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A new class of multi-dimensional (m-D,m=3,4) spectral estimation algorithms is introduced based on the minimum variance representations (MVR) of m-D (m=3,4) data fields. These representations are defined in the framework of linear prediction where it is shown that they may be classed into general categories depending upon the geometry of the prediction space. For example, in the 3-D case it is shown that there are four possible models: causal, semicausal I, semicausal II, and non-causal. The m-D (m=3,4) model formalisms and their linear-predictive and spectral interpretations are derived. The admissibility conditions of the spectral density function are also discussed. To obtain high-resolution spectral estimates from finite length m-D (m=3,4) data fields, the models are fitted to the data optimally in the sense of minimizing the covariance recursion errors within the prediction space considered. Computer-simulated short data fields consisting of two travelling waves embedded in noise are employed to demonstrate experimentally that the class of algorithms developed in this paper improves on the standard techniques for high-resolution and robustness in the presence of nonstationarities, such as envelope modulation.