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## Least-squares amplitude estimation

I assume that a collection of seismic data consists of signal and noise so that
 (99)
The variable t corresponds to time and x corresponds to the offset of a recorded sample for the shot data considered in this chapter. For simplicity, since I assume the signal varies by approximately t2, a scaled signal is defined as
 (100)
so that
 (101)
The use of the t2 in equation () indicates that a sample of is multiplied by the time of that sample squared to get the corresponding output sample in .Although the function t2 is actually a matrix with the corresponding gain values of t2 on the diagonal, for simplicity I will continue to use this notation for amplitude terms that are connected with time.

In this chapter I assume the amplitude of the signal varies by approximately t2 and is zero above the start time. This amplitude variation is represented in this chapter as the function T(t,x,v). T(t,x,v) is not necessarily fixed as the t2 scalar, or even fixed in x, but may be modified to fit the data being considered. For simplicity, I assume in this discussion that the data require a t2 scalar. The v in T(t,x,v) represents the start time velocity. It is unlikely that an exact amplitude representation for the signal is needed, since I am looking only for an approximation of the amplitude response, but the scaling for T(t,x,v) should fit the data at least approximately.

Replacing t2 in equation () with a generalized time scalar T'(t,x,v) gives
 (102)
where T'(t,x,v) is just T(t,x,v), except that T' produces large values before the start time, so that 1/T'(t,x,v) is effectively zero there. Thus, in equation (), becomes before the start time as 1/T'(t,x,v) goes to zero.

I further assume that the RMS amplitude of the noise is independent of time, while the RMS amplitude of , the signal scaled by time squared, is constant. Here, is a function of trace position, or x, making it ,while is a constant for the entire shot record. Both and are calculated over zones of the input that are considered typical of the noise and signal, respectively. Each is calculated over the part of the trace before the first arrival as
 (103)
where N is the number of samples between t=0 and x/v is the start time for that offset, x being the offset and v being the start time velocity. For traces with small offsets and little data before the first breaks, it may be possible to get a measure of from the deeper data, after the signal has died off.

Next, is approximated from the shot gather using the data after the first breaks, assuming that the reflection amplitudes scale as t2, so
 (104)
where N is the total number of samples after the first break time on all traces, and is the end of trace time. If the mean value of each trace is zero, and are the estimated variances of the signal and noise.

To make an amplitude estimation of the signal and noise, I scale the noise and signal by the corresponding RMS amplitudes and minimize the result, so that
 (105)
and
 (106)
where and .Adding the condition that
 (107)
or
 (108)
where ,the regression in equation () becomes
 (109)
making the system of regressions in equations () and ()
 (110)
Rearranging this expression produces
 (111)
Given that the least-squares solution for the expression is , the least-squares solution for is
 (112)
which, when multiplied, gives
 (113)
or expressed as the estimated RMS amplitudes is
 (114)
The data and signal are functions of and t and may be expressed as and , but for the rest of this discussion they will be expressed as simply and to avoid some complexity. The original unscaled signal becomes
 (115)
The expression for the noise is
 (116)
It can been seen from equations () and () that , which is consistent with equation ().

In the present case, S and are constant over all traces, while N(x) and are a function of space only. In the more general case, these values may be calculated from some spatial separation of the signal and noise on the data.

The scale factors in equations () and () only change the amplitude of the data to what I assume would be the correct result if there existed only noise or signal. I have done no real separation of noise and signal other than estimating the amplitudes. I attempt to take the separation one step farther in sections , , and   where the spectral characteristics of the noise and signal are included.

Next: Examples of amplitude estimation Up: Amplitude estimation of signal Previous: Amplitude estimation of signal
Stanford Exploration Project
2/9/2001