Multifractals and resolution-independent remote sensing algorithms: the example of ocean colour

We argue that geophysical and geographical fields are generally characteris ed by wide range scaling implying systematic, strong (power law) resolution dependencies when they are remotely sensed. The corresponding geometric st ructures are fractal sets; the corresponding fields are multifractals. Math ematically, multifractals are measures that are singular with respect to th e standard Lebesgue measures; therefore, they are outside the scope of many of the methods of classical geostatistics. Because the resolution of a mea surement is generally (due to technical constraints) much larger than the i nner scale of the variability/scaling; the observations will be fundamental ly observer dependent; hence, standard remote sensing algorithms that do no t explicitly take this dependence into account will depend on subjective re solution-dependent parameters. We argue that, on the contrary, the resoluti on dependence must be systematically removed so that scale-invariant algori thms independent of the observer can be produced. We illustrate these ideas in various ways with the help of eight-channel, 7 m resolution remote ocea n colour data (from the MIES II sensor) over the St Lawrence estuary. First , we show that the data is indeed multiscaling over nearly four orders of m agnitude in scale, and we quantify this using universal multifractal parame ters. With the help of conditional multifractal statistics, we then show ho w to use multifractals in various practical ways such as for extrapolating from one resolution to another or from one location to another, or to corre cting biases introduced when studying extreme, rare phenomena. We also show how the scaling interrelationship of surrogate and in situ data can be han dled using vector multifractals and examine the resolution dependence of pr inciple components in dual wavelength analyses. Finally, we indicate why th e standard ocean colour algorithms have hidden resolution dependencies. and we show how they can (at least in principle) be removed.