# 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

${\frac {\partial ^{2}P}{\partial t^{2}}}=V^{2}\nabla ^{2}P$ .

When the velocity is spatial variant or even anisotropic, estimation of $V$ becomes a complicated inversion problem. In this case, the following plane wave approximation can be used for wave modeling

${\frac {\partial P}{\partial t}}=p{\frac {\partial P}{\partial x}}$ where $p$ 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.

## History

Plane wave model was firstly introduced by Jon Claerbout in 1992 . In 2002, Sergey Fomel  proposed to model the plane waves in frequency domain, where, the apparent velocity (local slopes $p$ ) becomes nonlinear function of wavefield. A wideband linear phase approximation was introduced to plane wave model by Zhonghuan Chen in 2013  to optimize the frequency band of plane wave model. To improve performance near steep structures, an omnidirectional plane wave model is proposed .

## 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.

$p={\frac {\frac {\partial P}{\partial t}}{\frac {\partial P}{\partial x}}}$ 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

$(1-Z_{x}Z_{t}^{p})P=0.$ where $Z_{t}^{p}$ 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.