What is the null space?
Mathematically, if there is a vector δp and a matrix X such that Xδp = 0 then the set of all δp vectors for which the above equation holds defines the null space of X. Now suppose that X represents a model under calibration conditions. (The Jacobian matrix is such a representation.). Suppose further that we have calibrated that model and determined a set of parameters p which gives rise to a good fit between model outputs and field measurements. Then we can add any δp within the null space of X to p to obtain a parameter set which also calibrates the model. It follows that if a model has a null space, its parameters are not uniquely estimable. Regularisation (of one kind or another) must then be used to calibrate that model. This will find just one of the millions of parameter sets which fit the data equally well. Which one will it find? Hopefully, if regularisation is properly done, it will find something approaching the “minimum error variance solution” to the inverse problem of mode
