Inverse Problem & Retrieval Theory

Introduction and Objective

The goal of this section is to formally describe how atmospheric CO₂ is retrieved from measured satellite spectra using inverse theory.

In Part I: Fundamental Radiometric Quantities, we derived the forward radiative transfer model: $$ I_\nu = I_{\nu,0} \exp(-\tau_\nu(x)) $$

This equation describes how atmospheric properties determine the measured radiance. However, satellite retrieval operates in the opposite direction:

We measure radiance \( I_\nu \) and seek to determine the atmospheric state that produced it.

This constitutes an inverse problem.

So fundamental retrival question is how from a given vector of measured radiance \(y\) and a radiative trasfer model \(F\), we calculate atmospheric state vecotr \(x\), such that $$y = F(x)$$

For CO₂ missions like MicroCarb, the primary quantity of interest is:

$$ \text{XCO}_2 = \frac{\int n_{\text{CO}_2}(z), dz}{\int n_{\text{dry}\ \text{air}}(z), dz} $$ This is the column-averaged dry air mole fraction of CO₂. The inversion is not trivial because:
  • The forward model \( F(x) \) is nonlinear.
  • Measurements contain noise.
  • Multiple atmospheric parameters influence the spectrum simultaneously.
  • Different parameters may produce similar spectral effects (degeneracy).
  • The problem can be ill-conditioned.
Thus, direct inversion is unstable.

Retrieval as a Statistical Estimation Problem: Because measurements contain uncertainty, the solution must be formulated probabilistically. Instead of solving:

$$ y = F(x) $$ we solve: $$ y = F(x) + \epsilon $$ where \( \epsilon \) represents measurement error. The goal becomes: 
Estimate the most probable atmospheric state consistent with the measurements and prior knowledge.
This leads naturally to a Bayesian framework and the Optimal Estimation method.

What This Section Will Establish?

This section will:
  • Formulate the forward model in vector form
  • Define the state vector and measurement vector
  • Linearize the forward model
  • Derive the Optimal Estimation retrieval equation
  • Define the gain matrix
  • Derive the averaging kernel
  • Derive posterior error covariance
  • Decompose retrieval error into noise and smoothing components
By the end of this section, we will have a complete mathematical description of how CO₂ is retrieved from measured spectra and how the information content of the retrieval can be quantified.

Forward Model Formulation

The forward model describes how the atmospheric state determines the radiance measured by the satellite instrument. In retrieval problems, we define $$y = F(x)$$ where

  • \(x =\) atmospheric state vector,
  • \(y=\) measurement vector (radiances),
  • \(F=\)radiative transfer operator.

The forward model must include:

