Methods of Analysis

Lesson

Theoretical estimation

The Double Trace Moment DTM developed by [Lavallée, 1991]^[1] is a generalization of the Trace Moment technique, this method was introduced in order to estimate the UM parameter . We can present the DTM technique in two steps: the first step consist to arise the conservative field to the power at the highest resolution, then the obtained field is normalized. We calculate the statistical moment for the different resolution , for different and different moments :

$\left\langle \left(\varepsilon^{(\eta)}_{\lambda}\right)^{q}\right\rangle \approx \frac{\left\langle \left(\varepsilon_{\lambda}\right)^{\eta q}\right\rangle}{\left\langle \left(\varepsilon_{\lambda}\right)^{\eta}\right\rangle^{q}}\left\langle \left(\varepsilon_{\Lambda}\right)^{\eta}\right\rangle^{q} \approx \frac{\lambda^{K(q\eta)}}{\lambda^{qK(\eta)}}\left\langle \left(\varepsilon_{\Lambda}\right)^{\eta}\right\rangle^{q}\approx \lambda^{K(q,\eta)}\\ \quad \\ \left\langle \left(\varepsilon^{(\eta)}_{\lambda}\right)^{q}\right\rangle \approx \lambda^{K(q \eta) - qK(\eta)}$

Using the universality of the function (theoretical expression), one can obtain the expression of :

$K(q,\eta) = \eta^{\alpha}K(q)$

The next step consist on the estimation of the UM parameters, To do that we represent the as a function of in log-log plot for different and for fixed , this allows to estimate the value of :

Illustration of the estimation of the UM parameter using the DTM technique

For we can obtain:

[Lavallee, 1991]
Lavallee, D., Lovejoy, S., & Schertzer, D. (1991, November). Universal multifractal theory and observations of land and ocean surfaces, and of clouds. In San Diego,'91, San Diego, CA (pp. 60-75). International Society for Optics and Photonics.

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