Imposing a total-order on a regular 2-D discrete random eld induces an orthogonal decomposition of the random eld into two components: A purely-indeterministic eld and a deterministic one. The deterministic component is further orthogonally decomposed into a half-plane deterministic eld and a countable number of mutually orthogonal evanescent elds. Each of the evanescent elds is generated by the column-to-column innovations of the deterministic eld with respect to a di erent non-symmetrical-half-plane total-ordering de nition. The half-plane deterministic eld has no innovations, nor column-to-column innovations, with respect to any non-symmetrical-half-plane total-ordering de nition. This decomposition results in a corresponding decomposition of the spectral measure of the regular random eld into a countable sum of mutually singular spectral measures.
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