A computer-implemented method of improving a geologic model of a subsurface region, comprising: (a) selecting one or more sets of parameter values, each parameter directly or indirectly representing a geologic property; (b) inputting each of the one or more sets of parameter values into a numerical model to produce a predicted outcome for each set of parameter values; (c) comparing the predicted outcome for each of the one or more sets of parameter values with observations of the subsurface region, the difference there between defined as a cost associated with each of the one or more sets of parameter values; (d) obtaining a gradient of the cost associated with each of the one or more sets of parameter values; (e) constructing a geometric approximation of a parameter space defined by one or more formations, wherein a formation as a part of the geometric approximation is a polytope or a hypertriangle, each of the one or more formations corresponding to one of the one or more sets of parameter values for which the cost and the gradient of the cost have been evaluated; (f) generating a response surface model based on the geometric approximation, the response surface model expressing the cost and the gradient of the cost associated with each formation; (g) when a finishing condition is not satisfied, selecting at least one additional set of parameter values based at least in part on the response surface model associated with previously selected sets of parameter values; (h) repeating parts (b), (c), (d), (e), (f) and (g) using successively selected additional sets of parameter values to update the geometric approximation and the response surface model until the finishing condition is satisfied; (i) outputting at least one of the sets of parameter values having a predetermined level of cost to a geologic model of the subsurface region; and (j) updating the geologic model, taking into account at least one of the outputted at least one of the sets of parameter values, and the predicted outcome corresponding to the outputted at least one of the sets of parameter values; CHARACTERIZED IN THAT: each of the additional sets of parameter values are selected by: drawing a number from a probability function; when the number drawn from the probability function is greater than a predefined threshold, selecting an additional set of parameter values within a formation having associated therewith an N th best cost, compared to costs associated with other formations in the geometric approximation, where N is a positive integer; and when the number drawn from the probability function is not greater than the predefined threshold, selecting the additional set of parameter values in a randomly selected formation in the geometric approximation. (a) selecting one or more sets of parameter values, each parameter directly or indirectly representing a geologic property; (b) inputting each of the one or more sets of parameter values into a numerical model to produce a predicted outcome for each set of parameter values; (c) comparing the predicted outcome for each of the one or more sets of parameter values with observations of the subsurface region, the difference there between defined as a cost associated with each of the one or more sets of parameter values; (d) obtaining a gradient of the cost associated with each of the one or more sets of parameter values; (e) constructing a geometric approximation of a parameter space defined by one or more formations, wherein a formation as a part of the geometric approximation is a polytope or a hypertriangle, each of the one or more formations corresponding to one of the one or more sets of parameter values for which the cost and the gradient of the cost have been evaluated; (f) generating a response surface model based on the geometric approximation, the response surface model expressing the cost and the gradient of the cost associated with each formation; (g) when a finishing condition is not satisfied, selecting at least one additional set of parameter values based at least in part on the response surface model associated with previously selected sets of parameter values; (h) repeating parts (b), (c), (d), (e), (f) and (g) using successively selected additional sets of parameter values to update the geometric approximation and the response surface model until the finishing condition is satisfied; (i) outputting at least one of the sets of parameter values having a predetermined level of cost to a geologic model of the subsurface region; and (j) updating the geologic model, taking into account at least one of the outputted at least one of the sets of parameter values, and the predicted outcome corresponding to the outputted at least one of the sets of parameter values; each of the additional sets of parameter values are selected by: drawing a number from a probability function; when the number drawn from the probability function is greater than a predefined threshold, selecting an additional set of parameter values within a formation having associated drawing a number from a probability function; when the number drawn from the probability function is greater than a predefined threshold, selecting an additional set of parameter values within a formation having associated therewith an N th best cost, compared to costs associated with other formations in the geometric approximation, where N is a positive integer; and when the number drawn from the probability function is not greater than the predefined threshold, selecting the additional set of parameter values in a randomly selected formation in the geometric approximation.