NG33A-1206
Multi-scale Analysis of Radar Reflectivity Maps
This study focuses on the modelling of spatial variability of rain by a multifractal approach. We consider the framework of Universal Multifractals. The data are radar reflectivity maps from the AMMA campaign (Analyse Multidisciplinaire de la Mousson Africaine, Djougou, Benin, 2006) and Meteo France's operational network (Rennes, France, 2007-2008). Since rain-no rain intermittency has been shown to bias the parameter estimates, we perform our analysis only on non-intermittent square fields. The fundamental parameters are found to be close to α=1.7 and C1=0.15, regardless of the dataset. It is not possible to conclude about the non-stationarity parameter H, because its estimations are significantly different in the two cases (0.23 and 0). However, some discussion about data quality indicates that the first value is more likely to be correct, meaning that rain fields would be non-stationary. Then, simulations have been used to estimate the bias due to rain intermittence. This effect is to underestimate α and this explains the difference between the estimated α and the values usually found in the literature (1.3-1.5). Finally, these results are compared to those determined by analysis of non-intermittent temporal series of disdrometer rain rates, corresponding to AMMA campaign data. We show the similarity between the estimated values and propose some explanations.
NG33A-1207
Turbulent Flux based approaches to satellite precipitation
Analysis of thousands of orbits of TRMM radar reflectivities have shown that over the range 4 to 20000 km, that their statistics are very close to those predicted for turbulent fluxes, i.e. they are conserved scale by scale and are therefore the outcomes of a pure multiplicative cascade processes (with deviations of the order of a few percent). If the rain rate is a power of the radar reflectivity (as is usually assumed), then this implies the same property for the rain rate. In contrast, analysis of the TRMM satellite passively sensed radiances in the visible, infra red and microwave bands show that they are qualitatively different, being quantitatively quite close to the statistics of turbulent passive "tracers". This difference implies that the conventional satellite rain techniques - in which the radiances are used directly for rain rate estimates – can at most be correctly calibrated at a single (subjectively chosen) scale/resolution. On the contrary if the (absolute) radiance gradients are used as estimates of the corresponding turbulent fluxes, then products of powers of the fluxes from different radiances will have qualitatively the correct type of statistics and it is straightforward to constrain them to have the same statistics as the precipitation over the entire scaling range. These flux based techniques are physically based on the underlying atmospheric cascade structure and have the advantage of potentially giving accurate precipitation estimates at any resolution over a wide range. Using the TRMM data, we compare and contrast various flux based techniques and compare them with conventional ones.
NG33A-1208
The Space-Time Statistical Structure of TRMM and MTSAT Precipitation and Radiances
Up until now, attempts to systematically understand the space-time statistical structure of the atmosphere
(including precipitation) have been hampered by both inappropriate theoretical frameworks and inadequate,
problematic data. On the one hand, the theories have concentrated on classical turbulent fluxes especially
the energy and enstrophy fluxes which are only justified with strong but unrealistic isotropy assumptions: on
the contrary, the real atmosphere is strongly anisotropic (stratified) but nevertheless scaling. On the other
hand, if we restrict our attention to the wind, temperature and other standard meteorological fields, then only
very narrow ranges of space-time scales are empirically accessible.
If the stratification is scaling, then atmospheric dynamics can be governed by anisotropic cascades of
nonstandard turbulent fluxes. In this generalized scaling framework we expect scaling relations of the
(generalized) Kolmogorov form to hold:
F(L) = e(L)LH
where F(L) is the fluctuation in a field at scale L and H is a scaling exponent and e(L) is the underlying
resolution L flux. We use this approach to estimate e(L) and then to systematically degrade it to lower and
lower resolutions. The cascade hypothesis predicts that :
⟨e(L)q⟩ = (Louter/L)K(q)
where here L is the resolution of the flux, Louter is the outer scale of the cascade (where it starts) and
K(q) is a scaling exponent function describing all the statistical properties as a function of scale.
In order to exploit the space-time data with highest resolution possible, we analyse satellite radar and
radiance data spanning the visible, infra red, and microwave regions of the spectrum from the TRMM and
MTSAT satellites over the region ±40o latitude. The temporal resolutions are respectively 2 -4 days and 1
hour and both have spatial resolutions of 5km. We analysed the spatial properties over all these radiance
channels and found that scaling behaviour is obeyed to within about ±1% from planetary scales down to the
smallest available.
Since the dynamical (velocity) field is strongly coupled to the radiances (via the cloud field), and the velocity
couples the spatial and temporal statistics, we expect that the radiances will also be scaling in time. We
therefore determined the statistical lifetime-size relation for structures defined by the various fields. Up to
about 20-30 days (the lifetime of planetary size structures), we found that the relation is quite with an
effective velocity of about 6 m/s. For longer (climate scale) periods there seems to be another scaling
regime with outer scale around 1200 days with corresponding characteristic velocity of about 0.5 m/s. This
apparently corresponds to the temporal precipitation cascade as shown by space-time analyses of the TRMM
precipitation radar data. We discuss the implications for measuring and modeling atmospheric fields
especially precipitation.
