Forward Sensitivity Approach to Dynamic Data Assimilation

Table 1

Symbolism and nomenclature.

Symbol	Nomenclature
𝑛	Dimension of state vector
𝑚	Dimension of observation vector
𝑝	Dimension of parameter vector
𝑞=𝑛+𝑝	Dimension of control vector
𝑁	Number of observation vectors
𝑡	Time
𝑥(𝑡)∈𝑅^𝑛	True model state vector
𝑥(𝑡)=(𝑥1,𝑥2,…𝑥𝑛)^𝑇∈𝑅^𝑛	Model state vector
𝑓∶𝑅^𝑛×𝑅^𝑝	Vector field of the model
𝑥(0)∈𝑅^𝑛	Initial condition for model state vector
𝛼∈𝑅^𝑝	Parameter vector
𝑐=(𝑥(0),𝛼)∈𝑅^𝑛×𝑅^𝑝	Control vector
𝑧(𝑡)∈𝑅^𝑚	Observation vector
ℎ(𝑥(𝑡))∶𝑅^𝑛→𝑅^𝑚	Forward operator relating model state and observations
ℎ(𝑥(𝑡))∈𝑅^𝑚	Model counterpart of the observation
𝑣(𝑡)∈𝑅^𝑚	Observation error vector
𝐑(𝑡)∈𝑅^𝑚×𝑛	Covariance of observation error vector 𝑣(𝑡)
𝑒𝐹	Forecast error
𝑏(𝑡)	Systematic component of forecast error
𝐷𝑥(𝑓)=[𝜕𝑓𝑖/𝜕𝑥𝑗]∈𝑅^𝑛×𝑛	Model Jacobian w. R. T. 𝑥
𝐷𝛼(𝑓)=[𝜕𝑓𝑖/𝜕𝛼𝑗]∈𝑅^𝑛×𝑝	Model Jacobian w. R. T. 𝐴
𝐷𝛼(𝑥)=[𝜕𝑥𝑖/𝜕𝛼𝑗]∈𝑅^𝑛×𝑝	Sensitivity matrix w. R. T. Parameters
𝐷𝑥(0)(𝑥)=[𝜕𝑥𝑖/𝜕𝑥𝑗(0)]∈𝑅^𝑛×𝑛	Sensitivity matrix w. R. T. Initial condition
𝐷𝑥(ℎ)∈𝑅^𝑚×𝑛	Jacobian of the forward operator w. R. T. X
𝐻1(𝑡)=𝐷𝑥(ℎ)𝐷𝑥(0)(𝑥)∈𝑅^𝑚×𝑛	Sensitivity matrix accounting for ℎ(𝑥(𝑡))
𝐻2(𝑡)=𝐷𝑥(ℎ)𝐷𝛼(𝑥)∈𝑅^𝑚×𝑝	Sensitivity matrix accounting for ℎ(𝑥(𝑡))
⟨𝑎,𝑏⟩	Inner product
𝐽(𝑐),𝐽1(𝑐)	Objective or cost functions
𝛿𝐽,𝛿𝐽1,𝛿𝑥(𝑡),𝛿𝛼	First variations
∇𝐽,∇𝐽1	Gradients of cost functions
𝑀(𝑡,𝑠)	Model state transition matrix (Appendix A)
𝐿(𝑡,𝑠)	Matrix that determines particular solution (Appendix A)