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# Seismic imaging and inversion

The ability of seismic reflection technology to image subsurface targets is possible largely through the geometry of sources and receivers. A method similar to triangulation is used to place reflections in their correct locations with (more-or-less) correct amplitudes, which can then be interpreted. The amplitudes are indicative of relative changes in impedance, and the seismic volume can be processed to yield impedances between the reflecting boundaries.

## Stacking and interval velocities

The geometry of sources and receivers in a typical reflection seismic survey yields a number of seismic traces with common midpoints or central bins for stacking. These traces were recorded at different offset distances, and the travel times for seismic waves traveling to and from a given reflecting horizon varies with that distance (Fig. 1). If the overburden through which the seismic waves pass is of constant velocity, then the time-variation with distance is a simple application of Pythagorean geometry, and the shape of the reflector on a seismic “gather” of traces is hyperbolic.[1] As the overburden velocity structure becomes more complex, the shape is less perfectly hyperbolic, but most standard processing routines still assume a hyperbolic “moveout” of each reflector. An analysis is then made of selected seismic gathers to establish the ideal moveout required to “flatten” each reflection in the gather. This moveout is expressed in terms of a velocity and represents the seismic velocity that the entire overburden, down to the point of each particular reflection, would have to result in the idealized hyperbolic shape observed. This velocity analysis is usually conducted by examining the semblance (or some other measure of similarity) across all the traces, within a moving time window, and for all reasonable stacking velocities (Fig. 2). The seismic processor then selects the best set of velocities to use at a variety of reflectors and constructs a velocity function of two-way travel time. These velocity functions are interpolated, both spatially and in two-way travel time, and all seismic gathers are then “corrected for normal moveout” using them. Each moveout-corrected gather is then summed or “stacked” after eliminating (“muting”) those portions of the traces that have been highly distorted by the moveout process.

The final stacked traces exhibit a considerably better signal-to-noise ratio than the individual seismic traces recorded at zero-offset, but the improvement is better than just the square root of the number of traces that might be expected because of the systematic removal of coherent noise. Much of the noise present in individual seismic traces is not random but represents unwanted events, including surface waves or ground roll and multiply-reflected arrivals from shallow horizons; both of these can usually be identified in the velocity analysis and selected against. The stacking process then removes most of the unaligned energy associated with these types of coherent noise.

The velocities obtained in the analysis previously described are not true seismic velocities—they are simply those velocities which provided the best stack of the data and may or may not truly reflect the actual root mean square (RMS) velocities that approximate the accumulated effect of the stack of layers above the reflector (the name RMS is derived from the arithmetic used to define this overall velocity). If we assume, however, that the stacking velocities do in fact provide a reasonable approximation to the aggregate effect of the layers overlying each reflector, the actual velocities of each layer can be obtained through a set of equations because of Dix[2] (see Fig. 3[3]). These “interval” or “Dix” velocities can sometimes be used to characterize the rocks in each layer and may be sufficiently precise to enable differentiation of gross rock types, although the errors associated with interval velocities can be fairly large.

## Time and depth migration

Even after accounting for normal moveout and stacking the gathered traces to a common zero-offset equivalent set of traces, the locations of the reflected events are not usually correct because of lateral variations in velocity and dipping interfaces. Fig. 4 shows a simple 2D example of a dipping interface from which we observe a reflection. Each seismic trace is plotted directly beneath the respective midpoint or bin location used for stacking, but the reflection from any given interface may not have come from that location. The events have been shifted downdip to deeper locations, and the dip of the interface is less steep. To correct for this shift, the seismic processor “migrates” each sample to its appropriate position. In the simple case shown in the figure, we need only know the velocity of the one overlying layer, but in more realistic cases, the velocity function may be quite complex and is derived through a trial-and-error approach guided by statistical tests of lateral coherence, knowledge of expected geologic structure, and other constraints such as interval velocities and well log data. The problem can become quite difficult in complicated 3D data sets, and software has been developed to manage and visualize the velocity volume. The result of this model-driven 3D migration can be somewhat subjective, and, although it is possible to create structures where none really exist through this process, migration should be performed on all seismic data sets for appropriate imaging of structures. 3D migration can drastically improve the imaging of virtually any target by improving the accuracy of the spatial location of various features and by sharpening the image itself, allowing finer resolution than either migrated 2D data or unmigrated 3D data[1] (see Fig. 5). The results can occasionally be quite dramatic for interpretation; for example, a locally high feature on an unmigrated data set may move to a significantly different map location after migration. In general, the more dramatic the structure, or the larger the velocity contrasts between layers, the more important 3D migration is for proper imaging.

