On the Long-Period Statics Problem in Seismic Investigations

Abstract

One of the important points in performing research is that even for marine data acquired with ideal surface conditions, there are variations in the form of the initial seismic wavelet. Various kinds of corrections to seismic data can be made using factor models. Such models, used for the correction of residual statics, surface-consistent amplitudes, and seismic waveforms, lead to solutions of several problems. In particular, it is possible to understand the problem of long-period statics and its relationship with features of the structure of observation systems in seismic investigations, as well as with the algorithms that are used in the data processing. At the same time, the problem can be correctly solved by using a priori information. The latter is especially relevant for the analysis of the dynamic characteristics of signals. In the study of the corresponding linear systems, two approaches are realized: (i) an iterative process of sequential refinement for computing factors, and (ii) the introduction of heuristic conditions that guarantee uniqueness and stability in the solutions.

Appendices

Appendix A: Interactions Between Factors

From a theoretical point of view, the study of the uniqueness of the estimation of the parameters of model (1) can be considered as the problem of determining the values of $\gamma _p\left( c_1^pi + c_2^pj\right)$, $i = 1, 2$, from $\mathbf {z}$, i.e., from values of observations $z_{ij}$. In this case, the observations can be considered as the values of a certain function in some points on a discrete grid. Such sets of points will be called the observation plan and denoted by $Q = \left\{ (i_s, j_s); s = 1, 2, \dots , N \right\}$, where $(i_s, j_s)$ is a fixed grid node. Obviously, the values of $\gamma _p\left( c_1^pi + c_2^pj\right)$ will also be determined on discrete sets of points.

Thus, we need to determine the values of several unknown functions (factors) from the known values of one function (observations). It is obvious that such a problem may have a non-unique solution—for example, when we want to determine the values of $\gamma _1\left( c_1^1i + c^1_2j\right)$ and $\gamma _2\left( c_1^2i + c_2^2j\right)$ with $c_1^1 = c^1_2$ and $c^1_2 = c^2_2$. It is intuitively clear that the non-uniqueness of the solution to the problem may be associated with the type of functions $\gamma _p\left( c_1^pi + c_2^pj\right)$ and the structure of the observations plan Q. Since the early works, where the simplest factor models were considered, it was shown that it is impossible to fully define functions $\gamma _p\left( c_1^pi + c_2^pj\right)$ (Taner et al., 1974; Mitrofanov, 1975), both in the problems of time correction and signal shape. In addition, the influence of the structure of the observation plan on the non-uniqueness of the obtained solution was shown (Goldin and Mitrofanov, 1975; Wiggins et al., 1976).

We will be interested in the non-uniqueness of the solution, determined by the type of functions $\gamma _p\left( c_1^pi + c_2^pj\right)$, $i = 1, 2$, and the interaction between the factors. With regard to the observation plan, we will assume that it is admissible, i.e., allows us to determine the values $\gamma _p\left( c_1^pi + c_2^pj\right)$ for those components that can be determined in the absence of interaction between factors.

To study the relationship between factors, we will use the following constructions. We represent each of the factors in the form

$$\begin{aligned} \gamma _p\left( c_1^pi + c_2^pj\right) = \sum _{s = 0}^{N_p - 1}d^p_s\psi _s \left( c_1^pi + c_2^pj\right) . \end{aligned}$$

(14)

Here, $\left\{ \psi _s(l_p)\right\} _{s = 0}^{s = N_p-1}$ is a set of linearly independent functions, and $l_p = c_1^pi + c_2^pj$ is a discrete variable.

Using (14), we represent the initial factor model (1) in the following form:

$$\begin{aligned} z_{ij} = \sum _{p=1}^{P}\sum _{s=0}^{N_p -1}d_s^p\psi _s \left( c_1^pi + c_2^pj\right) \cdot \phi _s \left( c_3^pi + c_4^pj\right) + \xi _{ij}. \end{aligned}$$

(15)

Then we use the expansion of observations $z_{ij}$ and functions $\psi _s, \phi _p$ in terms of a system of linearly independent functions $\left\{ f_r(i, j)\right\} _{r=0}^{r=N-1}$, $(i, j)\in Q$, i.e.,

$$\begin{aligned}&z_{ij} = \sum _{r=1}^{N-1}q_r f_r(i, j), \nonumber \\&\psi _s \left( c_1^pi + c_2^pj\right) = \sum _{r=1}^{N-1}m^p_{sr} f_r(i, j), \nonumber \\&\phi _s \left( c_3^pi + c_4^pj\right) = \sum _{r=1}^{N-1}n^p_{sr} f_r(i, j). \end{aligned}$$

(16)

Substituting these expressions into (15) and equating the expansion coefficients for the same functions, we obtain the equivalent to model (1):

$$\begin{aligned} q_r = \sum _{p=1}^{P}\sum _{s=0}^{N_p -1}K_{sr}^p d^p_s + \nu _r, \end{aligned}$$

(17)

where $q_r$ and $\nu _r$ are the coefficients of the expansion of $z_{ij}$ and $\xi _{ij}$, respectively, in terms of functions $f_r(i ,j)$, and $K^p_{sr}$ are the coefficients obtained in the process of constructing this model.

