The specific definition of retarded coordinates is a matter of convenience. Often the retardation is based on hypothetical rays moving straight down with velocity .The definition of these coordinates has utility even in problems in which the earth velocity varies laterally, say v(x,z), even though there may be no rays going exactly straight down. In principle, any coordinate system may be used to describe any circumstance, but the utility of the retarded coordinate system generally declines as the family of rays defining it departs more and more from the actual rays.
Despite the simple case at hand it is worthwhile to be somewhat formal and precise. Define the retarded coordinate system (t' ,x' ,z' ) in terms of ordinary Cartesian coordinates (t,x,z) by the set of equations