NG33A-1209
Practical challenges on the estimation of universal multifractal parameters of rainfall observations
The Double Trace Moment (DTM, Lavall¨¦e et al., 1991) and Modified Double Trace Moment (MDTM, Veneziano and Furcolo, 1999) methods were used to estimate the universal multifractal parameters of both synthetically generated and observed time-series of rainfall with varying temporal resolutions. Their robustness to noise and the finite sampling bias effect were also investigated. Although the theoretical underpinnings of the MDTM suggest its application should be more general, this study shows that it works well for synthetic data can be modeled exactly by a Levi-stable distribution, but its systematic application to actual rainfall observations, inherently noisy and with relatively coarse temporal resolution, does not yield reliable results. Given the utility of multifractal theory to describe rainfall extremes as well as other nonlinear processes (e.g. turbulence, cloudiness, landform and river flows, etc), specific recommendations for the reliable practical estimation of multifractal parameters of observations are provided.
NG33A-1210
Techniques of Multifractal Nowcasting With Rada Data
In deterministic weather forecast models, the rain and many other physical processes are highly parameterized by rather ad-hoc sub-grid modelling. A consequence is the poor rain forecast; and further disadvantage is the long spin-up and computational time of these models prevents them to deliver nowcasting, which are indispensable in emergency situations. We have developed techniques of nowcasting based on the multifractal approach, applying cascade models which include continuous scales, scaling space-time anisotropy and causality. These models have the advantage of requiring a very limited number of theoretical or empirical parameters. Due to these properties, the multifractal models have been more and more used for analysing and simulating rainfall. To produce multifractal forecasts, we exploit the fact that at the core of a multifractal process there is a Levy white-noise -its "sub-generator"- whose future is therefore independent of its past. A first step corresponds to a backward simulation in order to estimate the past sub-generator from past observations (e.g. rain radar data). This raises several technical issues due to the fact it involve inversions. The second step corresponds to a forward simulations based on the sub-generator extended to a future period. This extension can be done along several modes and we discuss those corresponding respectively to deterministic and stochastic sub- grid modelling, as well as the question of stochastic forecast versus ensemble forecast. We focus on a case study of the extreme rainfall that occurs on the 8th and 9th of September 2002 in the Gard basin, France.
NG33A-1211
Comparison of two Stochastic Methods to Simulate Heavy Rainfalls
We discuss and compare two stochastic methods to simulate heavy rainfalls for accumulation durations ranging from 1 hour to 72 hours. The first one, the SHYPRE model (Simulation d'Hydrogramme pour la PREdétermination des crues) developed by CEMAGREF, uses stochastic simulations to reconstruct the high resolution time series, whereas the second one, the MultiFractal Method (MFM), uses the theoretical results obtained from the theory of stochastic processes. Thus the first step of both methods corresponds to a given statistical analysis of time series of a limited length to estimate the model parameters; then the stochastic modeling is used to infer the quantiles for the return periods ranging from 1 to 1000 years. It is frequently mentioned that for larger return periods, the empirical quantiles estimated from the long hydro- meteorological records are often much more important than those inferred from Gumbel law. It can be shown that MFM yields generally a power-law probability of the extremes, often called 'fat tail' distribution. The SHYPRE model leads to a hyper-exponential asymptotic behavior and thus might be comparable with the MFM. Based on the quantile estimations for different return periods, we have performed a detailed comparison of SHYPRE and MFM results for 252 French meteorological stations. A particular attention was given to Pyrenees, Rhone Valley, Cevennes and Alps - the four south France areas with homogeneous precipitation regimes. The results demonstrate that SHYPRE and MFM indeed yield quite comparable results for the larger return periods (100-1000 years). It is important to note that the spatial distributions of the rainfall quantiles of SHYPRE and MFM of any given duration and return period have strong space variability. In conclusion, we discuss consequences of such variability for floods, in particular flashfloods.
NG33A-1212
Spatial distribution of solute particles in longitudinal dispersion due to advection
Previously we reported how to obtain a distribution of solute arrival times, W(t) due to advection. This calculation involved using cluster statistics of percolation theory in the framework of critical path analysis to find the likelihood that a given system size is characterized by a particular controlling conductance value, g, i.e., W(g,x), or W(g) for any given value of x. Then we calculated deterministically the scaling of the arrival time of particles through any particular pore size distribution along one of the tortuous paths characterized by W(g,x). Then W(t) was obtained from W(g)/(dt/dg), including known scaling properties related to the mass fractal dimensionality of the backbone percolation cluster. We now use this framework to generate a self- consistent expression for W(g,t) that a particle is traveling along a cluster with minimum g and then divide by dx/dg to obtain the distribution of particle distances at any time. While the former result had approximate power law tails in time, the present result resembles more closely a "stretched exponential" in space.
