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This paper presents a solution for efficiently and accurately solving separable least squares problems with multiple datasets. These problems involve determining linear parameters that are specific to each dataset while ensuring that the nonlinear parameters remain consistent across all datasets. A well-established approach for solving such problems is the variable projection algorithm introduced by Golub and LeVeque, which effectively reduces a separable problem to its nonlinear component. However, this algorithm assumes that the datasets have equal sizes and identical auxiliary model parameters. This article is motivated by a real-world remote sensing application where these assumptions do not apply. Consequently, we propose a generalized algorithm that extends the original theory to overcome these limitations. The new algorithm has been implemented and tested using both synthetic and real satellite data for atmospheric carbon dioxide retrievals. It has also been compared to conventional state-of-the-art solvers, and its advantages are thoroughly discussed. The experimental results demonstrate that the proposed algorithm significantly outperforms all other methods in terms of computation time, while maintaining comparable accuracy and stability. Hence, this novel method can have a positive impact on future applications in remote sensing and could be valuable for other scientific fitting problems with similar properties. [ABSTRACT FROM AUTHOR]

Time series from Landsat and Sentinel-2 satellites have great potential for modeling vegetation seasonality. However, irregular time sampling and frequent data loss due to clouds, snow, and short growing seasons, makes this modeling a challenge. We describe a new method for modeling seasonal vegetation index dynamics from satellite time series data. The method is based on box constrained separable least squares fits to logistic model functions combined with seasonal shape priors. To enable robust estimates, we extract a base level (i.e., the minimum dormant season value) from the frequency distribution of clear-sky vegetation index values. A seasonal shape prior is computed from several years of data, and in the final fits local parameters are box constrained. More specifically, if enough data values exist in a certain time period, the corresponding local parameters determining the shape of the model function over this period are relaxed and allowed to vary freely. If there are no observations in a period, the corresponding local parameters are locked to the parameters of the shape prior. The method is flexible enough to model interannual variations, yet robust enough when data are sparse. We test the method with Landsat, Sentinel-2, and MODIS data over a forested site in Sweden, demonstrating the feasibility and potential of the method for operational modeling of growing seasons. [ABSTRACT FROM AUTHOR]