[24] Surface elevation increments are found to exhibit variability at all scales. For example, Figure 12 shows the temporal evolution of a given transect (transect A-A in Figure 2) that possesses a Devil's staircase–like structure. Notice that visually, the structure of evolution when viewed at different time scales (shown as inset plots in Figure 12) looks statistically similar. One common way of documenting the self-similar structure of a given time series is to look at the power spectral density of the time series. Figure 13 shows the power spectral density as a function of wavelength for the ensemble of temporal transects along Line 1.75. A power law decay, with an exponent of ϕ = −2.1 documents the presence of statistical scaling in the temporal evolution of elevation time series. The Hurst exponent (*H*) which is a measure of the “roughness” of the time series is related to the spectral density's power law decay exponent as −ϕ = 2*H* + 1 [*Turcotte*, 1992] resulting thus in *H* = 0.55. The fractal dimension *D*_{0} and the Hurst exponent relate as *D*_{0} = 2 − *H* [e.g., *Turcotte*, 1992], leading to an estimated fractal dimension for the elevation time series of *D*_{0} = 1.49.

[25] The power spectral density expresses the scaling of the second-order moment (variance) of the series and completely characterizes the scaling of Gaussian random variables. Since the pdf of elevation increments was documented to significantly deviate from the Gaussian form, it is important to test for scaling in higher-order statistical moments. We performed higher-order structure function analysis of the elevation time series to characterize the statistical scaling of the temporal evolution of the deltaic surface. Elevation increments in time were computed at different scales *r*, denoted by *δh*(*t*, *r*), as

where Δ*t* is the temporal resolution of the experimental data. The estimates of the *q*th-order statistical moments of the absolute values of elevation increments at scale *r*, also called structure functions, *M*(*q*, *r*), are defined as

where *N*_{r} is the number of data points of elevation increments at a scale *r*. Statistical scaling, or scale invariance, requires that *M*(*q*, *r*) be a power law function of the scale

where *ζ*(*q*) is the scaling exponent function. When the scaling exponent function has a linear dependence on the order of the statistical moments, that is, *ζ*(*q*) = *qH*, the series is called monofractal and *H* is the Hurst exponent discussed previously. If the scaling exponent function has a nonlinear dependence on the order of statistical moments then the series is called a multifractal. The simplest way to characterize the nonlinear dependence of *ζ*(*q*) on *q* is by using a quadratic approximation,

where *c*_{1} and *c*_{2} are constants parameterizing the scale invariance of the series over a range of scales [see *Arneodo et al.*, 1998; *Venugopal et al.*, 2006]. Note that from equation (11) the zero-order structure function *M*(0, *r*) is trivially equal to 1 and thus (from equation (12)) scale-independent. This approach therefore, does not allow us to characterize the possible fractality of the “sparseness” of the data series. However, as seen in section 4.2, the periods of inactivity exhibit a heavy-tailed distribution implying the existence of flat regions of all scales in the evolution of the elevation time series (see Figure 12) or sequences of zeroes of all scales in the time series of elevation increments (see Figure 3). Quantifying the nontrivial scaling of the zeroth-order moment of a data series would require relaxing the *ζ*(0) = 0 assumption in equation (13) and introducing a positive constant *c*_{0} in the characterization of the nonlinear dependence of the scaling exponents,

Figure 14a shows the estimated higher-order structure functions, *M*(*q*, *r*), as a function of scale *r*. The log-log linear relationship of *M*(*q*, *r*) on *r* over the range of scales 2–256 min (2^{1}–2^{8} in log scale) for the moments of order *q* = 0.1, 0.25, 0.5, 0.75, 1.0,…, 3.0 documents the scale invariance of the elevation increments. In Figure 14b, the scaling exponent function *ζ*(*q*) is plotted against the order of moments. The nonlinear dependence of *ζ*(*q*) on *q* documents the multifractal behavior of the elevation increments. Fitting the quadratic function of equation (14) to *ζ*(*q*) results in *c*_{0} = 0.192, *c*_{1} = 0.58 and *c*_{2} = 0.171. These three parameters *c*_{0}, *c*_{1} and *c*_{2} fully characterize the scaling of all statistical moments and thus the way the pdfs of elevation increments change over scales [e.g., see *Venugopal et al.*, 2006].

[28] As a final remark, we note that the scaling characterization presented above (via statistical moments or singularity spectrum) holds within a range of scales (see Figure 14a) whose upper bound coincides with the truncation parameter of the dominant time scale of the system, that is, periods of inactivity ( = 240 min), providing thus a much desired physical interpretation of the upper bound of the scaling regime.