Retrieval theory

This page describes the theoretical background of the inverse modelling capabilities implemented in JURASSIC. The retrieval framework follows the principles of optimal estimation and provides a consistent approach to deriving atmospheric state variables from infrared radiance measurements.

The implementation closely follows the formalism described by Rodgers (2000) and is fully integrated with the JURASSIC forward radiative transfer model.

Forward and inverse problems

In atmospheric remote sensing, the forward problem consists of computing radiances from a given atmospheric state using a radiative transfer model. In JURASSIC, this is performed by the forward model described in the Theory section.

The inverse problem aims to estimate the atmospheric state from a set of measured radiances. This problem is generally ill-posed because:

the measurements are finite and noisy,
the atmospheric state is high-dimensional,
different atmospheric states can produce similar radiances.

Optimal estimation provides a statistically well-defined framework to address these challenges.

State vector and measurement vector

State vector

The atmospheric state to be retrieved is represented by a state vector \(\mathbf{x}\). Depending on the application, the state vector may include:

temperature at selected altitude levels,
volume mixing ratios of one or more trace gases,
additional parameters such as background continua or scaling factors.

The mapping between the physical atmospheric representation used by JURASSIC and the state vector is handled internally by state-mapping operators.

Measurement vector

The measurement vector \(\mathbf{y}\) contains the observed radiances (or brightness temperatures) for the selected spectral channels and observation geometries.

Measurement noise and forward-model uncertainties are represented by a measurement error covariance matrix.

Forward model operator

The forward model is represented by an operator \(\mathbf{F}\) such that:

\[ \mathbf{y} = \mathbf{F}(\mathbf{x}) \]

In JURASSIC, \(\mathbf{F}\) corresponds to the radiative transfer model, including ray tracing, spectral approximations, and emissivity lookup tables.

For retrieval applications, the forward model must be differentiable with respect to the state vector. JURASSIC therefore provides analytic or semi-analytic Jacobians (also referred to as weighting functions or kernels).

Jacobians and sensitivity matrices

The Jacobian matrix \(\mathbf{K}\) describes the sensitivity of the measurements to changes in the state vector:

\[ K_{ij} = \frac{\partial y_i}{\partial x_j} \]

In JURASSIC, Jacobians are computed consistently with the forward model using the same numerical discretization and spectral approximations. This consistency is essential for stable and physically meaningful retrievals.

Jacobians can be computed for individual atmospheric quantities, such as temperature or trace gas concentrations, and are reused within the iterative retrieval algorithm.

Optimal estimation framework

Optimal estimation formulates the retrieval as the minimization of a cost function that combines measurement information and prior knowledge:

\[ J(\mathbf{x}) = (\mathbf{y} - \mathbf{F}(\mathbf{x}))^T \mathbf{S}_\epsilon^{-1} (\mathbf{y} - \mathbf{F}(\mathbf{x})) + (\mathbf{x} - \mathbf{x}_a)^T \mathbf{S}_a^{-1} (\mathbf{x} - \mathbf{x}_a) \]

where:

\(\mathbf{x}_a\) is the a priori state vector,
\(\mathbf{S}_a\) is the a priori covariance matrix,
\(\mathbf{S}_\epsilon\) is the measurement error covariance matrix.

The solution represents a compromise between fitting the measurements and remaining consistent with prior information.

Iterative solution and linearization

Because the forward model is generally non-linear, the retrieval problem is solved iteratively. At each iteration, the forward model is linearized around the current state estimate:

\[ \mathbf{F}(\mathbf{x}) \approx \mathbf{F}(\mathbf{x}_i) + \mathbf{K}_i (\mathbf{x} - \mathbf{x}_i) \]

JURASSIC employs a Levenberg–Marquardt-type scheme to ensure numerical stability and convergence. The damping parameter balances between Gauss–Newton and gradient-descent behaviour.

Iterations continue until convergence criteria based on state updates and cost-function changes are satisfied.

Averaging kernels and information content

The averaging kernel matrix \(\mathbf{A}\) describes the sensitivity of the retrieved state to the true atmospheric state:

\[ \mathbf{A} = \frac{\partial \hat{\mathbf{x}}}{\partial \mathbf{x}} \]

Averaging kernels provide insight into:

vertical resolution of the retrieval,
sensitivity to true atmospheric variations,
influence of the a priori information.

JURASSIC computes averaging kernels and related diagnostic quantities that can be used to assess retrieval performance and information content.

Error characterization

The retrieval error covariance matrix \(\mathbf{S}_x\) quantifies the expected uncertainty of the retrieved state:

\[ \mathbf{S}_x = (\mathbf{K}^T \mathbf{S}_\epsilon^{-1} \mathbf{K} + \mathbf{S}_a^{-1})^{-1} \]

This matrix accounts for both measurement noise and the imposed prior constraints. JURASSIC provides access to retrieval error estimates that can be used for scientific interpretation and validation.

Practical considerations

When performing retrievals with JURASSIC, users should consider:

appropriate choice of a priori state and covariance,
realistic measurement error estimates,
consistency between forward-model configuration and retrieval setup,
potential correlations between retrieved parameters.

Poorly chosen prior information or inconsistent configurations can lead to biased or unstable retrievals.

Summary

JURASSIC implements a physically consistent and well-established optimal estimation retrieval framework that is tightly integrated with its forward radiative transfer model. The use of analytic Jacobians and shared numerical infrastructure ensures stable and efficient retrieval calculations.

For a detailed theoretical treatment of optimal estimation, users are referred to Rodgers (2000) and the references listed in the References section.