This repository contains a Python implementation of a denoising algorithm for Sea Surface Height (SSH) data, particularly from SWOT (Surface Water and Ocean Topography) satellite observations. The algorithm uses the LBFGS optimization method to minimize a cost function that includes Laplacian and third-order regularization terms. This method helps to remove noise from SSH images, improving data quality for further analysis.

To run the code, you need the following Python libraries:

• jax
• jax.numpy
• jaxopt
• pandas
• matplotlib
• netCDF4
• xarray

You can install these libraries using pip:

pip install jax jaxopt pandas matplotlib netCDF4 xarray

Import Packages: The code begins with importing necessary packages for computation, data manipulation, and visualization.

Reading Data: The read_data function reads SSH data from a NetCDF file. This function requires the filename and the variables to be read as input arguments.

Adding Noise: The add_gaussian_noise function adds Gaussian noise to the original SSH image to simulate noisy observations.

Computing Derivatives and Laplacian:

• derivative function computes partial derivatives using a second-order centered scheme.
• gradient function computes the gradient of the input field.
• laplacian function computes the Laplacian of the input field.
• third_order_terms function calculates third-order terms inspired by the Quasi-Geostrophic model.

Cost Functions:

• cost_function calculates the cost function with Laplacian regularization.
• cost_function_third_order_terms calculates the cost function with both Laplacian and third-order regularization terms.

RMSE Calculation: The rmse function computes the Root Mean Square Error (RMSE) between two images.

Plotting: The splot function visualizes the SSH data, providing options to plot the