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**Chapter 11**

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### Inverse problem theory

### P-to-P AVA

### Instability in the reconstruction of shear-wave velocity

### Seismic response to an acoustic model

### The scattering problem

### Born approximation

### Geometry of the 1D model

### Born approximation

### Born approximation

### Born approximation

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Inverse problem theory
P-to-P AVA
Instability in the reconstruction of shear-wave velocity
Seismic response to an acoustic model
The scattering problem
Born approximation
Geometry of the 1D model
Born approximation
Born approximation
Born approximation
1D models
RMS velocities
RMS velocity
Nonhyperbolic
Nonhyperbolic moveouts
3D acquisition geometry
3D acquisition geometry (Cont'd)
Rose diagrams of azimuths distributions
Rose diagrams of azimuths distributions (Cont'd)
Rose diagrams of azimuths distributions (Cont'd)
3D acquisition geometry
Common azimuthal sections
Linearized inverse problem theory
Conditioning numbers
Conditioning numbers
Resolution
Histograms of wavenumbers associated with the multioffset synthetic data
Rays in a complex laterally heterogeneous medium
Rays through a simple and slightly laterally varying background velocity
Time migration and depth migration in the Green Canyon
A complex Rotliegend structure revealed by migration
Reconstructing the background velocity
Velocity spectrum
Velocity semblance plot
Velocity semblance plot
Migration velocity
Migration velocity (Cont'd)
Migration velocity (Cont'd)
Migration velocity (Cont'd)
Migration velocity (Cont'd)
Migration velocity (Cont'd)
Migration velocity (Cont'd)
Background velocity
Background velocity model
Snapshot of the pressure field
Snapshot of the pressure field (Cont'd)
VC survey
VC survey
VC survey (Cont'd)
Streamer survey
Velocity semblance
Flowchart

Images - Chapter 11
Simplified map of the inverse problem theory.

Inverse problems are encountered in many scientific disciplines, such as quantum mechanics, medical imaging, etc. However, the inverse problem theory in the broad sense has been developed by people working with geophysical data. The reason is that geophysicists try to understand the earth's interior but have available only data collected at the earth's surface, at the sea surface, on the sea floor, or inside a borehole.

Irrespective of scientific disciplines, the solution of an inverse problem generally includes three steps:

- Parameterization of the model: This step consists of determining an optimal set of model parameters which can be reconstructed from the available data. Ideally, we would like this set of model parameters to be able to completely characterize the model. Unfortunately, this is often not possible due to limitations in our data.
- Forward modeling: This step consists of using the physical laws which allow us, for given model parameters, to make predictions of how data are to be.
- Inverse problem: This step consists of using the observed data to infer the values of the model parameters for which the data predicted by the forward modeling best fit the observed data under a specific criterion. A possible criterion for fitting observed data and data predicted for a given model of the subsurface using the forward problem is to minimize the sum of the squared errors between the observed and predicted data.

Images - Chapter 11
P-to-P AVA (amplitude variations with angles) response to two different models. For a typical seismic aperture, the two responses are almost identical.

Five fundamental issues are associated with solving inverse problems, especially those related to petroleum seismology: (i) uniqueness, or how to be sure that the model of the subsurface obtained from a given dataset is the only such model which can explain that dataset; (ii) instability, i.e., a ``small'' perturbation of data can lead to a ``large'' perturbation of the inverse problem solution; (iii) convergence when inverse problems are solved iteratively; (iv) uncertainties due to inaccuracies in the physical models which allow us to predict data for a given model parameter or due to the incompleteness of these physical models and uncertainties in the measurements; and (v) the cost of the forward-problem step in the inverse-problem solution. If this list may sound like an old mathematics class, just go through the examples that follow, and you will realize that these issues are not just academic; they are real petroleum exploration and production concerns.

To add some concreteness to the issues related to the inverse problem solutions that we have just raised, let us look at a couple of examples. This figure shows identical pre-critical seismic AVA (amplitude variations with angles) responses to two very different models before the 30-degree incident angle---thus the issue of uniqueness. Notice also that for angles beyond the critical angle (i.e., greater than a 30-degree incident angle), there are enough differences between the two AVA responses. These differences can be used to distinguish these two models. In other words, a substantial amount of nonuniqueness in petroleum seismology problems can be resolved just by improving our theory or eliminating some of our assumptions, and by improving our acquisition geometries to collect, for instance, long offsets so that data corresponding to angles of incidence greater than 30 degrees can be recorded.

Images - Chapter 11
An illustration of potential instability in the reconstruction of shear-wave velocity from the precritical P-to-P AVA response; the AVA responses are almost identical for 50 percent variations in the shear-wave velocity. However, P-to-S AVA responses or postcritical P-to-P AVA responses can be used to overcome this instability.

