Introduction to earth modeling in depth
Seismic representation of an earth model in depth usually is described by two sets of parameters — layer velocities and reflector geometries. Depth migration is the ultimate tool for delineation of reflector geometries. If layer velocities are determined accurately, reflector geometries can be recovered by iterative depth migration (2-D poststack depth migration). Difficulties in estimating layer velocities with a required level of accuracy make the earth model estimation a challenging task for the geophysicist.
In this chapter, first, we shall discuss practical inversion methods to estimate layer velocities (models with horizontal layers, model with low-relief structure, and model with complex overburden structure). Next, by using appropriate combinations of inversion methods, we shall develop strategies to build an earth model (model building). Finally, we shall use residual moveout analysis and reflection tomography to update an initial model (model updating) and thus obtain a final model that can be used to image the subsurface in depth as discussed in earth imaging in depth. By being equipped with methods and strategies for interval velocity estimation, model building, and updating, and depth migration, we shall then study in structural inversion complete case studies for earth modeling and imaging in depth.
Nearly all of the practical methods of layer velocity estimation are based on ray theory, and more specifically, on inversion of seismic traveltimes. Velocity estimation methods include Dix conversion of rms velocities, inversion of stacking velocities, coherency inversion, and analysis of image gathers from prestack depth migration.
Velocity variations within the earth may be characterized in two ways — structure-dependent and structure-independent. A structure-dependent earth model comprises geological formations with interfaces that coincide with distinct velocity contrasts. We encounter structure-dependent earth models in areas with extensional and compressional tectonics, and especially in areas with salt and overthrust tectonics. A structure-dependent earth model usually requires a layer-by-layer estimation of layer velocities and delineation of reflector geometries that coincide with the layer boundaries themselves.
A structure-independent earth model comprises geologic formations with interfaces that do not necessarily coincide with distinct velocity contrast. Structure-independent earth models often are associated with low-relief structures or stratigraphic plays involving a depositional sequence with facies changes. A structure-independent earth model may be estimated initially by Dix conversion of rms velocities without requiring a layer-by-layer analysis.
The simplest method for estimating layer velocities is Dix conversion of rms velocities . The method requires the rms velocities associated with the layer boundaries that are included in the earth model to be constructed. The rms velocities ideally are estimated by prestack time migration (prestack time migration). Alternatively, a smoothly varying form of stacking velocities estimated from dip-moveout corrected data may be a reasonable substitute for rms velocities (dip-moveout correction in practice). Less desirably, stacking velocities themselves with a fair degree of smoothing applied may be used in lieu of rms velocities. The Dix conversion formula (Section J.4) is valid for horizontally layered earth models with constant layer velocities and small offsets. For an earth model with dipping layer boundaries and layer velocities with vertical and lateral variations, more accurate methods are required such as stacking velocity inversion, coherency inversion and image-gather analysis.
Stacking velocity inversion (Thorson et al., 1985) requires time horizons picked from unmigrated CMP-stacked data and stacking velocities at analysis locations. Assume that a velocity-depth model already has been estimated for the first n − 1 layers, and that we want to estimate the layer velocity for the nth layer below a CMP location. For a trial constant velocity assigned to the nth layer, the method involves normal-incidence time-to-depth conversion of the time horizon associated with the base of the nth layer, then modeling of the nonzero-offset traveltimes associated with the CMP reflection event that corresponds to the base of the nth layer and determining the moveout velocity by fitting a hyperbola to the modeled traveltime trajectory. This procedure is repeated for a range of constant trial velocities, and the velocity that yields the minimum discrepancy between the actual stacking velocity and the modeled moveout velocity is assigned to the nth layer below the CMP location where the stacking velocity inversion is being performed.
Coherency inversion  also requires time horizons picked from unmigrated CMP-stacked data. However, in lieu of stacking velocities as for stacking velocity inversion, coherency inversion requires analyzing CMP gathers themselves. Again, assume that a velocity-depth model already has been estimated for the first n − 1 layers, and that we want to estimate the layer velocity for the nth layer below a CMP location. For a trial constant velocity assigned to the nth layer, coherency inversion involves normal-incidence time-to-depth conversion of the time horizon associated with the base of the nth layer, then modeling of the nonzero-offset traveltimes associated with the CMP reflection event that corresponds to the base of the nth layer, and computing the semblance (velocity analysis) within a CMP data window that follows the modeled traveltime trajectory. This procedure is repeated for a range of constant trial velocities and the velocity that yields the highest semblance value is assigned to the nth layer below the CMP location where the coherency inversion is being performed.
