Propagator-matrix method

For a stack of horizontal and uniform layers, Gilbert and Backus (1966) proposed a method to solve the differential equation defined by

$$\frac{df(z)}{dz}=A(z)f(z)$$ (3.1)

where $f$ is a vector of n components and $A$ is a n*n matrix. If $A$ is independent of $z$ , which is valid inside a layer, the solution is given by

$$f(z)=G(z,z_0) f(z_0)$$ (3.2)

where,

$$G(z,z_0)=e^{(z-z_0)A(z)}$$ (3.3)

Equation (3.3) can be developed to find the elements of matrix $G$ using an eigenvalue decomposition of matrix $A$ (Aki and Richards (2002)). Because of the continuity of the displacement and the stresses at all interfaces between two layers, the following property is easily deduced from equation (3.2):

$$f(z_2)=G(z_2,z_1)f(z_1)=G(z_2,z_1)G(z_1,z_0)f(z_0)$$ (3.4)

Hence, the vector $f(z)$ at depth $z$ , inside layer $n$ is:

$$f(z)=G(z,z_n)G(z_n,z_{n-1}) \ldots{} G(z_1,z_0)f(z_0)$$ (3.5)

The propagator matrices $G$ are functions of the depth at the top and at the bottom of each layer, and of the matrix $A$ which depends upon layer properties. For Love and Rayleigh, vector $f(z)$ is called the motion-stress vector defined in sections 3.1.3 and 3.1.4, respectively.