Since all ocean wave components contribute to the second-order scattering of a high-frequency radio wave by the sea surface, it is theoretically possible to estimate the ocean wave spectrum from first- and second-order scattering in the Doppler spectrum measured with an HF ocean radar. To extract the wave spectral information, however, it is necessary to solve a nonlinear integral equation. This paper describes in detail how to solve the nonlinear integral equation without linearization or approximation. We show that the problem of solving the nonlinear integral equation can be converted into a nonlinear optimization problem. An algorithm to find the optimal solution is described. Examples of the algorithm applied to simulated data and measured data are shown. The wave frequency spectrum can be estimated even if the Doppler spectrum is available in only a single direction. In this case, however, the solution of the two-dimensional wavenumber spectrum tends to converge to a spectrum that is symmetrical to the beam direction. Even if the wave spectrum is dominant in a single direction, the solution may give two peaks in the wavenumber spectrum. One of them is the true peak and the other is the mirror image of it with respect to the beam direction. This ambiguity can be avoided by using Doppler spectra measured in at least two different directions. Although there is still some room for improvement in the practical application of this method, it can be applied to estimate the wave directional spectrum up to a rather high frequency, or Bragg frequency. ¿ American Geophysical Union 1996