|z = a + bx + cy + dx2 + exy + fy2 + gx2 y2,||(13)|
Since the surface has seven terms, we must know the function z=f(x,y) at seven points in order to find the Laplacian at one of the seven. This means solving a linear system to find the coefficients a, b, c, d, e, f and g, given three pairs of (x, y, z) points. It seems frustrating that we have to compute all the seven coefficients while using only the values of d, f, and g. Fortunately, we do not have to solve innumerable amounts of systems for each downward continuation step: the x and y values depend only on the geometry of the spatial mesh. For each point, the matrix inversion can be done only once: in the beginning, and after that for each point. We only need to multiply the vector of z values with three rows of the precomputed inverted matrix in order to find out the values of d, f, and g.
On a Cartesian grid, only five values are needed to compute the Laplacian. In that case, we also deal with two extra ``hidden'' relationships that state the particular geometrical relationships between the five points. In the general case, we do not have that information and, therefore, need more points.
The Laplacian may be found in other ways as well; perhaps interpolating with splines or other basis functions the entire wavefield - not just local neighbourhoods - at each depth step. The fastest and most elegant approach would nevertheless not involve finding a complete analytical expression of the wavefield function, but only its curvature information represented by the Laplacian.