The process of imaging through modeling the velocity structure is a form of inversion[4] of seismic data, and the term inversion is often used to imply building a velocity model which is iteratively improved until it and the seismic data are optimally in agreement.[5] Improvements in imaging are continually being made, and research in this area is one of the most fruitful in reservoir and exploration geophysics.[6] The current methods of migration involve operating in two-way travel time (as previously described), or in depth (using the model velocities to convert from travel time to depth), and either method can be performed prestack or post-stack.[7] In addition, there have been a number of shortcuts developed over the years to provide reasonable results in a short time; all of the methods are quite computation-intensive, and the technology has benefited greatly from improved computing capacity. The finest results can usually be obtained from prestack depth migration, in which each sample of each trace, prior to gather, is migrated using the velocity function to a new location then stacked and compared with various tests for model improvement; the model is changed, and the process is repeated.

In areas where it is important to image beneath layers of high velocity contrasts, such as beneath salt bodies, prestack depth migration is required. The example[8] shown in Fig. 6 shows the possible improvements that can be obtained using prestack depth migration. The process required to create the final stack is as follows: a velocity model is first constructed through the water and sediment layers to the top of salt, and prestack depth migration is used to optimize that model. Then, the salt velocity (which is fairly constant and typically much higher than the surrounding sediments, resulting in severe bending of seismic ray paths) is used for the half-space beneath the top of salt. The reflections from the base of the salt body then appear, although the underlying sediments are very poorly imaged. Finally, the velocity model within these sediments is modified until an acceptable image is obtained.

## Trace inversion for impedance

Seismic reflections at zero offset result from contrasts in acoustic impedance, involving just the P-wave velocity and density of the layers at the interface. If we can identify the seismic wavelet that propagated through the earth and reflected from the layer contrasts, we can then remove the effect of that wavelet and obtain a series of reflection coefficients at the interfaces. Then, we can simply integrate these reflection coefficients and determine the acoustic impedance in the layers between the interfaces. This “inversion” procedure leads us to a seismic volume that portrays layer properties (in terms of impedance) rather than interface characteristics, and assumes that the reflecting horizons have already been properly migrated to their appropriate positions.[9] (Note that in the strict sense, the inversion described for migration or imaging and the inversion described in this section have equivalent goals: they both attempt to model the velocity and/or density structure of the earth that best fits and images the seismic data set. However, the approaches used are quite different, and the two processes should not be confused. Future research developments may tend to blur this distinction by integrating appropriate aspects of both techniques into one method.)

### Acoustic impedance

If the seismic data were noise-free and contained all frequencies, from zero frequency (infinite wavelength) to very high frequencies (short wavelengths), the solution should be unique, but seismic data are noisy and band-limited and do not contain the very lowest frequencies nor the higher frequencies that are often of interest. A number of methods have been developed to overcome these shortcomings, including a “sparse-spike” inversion,[10] in which the trade-off between the number of reflecting horizons and “noise” is chosen by the investigator (a technique that simultaneously solves for the “background” velocity trend and the impedance contrasts[11]) and statistical or neural-network techniques that relate seismic features to properties inferred from borehole data.[12] To a greater or lesser degree, these techniques rely on borehole sonic logs or on other velocity information or assumptions to incorporate long-wavelength velocity models (the background velocity trend). In general, a calibrated and competently processed inversion volume can be of considerable use to the interpreter or the engineer, providing insight to layer properties and continuity, which may not be apparent from the traditional reflection-seismic display; in particular, the thinner beds are usually more distinctly identified through removal of wavelet tuning (interference of reflections from the top and bottom of the bed) and subtle changes in impedance that are not easily recognized in the reflection image that can be seen in the inverted volume. Because the inversion process results in volume properties, rather than interface properties, it is possible to isolate and image individual bodies within certain impedance ranges. An example of the results of body-capture after a sparse-spike inversion, intended to identify hydrocarbon reservoirs, is shown in Fig. 7.

In general, it is appropriate to invert only true zero-offset seismic data for acoustic impedance because the nonzero offsets are influenced by other parameters, notably the ratio between the P-wave velocity and the S-wave velocity (or, alternatively, Poisson’s ratio; see Seismic attributes for reservoir studies). Yet typical seismic data has been processed by stacking all appropriate offsets after correcting for normal moveout and muting, and the amplitude of each reflection represents a sort of average amplitude over all of the offsets used. In many cases, this distinction is not important because the amplitude normally decays slightly with offset (after routine correction for geometric spreading) and affects all stacked samples similarly, but for many cases, and especially those of most interest, the amplitudes vary with offset. Inverting a seismic section containing stacked data does not always yield a true acoustic impedance volume. (Note: the term “acoustic” refers to compressional-wave effects only, and acoustic models assume that the material does not propagate shear waves or that shear waves are not of any significance in wave transmission.) In practice, this is true for seismic compressional waves at normal incidence but is not valid for compressional waves at nonnormal incidence in a solid material because of partial conversion to reflected and refracted shear waves. The term “elastic” is used to describe models incorporating compressional and shear effects.) Thus, if we interpret a stacked seismic volume that has been inverted for acoustic impedance, we have implicitly assumed that the offsets used in stacking were small and/or that the offset-dependence of amplitudes is negligible. In the cases where these assumptions are not true, we must recognize that the values of acoustic impedance resulting from the inversion process are not precise; in fact, the disagreement of the acoustic inversion results, with a model based on well logs, is often an indication of AVO effects and can be used as an exploration tool.

