Geophysical and geologic data, e.g., results obtained using rock cores, outcrops, and borehole images, reveal the presence of multiple sets of fractures in rock. To analyze how fracture interactions affect seismic anisotropy and dispersion, we consider an effective model that contains two sets of orthogonal fractures. In seismic amplitude analysis, we focus on the case of subsurface target zones containing a fracture network composed of primary and secondary fracture sets. Focusing on gas-bearing rocks with small fracture densities, we formulate simplified stiffness parameters in terms of two groups of fracture weaknesses, which are related to the primary and secondary fractures, respectively. Using the simplified stiffness parameters, we derive the PP-wave reflection coefficient and the azimuthal elastic impedance (AEI) as functions of the two groups of normal and tangential fracture weaknesses. Based on the derived PP-wave reflection coefficient and AEI, we propose a method and workflow in which the azimuthal variations of the partial incidence-angle-stacked seismic data are used to estimate the AEI and differences in AEI (which we refer to as DEI) is input to a nonlinear inversion for two groups of fracture weaknesses. In the nonlinear inversion, the initial values of fracture weaknesses are obtained based on a two-term approximation of the PP-wave reflection coefficient. First- and second-order derivatives of DEI with respect to fracture weakness parameters are calculated to generate the update in the unknown parameter vector. Synthetic seismic gathers with varying signal-to-noise ratios are used to validate the robustness of the estimation. Applying the inversion method to a real data set, we obtain what we interpret to be reliable fracture weakness estimates that produce azimuthal amplitude differences matching those extracted from real seismic data. The results provide a potential tool for determining if two sets of fractures have developed in a reservoir.