Table 1: Symbolism and nomenclature.

SymbolNomenclature

Dimension of state vector
Dimension of observation vector
Dimension of parameter vector
Dimension of control vector
Number of observation vectors
Time
True model state vector
Model state vector
Vector field of the model
Initial condition for model state vector
Parameter vector
Control vector
Observation vector
Forward operator relating model state and observations
Model counterpart of the observation
Observation error vector
Covariance of observation error vector
Forecast error
Systematic component of forecast error
Model Jacobian w. R. T.
Model Jacobian w. R. T.
Sensitivity matrix w. R. T. Parameters
Sensitivity matrix w. R. T. Initial condition
Jacobian of the forward operator w. R. T. X
Sensitivity matrix accounting for
Sensitivity matrix accounting for
Inner product
Objective or cost functions
First variations
Gradients of cost functions
Model state transition matrix (Appendix A)
Matrix that determines particular solution (Appendix A)