# Plane wave modeling

Plane wave model is a far field approximation of wave propagation. Wave propagating in the Earth follows the second order wave equation

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \frac{\partial^2 P}{\partial t^2} = V^2 \nabla^2 P}**.

When the velocity is spatial variant or even anisotropic, estimation of **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle V}**
becomes a complicated inversion problem. In this case, the following plane wave approximation can be used for wave modeling

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \frac{\partial P}{\partial t} = p \frac{\partial P}{\partial x}}**

where **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle p}**
is the local slopes of event and appearent velocity of the wavefield.
Phisical velocity inversion can be simplified by slope estimation in plane wave model.

## Contents

## History

Plane wave model was firstly introduced by Jon Claerbout in 1992 ^{[1]}.
In 2002, Sergey Fomel ^{[2]} proposed to model the plane waves in frequency domain,
where, the apparent velocity (local slopes **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle p}**
) becomes nonlinear function of wavefield.
A wideband linear phase approximation was introduced to plane wave model by Zhonghuan Chen in 2013 ^{[3]} to optimize the frequency band of plane wave model. To improve performance near steep structures, an omnidirectional plane wave model is proposed ^{[4]}.

## Slope inversion

One of the most significant application of plane wave modeling is local slope estimation. In time domain, the slope can be estimatied from the wave field directly by finite difference methods.

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle p = \frac{\frac{\partial P}{\partial t}}{\frac{\partial P}{\partial x}}}**

A least squares or total least squares method is used to obtain a robust estimation here in noisy cases. The time domain plane wave model is straightforward and it can work well in single slope case. In frequency domain, the plane wave model becomes

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle (1-Z_xZ_t^p)P = 0.}**

where **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle Z_t^p}**
is a linear phase operator.
With the polynomial-coefficients maxflat linear phase approximation, the above equation can be seen as a polynomial equation of the slope.
The frequency domain plane wave model is suitable for multiple slope estimation,
each slope becomes a solution of the polynomial equation.

## Applications

### Nonstationary denoise

### Velocity analysis

### Geometric interpretation

## References

- ↑ Claerbout, J. F., 1992, Earth soundings analysis: Processing versus inversion: Blackwell Scientific Publications
- ↑ Fomel, S., 2002, Applications of plane-wave destruction filters: Geophysics, 67, 1946–1960
- ↑ Chen, Z. and et., 2013, Accelerated plane wave destruction: Geophysics, 78(1), V1-V9
- ↑ Chen, Z. and et., 2013, Omnidirectional plane wave destruction: Geophysics, 78(5), V171-V179