A Hybrid MPI-OpenMP Parallel Algorithm for the Assessment of the Multifractal Spectrum of River Networks

Leonardo Primavera,Emilia Florio

doi:10.3390/w13213122

Abstract

The possibility to create a flood wave in a river network depends on the geometric properties of the river basin. Among the models that try to forecast the Instantaneous Unit Hydrograph (IUH) of rainfall precipitation, the so-called Multifractal Instantaneous Unit Hydrograph (MIUH) rather successfully connects the multifractal properties of the river basin to the observed IUH. Such properties can be assessed through different types of analysis (fixed-size algorithm, correlation integral, fixed-mass algorithm, sandbox algorithm, and so on). The fixed-mass algorithm is the one that produces the most precise estimate of the properties of the multifractal spectrum that are relevant for the MIUH model. However, a disadvantage of this method is that it requires very long computational times to produce the best possible results. In a previous work, we proposed a parallel version of the fixed-mass algorithm, which drastically reduced the computational times almost proportionally to the number of Central Processing Unit (CPU) cores available on the computational machine by using the Message Passing Interface (MPI), which is a standard for distributed memory clusters. In the present work, we further improved the code in order to include the use of the Open Multi-Processing (OpenMP) paradigm to facilitate the execution and improve the computational speed-up on single processor, multi-core workstations, which are much more common than multi-node clusters. Moreover, the assessment of the multifractal spectrum has also been improved through a direct computation method. Currently, to the best of our knowledge, this code represents the state-of-the-art for a fast evaluation of the multifractal properties of a river basin, and it opens up a new scenario for an effective flood forecast in reasonable computational times.

Highlights

The idea that the branched structures of river networks are multifractal and non-plane filling objects was first suggested by several authors [1,2,3]
The model was further refined by De Bartolo et al (2003) [5], who substituted the fractal dimension D of Fiorentino et al (2002) with the generalized fractal dimension D−∞ under the assumption that a river basin could be better represented as a multifractal object than as a simple fractal, proposing the so-called Multifractal Instantaneous Unit Hydrograph (MIUH)
The problem of determining the fractal dimensions of the basin can be summarized in the following steps: first, the map of the river basin is projected onto a plane, and its structure is represented by picking up a meaningful set of points lying on the branches of such 2D maps, the so-called net-points; second, a multifractal analysis is carried out in order to derive the generalized fractal dimensions Dq, the sequence of Lipschitz–Hölder exponents αq and the multifractal spectrum f

Summary

Introduction

The idea that the branched structures of river networks are multifractal and non-plane filling objects was first suggested by several authors [1,2,3]. The model was further refined by De Bartolo et al (2003) [5], who substituted the fractal dimension D of Fiorentino et al (2002) with the generalized fractal dimension D−∞ under the assumption that a river basin could be better represented as a multifractal object than as a simple fractal, proposing the so-called Multifractal Instantaneous Unit Hydrograph (MIUH). The problem of determining the fractal dimensions of the basin can be summarized in the following steps: first, the map of the river basin is projected onto a plane, and its structure is represented by picking up (manually, or through automatic algorithms) a meaningful set of points lying on the branches of such 2D maps, the so-called net-points; second, a multifractal analysis is carried out in order to derive the generalized fractal dimensions Dq , the sequence of Lipschitz–Hölder exponents αq and the multifractal spectrum f (αq )

Methods

Results

Discussion

Conclusion