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<h1>Inverse Problem & Retrieval Theory</h1>
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<section>
<h3>Introduction and Objective</h3>
<p>The goal of this section is to formally describe how atmospheric CO₂ is retrieved from measured satellite spectra using inverse theory.</p>
In <a href="co2_1.html" target="_blank">Part I: Fundamental Radiometric Quantities</a>, we derived the forward radiative transfer model:
$$
I_\nu = I_{\nu,0} \exp(-\tau_\nu(x))
$$
<p>This equation describes how atmospheric properties determine the measured radiance. However, satellite retrieval operates in the opposite direction:</p>
<p><b>We measure radiance \( I_\nu \) and seek to determine the atmospheric state that produced it.</b></p>
<p>This constitutes an inverse problem.</p>
<p><b>So fundamental retrival question is</b>
how from a given vector of measured radiance \(y\) and a radiative trasfer model \(F\),
we calculate atmospheric state vecotr \(x\), such that
$$y = F(x)$$
</p>
<p>For CO₂ missions like MicroCarb, the primary quantity of interest is: </p>
$$
\text{XCO}_2 = \frac{\int n_{\text{CO}_2}(z), dz}{\int n_{\text{dry}\ \text{air}}(z), dz}
$$
This is the column-averaged dry air mole fraction of CO₂. The inversion is not trivial because:
<ul>
<li>The forward model \( F(x) \) is nonlinear.</li>
<li>Measurements contain noise.</li>
<li>Multiple atmospheric parameters influence the spectrum simultaneously.</li>
<li>Different parameters may produce similar spectral effects (degeneracy).</li>
<li>The problem can be ill-conditioned.</li>
</ul>
Thus, direct inversion is unstable.
<p><b>Retrieval as a Statistical Estimation Problem: </b>Because measurements contain uncertainty, the solution must be formulated probabilistically.
Instead of solving:
</p>
$$
y = F(x)
$$
we solve:
$$
y = F(x) + \epsilon
$$
where \( \epsilon \) represents measurement error. The goal becomes:
<pre><code>Estimate the most probable atmospheric state consistent with the measurements and prior knowledge.</code></pre>
This leads naturally to a Bayesian framework and the Optimal Estimation method.
<p><b>What This Section Will Establish?</b></p>
This section will:
<ul>
<li>Formulate the forward model in vector form</li>
<li>Define the state vector and measurement vector</li>
<li>Linearize the forward model</li>
<li>Derive the Optimal Estimation retrieval equation</li>
<li>Define the gain matrix</li>
<li>Derive the averaging kernel</li>
<li>Derive posterior error covariance</li>
<li>Decompose retrieval error into noise and smoothing components</li>
</ul>
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.
<!-- Section started -->
<h3>Forward Model Formulation</h3>
<p>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
<ul>
<li>\(x =\) atmospheric state vector,</li>
<li>\(y=\) measurement vector (radiances),</li>
<li>\(F=\)radiative transfer operator.</li>
</ul>
</p>
<p>The forward model must include:</p>
<ul>
<li>Radiative transfer physics</li>
<li>Atmospheric composition</li>
<li>Scattering processes</li>
<li>Surface reflection</li>
<li>Instrument effects</li>
</ul>
<!-- subsection -->
<h5>Measurement Geometry</h5>
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:
<ul>
<li>Sun → atmosphere (downward path)</li>
<li>Surface reflection</li>
<li>Atmosphere (upward path) → satellite</li>
</ul>
Thus total optical depth is two-way.
