The evaluation of spatial ambiguity functions is required for several acoustical signal processing applications including matched-field processing (MFP) and computing bounds on array localization error. In both of these applications, the ambiguity function is usually evaluated repeatedly over a region of interest for a specific acoustic array. When either the region or the array is large, computational efficiency is an issue. In this paper, two algorithms are presented for efficiently evaluating the spatial ambiguity function over weakly range-dependent regions utilizing an adiabatic normal mode signal model. The algorithms are referred to as the direct method and the indirect method. The direct method involves a recursive update in range of the so-called replicant vector. The computational complexity of the direct method for each range step is proportional to the product of the number of array sensors and the number of modes. In contrast, the indirect method recursively updates the entire ambiguity function and has an incremental computational complexity proportional to the number of modes squared. Thus, for low-frequency operation with a large array, the indirect method is more efficient. Computational results are presented in which the relative efficiency of the two algorithms is compared.