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A new model for rough surface scattering is presented; the model has a form similar to the small slope approximation (SSA) of Voronovich, but with modified kernel functions. As with the SSA, when including two field series terms in the solution the model matches the first- and second-order small perturbation method in the low-frequency limit. Unlike the SSA, the model also achieves agreement with the Kirchhoff approximation in the high-frequency limit even for penetrable surfaces. It is also shown that the new model achieves first-order tilt invariance for first-order SPM predictions. The new model is derived based on a previous extension of the local curvature approximation (LCA) to third order; the new model is termed the "reduced local curvature approximation of third order" (RLCA3) for this reason. Sample results for scattering from dielectric surfaces are presented to illustrate the new model and its relationship with other theories of rough surface scattering.