Model (17) makes it easy to analyze the non-uniqueness of the definition of the components of the factors $\psi _s \left( c_1^pi + c_2^pj\right)$ and understand the degree of interaction between them. To do this, it is necessary to analyze which of the coefficients $d^p_s$ cannot be determined from the coefficients $q_r$.

For the simplest two-factor model (4) and in the case of power functions $f_r(i,j)$, we have one coefficient, $q_0$, and two unknown coefficients, $d^1_0$ and $d^2_0$. Therefore, only the constant component of factors is uncertain. For the case of three-factor models (5) and (6), there is also no possibility of determining the constant component of the factors. But in addition, the linear component cannot be determined. There are three unknown coefficients: $d^1_0$, $d^2_0$, and $d^3_0$.

Appendix B: Fixing Components Using Finite Differences

Let non-unique factors be polynomial. This usually happens in the case of power functions $\phi _p(c_3^pi + c_4^pj)$ in model (1). All of the above particular cases, (2)–(6), refer to such models. Then, structures similar to finite differences can be used to fix such components.

It is known that the finite differences of the kth order of a function f(x) can be represented in terms of its values in the form

$$\begin{aligned} f^k_i = f_{i + k/2} - \left( {\begin{array}{c}k\\ 1\end{array}}\right) f_{i - 1 + k/2} + \dots + (-1)^m\left( {\begin{array}{c}k\\ m\end{array}}\right) f_{i - m + k/2} + \dots + (-1)^kf_{i - k/2}. \end{aligned}$$

(18)

In particular,

$$\begin{aligned}&f^1_{i + 1/2} = f_{i+1} - f_i,\nonumber \\&f^2_i = f^1_{i+1/2} - f^1_{i-1/2} = f_{i+1} - 2f_i + f_{i+1}, \nonumber \\&f^3_{i+1/2} = f^2_{i+1} - f^2_{i} = f_{i+2} - 3f_{i+1} + 3f_{i} + f_{i-1}. \end{aligned}$$

(19)

Here, $f_i$ denotes the value of the function at the point $x=x_i$. It can be shown that the coefficients of the kth degree polynomial are equal to zero when the differences up to the kth order inclusive are identically zero, while the values $f_i$, involved in the formation of these differences, do not correspond to the roots of this polynomial.

We suggest modifying expression (19) to obtain conditions that eliminate linear relationships between factors and good conditioning of the matrices $\mathbf {X}$ and $\mathbf {X}^T\mathbf {X}$. Following a statistical approach for constructing difference relations, we do not use the initial values of the function, but their averaged values. This averaging is carried out as follows. The entire domain of the function is divided into the number of intervals required to construct the difference of the kth order. It is easy to show that this requires $2^k$ intervals. At each of the intervals, the average value of the function is formed, which is involved in constructing the difference.

Suppose that the function is defined on a sufficient number of discrete values that are multiples of $2^k$. Then each of the intervals contains $N = N_f/2^k$ values of the function, where $N_f$ is the total number of values. Having completed the process of forming the differences by the averaged values, we obtain the following analogue of the representation (18)

$$\begin{aligned} f^k_i = \hat{f}_{i + k/2} - \hat{f}_{i - 1 + k/2} + \dots + (-1)^m\hat{f}_{i - m + k/2} + \dots + (-1)^k\hat{f}_{i - k/2}, \end{aligned}$$

where to obtain the averaged values of the function $\hat{f}_{i - m + k/2}$, intervals of various durations are used that contain $N_m = \left( {\begin{array}{c}k\\ m\end{array}}\right) N$ function values. Thus, multiple values of the function in representation (18) will correspond to multiple extensions of the intervals over which the function is averaged. As a result, we arrive to expressions similar to (19):

$$\begin{aligned} \begin{aligned}&f^0 = \frac{1}{N}\sum _{i=1}^N f_i, \\&f^1 = \frac{1}{N}\left( \sum _{i=N+1}^{N_f} f_i - \sum _{i=1}^N f_i \right) , \\&f^2 = \frac{1}{N}\left( \sum _{i=3N+1}^{N_f} f_i - \sum _{i=N+1}^{3N} f_i + \sum _{i=1}^N f_i\right) , \\&f^3 = \frac{1}{N}\left( \sum _{i=7N+1}^{N_f} f_i - \sum _{i=4N+1}^{7N} f_i + \sum _{i=N+1}^{4N} f_i - \sum _{i=1}^N f_i\right) . \end{aligned} \end{aligned}$$

(20)

The first two expressions in (20) were used to form conditions (12) and (13). In this case, condition (12) promotes a zero value of the constant component for each factor, and condition (13) promotes a zero value of the linear component.