# Coherent Energy, Total Energy, Energy Ratio Similarity, Outer Product Similarity and Sobel Filter Similarity

**Attribute Description:** Similarity is a measurement between traces based on the coherence of the waveforms. Some forms of similarities may be more sensitive to amplitudes than others while the results of the energy ratio volume are designed to not be sensitive to the amplitude changes.

**Interpretation Use:** The similarity attribute identifies lateral changes in waveforms such as stratigraphic features and faults/fractures. This attribute highlights the abrupt boundaries or discontinuities in the waveform caused by geological features. The linearity is better seen on the time/depth or horizon slices. Coherent Energy and Total Energy are two volumes to show regional energy trends within the data.

**Recommended Color Palette:** A colorbar where one end of the color is black and is gradationally transitioned to white on the other end, is recommended for energy ratio similarity display. The idea is to highlight the discontinuity by the dark color and makes the coherent background white. Gray and purple are the common intermediate colors. Coherent energy and total energy can be displayed with the gradational color bar, or alternatively, add red-yellow gradational color to emphasize the energy trend.

**Example:**

**a)**

b)

c)

d)

e)

**Computation:**

**Energy ratio similarity** is the ratio of the coherent energy over the total energy of the input data within the analysis window. Coherence energy here is estimated based on eigenstructure methodology by Gersztenkorn and Marfurt (1999). The first step is to capture the trace information by first constructing a covariance matrix centered around the analysis point **d**. The matrix contains the cross-correlation value from the data matrix within the analysis window. The covariance matrix can be written as

C: M by M square covariance matrix

M: number of traces in the analysis window

λk: the kth eigenvalue

ν(k): the kth eigenvector

Then the eigenstructure coherence was derived by calculating the eigenvalues of the covariance matrix. The assumption is that the first eigenvector is sufficient to represent the coherent energy of all traces

J, referring to the total number of eigenvalue-eigenvector pairs, equals to M. Ceigen is the eigenstructure coherence based on the first eigenvalue (λ1).

Instead of calculating use the first eigenvalue to represent the coherent energy, Chopra and Marfurt (2007) offer a closely related method to calculate **energy ratio similarity** defined as

Where **Coherence energy** (Kcoh) is the sum of the energy of the corresponding weighted PC-filtered traces.

and

*d H=* Traces after Hilbert transform

d(j)pc= Filtered jth principal component centered at the point (

**t**0,

**x**n,

**y**n)

The **Total Energy** (Ktot) is the sum of the energy of the weighted analytic traces used to compute the covariance matrix:

On the other hand, Marfurt *et al.* (1999) define **Outer Product Similarity** to be the ratio of the energy of the average trace to the average of the energies of each of the traces.

The calculation adopts the idea from Luo et al. (1996) which normalized Sobel filter by dividing the inline and crossline component by the energy.

**Reference**

- AASPI Document: http://mcee.ou.edu/aaspi/documentation/Volumetric_Attributes-similarity3d.pdf
- Gersztenkorn, A., and K. J. Marfurt, 1999, Eigenstructure-based coherence computations as an aid to 3-D structural and stratigraphic mapping: Geophysics, 64, no. 5, 1468-1479, http://dx.doi.org/10.1190/1.1444651
- Chopra, S. and K. J. Marfurt, 2007, Seismic attributes for prospect identification and reservoir characterization: SEG Geophysical development series,
**11** - Luo, Y., W. G. Higgs, and W. S. Kowalik, 1996, Edge detection and stratigraphic analysis using 3-D seismic data: 66th Annual International Meeting, SEG, Expanded Abstracts, 324–327