## Model Structure Error Parameter Structure Error

Sun et al. (1998) presented a procedure whereby the model structure error of using one model structure to replace another model structure is defined by a max -min problem that is based on the distance between the two models measured in the parameter, observation, and prediction/management space. Parameter structure error resulting from a different level of parameterization is a special case of model structure error. Without losing generality, we will use parameter structure error to represent model structure error.

The parameter structure error, SE(G/(, GB), of using parameter structure GB to replace parameter structure GA can be defined by the following max-min problem (Sun, 1994a, 1996):

Pa Pb

where d is the distance (to be defined later) between the two models, MA(GA, pA ) and Mb(Gb, pB); and parameters pA and pB must be in their admissible regions PA and PB. In general, SE(G/(, GB) / SE(Gfl, G/( ). When GA is a simplification of GB, we have SE(G/(. Gs) = 0.

The distance between the two models, MA(GA, pA) and MB(GB, pB), as generalized by Sun et al. (1998), can be defined as:

where dE(MA, Mb) = \\gE(MA) - gE(MB)\\E dD(MA, Mb) = \\hD(MA) - hD(MB)\\D dP(MA,MB) = ||p^ -PbIIg

where subscript E denotes a prediction/management alternative and its associated prediction space; || • ||£ is a norm defined in the prediction space; subscript D denotes an observation design and its associated observation space; hD(MA) and hD(MB) are "observations" based on the same observation design but generated from difference models, MA and MB

is a norm defined in the observation space; G is a parameter space having a common overparameterization structure of GA and GB; pA and pB are spans of pA and pB; || • || g is a norm defined in G; ¡j. and /. are weighting coefficients. It is clear that by varying the weighting coefficients, one can emphasize the importance of each distance in the parameter, observation, or prediction/man-management space. As a result, this will influence the inverse solution.