Sensitivity Studies (1): Motivation Theoretical background. Sensitivity of the Lorenz model. Thomas Jung ECMWF, Reading, UK

Time Evolution of Forecast Errors

Initial Conditions: Error Growth Initial Time Final Time FC Error Reduced FC Error

Model Error Growth Initial Time FC Error Reduced FC Error Final Time Time

Linear Growth of Errors Nonlinear model equationLinearized model equation Solution to the linearized model equation

Linear Initial Perturbation Growth For a perfect model with f=0 we have:

Linear Model Perturbation Growth For perfect initial conditions we have:

Gradient of the Forecast Error w.r.t. Initial Conditions (Sensitivity)

Sensitivity Gradient and Norms

Interpretation of the Gradient

Key-Analysis Errors

Key-Analysis Errors: Schematic

Lorenz System

Discrete Lorenz System: Tangent Linear Approximation Discrete Lorenz System: Tangent Linear Lorenz System:

Discrete Lorenz System: Tangent Linear Approximation

Discrete Lorenz System: The Adjoint Discrete adjoint Lorenz system:

Testing the Tangent Linear Hypothesis: Lorenz Model Hypothesis: G: Nonlinear Model R: Tangent Linear Propagator

Experimental Design First the Lorenz model is run for a long period. This forecast serves a truth (nature). Then, the Lorenz model is run from the truth using slightly perturbed initial conditions. This is meant to mimic analysis error. Random number were used to generate the initial conditions. To mimic model error the model parameters of the Lorenz model were perturbed and weather forecasts with the erroneous model were carried out using perfect initial conditions.

Sensitivity Gradients

Analysis vs. Key-Analysis Errors Key-Analysis ErrorsAnalysis Errors

Analysis vs. Key-Analysis Errors

Sensitivity: Initial vs Model Error Initial Error OnlyModel Error Only

Sensitivity: Initial vs Model Error

Cost Function Reduction FC Error SFC Error

Forecast Errors

Skill: Regular vs. Sensitivity Forecast

Conclusions I Sensitivity gradients are being determined by integrating the short-term forecast error backwards in time using the adjoint model. It is possible to determine the sensitivity of the short-term forecast error w.r.t. the initial conditions and model tendencies. Key-analysis errors are obtained through minimization of the cost function. They are supposed to represent the part of the analysis and model errors, respectively, which are largely responsible for the forecast error.

Conclusions II The sensitivity technique has been further illustrated using the Lorenz system. For this simple system it has been shown that key-analysis errors reflect growing parts of analysis errors. A cautionary example has been presented showing that key-analysis errors might be misleading if model errors significantly contribute to forecast errors.

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Interpretation of Sensitivity Gradients Therefore, if the analysis error has a white spectrum in the expansion, then the sensitivity pattern is dominated by the singular vectors with largest amplification factors.