NG33A-1213
Towards Efficient Encoding of Hydrologic Information via a Fractal Geometric Approach
Hydrologic data series typically show complex features that are difficult to represent as a whole using classical stochastic models. While these models reproduce with a certain degree of confidence important statistical qualifiers (e.g., low order moments, power and multifractal spectra), they typically do not allow an exact representation of the fine details of a given realization (e.g., relative position of the major peaks, periods of no activity). In recent years, we have developed a fully deterministic procedure aimed at reproducing not only the major statistical qualifiers, but also the fine details of individual data sets as found in geophysical applications. Our procedure is based on a natural combination of fractal functions and multifractal measures via iterations of affine mappings, and yields an amazingly vast set of patterns that are indistinguishable from real world data sets. While the generation of these patterns is relatively straightforward, matching to real data measurements has proven to be a challenging task. In this work we report on the progress that we have achieved on the inversion of synthetic data sets generated by means of our procedure, which are taken as a close proxy of real world data sets. As it is shown, through a combination of heuristic procedures which include Artificial Neural Network Pattern Recognition and Particle Swarm and Differential Evolution searches, we have been able to achieve successful inversion of a relatively large class of synthetic data sets. These patterns are described by a very small (less than 15) set of parameters, leading to compression ratios exceeding 100:1. We envision an efficient implementation of this inverse problem procedure for general data sets, at comparable compression ratios, by improving our computational capabilities, using hardware acceleration (e.g., Field Programmable Gate Arrays boards) that should result in computational speed-ups of the order of x1000. This way our fractal geometric procedure would not only be a useful tool to simulate geophysical information but also an elegant tool for data encoding and compression.
NG33A-1214
Thermalization and scalings in turbulence
It was argued that application of hyperviscosity in turbulence simulation will lead to artifacts caused by partial thermalization --- the tendency to the physical behavior corresponding to the equilibrium of the Galerkin-truncated inviscid system [U. Frisch et al., Phys. Rev. Letts. in press; or, Arxiv:0803.4269]. We study the partial thermalization physics in three dimensional high-resolution numerical hyperviscous turbulence. Special attention is given to the effect of thermalization on intermittency growth, and, the inertial- and dissipation-range scalings. Hierarchical flow visualizations [J. Clyne et al., New Journal of Physics 9 (2007) 301] are made, discovering coherent structures embedded in the thermalized fluctuations. We will address relevant issues in modeling and simulation of geophysical and atmospheric flows.
NG33A-1215
Persistence analysis in geophysical spatial series: an application to well log data
Fractal geometry is the field of Mathematics that studies the properties and behavior of fractals. It was already applied to several areas of Science, explaining many situations that cannot be described by classical geometry. A fractal is a geometric object that can be divided into several parts, each one similar to the original object. Fractal geometry has been used frequently to characterize and describe natural models. Its applications range from microscopic dimensions to the understanding of macroscopic processes. Many geophysical variables seem to have a scale behavior, which means that their power spectrum, P(f), seem to be proportional to some frequency f, that is, ln[P(f)] = - b ln(f), where b is called the spectral exponent. Following this principle, we propose to study a set of well log data, collected in the Jequitinhonha basin (Brazil), through the methods of spectral analysis and re-scaled analysis. When the spectral exponent varies from 0.5 to 1.5, the physical property is said to have a 1/f noise. One method to verify the long range persistence is through the re-scaled (R/S) analysis, with computes the Hurst coefficient H. H and the exponent b are related by bcum = 2H + 1, where bcum is cumulative spectral exponent of the sequence with exponent b. The spatial series are the following well data: acoustic velocity, gamma ray, induction deep resistivity, porosity, density, resistivity average, and spontaneous potential. For each parameter, we computed the spectral exponent b, and the coefficient H, attesting the variability of the available data. The results for H were coherent with other works mentioned in the literature. We obtained 0.6 < H < 0.9 for both methods, in such a way that all the parameters behave as persistent. The importance of such study is to show that different methods provided the same results, which is a significant fact for complex systems. The knowledge of H also provides the computation of the fractal dimension, as well as to understand the complexity of layer sedimentation phenomenon.
NG33A-1216
Multifractal Analysis of Rainfall Change Over France in a Climate Scenario
In this presentation the scaling properties of rainfall time-series generated by a climate model are analysed by means of a multifractal characterization based on the "universal multifractal" formalism. The analysed data are the rainfall daily time-series over France computed in a simulation over the period 1860-2100 by the climate model CNRM-CM3 of Meteo-France in a coupled IPCC climate scenario (A2). We quantify the scaling variability of the simulated rainfall with the help of a few relevant multifractal exponents characterizing the intermittency and multifractality of the field. These multifractal exponents are determined by the Double Trace Moment (DTM) which shows a scaling range from one day to about 16 days. The opposite trends found in the evolution of the intermittency and multifractality exponents, have contradictory effects on the evolution of the extremes. However, a refined analysis shows that due to the dominant effect of intermittency increase, we may expect an effective increase of rainfall extremes for the next hundred years.