This figure shows an illustration of the instability in the reconstruction of the shear velocity using the precritical AVA response of the P-to-P reflection. We can see that for 50-percent variations in the shear velocity, the AVA response of P-to-P reflection is almost unchanged. The consequence of this instability is that we may not be able to discriminate between some unconsolidated and consolidated rock formations from the P-to-P reflection data. However, as we noticed for the issue of uniqueness, a substantial number of the instabilities in petroleum seismology problems can be resolved just by improving our theory or eliminating some of our assumptions, and by improving our acquisition geometries to collect, for instance, long offsets so that our data can include significant P-to-S reflection energy. In fact, we can see in this figure that the instability in the reconstruction of the shear velocity can be resolved by using the postcritical AVA response of a P-to-P reflection or by using a P-to-S AVA response in addition to a P-to-P response.

Images - Chapter 11
(a) Seismic response to an acoustic model, (b) seismic response to an elastic model, and (c) the difference between the two responses. Notice that the same physical quantity is displayed in these three plots, and that the same explosive source was used in both experiments.

Suppose that the physical models used in our forward problem for predicting P-P reflections are based on acoustic equations of wave motion instead of elastic ones (see Chapters 2 and 6). As we can see in this figure, although the traveltimes are well predicted, the amplitudes are quite erroneous, especially in large offsets. Again, the consequence of these inaccuracies is that we may not be able to discriminate between some lithologies.

Images - Chapter 11 Configuration of the scattering problem: the medium is decomposed into a background medium and a scatterer. The wavefield can also be decomposed into a direct wavefield, which is made of waves which travel from source and receiver without interaction with the scatterer; and the scattered wavefield, which contains waves with interactions with the scatterer. Notice that the sources and receivers are located in the background medium in this configuration. Notice also that we did not include the free surface in the configuration because we assume that our data contained only primaries.

Images - Chapter 11 Examples of scattering events that can be predicted by the Born approximation for a medium with an infinite water layer, as illustrated in Figure 11-22. These events consist of primaries and of internal multiples. Notice that if the background medium, located between the source-receiver surface and the scattering point, is smooth (i.e., the seismic response of the background medium includes only direct-wave arrivals; no reflections), only the primary event in (a) will be predicted by the Born approximation.

Images - Chapter 11 Geometry of the 1D model used to generate the data in Figures 11-10 and 11-11. This model consists of a slab imbedded in a homogeneous infinite background medium. The source and receivers are located at z=0, and the background velocity is 2.0 km/s. The variables in the geometry are the thickness and the velocity of the slab. (PP0 is the primary reflection at the top of the slab, PP1 is the primary reflection at the bottom of the slab, and mi is an internal multiple due reflection within the slab).

Images - Chapter 11
A comparison of modeling results based on the Born approximation and those on an exact solution for a slab velocity of 2.2 km/s and for various slab thicknesses (PP0 is the primary reflection at the top of the slab, and PP1 is the primary reflection at the bottom of the slab).

This figure and the next two figures show comparisons of the Born solution and the exact solutions for different values of slab thickness and of slab velocities. We can see for the cases in which the wavelength of the signal propagating in the slab is much greater compared to the thickness of the slab (i.e., h << λ_{1}), and in which the relative perturbation is less that 0.36, the Born approximation solution is almost identical to the exact solution. However, as the thickness of the slab increases, we can see that the reflection at the bottom of the slab in the Born approximation solution starts departing from the exact solution, even when ΔK is quite small. We earlier alluded to the reason for the differences between the Born approximation solution and the exact solution, which is that the Born approximation solution propagates in the slab with the velocity of the background medium instead of the slab velocity. If the difference between the velocity of the background medium and the slab velocity is large, or the duration of propagation through the slab is long enough, then the arrival time of the reflection from the bottom of the slab cannot be accurately predicted by the Born approximation.

Notice that the arrival time of the reflection from the top of the slab in the Born approximation solution is not affected by the slab velocity or slab thickness. However, the amplitudes of this reflection are affected by the relative perturbation between the slab and the background medium, ΔK. We will see in the next figures that for ΔK = 0.55, the amplitude in the Born approximation solution can be quite inaccurate.

Notice also that the shape of the Born approximation solution for the case in which the slab thickness is 25 m and the slab velocity is 3.0 km/s appears quite different from that of the exact solution. The reason for this difference in shape is that the Born approximation, which assumes that the waves propagate in the slab with the background velocity, predicts two events with a small overlap between them, whereas in the exact solution, which is based on waves propagating with the actual velocity, the two events totally overlap due to the fast slab velocity.

Finally, we can also notice that for the cases in which ΔK = 0.55 and h ≥ 100 m, the internal multiple corresponding to two bounces in the slab is visible in the exact solution and not in the Born approximation, because the background medium is assumed to be homogeneous. The internal multiples are not predicted by the Born approximation if the medium is smooth; the homogeneous medium is an ideal example of a smooth medium.

Images - Chapter 11 A comparison of modeling results based on the Born approximation and on the exact solution for a slab velocity of 2.5 km/s and for various slab thicknesses.

Images - Chapter 11 A comparison of modeling results based on the Born approximation and on the exact solution for a slab velocity of 3.0 km/s and for various slab thicknesses (mi is an internal multiple due to reflection within the slab).