Fitting the Multitemporal Curve: A Fourier Series Approach to the Missing Data Problem in Remote Sensing Analysis

Abstract

With the advent of free Landsat data stretching back decades, there has been a surge of interest in utilizing remotely sensed data in multitemporal analysis for estimation of biophysical parameters. Such analysis is confounded by cloud cover and other image-specific problems, which result in missing data at various aperiodic times of the year. While there is a wealth of information contained in remotely sensed time series, the analysis of such time series is severely limited due to the missing data. This paper illustrates a technique which can greatly expand the possibilities of such analyses, a Fourier regression algorithm, here on time series of normalized difference vegetation indices (NDVIs) for Landsat pixels with 30-m resolution. It compares the results with those using the spatial and temporal adaptive reflectance fusion model (STAR-FM), a popular approach that depends on having MODIS pixels with resolutions of 250 m or coarser. STAR-FM uses changes in the MODIS pixels as a template for predicting changes in the Landsat pixels. Fourier regression had an <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">R</i> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> of at least 90% over three quarters of all pixels, and it had the highest <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">R</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Predicted</sub> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> values (compared to STAR-FM) on two thirds of the pixels. The typical root-mean-square error for Fourier regression fitting was about 0.05 for NDVI, ranging from 0 to 1. This indicates that Fourier regression may be used to interpolate missing data for multitemporal analysis at the Landsat scale, especially for annual or longer studies.