# Inversion of seismic data

Series Investigations in Geophysics Öz Yilmaz http://dx.doi.org/10.1190/1.9781560801580 ISBN 978-1-56080-094-1 SEG Online Store

A narrow meaning of seismic inversion — commonly referred to as trace inversion, is acoustic impedance estimation from a broad-band time-migrated CMP-stacked data. A broad meaning of seismic inversion — commonly referred to as elastic inversion, is the grand scheme of estimating elastic parameters directly from observed data. Nevertheless, in practice, applications of inversion methods can be grouped in two categories — data modeling and earth modeling. Much of what we do in seismic data processing described in fundamentals of signal processing through 3-D seismic exploration is based on data modeling.

Applications of seismic inversion for data modeling include deconvolution, refraction and residual statics corrections (velocity analysis and statics corrections) and the discrete Radon transform (noise and multiple attenuation). The discrete Radon transform is an excellent example to demonstrate the benefits of data modeling in seismic data processing. Consider a 2-D operator LT that corresponds to moveout correction to a CMP gather using a range of constant velocities and summing the trace amplitudes along the offset axis. (T stands for matrix transpose.) As a result, the data represented by the CMP gather is transformed from the offset space (offset versus two-way traveltime) to velocity space (velocity versus two-way zero-offset time). The gather in the output domain is called the velocity stack. The stack amplitudes on the velocity-stack gather exhibit smearing along the velocity axis. This is caused by discrete sampling along the offset axis and finite cable length. The operator LT alone does not account for these effects. Instead, we must use its generalized linear inverse (LTL)−1LT. Application of the operator LT is within the framework of conventional processing, whereas application of the operator (LTL)−1LT is within the framework of seismic inversion. The CMP gather can be reconstructed by applying inverse moveout correction and summing over the velocity axis. This inverse transformation is represented by the operator L. Reconstruction of the CMP gather from the velocity-stack gather is one example of data modeling. Data modeling using the velocity-stack gather computed by the processing operator LT does not faithfully restore the amplitudes of the original CMP gather, whereas data modeling using the inversion operator (LTL)−1LT does.

Just as there is a difference between processing and inversion in data modeling, also, there exists a difference between processing and inversion in earth modeling. The primary objective in processing is to obtain an earth model in time with an accompanying earth image in time — a time-migrated section or volume of data (Figure I-13). Representation of an earth model in time usually is in the form of a velocity field, which has to be smoothly varying both in time and space. Whereas the primary objective in inversion is to obtain an earth model in depth with an accompanying earth image in depth — a depth-migrated section or volume of data (Figure I-14). Representation of an earth model in depth usually is in the form of a detailed velocity-depth model, which can include layer boundaries with velocity contrast (Figure I-14).

Earth imaging in depth and earth modeling in depth are devoted to earth imaging and modeling in depth, respectively. Results of conventional processing of seismic data often are displayed in the form of an unmigrated (Figure I-15a) and migrated CMP-stacked section (Figure I-15b), with the vertical axis as time, which is different from the recording time of seismic wavefields. For unmigrated data, the vertical axis of the CMP-stacked section represents times of reflection events in the unmigrated position in the sub-surface. These event times are associated with normal-incidence raypaths from coincident source-receiver locations at the surface to reflectors in the subsurface and back. For migrated data, the vertical axis represents times of reflection events in the migrated position. These event times are associated with vertical-incidence raypaths from coincident source-receiver locations at the surface to reflectors in the subsurface and back. As long as there are no lateral velocity variations, seismic imaging of the subsurface can be achieved using time migration techniques and the result can be displayed in time. This time-migrated section can then be converted to depth along vertical raypaths.

When there are mild to moderate lateral velocity variations, time migration can still yield a reasonably accurate image of the subsurface. Nevertheless, depth conversion must be done along image rays to accommodate for the lateral mispositioning of the events as a result of time migration.

In the presence of strong to severe lateral velocity variations, however, time migration no longer is valid. Instead, seismic imaging of the subsurface must be done using depth migration techniques so as to properly account for lateral velocity variations and the result must be displayed in depth.

The depth-migrated section (Figure I-15c) can be considered a close representation of the structural cross-section of the subsurface only if the velocity-depth model is sufficiently accurate. In the example shown in Figure I-15, the picked horizons correspond to layer boundaries with significant velocity contrast. The zone of interest is base Zechstein (the red horizon) and the underlying Carboniferous sequence. The green horizon just below 2 km is the top Zechstein. This formation consists of two units of anhydrite-dolomite with a thickness of approximately 100 m — the shallow unit very close to top-Zechstein and concordant with it, and the deeper unit which manifests itself with a very complex geometry as seen in the migrated sections.

An earth model in depth usually is described by two sets of parameters — layer velocities and reflector geometries (Figure I-16). Practical methods to delineate reflector geometries described in earth imaging in depth, and to estimate layer velocities described in earth modeling in depth can be appropriately combined to construct earth models in depth from seismic data.

