Calibrating river bathymetry via image to depth quantile transformation

Abstract:

Remote sensing has emerged as a powerful means of measuring river depths, but standard algorithms such as Optimal Band Ratio Analysis (OBRA) require field measurements to calibrate image-derived estimates. Such reliance upon field-based calibration undermines the advantages of remote sensing. This study introduces an alternative approach based on the probability distribution of depths $d$ within a reach. Provided a quantity $X$ related to $d$ can be derived from a remotely sensed data set, image-to-depth quantile transformation (IDQT) infers depths throughout the image by linking the cumulative distribution function (CDF) of $X$ to that of $d$. The algorithm involves determining, for each pixel in the image, the CDF value for that particular value of $X/\bar{X}$ and then inferring the depth at that location from the inverse CDF of the scaled depths $d/\bar{d}$, where the overbar denotes a reach mean. For $X/\bar{X}$, an empirical CDF can be derived directly from pixel values or a probability distribution fitted. Similarly, the CDF of $d/\bar{d}$ can be obtained from field data or from a theoretical model of the frequency distribution of $d$ within a reach; gamma distributions have been used for this purpose. In essence, the probability distributions calibrate $X$ to $d$ while the image provides the spatial distribution of depths. IDQT offers a number of advantages: 1) direct field measurements of $d$ during image acquisition are not absolutely necessary; 2) because the $X$ vs.\ $d$ relation need not be linear, negative depth estimates along channel margins and shallow bias in pools are avoided; and 3) because individual pixels are not linked to specific depth measurements, accurate geo-referencing of field and image data sets is not critical. Application of OBRA and IDQT to a gravel-bed river indicated that the new, probabilistic algorithm was as accurate as the standard, regression-based approach and lead to more hydraulically reasonable bathymetric maps.