Optimum Boundaries For Finite-difference Modelling of Waves

A technique is described to generate an optimum set of points beyond the boundary of a finite-difference model; points which can be used for finitedifference operations, and which eliminate reflections from that boundary. The rationale is explained, the mathematics are developed, and the final formulae are given. Examples are shown for the case of a two-dimensional elastic pressure wave. The limitations of the technique are shown to be consistent with the general limitations of simulating continuous waves by finite-difference techniques. INTRODUCTION There have been many techniques developed to reduce the reflections from the boundaries of finite-difference models, and simulate the infinite real earth. These techniques are useful for economic reasons, so that a model size can remain small and yet simulate the effects of specified internal boundaries without interference from the model edges. The basic technique is to provide extra rows and columns of points around the edges of a model. The amplitudes at these points are needed to allow the finitedifference operations to be executed within the model, but cannot themselves be generated by the same techniques because of their edge position. Unique algorithms, or in some cases unique conditions, must be used to calculate these amplitudes. The earlier techniques used to reduce boundary effects were called absorbing boundaries, and simulated the effects of having a highly attenuating material around the model. This technique is very practical where the modelling already accounts for viscous effects on the particle motion (Kelly and Marfurt, 1990). The viscosity is simply made very high for several rows and columns around the model’s area of interest. Another absorbing technique that can be used is to taper, at each time step, the amplitudes toward the model edge by a minimal amount. Cerjan et al. (1985) got very successful results by tapering to a maximum of 0.92 across a boundary zone of 20 points in width. With absorbing boundaries, the edge-point amplitudes are calculated by an approximate algorithm, but any errors that this introduces is shielded by the attenuating zone. The increased overhead caused by providing the attenuating zone is usually not a major barrier with modern computers. An alternative to absorbing boundary conditions can be called transmitting boundary conditions. Reynolds (1978) called his boundaries transmitting, and although Clayton and Engquist (1977) called their boundaries absorbing, they used algorithms similar to Reynolds. These algorithms project amplitudes into the boundary zones from the values already calculated for the zone of interest. Clayton and Engquist adapted a migration algorithm to project boundary values. Reynolds factored the wave equation and then used approximations for finitedifferencing. A requirement of these techniques is to select only those solutions that advance into the boundary, and suppress solutions that advance out of the boundary (the reflections). They are found to work very well with waves moving directly toward the boundary, but not so well with waves approaching the boundary at an acute angle. This paper describes a transmitting boundary solution for the second-order finitedifference elastic wave equation. In this space of digital values, two unknowns must be found. The first unknown is the extra boundary value amplitude, and the second unknown is the advanced time-step amplitude that is calculated using the extra boundary value. The first of the two equations that is required for a solution is, of course, the time stepping equation. We have found that the second required equation is the one that relates all the first derivatives of an unimpeded advancing wave (the eikonal equation). Any solution that does not satisfy this equation must involve some reflected energy. The above simultaneous solution takes the form of a quadratic. The root of the quadratic must be chosen so that the slope of the wave toward the boundary is consistent with the slope in time of an advancing wave. In particular, a slope down toward the boundary must accompany more positive amplitudes with time, and vice-versa. THEORY The development of the theory starts with the definition of a scalar plane-wave, which may be chosen to advance with time ( ) ( ) t k x z F P ω θ θ − + = sin cos . (1) Then an equation relating the derivatives of the function may be shown to be