![]() For example, it is clear that the atmospheric barometric pressure induces a variation in both the ocean water levels and the groundwater levels, but the barometric pressure is not included in the system model as an input variable. Another common mistake is to assume a causal input/output relation between observed variables, when in fact the causative mechanism is not in the system model. If the relation ( transfer function) between the input and output is nonlinear, then values of the coherence can be erroneous. However, one must exercise caution in attributing causality. The computed coherence (figure 1) indicates that at most of the major ocean tidal frequencies the variation of groundwater level at this particular site is over 90% due to the forcing of the ocean tides. We further assume that the ocean surface height controls the groundwater levels so that we take the ocean surface height as the input variable, and the groundwater well height as the output variable. Let us assume that there is a linear relationship between the ocean surface height and the groundwater levels. To estimate the extent at which the groundwater levels are influenced by the ocean surface levels, we compute the coherence between them. It is clear that variation of the groundwater levels have significant power at the ocean tidal frequencies. The coherence (sometimes called magnitude-squared coherence) between two signals x(t) and y(t) is a real-valued function that is defined as: C x y ( f ) = | G x y ( f ) | 2 G x x ( f ) G y y ( f ) provides a spectral quantification of the output power that is uncorrelated with noise or other inputs.įigure 4: Autospectral density of groundwater well level. ![]()
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