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Theoretical and empirical scale dependency of Z‐R relationships: Evidence, impacts, and correction
Author(s) -
Verrier Sébastien,
Barthès Laurent,
Mallet Cécile
Publication year - 2013
Publication title -
journal of geophysical research: atmospheres
Language(s) - English
Resource type - Journals
eISSN - 2169-8996
pISSN - 2169-897X
DOI - 10.1002/jgrd.50557
Subject(s) - disdrometer , scale (ratio) , radar , power law , statistical physics , exponent , dependency (uml) , mathematics , spectral density , range (aeronautics) , econometrics , statistics , meteorology , environmental science , physics , computer science , telecommunications , linguistics , philosophy , software engineering , materials science , quantum mechanics , rain gauge , composite material
Estimation of rainfall intensities from radar measurements relies to a large extent on power‐laws relationships between rain rates R and radar reflectivities Z , i.e., Z = a * R ^ b . These relationships are generally applied unawarely of the scale, which is questionable since the nonlinearity of these relations could lead to undesirable discrepancies when combined with scale aggregation. Since the parameters ( a , b ) are expectedly related with drop size distribution (DSD) properties, they are often derived at disdrometer scale, not at radar scale, which could lead to errors at the latter. We propose to investigate the statistical behavior of Z‐R relationships across scales both on theoretical and empirical sides. Theoretically, it is shown that claimed multifractal properties of rainfall processes could constrain the parameters ( a , b ) such that the exponent b would be scale independent but the prefactor a would be growing as a (slow) power law of time or space scale. In the empirical part (which may be read independently of theoretical considerations), high‐resolution disdrometer (Dual‐Beam Spectropluviometer) data of rain rates and reflectivity factors are considered at various integration times comprised in the range 15 s – 64 min. A variety of regression techniques is applied on Z‐R scatterplots at all these time scales, establishing empirical evidence of a behavior coherent with theoretical considerations: a grows as a 0.1 power law of scale while b decreases more slightly. The properties of a are suggested to be closely linked to inhomogeneities in the DSDs since extensions of Z‐R relationships involving (here, strongly nonconstant) normalization parameters of the DSDs seem to be more robust across scales. The scale dependence of simple Z = a * R ^ b relationships is advocated to be a possible source of overestimation of rainfall intensities or accumulations. Several ways for correcting such scaling biases (which can reach >15–20% in terms of relative error) are suggested. Such corrections could be useful in some practical cases where Z‐R scale biases are significant, which is especially expected for convective rainfall.