  • Radiative transfer physics
  • Atmospheric composition
  • Scattering processes
  • Surface reflection
  • Instrument effects
Measurement Geometry
For reflected solar missions (SWIR CO₂ retrieval), radiance measured at the sensor depends on Solar zenith angle \(\theta_s\), \(\theta_v\), Relative azimuth angle \(\phi\). Therefore radiance is a function of: $$I_\nu = I_\nu(\theta_s, \theta_v, \phi)$$ Photon path consists of:
  • Sun → atmosphere (downward path)
  • Surface reflection
  • Atmosphere (upward path) → satellite
Thus total optical depth is two-way.
Monochromatic Radiance Expression
Ignoring polarization for now, the radiance at top of atmosphere (TOA) can be written as: $$ I_\nu^{\rm TOA} = \frac{\mu_s}{\pi} E_\nu^{\rm sun}T_\nu^{\rm down}R_\nu^{\rm surf}T_\nu^{\rm up} + I_\nu^{\rm path} $$ where:
  • \( \mu_s = \cos\theta_s \)
  • \( E_\nu^{sun} \) = solar irradiance
  • \( T_\nu^{down} \) = transmission along downward path
  • \( T_\nu^{up} \) = transmission along upward path
  • \( R_\nu^{surf} \) = surface reflectance
  • \( I_\nu^{path} \) = atmospheric path radiance (scattering contribution)
Transmission Terms
Transmission is exponential of optical depth: $$ T_\nu^{down} = \exp(-\tau_\nu^{down}) $$ $$ T_\nu^{up} = \exp(-\tau_\nu^{up}) $$ For slant geometry: $$ \tau_\nu^{down} = \frac{1}{\mu_s} \int_0^\infty \beta_\nu(z) dz $$ $$ \tau_\nu^{up} = \frac{1}{\mu_v} \int_0^\infty \beta_\nu(z) dz $$ Thus total transmission: $$ T_\nu = \exp\left(-\left(\frac{1}{\mu_s} + \frac{1}{\mu_v}\right) \int_0^\infty \beta_\nu(z) dz\right) $$
Optical Depth Components
Extinction coefficient: $$ \beta_\nu(z) = \sum_g n_g(z) \sigma_{\nu,g}(T,p) + \beta_\nu^{aer}(z) + \beta_\nu^{cloud}(z) $$ So optical depth becomes: $$ \tau_\nu = \tau_\nu^{gas} + \tau_\nu^{aer} + \tau_\nu^{cloud} $$ Gas optical depth: $$ \tau_\nu^{gas} = \sum_g \int_0^\infty n_g(z) \sigma_{\nu,g}(T,p) dz $$ For CO₂ retrieval, we isolate the CO₂ term: $$ \tau_\nu^{CO_2} = \int_0^\infty n_{CO_2}(z) \sigma_\nu^{CO_2}(T,p) dz $$
Surface Reflectance Model
Surface reflectance may be modeled as: Lambertian: $$ R_\nu^{surf} = A_\nu $$ Or BRDF model: $$ R_\nu^{surf} = f(\theta_s, \theta_v, \phi) $$ This becomes part of state vector.
Full Forward Model Expression
Putting all terms together: $$ I_\nu^{TOA} = \frac{\mu_s}{\pi} E_\nu^{sun} A_\nu \exp(-\tau_\nu^{down} - \tau_\nu^{up}) + I_\nu^{path} $$ In simplified pure absorption case (neglecting path radiance): $$ I_\nu^{TOA} = C_\nu \exp(-\tau_\nu^{total}) $$ Where: $$ C_\nu = \frac{\mu_s}{\pi} E_\nu^{sun} A_\nu $$ This is often sufficient for conceptual retrieval explanation.

Vector Form of Forward Model

We write: \(y =F(x)+\epsilon\), where \(F(x)\) is the non-linear radiative transfer model and \(\epsilon\) is the measurement noise. This is a nonlinear inverse problem. Discretize wavelengths: $$ y = \begin{bmatrix} I_{\nu_1} \\ I_{\nu_2} \\ \vdots \\ I_{\nu_m} \end{bmatrix} $$ State vector: $$ x = \begin{bmatrix} \text{XCO}_2 \\ \text{Aerosol parameters} \\ \text{Surface albedo} \\ \text{Temperature scaling} \\ \text{Wavelength shift} \end{bmatrix} $$ Here dimension of this matrix is: \(n\) Define nonlinear operator: $$ F(x) = \begin{bmatrix} F_{\nu_1}(x) \\ F_{\nu_2}(x) \\ \vdots \\ F_{\nu_m}(x) \end{bmatrix} $$ Here dimension of this matrix is: \(m\) and usually \(m \gg n \). We expand around a reference state \(x_a\): $$F(x)\approx F(x_a) + K(x-x_a).$$ Where Jacobian matrix: $$K = \frac{\partial F}{\partial x}~~~~~~~~~~~~ (\text{Dimension}: K \in \mathbb{R}^{m \times n})$$ Each elements $$K_{ij} = \frac{\partial F_i}{\partial x_j}$$ physically it means Sensitivity of radiance at wavelength \(\nu_i\) to state parameter \( x_j\).