### Elastic impedance

In order to separate the acoustic model (compressional-wave only) from the elastic model (including shear effects), the inversion process can be conducted on two or three different stacked seismic volumes, each composed of traces that resulted from stacking a different range of offsets. The volume created from traces in the near-offset range (or a volume made by extrapolating the AVO behavior to zero offset at each sample) is inverted to obtain the acoustic impedance volume. A volume created from traces in the far-offset range is inverted to obtain a new impedance volume called the “elastic impedance.”[3] The elastic impedance volume includes the effects of the compressional impedance and the AVO behavior resulting from the Vp / Vs ratio; the two volumes can be interpreted jointly to obtain the desired fluid or lithology indicator sought. Just as in AVO studies, one can also try to obtain a three-parameter inversion, using three different offset ranges and, for example, solve for compressional/shear velocities and density. Converted-wave data can also be inverted for shear impedance.[13]

## Nomenclature

 Vp = P-wave velocity Vs = S-wave velocity

## References

1. Yilmaz, O. 2001. Seismic Data Analysis: Processing, Inversion, and Interpretation of Seismic Data. No. 10, Investigations in Geophysics, Soc. of Exploration Geophysicists, Tulsa.
2. Dix, C.H. 1955. Seismic Velocities from Surface Measurements. Geophysics 20 (1): 68. http://dx.doi.org/10.1190/1.1438126
3. Connolly, P. 1999. Elastic Impedance. The Leading Edge 18 (4): 438. http://dx.doi.org/10.1190/1.1438307
4. Treitel, S. and Lines, L. 2001. Past, Present, and Future of Geophysical Inversion—A New Millennium Analysis. Geophysics 66 (1): 21. http://dx.doi.org/10.1190/1.1444898
5. Weglein, A.B. and Stolt, R.H. 1999. Migration Inversion Revisited. The Leading Edge 18 (8): 950, 975. http://dx.doi.org/10.1190/1.1438416
6. Gray, S.H. 2001. Seismic Imaging. Geophysics 66 (1): 15. http://dx.doi.org/10.1190/1.1444892
7. Gray, S.H. et al. 2001. Seismic Migration Problems and Solutions. Geophysics 66 (5): 1622. http://dx.doi.org/10.1190/1.1487107
8. Liro, L. et al. 2000. Application of 3D Visualization to Integrated Geophysical and Geologic Model Building: A Prestack, Subsalt Depth Migration Project, Gulf of Mexico. The Leading Edge 19 (5): 466. http://dx.doi.org/10.1190/1.1438628
9. Oldenburg, D.W., Scheuer, T., and Levy, S. 1983. Recovery of the Acoustic Impedance from Reflection Seismograms. Geophysics 48 (10): 1318. http://dx.doi.org/10.1190/1.1441413
10. Debeye, H.W.J. and van Riel, P. 1990. LP-Norm Deconvolution. Geophysical Prospecting 38 (4): 381. http://dx.doi.org/10.1111/j.1365-2478.1990.tb01852.x
11. Cao, D. 1990. A Simultaneous Inversion for Background Velocity and Impedance Maps. Geophysics 55 (4): 458. http://dx.doi.org/10.1190/1.1442855
12. Hampson, D.P., Schuelke, J.S., and Quirein, J.A. 2001. Use of Multiattribute Transforms to Predict Log Properties from Seismic Data. Geophysics 66 (1): 220. http://dx.doi.org/10.1190/1.1444899
13. Duffaut, K. et al. 2000. Shear-Wave Elastic Impedance. The Leading Edge 19 (11): 1222. http://dx.doi.org/10.1190/1.1438510

## Noteworthy papers in OnePetro

Berkhout, A.J. Seismic Imaging Beyond Depth Migration. Presented at the 1999/1/1/.

Coléou, T., Allo, F., Bornard, R. et al. Petrophysical Seismic Inversion. Presented at the 2005/1/1/.

French, W.S., Pan, N., and Perkins, W.T. Practical Seismic Imaging. Presented at the 1988/1/1/.

Macdonald, C., Jackson, D.D., and Davis, P.M. Nonlinear Seismic Inversion. Presented at the 1984/1/1/.

Nur, A.M. Seismic Imaging in Enhanced Recovery. Presented at the 1982/1/1/. http://dx.doi.org/10.2118/10680-MS.

Pedersen, J.M., Vestergaard, P.D., and Zimmerman, T. Simulated Annealing-Based Seismic Inversion. Presented at the 1991/1/1/.

Versteeg, R.J. and Symes, W.W. Geologic Constraints On Seismic Inversion. Presented at the 1994/1/1/.