High frequency approximations in the solution of an acoustic wave equation

6 High frequency approximations in the solution of an acoustic wave equation The reader, familiar with the background of seismic exploration methods, should recall that many successful seismic interpretation algorithms are based on the simple principles of geometrical seismics, which resembles the ideas of geometrical optics. The question is how this simple but powerful approach is connected with the 

To understand better the connection between the geometrical optics approach and wave equation solutions, we will discuss in this section the basic equations describing high frequency scalar wavefield propagation. Following Bleistein (1984) and Bleistein et al. (2001), we represent the solution of the scalar wave equation (13.56) outside of the source in the form of the Debye series 

We now substitute the Debye series into the left-hand side of the scalar wave-field equation (13.56)  

Outside the source, the series on the right hand side of expression (13.59) must be equal to zero. Obviously, the terms with different powers of u cannot cancel each other, which requires that all coefficients of the series at different powers of w must vanish. In particular, from the first two terms (n = 0,1), we find all expressions containing the second and the first power of frequency  

As a result, we arrive at the following two equations for travel time r (r) and coefficient 

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