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The Orthogonal Polynomial Transform module decomposes a prestack NMO-corrected CMP gather into its Legendre polynomial components as a function of offset. Rather than representing the gather as a collection of individual traces, this transform expresses the amplitude-versus-offset (AVO) behavior at every time sample as a sum of orthogonal polynomial basis functions. The zeroth-order term corresponds to the AVO intercept (P) and the first-order term to the AVO gradient (G), providing a direct link to standard AVO analysis.
The algorithm normalizes trace offsets to the range [-1, 1] and then projects the gather amplitude data onto each Legendre polynomial using numerical integration. The resulting polynomial coefficients are stored as separate traces in the Transform Gather output, one trace per polynomial term. A Reconstruction Gather is also produced by summing back a user-selected number of terms, allowing you to assess how well a truncated polynomial series reproduces the original data. The difference between the original and reconstructed gathers is written to the Error Gather output, which is useful for quality control of the approximation.
This module is designed for use in AVO workflows after NMO correction. It provides an alternative, noise-robust representation of gather kinematics that can reveal AVO anomalies and support attribute extraction. An interactive time-slice display is also available, showing observed amplitude versus offset and the polynomial fit at a user-selected time level.
Sets the maximum source-receiver offset (in meters) to include in the polynomial transform. Traces with absolute offsets greater than this value are excluded from the computation. Set this to -1 (the default) to include all available offsets in the gather. Use a positive value to restrict the analysis to near or mid offsets, for example to avoid far-offset noise or stretch artifacts introduced by NMO correction.
Controls how many Legendre polynomial basis functions are computed during the transform. The default value is 3, and valid values range from 1 to 8. Each computed term appears as a separate trace in the Transform Gather output. The zeroth-order term captures the average amplitude level (AVO intercept), the first-order term captures the linear variation with offset (AVO gradient), and higher-order terms capture increasingly complex curvature in the AVO response. Use a higher Approximation Order when the amplitude-versus-offset relationship is non-linear and you need to characterize higher-order AVO behavior. For standard two-term AVO analysis, an order of 2 is typically sufficient.
Specifies how many of the computed polynomial terms are summed back together to reconstruct the output gather. The default value is 3, and valid values range from 1 to 8. This value must not exceed the Approximation Order, since you can only reconstruct from terms that were computed. Using a Reconstruction Order lower than the Approximation Order acts as a low-order filter on the AVO curve, suppressing contributions from higher-degree polynomial terms. This can be useful for noise attenuation: by reconstructing with only the lowest-order terms, random scatter in the offset domain is removed while the dominant AVO trend is preserved. The Error Gather output shows the signal not captured by the reconstruction, which helps you judge the quality of the approximation.
The NMO-corrected prestack CMP gather to be decomposed. Traces must be sorted by offset. The module requires at least three live traces in the gather to perform the polynomial fit. The offset values stored in the trace headers are used to normalize the offset axis before the Legendre projection is computed. Typically this gather comes directly from an NMO correction module applied earlier in the processing sequence.
A single-trace gather containing the zeroth-order Legendre polynomial coefficient at every time sample. This coefficient represents the AVO intercept P — the amplitude extrapolated to zero offset. It is equivalent to the normal-incidence reflection amplitude and is one of the primary AVO attributes used to distinguish lithology and fluid effects.
A single-trace gather containing the first-order Legendre polynomial coefficient at every time sample. This coefficient captures the rate of change of amplitude with offset — the AVO gradient G. Together with the intercept P, it forms the basis of standard two-term AVO classification (Class I, II, III, IV anomalies).
A gather containing all computed Legendre polynomial coefficients as separate traces. The number of traces in this gather equals the Approximation Order. The first trace is the intercept P, the second is the gradient G, and subsequent traces represent higher-order polynomial terms. This gather is the core output of the OPT decomposition and can be used for advanced AVO attribute analysis.
The gather reconstructed by summing the first N polynomial terms back to the offset domain, where N is the Reconstruction Order. This gather has the same offset geometry as the input and represents the smooth polynomial approximation of the original data. Comparing this gather with the input visually reveals how much of the AVO signal is captured by the polynomial model.
The residual gather, computed as the difference between the original input gather and the Reconstruction Gather. A small, noise-like error gather indicates that the polynomial model fits the data well. A coherent signal in the error gather may indicate that the Approximation Order or Reconstruction Order is too low to capture the true AVO response, and should be increased.