The problem of determining subsurface structure of a material from the measured dispersion of Rayleigh surface waves is obviously important in nondestructive testing and in the operation of surface acoustic wave devices. As is typical of many inverse problems this one is ill-posed and thus the use of special approaches is indicated. One approach is to represent possible profiles of the subsurface structure by trial functions containing a finite set of adjustable parameters and then find the parameter values giving the best fit to the dispersion data. Here we present an approach using a nonparametric version of estimation theory. This involves a mathematical model representing measurement errors and all possible profiles with reasonable a priori probability weightings. The present paper is limited to the case in which the dispersion data are dense (i.e., in a practical sense the data points are sufficiently dense on the frequency axis that interpolation can be performed with impunity). Later related papers will deal with the sparse data case. To elucidate the nature of the ill-posedness, a formal solution of the inverse problem is given in closed form. The application of estimation theory yields tractable estimator which corresponds to the inclusion of a convergence factor in the formal solution. The properties of the estimator are given by a set of auxiliary measures relating to statistical bias, model vs data dominance, resolution and reduction of variance. Because of the present lack of actual data sufficiently dense in a practical sense, synthetic test data, with and without simulated measurement noise, are used to investigate the performance of the estimator.