[Paper Review] On Backus average for oblique incidence

David R. Dalton, Michael A. Slawiński|arXiv (Cornell University)|Jan 12, 2016

Seismic Imaging and Inversion Techniques4 references3 citations

TL;DR

This paper proposes a modified Backus average that weights layer contributions by the raypath distance traveled in each layer rather than by thickness, significantly improving traveltimes for oblique incidence. The modified average reduces traveltime error from 13.3 ms to 1.9 ms in a 10-layer synthetic model at 30° takeoff angle, demonstrating superior accuracy for non-vertical seismic waves.

ABSTRACT

We postulate that validity of the Backus (1962) average, whose weights are layer thicknesses, is limited to waves whose incidence is nearly vertical. The accuracy of this average decreases with the increase of the source-receiver offset. However, if the weighting is adjusted by the distance travelled by a signal in each layer, such a modified average results in accurate predictions of traveltimes through these layers.

Motivation & Objective

• Address the limitation of the standard Backus average, which assumes nearly vertical incidence and becomes inaccurate for increasing source-receiver offsets.
• Investigate whether raypath distance traveled in each layer provides a better weighting criterion than layer thickness for equivalent medium modeling.
• Improve the accuracy of traveltimes in layered anisotropic media for non-vertical, oblique incidence waves.
• Assess the validity and performance of the modified average under varying ray angles and offset conditions.
• Explore the implications of offset-dependent equivalent elastic parameters for qP, qSV, and SH waves in Hookean solids.

Proposed method

• Apply the standard Backus average using layer thickness as weights, defined by equation (8), to compute equivalent transversely isotropic elasticity parameters.
• Introduce a modified Backus average where weights are proportional to the raypath distance traveled in each layer, as defined in equation (9).
• Use Snell’s law and ray tracing to compute the horizontal distance traveled in each layer for a given takeoff angle.
• Calculate Fermat traveltimes through the layered model using raytracing for comparison with equivalent medium predictions.
• Compute equivalent medium traveltimes using the modified elasticity parameters and compare them to Fermat traveltimes to assess accuracy.
• Evaluate performance across different takeoff angles (e.g., π/6, π/4, π/3) and extreme oblique cases to test robustness.

Experimental results

Research questions

• RQ1 Does weighting the Backus average by raypath distance instead of layer thickness improve traveltimes for oblique incidence waves?
• RQ2 How does the error in predicted traveltimes change with increasing source-receiver offset when using thickness-weighted versus distance-weighted averaging?
• RQ3 Can a single equivalent medium with offset-dependent elasticity parameters accurately represent both near- and far-offset wave propagation?
• RQ4 Why does the standard Backus average overpredict traveltimes for oblique incidence, and is this due to the underlying mathematical approximation?
• RQ5 Is the Fermat traveltime a valid benchmark for evaluating the accuracy of the Backus average in heterogeneous layered media?

Key findings

• The standard Backus average, weighted by layer thickness, overpredicts traveltimes by 13.3 ms for a 30° takeoff angle (π/6), compared to Fermat’s principle.
• The modified Backus average, weighted by raypath distance traveled in each layer, reduces the traveltime error to only 1.9 ms for the same 30° case.
• For a 45° takeoff angle (π/4), the thickness-weighted average overpredicts by 25.7 ms, while the distance-weighted average reduces this error to 10.8 ms.
• At a 60° takeoff angle (π/3), the thickness-weighted average overpredicts by 35.8 ms, and the distance-weighted average reduces this to 14.2 ms.
• For extreme oblique incidence (approaching horizontal propagation), the distance-weighted average still underperforms, overpredicting by 9.9 ms, indicating limitations at very high offsets.
• The modified average results in equivalent elasticity parameters that depend on ray angle, implying distinct effective media for qP, qSV, and SH waves, though mathematically consistent in the Hookean framework.