In the previous chapter we introduced the vertical coordinate η to handle orographic effects in mesoscale models. In the synoptic-scale models we are going to replace the height coordinate z which extends to infinity by a generalized vertical coordinate ξ. The introduction of ξ is motivated by the fact that we cannot integrate the predictive equations using z as a vertical coordinate to infinitely large heights. Replacing z by the atmospheric pressure p, for example, results in a finite range of the vertical coordinate. We will see that another advantage of the (x, y, p)-coordinate system is that the continuity equation is time-independent. There are other specific coordinate systems that we are going to discuss. Therefore, it seems of advantage to first set up the atmospheric equations in terms of the unspecified generalized vertical coordinate ξ. Later we will specify ξ as desired. We wish to point out that the introduction of the generalized coordinate is of advantage only if the hydrostatic equation is a part of the atmospheric system.

We will briefly state the consequences of the transformation from the stereographic (x, y, z)-coordinate system to the stereographic (x, y, ξ)-coordinate system, which henceforth will be called the ξ system.

Introduction

So far we have treated the so-called primitive equations of baroclinic systems which, in addition to typical meteorological effects, automatically include horizontally propagating sound waves as well as external and internal gravity waves. These waves produce high-frequency oscillations in the numerical solutions of the baroclinic systems, which are of no interest to the meteorologist. Thus, the tendencies of the various field variables are representative only of small time intervals of the order of minutes while the predicted weather tendencies should be representative of much longer time intervals.

In order to obtain meteorologically significant tendencies we are going to eliminate the meteorological noise from the primitive equations b y modifying the predictive system so that a longer time step in the numerical solution becomes possible. We recall that the vertically propagating sound waves are no longer a part of the solution since they are removed by the hydrostatic approximation. The noise filtering is accomplished by a diagnostic coupling of the horizontal wind field and the mass field while in reality at a given time these fields are independent of each other. The simplest coupling of the wind and mass field is the geostrophic wind relation. The mathematical systems resulting from the artificial inclusion of filter conditions are called quasi-geostrophic systems or, more generally, filtered systems. For such systems at the initial time t0 = 0 only one variable, usually the geopotential, is specified without any restriction.

Only the simplest seismic wave propagation problems are amenable to a direct analytic solution. As we have seen in Part II, for stratified media the methods of attack have been based on a semi-analytic approach in which transform methods are used to simplify the equations so that attention can be concentrated on behaviour in the frequency-slowness domain; the resulting integrals need to be evaluated numerically. Once we face a three-dimensionally varying medium the simplicity of such transform methods are lost, and coupling between different slowness components has to be taken into account to describe the passage of waves through the 3-D structure (see, e.g., Haines, 1988). This approach can be quite successful for some classes of simple problems where the the medium remains quasi-stratified but the shape of interfaces are distorted (Koketsu et al., 1991). Interaction of the seismic wavefield with isolated heterogeneities can also be tackled by using specific developments based on a multipole representation of the resulting scattered wavefield as in the T-matrix methods of Boström & Karlsson (1984) and Bostock & Kennett (1992). However, multiple interactions between ‘scatterers’ or between the scattered field and interfaces rapidly leads to challenging computational issues.

For fully three-dimensional problems a more direct attack is needed. At high frequencies the main tool is asymptotic ray theory, as introduced in Chapter I:9. These techniques have been progressively developed to allow for full 3-dimensional variations including the influence of anisotropic structures. Červený (2001) provides a comprehensive development of the current state of the theory for local and regional scale problems. The results for global models for both body waves and surface waves are presented by Dahlen & Tromp (1998, Chapters 15, 16). The details of the implementation of ray methods are quite dependent on the way in which the three-dimensional structure is described, and in particular on the specification of interfaces and their interaction.

Ray methods provide travel times and, with extra effort, amplitude information in a high frequency approximation. They are therefore a fundamental tool in understanding the nature of the main propagation processes, in the same way as we have used them for stratified structures.