As discussed in section4.2, the singularity exponents provide information about the geometry of the underlying flow and thus should be independent of the particular scalar from which they are derived, i.e., they are basically related to a component of the signal which is common to all ocean scalars (the advection term). In practice, due to errors, noise and limitations of the algorithm used to estimate the singularity exponents, certain singularity exponents associated with the streamlines of the flow are lost and the correspondence is not perfect (Turiel et al., 2009). Previous studies show that the singularity fronts (bright white lines) of high-quality ocean surface remote sensing maps are highly aligned with altimeter-derived surface currents.
An example of the comparison of singularity lines from ADT(left) and SST(right) with OSCAR
velocity product (Figure 6.17), shows an accurate qualitatively correspondence with the flow streamlines: OSCAR velocities align well with the lines of singularity exponents, especially for the case of ADT derived singularity exponents, as OSCAR product is mainly nourished of altimeter information. The singularity maps shown in Turiel et al. (2009) and in Figure 6.17 right are certainly similar to those in Figure 6.17 left, indicating that AVISO ADT is indeed a geophysically consistent product.
Figure 6.17: Singularity maps derived from AVISO ADT (left) and AVHRR SST (right) for September 11th 2012, both compared with OSCAR velocities for the same date (red arrows).
Maps of singularity exponents (calculated as inTuriel et al.(2008b)) generated from SMOS L3, AVISO ADT, AVHRR SST, L4-ADT and L4-SST maps are shown in Figure 6.18 for the Gulf Stream region in September 11 and in Figure6.19 for the Brazil Current in November 7, 2012.
As revealed by its singularity map, the SMOS L3 product (Figures 6.18and 6.19b) is quite noisy and leads to unstructured singularity maps, where only few parts of the large-scale struc-tures (such as the salinity front associated with the Gulf Stream) can be recognized and many sampling artifacts are noticeable.
The singularity maps of the L4-ADT product (Figure 6.18c and 6.19c) show that the fu-sion algorithm is able to restore much of the singularity structure present in the ADT map (Figures 6.18e and 6.19e). Similarly, the L4-SST singularity maps (Figures 6.18d and 6.19d) incorporates much of the singularity structure of the SST singularity maps (Figure 6.18f and 6.19f). The singularity exponents of L4-ADT and L4-SST are thus better defined than in the L3 map but are not as rich as in the original ADT or SST maps, due to the known limitations of the present fusion algorithm. Both in the Gulf Stream and more notably in the Brazil current area, the singularity exponents of both L4 products are really close, able to resolve the large-scale western boundary currents structures with their meandering and associated ring structures. The analysis of singularity exponents also demonstrates that the extrapolated areas in the Level 4 products are geophysically consistent.
Figure 6.18: Singularity spectra for L3 and L4 products corresponding to the Gulf Stream area days during the year 2012 (a). Maps of singularity exponents derived from L3 (b), L4-ADT (c), L4-SST (d), AVISO ADT (e) and AVHRR SST (f) in September 11 2012.
Going further in the characterization of the signals through the analysis of their singularity exponents, we can fully characterize the statistical behavior of changes of scale in multifractal systems by means of the so-called singularity spectrum (Parisi and Frisch(1985);Frisch(1995);
Pont et al.(2009)). The singularity spectrum of a multifractal system, D(h), is a scale-invariant function that for each singularity exponent informs us about the fractal dimension of the as-sociated fractal component in the multifractal hierarchy. The fractal components are sets of points at which the variable under study presents different scaling properties.The multifractal spectrum can be calculated from the distribution of the singularity exponents at a given scale
Figure 6.19: Singularity spectra for L3 and L4 products corresponding to the Brazil Current area during the year 2012 (a).
Maps of singularity analysis exponents derived from L3 (b), L4-ADT (c), L4-SST (d), AVISO ADT (e) and AVHRR SST (f) in the November 7 2012.
(Pont et al., 2009). Its shape is linked to the energy cascade dissipation (Turiel et al., 2008b) and it should be the same for all parameters advected by the same flow considering it is as-sumed that all of them have the same underlying multifractal hierarchy. The computation of the singularity spectra is done as inTuriel et al.(2006): the empirical histogram of the values of singularity exponents h(~x) is log-transformed and normalized to compute D(h). The singularity spectra associated with the products used in this chapter can be found in Figures 6.18a and 6.19a. The method also allows evaluating the errorbars associated with sampling size (note that errorbars are shown in Figures6.18a and6.19a). These errorbars do not account though for the
uncertainty associated with the method itself, which is estimated to be about 0.1 (Turiel et al., 2006,2008b).
The right part of the spectrum (positive singularity exponents) corresponds to the less sin-gular values, which are represented using dark colors in the maps of Figures6.18b-f and6.19b-f.
Large differences exist among the SSS products in this part of the spectrum as a result of the less singular (i.e., positive) values are associated with the more regular and smooth parts of the function, and are therefore more sensitive to noise and artifacts. The left part of the spectrum corresponds to the most singular (negative) values, corresponding to the brightest (whitest) ar-eas in the maps of singularity exponents of Figures6.18b-f and6.19b-f. Negative values identify regions of abrupt changes in the flow, such as fronts or filaments, but also, as mentioned earlier, potential sampling artifacts. In this part of the spectrum, the spectra can present an excess bump just above -0.5 which is the typical signature of sharp edges, e.g. coastlines and, in spe-cific products, artifacts like orbital gaps. To remove this statistical excess of sharp transitions, all points side by side of a boundary are discarded from the statistics.
The spectra of L4-ADT and L4-SST are close on the full range of values of the spectrum, but the difference with L3 is larger than the error bar associated with both the sampling size and the uncertainty in the method used to calculated singularity exponents (which accounts for a horizontal displacement of the full spectra by ±0.1). This means that the singularity exponents of L3 do not follow the same distribution as the fused products. A close inspection of the spectra shows that the spectra associated with L3 products are a bit narrower than those associated with L4-ADT and L4-SST. Notice that the spectrum of L3 in Figure 6.19a is slightly shifted to the left: we know this because of the constraint imposed by translational invariance on the singularity exponent at which the tangent line has a slope exactly equal to 1, see Turiel and Parga (2000) for a deeper discussion. Corrected by this shift, the spectra in Figures 6.18 and 6.19 evidence a relative lack of singularity exponents on the negative part of the singularity spectrum of L3 with respect to the spectra of L4-ADT and L4-SST. This means that the many singularity fronts associated with current lines have been destroyed as a result of the noise and only those associated with the most evident features are still present in L3 maps, as is confirmed by the visual inspection of singularity maps.