A computer-implemented method for reducing artifacts in a subsurface physical property model caused by an approximation, other than source encoding, in an iterative, computerized geophysical data inversion process, said method comprising varying the approximation as the iterations progress, wherein said method comprises: (a) assuming a physical property model of a subsurface region, said model providing values of at least one physical property at locations throughout the subsurface region; (b) selecting an iterative data inversion process having a step wherein a calculation is made of an update to the physical properties model that makes it more consistent with measured geophysical data acquired from the subsurface region; (c) making in said calculation an approximation that either speeds up the selected iterative inversion process other than by source encoding, or that works an accuracy tradeoff; (d) executing, using a computer, one cycle of the selected iterative inversion process with said approximation and using the physical properties model; (e) executing, using the computer, a next iterative inversion cycle, wherein a selection is made to either change some aspect of the approximation or not to change it; (f) repeating (e) as necessary, changing the approximation in some or all of the iteration cycles, until a final iteration wherein a selected convergence criterion is met or another stopping condition is reached; and (g) downloading the updated physical properties model from the final iteration or saving it to computer storage; CHARACTERISED IN THAT: one or more artifact types are identified in an updated subsurface physical property model as being caused by the approximation; the aspect of the approximation that is changed in some or all iteration cycles is selected for having an effect on artifacts of the one or more identified artifact types;and said effect is such that artifacts from one approximation do not add constructively with artifacts from another iteration cycle that uses an approximation with a changed aspect. (a) assuming a physical property model of a subsurface region, said model providing values of at least one physical property at locations throughout the subsurface region; (b) selecting an iterative data inversion process having a step wherein a calculation is made of an update to the physical properties model that makes it more consistent with measured geophysical data acquired from the subsurface region; (c) making in said calculation an approximation that either speeds up the selected iterative inversion process other than by source encoding, or that works an accuracy tradeoff; (d) executing, using a computer, one cycle of the selected iterative inversion process with said approximation and using the physical properties model; (e) executing, using the computer, a next iterative inversion cycle, wherein a selection is made to either change some aspect of the approximation or not to change it; (f) repeating (e) as necessary, changing the approximation in some or all of the iteration cycles, until a final iteration wherein a selected convergence criterion is met or another stopping condition is reached; and (g) downloading the updated physical properties model from the final iteration or saving it to computer storage; CHARACTERISED IN THAT: one or more artifact types are identified in an updated subsurface physical property model as being caused by the approximation; the aspect of the approximation that is changed in some or all iteration cycles is selected for having an effect on artifacts of the one or more identified artifact types;and said effect is such that artifacts from one approximation do not add constructively with artifacts from another iteration cycle that uses an approximation with a changed aspect. one or more artifact types are identified in an updated subsurface physical property model as being caused by the approximation; the aspect of the approximation that is changed in some or all iteration cycles is selected for having an effect on artifacts of the one or more identified artifact types;and said effect is such that artifacts from one approximation do not add constructively with artifacts from another iteration cycle that uses an approximation with a changed aspect.