Since the cost of solving a dense matrix grows as the cube of the number of unknowns and gets burdensome at about 103 unknowns, many approximations are necessary for the 109 unknowns in Figure 1.
Familiar formulas from inverse theory such as
must have the inverse matrix approximated by an identity, a diagonal, or a band matrix (hence industrial allergy to the word ``inversion''). Reality is that the images are created with weights and filters both before and after applying the adjoint, .No one would ever express the basic operator as a matrix, for it would have 109+12 elements.
The most basic and widely used imaging techniques involve hyperbolas. An early stage of most data processing reduces the dimensionality of the data by stacking (summing over shot-receiver separation). The summing is done with time shifts (called Normal MoveOut (NMO) and Dip Moveout (DMO)) that attempt to mimic zero-offset signals from the nonzero-offset ones. (Offset is the shot-receiver separation.) This enhances signal-to-noise ratio and reduces the data dimensionality to that of the final image cube, (103)3.
An impulse response in the exploding-reflector model (see Figure 2)
is a point at (x0,y0,z0) in the earth making a spherical wave
v2t2= (z-z0)2+(x-x0)2+(y-y0)2seen at the earth surface z=0 which is a hyperbola of revolution around the t-axis. The impulse response, a delta function on this hyperboloid, is a column vector in the earth response matrix .Our approximation to an inverse is the transpose matrix, or adjoint .Thus we sum all values on the hyperboloid to find each point in the earth at the location of the exploding reflector. Considering all points in the earth, this summing gives us the image of reflectivity in Figure 1.
Fourier transforms are useful over the time axis because the earth's velocity and reflectivity are time invariant. Unfortunately, the space axes are more troublesome because the velocity varies rapidly with depth and somewhat rapidly horizontally.
Before forming the reflectivity image cube of Figure 1, we need the earth velocity as a function of (x,y,z), since in reality we will need a traveltime function that is considerably more accurate than the constant velocity expression
v2t2= z02+(x-x0)2+(y-y0)2. We often make the ``Dix approximation'' which (in 2-D) says that the observed data can be fit to the traveltime equation
where h is the offset and where is the vertical traveltime depth. To do this data fitting, we sum the CMP gather over this trajectory for many constant values of v and then for each ,select the maximum sum (actually, the maximum coherence). See Figure 3.
The function that emerges, estimates the Root Mean Square (RMS) integral of the earth's velocity from the surface to the reflector point beneath. Given these values of RMS velocity, differentiation gives us local velocity as a function of depth, and midpoint (horizontal location).
Most interpretors of reflection seismic data ignore the shear waves and handle the seismic data as though it were from a single scalar wave equation for the compressional waves. Thus you might expect that we would set up an inverse problem to seek the density and the velocity. This is not the case. We can find the reflectivity (impedance gradient) and velocity but not the density. Here is why: Our most unambiguous measurement is reflectivity between about 10-50 Hz. Our measurements of velocity do not measure velocity directly, but its integral from one significant reflector to a deeper one. See Figure 4
Where we are lucky, there are coherent reflectors almost everywhere, but the process of subtracting the integrated velocity from one layer to the next deeper one introduces so much error that our measurement of velocity must be regarded as having a scale significantly larger than the wavelengths of our 10-50 Hz waves. Expressing the velocity as a function of vertical travel time, the bandwidth of our velocity measurements is well below 10 Hz. Thus we measure reflectivity from 10-50 Hz and velocity below 10 Hz, so we cannot divide the velocity out of the impedance to find the density. Reflectivity gives us shapes of objects but tells us little about what they are made from. For this we study polarity and amplitude versus angle, but these are not especially reliable. Where possible we use logs of a not-too-distant well.
Some smaller problems that some geophysicists attack with inverse theory are: (1) suppress strong multiple reflections (2) estimate large lateral velocity changes in the shallow weathered layers. (3) interpolate spatially aliased data.
Our most exciting advance for the coming decade promises to be what we grandly call 4-D, what in reality is merely subtracting 3-D surveys done about a year apart. In several pilot studies, 4-D has given a reliable indication of where the fluid pressure has changed because of gas going in or out of solution with the liquid. It is the key to hydrocarbon reservoir delineation and management.