<!-- subsection -->
<h5>Monochromatic Radiance Expression</h5>
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:
<ul>
<li>\( \mu_s = \cos\theta_s \)</li>
<li>\( E_\nu^{sun} \) = solar irradiance</li>
<li>\( T_\nu^{down} \) = transmission along downward path</li>
<li>\( T_\nu^{up} \) = transmission along upward path</li>
<li>\( R_\nu^{surf} \) = surface reflectance</li>
<li>\( I_\nu^{path} \) = atmospheric path radiance (scattering contribution)</li>
</ul>
<!-- subsection -->
<h5>Transmission Terms</h5>
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:
$$
T_\nu = \exp\left(-\left(\frac{1}{\mu_s} + \frac{1}{\mu_v}\right) \int_0^\infty \beta_\nu(z) dz\right)
$$
<!-- Subsection -->
<h5>Optical Depth Components</h5>
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:
$$
\tau_\nu^{CO_2} = \int_0^\infty n_{CO_2}(z) \sigma_\nu^{CO_2}(T,p) dz
$$
<!-- subsection -->
<h5>Surface Reflectance Model</h5>
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.
<h5>Full Forward Model Expression</h5>
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.
<!-- section -->
<h3>Vector Form of Forward Model</h3>
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 \( x_j\).
<h3>Bayesian Formulation</h3>
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:
<ul>
<li>Likelihood: \( P(y|x) \)</li>
<li>Prior: \( P(x) \)</li>
</ul>
<p><b>Assume Gaussian Statistics:</b></p>
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:
$$
P(x) \propto
\exp
\left(
-\frac{1}{2}
(x - x_a)^T
S_a^{-1}
(x - x_a)
\right)
$$
<h4>Cost Function Derivation</h4>
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.
<h4>Derivation of Retrieval Equation (Linear Case)</h4>
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:
$$
J =
(\delta y - K\delta x)^T
S_\epsilon^{-1}
(\delta y - K\delta x)
+
\delta x^T
S_a^{-1}
\delta x
$$
<h5>Minimize Cost Function</h5>
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.
<h4>Gain Matrix</h4>
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.
<h4>Averaging Kernel Derivation</h4>
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:
<ul>
<li>If \( A = I \) → perfect retrieval</li>
<li>If \( A < I \) → smoothed by prior</li>
</ul>
<h4>Degrees of Freedom for Signal (DOFS)</h4>
<h5>Instrument Effects</h5>
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.
<div class="important-box">
<h3>Final Forward Model Definition</h3>
Complete forward operator:
$$
y = \mathcal{C}
\left(
\mathcal{R}(x)
\right)
$$
Where:
<ul>
<li>\( \mathcal{R} \) = radiative transfer operator</li>
<li>\( \mathcal{C} \) = convolution with instrument response</li>
</ul>
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.
</div>
<br>
<p></p>
<p></p>
<h3>Reference</h3>
<ul>
<li><a href="https://arunp77.github.io/Remote-sensing.html" target="_blank">Fundamentals of Remote sensing</a></li>
<li><a href="https://arunp77.github.io/electromagnetic-waves.html" target="_blank">Relevance of Electromagnetic waves in the context of earth observation</a></li>
<li><a href="https://arunp77.github.io/Remote-sensing-content.html" target="_blank">Concept of the orbits for a satellite (non scientific discussion)</a></li>
<li><a href="https://arunp77.github.io/Ground-segment-systems.html" target="_blank">How various teams works in close collaboration for the ground data processing?</a></li>
<li><a href="https://arunp77.github.io/Satellite-data.html" target="_blank">How raw satellite data is processed to do a level where you do your scientifc research?</a></li>
<li><a href="https://arunp77.github.io/toa-reflectance.html" target="_blank">In depth understandingof the satellite data (op-of-atmosphere reflectance)</a></li>
<li><a href="https://arunp77.github.io/Satellites-sensors.html" target="_blank">Resolution and calibration</a></li>
<li><a href="https://arunp77.github.io/OLCI.html" target="_blank">Understanding how OLCI data is processed</a></li>
<li><a href="https://www.nesdis.noaa.gov/news/transforming-energy-imagery-how-satellite-data-becomes-stunning-views-of-earth" target="_blank">Transforming Energy into Imagery: How Satellite Data Becomes Stunning Views of Earth</a></li>
</ul>
</section>
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