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SVD analysis and the cost of Uniqueness of the calibrated parameter field.

The would be estimated parameter set is considered to be a unique solution because it is a projection, a shadow of real parameters on a smaller solution space.  

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Reinserting (h) in the preceding formulation, we reach a new two-fold error vector (^k).
From now on the already presented calibration noise (ε), gains a not new (or at best not  yet unanticipated) error component. 

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So this new estimation process can be seen carring  this kind of "structural" error, ... and the earlier noise, that we still don't know, because we don't know the real parameters. This is the cost of uniqueness of the scalar solution.

Finally, even not knowing the real parameters, comes the innate variability and spatial correlation between these parameters, ... and the noise in the dataset, by a covariance matrix of parameter error (as seen in propagation of variance rule page 13). 

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Beta μ

P.S. For sake of simplification, as stated in page 13, this example also assume that the noise C(ε) associated with each one and between the measurements are statistically independent and that it have a even standard deviation of σ2k

P.S. Similarly, the covariance matrix C(k) says that the parameters are statistically independent, and that it posses a even standard deviation of σ2ε

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Next page comes the definition of the covariance matrix of propensity for a prediction to be in error, given the uncertainty imbued in the covariance matrix of estimated parameter error. 

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