In practice, smoothness of earth models derived from processing means that we can make a straightray assumption and usually do not have to honor ray bending at layer boundaries. In contrast, detailed definition of earth models derived from inversion with a more stringent requirement in accuracy means that we do have to honor ray bending at layer boundaries and account for vertical and lateral velocity gradients within the layers themselves. Hence, to a large extent, processing can be automated, while inversion requires interpretive pause at each layer boundary.

There is a fundamental problem with inversion applied to earth modeling in depth — velocity-depth ambiguity. This means that an error in depth is indistinguishable from an error velocity. To resolve velocity-depth ambiguity as much as possible, one needs to do an independent estimate of layer velocities and reflector geometries using prestack data. As a result of velocity-depth ambiguity, an output from inversion is an estimated velocity-depth model with a measure of uncertainty in layer velocities and reflector geometries. It is now widely accepted in the industry that results of inversion are geologically plausable only when there is a sound interpretation effort put into the data analysis.

It is the limited accuracy in velocity estimation that has led to the acceptance of time sections to be the standard mode of display in seismic exploration. Facing the challenge of improving the accuracy in velocity estimation should make the depth sections increasingly more acceptable. Specifically, improving the accuracy means the ability to resolve detailed velocity variations in the vertical and lateral directions, associated with both structural and stratigraphic targets.

Earth modeling in depth usually involves implementation of an inversion procedure layer by layer starting from the top (Figure I-17). First, estimate a velocity field (the color-coded surface and the vertical cross-section) for the first layer, for instance, using 3-D coherency inversion. Then delineate the reflector geometry (the silver surface) associated with the base of the layer, for instance, using 3-D poststack depth migration (Figure I-17a). Next, estimate a velocity field for the second layer and delineate the reflector geometry associated with the base of the layer (Figure I-17b). Alternate between layer velocity estimation and reflector geometry delineation, one layer at a time, to complete the construction of the earth model in depth (Figure I-17c). This layer-by-layer, structure-dependent estimation of earth models in depth is needed when there are distinct layer boundaries with significant velocity contrast (as in many parts of the North Sea). In practice, an iterative, structure-independent estimation of earth models in depth also is used in the case of a background velocity field with not-so-distinct layer boundaries (as in the Gulf of Mexico).

Practical methods of layer velocity estimation include Dix conversion and inversion of stacking velocities, coherency inversion, and analysis of image gathers from prestack depth migration (earth modeling in depth). Velocity nodes at analysis locations for the layer under consideration (Figure I-18a) are assigned to the normal-incidence reflection points over the surface associated with the base of the layer (Figure I-18b). A velocity field for the layer is then created by spatial interpolation of the velocity nodes. This layer velocity field is assigned to the layer together with a similar field for a vertical velocity gradient whenever it is available from well data.

Practical methods of reflector geometry delineation include vertical-ray and image-ray depth conversion of time horizons interpreted from time-migrated data, commonly known as vertical stretch and map migration, respectively. Additionally, reflector geometries in depth can be delineated by interpreting post- and pre-stack depth-migrated data. By interpreting cross-sections from the volume of depth-migrated data at appropriate intervals, horizon strands are created (Figure I-19a). These strands then are interpolated spatially to create the surface that represents the reflector geometry associated with the layer boundary included in the earth model in depth (Figure I-19b).

In structural inversion, we present case studies for 2- and 3-D earth modeling and imaging in depth applicable to structural plays. These cases involve exploration and development objectives that require solving specific problems such as imaging beneath diapiric structures associated with salt tectonics, imaging beneath imbricate structures associated with overthrust tectonics, target reflectors below an irregular water-bottom topography, fault shadows, and shallow velocity anomalies.

A concise, but sufficiently rigorous, review of seismic wave propagation is given in reservoir geophysics. This also is intended to remind the reader of the two components of observed seismic data that can be used in inversion — traveltimes and amplitudes, to estimate the earth parameters. It is generally favorable to do the inversion of reflection traveltimes and amplitudes separately. The former is more robust and stable in the presence of noise. The latter is more sensitive to ambient noise and is prone to producing unstable solutions, and therefore, it may require more stringent constraints.

In reservoir geophysics, we review inversion of amplitudes of acoustic wavefields, specifically, prestack amplitude inversion to derive the attributes associated with amplitude variation with offset (AVO) and poststack amplitude inversion to estimate an acoustic impedance (AI) model of the earth. We broadly associate traveltime inversion with the estimation of a structural model of a reservoir that describes the geometry of the layer boundaries and faults. Whereas, we broadly associate amplitude inversion with the estimation of a stratigraphic model of the reservoir that describes the lateral and vertical variations of the AVO and AI attributes within the layers themselves. The latter can then be transcribed into petrophysical parameters — pore pressure, porosity, permeability, and fluid saturation, and it combined with the structural model to create a model of the reservoir. Therefore, seismic inversion is a true pronouncement of integration between petroleum geology, petroleum engineering, and exploration seismology. Only the exploration seismologists timespeak, while the peroleum geologists and engineers depthspeak. To achieve integration, they all must be fluent in the same language — depthspeak.