Bayesian Formulation

Bayesian Formulation is one of the best method to evaluate \(x\). In this case, we compute posterior probability \(P(x|y)\). Now using the Bayes theorem: $$P(x|y) \propto P(y|x) P(x)$$ where:
  • Likelihood: \( P(y|x) \)
  • Prior: \( P(x) \)

Assume Gaussian Statistics:

Measurement error: $$ \epsilon \sim \mathcal{N}(0, S_\epsilon) $$ Thus likelihood: $$ P(y|x) \propto \exp \left( -\frac{1}{2} (y - F(x))^T S_\epsilon^{-1} (y - F(x)) \right) $$ Prior: $$ P(x) \propto \exp \left( -\frac{1}{2} (x - x_a)^T S_a^{-1} (x - x_a) \right) $$

Cost Function Derivation

Posterior maximization = minimize negative log posterior. Define cost function: $$ J(x) = (y - F(x))^T S_\epsilon^{-1} (y - F(x)) + (x - x_a)^T S_a^{-1} (x - x_a) $$ This is the Optimal Estimation cost function.

Derivation of Retrieval Equation (Linear Case)

Assume linear model: $$ F(x) = F(x_a) + K(x - x_a) $$ Define: $$ \delta x = x - x_a $$ $$ \delta y = y - F(x_a) $$ So: $$ \delta y = K \delta x + \epsilon $$ Insert into cost function: $$ J = (\delta y - K\delta x)^T S_\epsilon^{-1} (\delta y - K\delta x) + \delta x^T S_a^{-1} \delta x $$
Minimize Cost Function
Take derivative wrt \( \delta x \): $$ \frac{\partial J}{\partial \delta x} = -2K^T S_\epsilon^{-1} (\delta y - K\delta x)+ 2 S_a^{-1} \delta x $$ Set equal to zero: $$ K^T S_\epsilon^{-1} (\delta y - K\delta x)= S_a^{-1} \delta x $$ Expand: $$ K^T S_\epsilon^{-1} \delta y - K^T S_\epsilon^{-1} K \delta x = S_a^{-1} \delta x $$ Rearrange: $$ (K^T S_\epsilon^{-1} K + S_a^{-1}) \delta x = K^T S_\epsilon^{-1} \delta y $$ Thus: $$ \delta x = (K^T S_\epsilon^{-1} K + S_a^{-1})^{-1} K^T S_\epsilon^{-1} \delta y $$ Final retrieval: $$ \hat{x} = x_a + (K^T S_\epsilon^{-1} K + S_a^{-1})^{-1} K^T S_\epsilon^{-1}(y - F(x_a)) $$ This is the core retrieval equation.

Gain Matrix

Define gain matrix: $$ G =(K^T S_\epsilon^{-1} K + S_a^{-1})^{-1} K^T S_\epsilon^{-1} $$ Then: $$ \hat{x} = x_a + G (y - F(x_a)) $$ Interpretation: Maps measurement residuals to state corrections.

Averaging Kernel Derivation

Define: $$ A =\frac{\partial \hat{x}}{\partial x} $$ Substitute linear model: $$ A = G K $$ So: $$ A = (K^T S_\epsilon^{-1} K + S_a^{-1})^{-1} K^T S_\epsilon^{-1} K $$ Interpretation:
  • If \( A = I \) → perfect retrieval
  • If \( A < I \) → smoothed by prior

Degrees of Freedom for Signal (DOFS)

Instrument Effects
Real measurement is convolution with instrument line shape (ILS): $$ I_\nu^{\rm meas} = \int I_{\nu'}^{\rm true} G(\nu - \nu') d\nu' $$ Where \( G \) = instrument spectral response. Thus forward model includes convolution step.

Final Forward Model Definition

Complete forward operator: $$ y = \mathcal{C} \left( \mathcal{R}(x) \right) $$ Where:
  • \( \mathcal{R} \) = radiative transfer operator
  • \( \mathcal{C} \) = convolution with instrument response
Expanded: $$ y = \text{ILS} \Big[\frac{\mu_s}{\pi}E_\nu^{\rm sun} A_\nu \exp\left(\tau_\nu^{{\rm CO}_2} \tau_\nu^{\rm aer} \tau_\nu^{\rm cloud}\right) \Big] $$ This is the fully defined forward model.

Reference

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