Seismic Resolution

Seismic resolution and fidelity are the two important measures of the quality of the seismic record and the seismic images. Seismic resolution quantifies the level of precision, such as the finest size of the subsurface objects detectable by the seismic data whereas the seismic fidelity quantifies the truthfulness such as the genuineness of the data or the level to which the imaged target position matches its true subsurface position.

Let us try to understand this by making a synthetic data and doing the analysis over it.

Seismic Resolution Analysis: 1

Estimation of the Width of the Fresnel’s Zone

The Fresnel’s resolution quantifies the resolvability of seismic wave perpendicular to the direction of wave propagation. Fresnel’s resolution is defined as the width of the first Fresnel’s zone due to interference of the spherical waves from the source and from the receiver.

Seismic Resolution Analysis: 2

Fidelity

To investigate the fidelity of our data, let us consider the technique of resampling. For our case we consider the method of “bootstrapping”. Bootstrapping basically relies on random sampling with replacement. The other popular method for resampling is “jackknifing” which predates “bootstrapping”. The jackknife estimator of a parameter is found by systematically leaving out each observation from a dataset and calculating the estimate and then finding the average of these calculations.

The principle behind “bootstrapping” is that a dataset is taken, the total dataset is divided into two by randomly sampling with replacement. The newly sampled data are now used to invert for the model using some kernel function. If the two models correlate high enough then we can say that the prominent features in the model come from consistent signals in the data.

We don’t have the data set to make the velocity model. So instead, we can take random Gaussian distribution data and play with it.

Let’s pose the null hypothesis that the two sets of data come from the same probability distribution (not necessarily Gaussian). Under the null hypothesis, the two sets of data are interchangeable, so if we aggregate the data points and randomly divide the data points into two sets, then the results should be comparable to the results obtained with the original data. So, the strategy is to generate random datasets, with replacement (bootstrapping), compute difference in means (or difference in medians or any other reliable statistic), and then compare the resulting values to the statistic computed from the original data.

Seismic Resolution Analysis: 3

wave_propagation

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Singular Value Decomposition

The inversion of a matrix is a very important part of data analysis. The quality of inverse solution depends on how well we can invert the matrix.

d=Gm where “d” is the N dimensional column vector consisting the data, “m” is the M dimensional column vector consisting the model parameters which we seek to invert for, “G” is the NxM dimensional kernel matrix to map the model vectors into the data matrix.

=>m = Ad, where A is the inverse of the matrix G.

There are several matrix decomposition techniques. These techniques are not only useful for the inverse problems, but also for applications like principal decomposition that are widely used in seismic attribute studies.

When a matrix is pre-multiplied to a vector, the resultant vector can be regarded as a linear transformed version of the original vector. Therefore the multiplication of matrices is a linear transformation. Any matrix can generally be decomposed into transformations of some other matrices.

The decomposition of a general rectangular matrix requires the use of singular value decomposition or SVD (Lanczos, 1961). The SVD decomposes any rectangular matrix A of m rows and n columns into a multiplication of three matrices of useful properties:

A = U L V’

Here, L is the m x n rectangular diagonal matrix containing p singular values (or principal values) of the matrix A.

U and V are the two square unitary matrices. The number of non-trivial singular values, p is called the trace or rank of the matrix.

We try to solve for the an example of

A = (10  2; -10 2)

Example 1

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Inverse Theory

We follow the Geophysical Data Analysis: Discrete Inverse Theory by William Menke book.

We solve the chapters and try to make the things more simpler and faster to read and understand.

Chapter 1: Probability Theory

Click the link below to download the notes:

Probability Theory

Chapter 2: Solution of the Linear, Gaussian Inverse Problem : The length method

Linear, Gaussian Inverse Problem 1

Chapter 3: Solution of the Linear, Gaussian Inverse Problem : Generalized Inverses

Linear, Gaussian Inverse Problem 2

Stay tuned for the other chapters…

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