Time horizons used in normal-incidence time-to-depth conversion as part of the stacking velocity inversion and coherency inversion procedures are picked from unmigrated CMP-stacked data. Alternatively, time horizons interpreted from the time-migrated volume of data can be unmigrated to obtain the time horizons equivalent to the time horizons picked from the unmigrated data. We circumvent the picking of prestack reflection traveltimes in coherency inversion by measuring the discrepancy between the modeled and actual traveltimes by way of semblance. Similarly, we avoid the picking of prestack reflection traveltimes in stacking velocity inversion by measuring the discrepancy between the modeled and actual stacking velocities.
Stacking velocity inversion and coherency inversion both take into account vertical velocity gradients which may be available from sonic logs. The methods also honor ray bending at layer boundaries. While Dix conversion assumes a hyperbolic moveout for the reflection event that corresponds to the base of the layer under consideration, both interval velocity estimate from coherency inversion and stacking velocity inversion are based on nonhyperbolic CMP traveltime modeling.
Both stacking velocity inversion and coherency inversion can be considered accurate for velocity-depth models with smoothly varying reflector geometries and lateral velocity variations greater than the effective cable length associated with the layer boundary under consideration. As for conventional stacking velocity estimation (velocity analysis), the accuracy in interval velocity estimation from Dix conversion, stacking velocity inversion, and coherency inversion are all influenced by the reflector depth, magnitude of the velocity, and the cable length. Specifically, the deeper the reflector, the larger the layer velocity above, and the shorter the cable length, the less accurate is the interval velocity estimate.
To estimate, update and verify velocity-depth models for targets beneath complex overburden structures, such as those associated with overthrust and salt tectonics, ultimately, we have to do image-gather analysis   , . An image gather is the output from prestack depth migration and is a true (common depth-point) CDP gather at a surface location. Stacking of image gathers yields an earth image in depth. If the velocity-depth model is correct, then events on an image gather are flat. In this respect, an image gather can be considered like a moveout-corrected CMP gather, except the vertical axis on an image gather is in depth.
An event on an image gather with a moveout indicates an erroneously too low or too high velocity. By examining a panel of image gathers from the same location but with different constant trial velocities for the layer under consideration, one can pick the velocity that yields a flat event and assign it as the velocity of the layer above. Image gathers also can be used to make residual corrections to velocity estimates at analysis locations. This normally is done by first converting the gather to the time domain, performing residual moveout velocity analysis, and converting back to the depth domain. The resulting residual correction should favorably improve the power of the stack obtained from image gathers and yield an updated velocity-depth model. The model updating based on the image-gather analysis usually is repeated until residual moveouts on image gathers are reduced to a minimum  .
In the following sections, we shall review the velocity estimation methods described above by using three synthetic model data sets with varying complexity. These are models with horizontal layers, low-relief structure and complex structure. Emphasis will be on practical aspects of the velocity estimation techniques.
- Dix, 1955, Dix, C. H., 1955, Seismic velocities from surface measurements: Geophysics, 20, 68–86.
- Landa et al., 1988, Landa, E., Kosloff, D., Keydar, S., Koren, Z., and Reshef, M., 1988, A method for determination of velocity and depth from seismic data: Geophys. Prosp., 36, 223–243.
- Faye and Jeannaut, 1986, Faye, J-P. and Jeannaut, J-P., 1986, Prestack migration velocities from focusing depth analysis: 56th Ann. Internat. Mtg., Soc. Expl. Geophys., Expanded Abstracts, 438–440.
- Reshef and Kessler, 1989, Reshef, M. and Kessler, D., 1989, Practical implementation of three-dimensional poststack depth migration: Geophysics, 54, 309–318.
- Reshef, 1997, Reshef, M., 1997, The use of 3-D prestack depth imaging to estimate layer velocities and reflector positions: Geophysics, 62, 206–210.
- Reshef, 2001, Reshef, M., 2001, Some aspects of interval velocity analysis using 3-D depth migrated gathers: Geophysics (Scheduled for publication in Jan.-Feb. 2001 issue).
- Al-Yahya, 1989, Al-Yahya, K., 1989, Velocity analysis by iterative profile migration: Geophysics, 54, 718–729.
- Deregowski, 1990, Deregowski, 1990, Common-offset migrations and velocity analysis: First Break, 8, 225–234.
- Inversion methods for data modeling
- Inversion procedures for earth modeling
- Velocity-depth ambiguity
- Model representation and visualization