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![]() | Model fitting by least squares | ![]() |
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In practice, might not be a perfect wave but an observed field of many waves that we might wish to fit to the idea of a single wave of a single
. We seek the parameter
. First, we need a method of discretization that allows the mesh for
to overlay exactly
. To this end, I chose to represent the
-derivative by averaging a finite difference at
with one at
.
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(15) |
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(16) |
Now, let us find the numerical value of that fits a plane wave
to observations
. Let
be an abstract
vector having components with values
taken
everywhere on a 2-D mesh in
. Likewise, let
contain
. Because we want
, we minimize the quadratic function of
,
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(17) |
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(18) |
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(19) |
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(20) |
To suppress noise, the quadratic functions
,
, and
were smoothed over time with a triangle filter.
puckin
Figure 3. Input synthetic seismic data includes a low level of noise. |
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residual5
Figure 4. Residuals, i.e., an evaluation of ![]() |
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puckout
Figure 5. Output values of ![]() |
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Subroutine puck2d shows the code that generated Figures 3-5. An example based on synthetic data is shown in Figures 3 through 5. The synthetic data in Figure 3 mimics a reflection seismic field profile, including one trace that is slightly delayed as if recorded on a patch of unconsolidated soil.
Figure 4 shows the residual. The residual is small in the central region of the data; it is large where the signal is not sampled densely enough, and it is large at the transient onset of the signal. The residual is rough because of the noise in the signal, because it is made from derivatives, and the synthetic data was made by nearest-neighbor interpolation. Notice that the residual is not particularly large for the delayed trace.
Figure 5 shows the dips. The most significant feature of this figure is the sharp localization of the dips surrounding the delayed trace. Other methods based on ``beam stacks'' or Fourier concepts might lead us to conclude that the aperture must be large to resolve a wide range of angles. Here, we have a narrow aperture (two traces), but the dip can change rapidly and widely.
Once the stepout is known between each of the signals, it is a simple matter to integrate to get the total time shift. A real-life example is shown in Figure 6.
twod
Figure 6. A seismic line before and after flattening. |
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A disadvantage, well known to people who routinely work with finite-difference solutions to partial differential equations, is that for short wavelengths a finite difference operator is not the same as a differential operator; therefore, the numerical value of is biased. This problem can be overcome in the following way. First, estimate the slope
between each trace. Then, shift the traces to flatten arrivals. Now, there may be a residual
because of the bias in the initial estimate of
. This process can be iterated until the data is flattened. Everywhere in a plane we have solved a least squares problem for a single value
.
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