Very few techniques have been proposed for estimating traffic matrices in the context of Internet traffic. Our work on POP-to-POP traffic matrices (TM) makes two contributions. The primary contribution is the outcome of a detailed comparative evaluation of the three existing techniques. We evaluate these methods with respect to the estimation errors yielded, sensitivity to prior information required and sensitivity to the statistical assumptions they make. We study the impact of characteristics such as path length and the amount of link sharing on the estimation errors. Using actual data from a Tier-1 backbone, we assess the validity of the typical assumptions needed by the TM estimation techniques. The secondary contribution of our work is the proposal of a new direction for TM estimation based on using choice models to model POP fanouts. These models allow us to overcome some of the problems of existing methods because they can incorporate additional data and information about POPs and they enable us to make a fundamentally different kind of modeling assumption. We validate this approach by illustrating that our modeling assumption matches actual Internet data well. Using two initial simple models we provide a proof of concept showing that the incorporation of knowledge of POP features (such as total incoming bytes, number of customers, etc.) can reduce estimation errors. Our proposed approach can be used in conjunction with existing or future methods in that it can be used to generate good priors that serve as inputs to statistical inference techniques.