## 5 Examples of Forward-Inverse Modeling
### Atmospheric Input & Output
#### 5.4.1 The OC-SMART Optimal Estimation Approach  
To address this situation, the OC-SMART (**O**cean **C**olor--**Simultaneous **M**arine and **A**erosol **R**etrieval **T**ool) approach was developed. The goal was to improve retrieval accuracy by use of AccuRT forward modeling and Optimal Estimation/Levenberg-Marquardt (OE/LM) inversion [108]:  
* AccuRT: accurate discrete-ordinates radiative transfer model for the _coupled_ atmosphere-ocean system; delivers a complete set of simulated radiances and Jacobians (weighting functions). * OE/LM inversion: is an iterative, nonlinear least squares cost function minimization with _a priori_ and Levenberg-Marquardt regularization. * For retrievals of aerosol and aquatic parameters from Ocean Color data, we define a 5-element state vector: \[\textbf{x}=\{\tau_{865},f_{a},\textbf{CHL},\textbf{CDM},\textbf{BBP}\}\]  
Figure 6: The ratio of the values for turbid water to those for clear water in Figure 5.  
consisting of
* _2 aerosol parameters_ (\(\tau_{865}=\) optical depth at 865 nm and \(f_{a}=\) bimodal fraction of particles),
* _3 marine parameters_ (chlorophyll concentration, **CHL**, combined absorption by detrital and dissolved material at 443 nm, **CDM**, and backscattering coefficient at 443 nm, **BBP**). At each iteration step, the next estimate of the state vector is given by the OE/LM inversion [70]  
\[\mathbf{x}_{n+1}=\mathbf{x}_{n}+[(1+\gamma_{n})\mathbf{S}_{a}^{-1}+\mathbf{J}_ {n}^{T}\mathbf{S}_{m}^{-1}\mathbf{J}_{n}]^{-1}\{\mathbf{J}_{n}^{T}\mathbf{S}_ {m}^{-1}(\mathbf{y}_{m}-\mathbf{y}_{n})-\mathbf{S}_{a}^{-1}(\mathbf{x}_{n}- \mathbf{x}_{a})\}. \tag{155}\]  
In Equation (155) \(\mathbf{y}_{m}\) is the vector of measured TOA radiances; \(\mathbf{y}_{n}=\mathrm{F}(\mathbf{x}_{n},\mathbf{b})\) is the vector of simulated TOA radiances generated by the AccuRT forward model; \(\mathbf{y}_{n}\) is a (non-linear) function of the state vector \(\mathbf{x}_{n}\) of retrieval elements, and \(\mathbf{b}\) represents model parameters; \(\mathbf{J}_{n}\) is a matrix of simulated radiance partial derivatives with respect to the state vector elements \(\mathbf{x}_{n}\) (the Jacobians); \(\mathbf{x}_{a}\) and \(\mathbf{S}_{a}\) are the _a priori_ state vector and covariance matrix, respectively, and \(\mathbf{S}_{m}\) is the measurement error covariance matrix. \(\gamma_{n}\) is the Levenberg-Marquardt (LM) regularization parameter, When \(\gamma_{n}\to 0\) Equation (155) becomes the standard Gauss-Newton Optimal Estimation (OE), and when \(\gamma_{n}\rightarrow\infty\) it tends to the steepest descent method. AccuRT returns simulated radiances (\(\mathbf{y}_{n}\)) and Jacobians (\(\mathbf{J}_{n}\)) required to update the state vector estimate (\(\mathbf{x}_{n}\)) according to Equation (155) above. One issue with the OC-SMART approach is that it is relatively slow due to the need to call the forward AccuRT model repeatedly in the iterative inversion to compute radiances and Jacobians. To deal with that problem, one could use AccuRT instead to create a training ensemble in order to construct a Radial Basis Function Neural Network. This approach typically leads to an increase in computational speed by a factor of about 1000. In this way one replaces the AccuRT forward model (thousands of lines of code) with the following single equation (similar to Equation (151)):  
\[I_{i}=\sum_{j=1}^{N}a_{ij}\exp[-b\sum_{k=1}^{K}(P_{k}-c_{jk})^{2}]+d_{i}\]  
where \(I_{i}\) it the TOA radiance in channel \(i=1,\dots,8\), \(K=\#\) of input parameters, and \(a_{i,j},b,c_{j,k},d_{i}\) are the coefficients to be optimized in the network training. The Jacobians (partial derivatives) \(\mathbf{J}\) are also required for the non-linear optimal estimation using Equation (155). These Jacobians can be calculated by taking partial derivatives of \(I_{i}\) respect to each input parameter:  
\[J_{k}=\frac{\partial I_{i}}{\partial P_{k}}=-2\sum_{j=1}^{N}a_{ij}b^{2}(P_{k} -c_{jk})\exp\{\sum_{k=1}^{K}(P_{k}-c_{jk})^{2}\}. \tag{156}\]  
[PERSON] et al.