 as the depth reached in the earth, and t z iso (x,y,t) V iso (x,y, )d (2) 0 as the corresponding depth in the isotropic model. The function m(x,y,t) is introduced to describe the discrepancy, or mis-tie: m(x,y,t) z iso ( x,y,t) z(x,y,t). (3) Differentiation with respect to time produces m( x,y,t) V iso (x,y,t) V t 0 ( x,y,t). (4) Below, I describe a method for estimating m(x,y,t) given its value at a sparse distribution of points. Since V iso is assumed known, equation (4) then gives an estimate of V 0. The estimate is the calibration velocity. Typically, the calibration velocity inherits high-frequency lateral variations from V iso, while the mis-tie derivative supplies a long wavelength adjustment. (See also MacKay et al., 2006.) Note that where V iso is determined using short-offset data, an estimate of the anisotropic

3 parameter follows from equation (4) and Thomsen s (1986) relationship V (x,y,t) V (x,y,t)(1 (x,y,t)) iso 0. Method Given, at each of M locations (x i,y i,,z i ), the traveltime t i for vertical propagation from the earth s surface, define the quantity d i z iso (x i,y i,t i ) z i. (5) Next introduce a vector m of length N to parameterize mis-tie throughout the volume (x,y,t). (Elements of m might, for example, be values of mis-tie on a regular grid.) Interpolating to obtain the mis-tie at (x i,y i,t i ) and equating to d i produces d i a i T m, i 1,,M. (6) Here, a i is a vector of interpolation coefficients. In matrix notation equation (6) becomes d A m, (7) ~ where a T i is now the ith row of matrix A. ~ All of the measured quantities in equation (5), namely x i, y i, z i and t i, are subject to measurement error, the aggregate effect of which is expressed as an uncertainty, i, in the ith datum d i. The uncertainties are inserted as weights into equation (7) using the matrix W diag(1 / ~ i ) as W d W A m. (8) ~ ~ ~ In the calibration problem, it is often the case that the number of unknowns, N, exceeds the number of data, M, so that a unique solution to equation (8) does not exist. A key issue is therefore one of regularization, that is, of the many possible solutions which one should be sought. It is an important issue since misfit is a map between isotropic and calibrated depth images. The solution I seek is the one that changes the isotropic image in the most cautious way while attaining a certain goodness of fit to the data. I choose most cautious to mean smoothest; thus, following Constable et al. (1987), I minimize the objective function O 2 D ~ m 2 W ~ m W ~ d 2, (9) where is a regularization parameter and D ~ is an N x N matrix of second derivatives. (Because it is particularly straightforward, the development here is in the time domain; strictly, though, the second derivatives should be applied in the isotropic depth domain.) Minimization of O yields the following linear system to be solved for m ( 2 D ~ T D~ (W ~ ) T (W ~ ))m (W ~ ) T W ~ d. (10) The parameter controls the trade-off between smoothness of the solution and fit to the data, and a suitable value for must be determined. Constable et al. (ref. cited) describe the procedure. First, with i set equal to the standard deviation of the ith datum, recognize the second term on the right hand side of equation (9) as the familiar quantity chi-square : 2 W A m W d 2. (11) ~ ~ ~

4 Next, assume independently random, zero mean, Gaussian errors in the data. The expected value of 2 is then M, the number of data. Finally, solve equation (10) repeatedly with a series of values for. The solution that yields a 2 closest to M is chosen. This is the smoothest possible solution for the target goodness of fit. Example The following example is from the Vermilion region of the Gulf of Mexico. Check shots from 59 wells were downloaded from the Minerals Management Service (MMS) web site. Figure 1 indicates the locations of the wells. Mis-ties were computed using a velocity model obtained from isotropic migration velocity analysis and tomography. The check shots yielded a total of 1846 mis-ties (M in equation (6)). All of the mis-ties are plotted as a function of time in Figure 2. Positive mis-tie indicates that the corresponding depth in an isotropic image will be greater than true depth in the earth (equation (3)). This is clearly the case for times greater than about 1500 ms and is characteristic of the occurrence of anisotropic, possibly pressured, shales. Interestingly, shallow times seem to indicate the reverse trend, with mis-tie becoming increasingly negative down to about 500 ms. Note that within a zone where the earth is isotropic the mis-tie trend will tend to level out in the plot since V iso (x,y,t) and V 0 (x,y,t) should be the same in an interval of isotropy. (See Tsvankin and Thomsen, Polar anisotropy is assumed.) I use this observation to estimate uncertainty in the mis-tie values, as follows. In Figure 2, I interpret the zone 500 ms t 1000 ms to be approximately isotropic across the study area. In addition, I assume that the shallow earth (t 500 ms) is approximately laterally homogeneous. The effect of shallow anisotropy is then a bulk shift of the misties below 500 ms, so that scatter in the zone of isotropy can be attributed to measurement errors in the check shot data. For 500 ms t 1000 ms the mis-ties in Figure 2 have a standard deviation of 70 ft. I used this value for the elements of the diagonal matrix W ~ in equation (8); that is, i = 70, all i. Mis-tie was parameterized on a three dimensional grid, with dx = dy = 3960 ft, and dt = 25 ms. Solving equation (10) repeatedly with trial values of the regularization parameter, the chi-squares shown in Figure 3 are observed. The value = 3500 gives a chi-square closest to the target and the mis-tie vector m obtained from this inversion is taken to be the solution. Differentiating m and inserting into equation (4) gives the calibration velocity. For comparison with Figure 2, Figure 4 shows the result of replacing V iso with the calibration velocity in the mis-tie calculation. The figure is useful as a prediction of mis-ties in a calibrated image. The mean value of the mis-ties shown is -4 ft; standard deviation is 69 ft. Samples of the calibration velocity at four well locations are plotted in Figure 5. Also shown in the plots are the corresponding

5 isotropic velocities V iso, and interval velocities computed directly from the check shot data. Consistent with Figure 2, notice the slower isotropic velocities at early times, followed by an intermediate zone of better agreement. A velocity inversion occurs at about 1300 ms; isotropic velocities at later times are too fast. Finally, Figure 6 shows Thomsen s parameter along the west to east line through well VR_99_1. Conclusions Of the many calibrations that fit the data, a method has been described that produces the smoothest, or most featureless, update to the uncalibrated image. Key aspects of the method are: (i) the problem is posed in terms of mis-tie; (ii) data for the problem are mis-ties at wells; (iii) uncertainties in the data are taken into account; (iv) regularization is applied directly to a mis-tie volume; (v) calibration velocity is obtained as the difference between isotropic velocity and the time derivative of the regularized mis-tie. References Constable, S. C., Parker, R. L., and Constable, C. G., 1987, Occam s inversion: A practical algorithm for generating smooth models from electromagnetic sounding data: Geophysics, 52,

6 P Ann. Robein, E., 2003, Velocities, time-imaging and depth-imaging in reflection seismics: Principles and Methods: EAGE. MacKay, S.,. Jimenez, H. R., Romero, J. S. M., and Morford, M., 2006, Calibrating prestack depth migration volumes with th well control: 76P Internat. Mtg., Soc. Expl. Geophys., Expanded Abstracts, Shumaker, N., Lindsay, R., and Ogilvie, J., 2007, Depth-calibrating seismic data in the presence of allochthonous salt: The Leading Edge, 26, Thomsen, L., 1986, Weak elastic anisotropy: Geophysics, 52, Tsvankin, I., and Thomsen, L., 1994, Nonhyperbolic reflection moveout in anisotropic media: Geophysics, 59,