# UVsq-Sat NG, a New CubeSat Pathfinder for Monitoring Earth Outgoing Energy and Greenhouse Gases
## 4 Methodology for Determining ERB and GHGs with Uvsq-Sat NG
Considering that C\({}_{1}\), C\({}_{2}\), C\({}_{3}\), and C\({}_{4}\) are all equal to 0.5, this results in initial corresponding mixing ratios  
Figure 3.0 **(top)** Dimensionless transmittance functions at the UVsq-Sat NG Spectrometer resolution assuming a spectral resolution of 1 nm (FWHM) for the spectrometer. A Gaussian convolution filter is used. **(bottom)** Relative variation between ’maximum’ mixing ratio and nominal mixing ratio. of 425 ppm for CO\({}_{2}\), 1.95 ppm for CH\({}_{4}\), 0.215 for O\({}_{2}\), and 1.05 precipitable cm for H\({}_{2}\)O. * Determination of the Uvsq-Sat NG NIR Spectrometer theoretical normalized dimensionless transmittance function. The Uvsq-Sat NG NIR Spectrometer has its own instrumental characteristics and a SNR that limit its performance. The 'true' dimensionless signal that will be observed (TR\({}_{n}^{Tot}(\lambda)\) in this scenario) will therefore be affected by the observations made via the spectrometer. In order to assess the anticipated performance of the Uvsq-Sat NG NIR Spectrometer, Gaussian noise is subsequently introduced to the simulated spectra (TR\({}_{n}^{Tot}(\lambda)\)) prior to conducting the Levenberg-Marquardt spectral inversion. This inversion is performed while considering an adjustable SNR within the range of 50 to 5000. Equation (5) allows us to determine the dimensionless signal (TR\({}_{Spectro}(\lambda)\)) as it can be observed via the Uvsq-SatNG NIR Spectrometer, considering its spectral resolution (between 1 and 6 nm) and SNR (around 2000, which would correspond to an excellent value). Figure 3.1 (5) where \(\mathcal{N}(0,1)\) represents a random Gaussian noise with a mean of 0 and a standard deviation of 1. * Utilization of the Levenberg-Marquardt algorithm to determine normalized coefficients C\({}_{1}\), C\({}_{2}\), C\({}_{3}\), and C\({}_{4}\), which correspond to the concentrations of the four gases as they could be observed via the Uvsq-Sat NG Spectrometer. From the dimensionless signal (TR\({}_{Spectro}(\lambda)\)), the interpolation coefficients (C\({}_{1}\), C\({}_{2}\), C\({}_{3}\), and C\({}_{4}\)) are determined using the Levenberg-Marquardt algorithm, which is commonly used for solving non-linear least squares problems. Here is a description of the method:
* Initialization: Start with an initial estimate of the interpolation coefficients \(C_{1}\), \(C_{2}\), \(C_{3}\), and \(C_{4}\) (all values are set to 1), as well as the parameter \(\beta\) (regularization factor). Choose a starting value for \(\beta\) (typically a small value). * Residual Calculation: Use the current interpolation coefficients to calculate the model predictions. Then, subtract these predictions from the observed data to obtain the residuals. * Jacobian Matrix Calculation: It represents the partial derivatives of the residuals with respect to the interpolation coefficients. It quantifies how each coefficient contributes to the residuals. * Hessian Matrix Calculation: Calculate the Hessian matrix, which is the second derivative matrix of the residuals with respect to the coefficients. It helps adjust the weighting between the model terms and regularization. * Coefficient Update Calculation: Calculate the coefficient update using the formula of the Levenberg-Marquardt algorithm: Figure 3.2 (6) where \(J\) is the Jacobian matrix and \(R\) is the residuals function. The \(\beta\cdot\text{diag}(J^{T}J)\) helps to enhance and stabilize the matrix, while noise will be incorporated into \(R\). * Evaluation of Improvement: Calculate the potential improvement by comparing the norm of residuals before and after the coefficient update. If the improvement is significant, accept the update. Otherwise, adjust \(\beta\) and repeat. * Iterative Repetition: Repeat stages 2 (Initialization) through 6 (Evaluation of Improvement) until the residuals either converge to a minimum or meet
a stopping criterion. Such criteria can include reaching a maximum number of iterations or observing only minor variations in the coefficients.