Least-squares amplitude estimation

I assume that a collection of seismic data $\sv d(t,x)$consists of signal $\sv s(t,x)$ and noise $\sv n(t,x)$ so that  

$$\sv d(t,x)=\sv s(t,x)+\sv n(t,x).$$ (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 $\sv s'$ is defined as  

$$\sv s' = t^2 \sv s.$$ (100)

so that  

$$\sv d(t,x)= \frac{1}{t^2} \sv s' +\sv n(t,x).$$ (101)

The use of the t² in equation () indicates that a sample of $\sv s$ is multiplied by the time of that sample squared to get the corresponding output sample in $\sv s'$.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  

$$\sv d(t,x)= \frac{1}{T'(t,x,v)} \sv s' +\sv n(t,x),$$ (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 (), $\sv n(t,x)$ becomes $\sv d(t,x)$ before the start time as 1/T'(t,x,v) goes to zero.

I further assume that the RMS amplitude $\sigma_n$ of the noise is independent of time, while the RMS amplitude $\sigma_s$ of $\sv s'$, the signal scaled by time squared, is constant. Here, $\sigma_n$ is a function of trace position, or x, making it $\sigma_n(x)$,while $\sigma_s$ is a constant for the entire shot record. Both $\sigma_n(x)$ and $\sigma_s$ are calculated over zones of the input that are considered typical of the noise and signal, respectively. Each $\sigma_n(x)$ is calculated over the part of the trace before the first arrival as

$$\sigma_n(x) = \sqrt{ \frac{1}{N} \sum_{t=0}^{x/v} \sv d(t,x)^2 } ,$$ (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 $\sigma_n(x)$ from the deeper data, after the signal has died off.

Next, $\sigma_s$ is approximated from the shot gather using the data after the first breaks, assuming that the reflection amplitudes scale as t², so

$$\sigma_s = \sqrt{ \frac{1}{N} \sum_{x} \sum_{t=x/v}^{t_{\rm end}} \left( t^2 \sv d(t,x) \right)^2},$$ (104)

where N is the total number of samples after the first break time on all traces, and $t_{\rm end}$ is the end of trace time. If the mean value of each trace is zero, $\sigma_s^2$ and $\sigma_n(x)^2$ 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  

$$0 \approx \frac{1}{\sigma_s} \sv s' = S \sv s'$$ (105)

and  

$$0 \approx \frac{1}{\sigma_n(x)} \sv n = N \sv n,$$ (106)

where $S=1/\sigma_s$ and $N = 1/\sigma_n(x)$.Adding the condition that

$$\sv d=\sv s+\sv n,$$ (107)

or

$$\sv d=\frac{\sv s}{T'} + \sv n,$$ (108)

where $\sv s' = T' \sv s$,the regression in equation () becomes

$$\sv 0 \approx N \left( \sv d - \frac{\sv s'}{T'} \right),$$ (109)

making the system of regressions in equations () and ()

$$\left( \begin{array} {c} \sv 0 \\ \sv 0 \end{array}\right) \... ... N \left( \sv d - \frac{\sv s'}{T'} \right)\end{array}\right).$$ (110)

Rearranging this expression produces

$$\left( \begin{array} {c} \sv 0 \\ N \sv d \end{array}\right... ...begin{array} {c} S \\ \frac{ N}{T'} \end{array}\right) \sv s'.$$ (111)

Given that the least-squares solution for the expression $\sv y= A\sv x$ is $\sv x = ( A^\dagger A)^{-1} A^\dagger \sv y$, the least-squares solution for $\sv s$ is

$$\sv s' = \left( \left(\begin{array} {cc} S & \frac{ N}{T'}... ... \left( \begin{array} {c} \sv 0 \\ N\end{array}\right) \sv d ,$$ (112)

which, when multiplied, gives

$$\sv s' = \frac{ \frac{ N^2}{T'^2} }{ S^2 + \frac{ N^2}{T'^4} } \sv d,$$ (113)

or expressed as the estimated RMS amplitudes is

$$\sv s'=\frac{ T(x,t,v) \sigma_s^2}{T(x,t,v)^2 \sigma_s^2 + \sigma_n^2(x) }\sv d.$$ (114)

The data $\sv d$ and signal $\sv s'$ are functions of $\sv x$ and t and may be expressed as $\sv d(x,t)$ and $\sv s'(x,t)$, but for the rest of this discussion they will be expressed as simply $\sv d$ and $\sv s'$to avoid some complexity. The original unscaled signal becomes  

$$\sv s=\frac{ T(x,t,v)^2 \sigma_s^2}{T(x,t,v)^2 \sigma_s^2 + \sigma_n^2(x) }\sv d.$$ (115)

The expression for the noise is  

$$\sv n=\frac{ \sigma_n(x)^2}{T(x,t,v)^2 \sigma_s^2 + \sigma_n^2(x) }\sv d.$$ (116)

It can been seen from equations () and () that $\sv s+\sv n = \sv d$, which is consistent with equation ().

In the present case, S and $\sigma_s$ are constant over all traces, while N(x) and $\sigma_n(x)$ 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.