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Downscaling Satellite Precipitation with Emphasis on Extremes: A Variational ℓ1-Norm Regularization in the Derivative Domain

  • E. Foufoula-Georgiou
  • A. M. Ebtehaj
  • S. Q. Zhang
  • A. Y. Hou
Chapter
Part of the Space Sciences Series of ISSI book series (SSSI, volume 46)

Abstract

The increasing availability of precipitation observations from space, e.g., from the Tropical Rainfall Measuring Mission (TRMM) and the forthcoming Global Precipitation Measuring (GPM) Mission, has fueled renewed interest in developing frameworks for downscaling and multi-sensor data fusion that can handle large data sets in computationally efficient ways while optimally reproducing desired properties of the underlying rainfall fields. Of special interest is the reproduction of extreme precipitation intensities and gradients, as these are directly relevant to hazard prediction. In this paper, we present a new formalism for downscaling satellite precipitation observations, which explicitly allows for the preservation of some key geometrical and statistical properties of spatial precipitation. These include sharp intensity gradients (due to high-intensity regions embedded within lower-intensity areas), coherent spatial structures (due to regions of slowly varying rainfall), and thicker-than-Gaussian tails of precipitation gradients and intensities. Specifically, we pose the downscaling problem as a discrete inverse problem and solve it via a regularized variational approach (variational downscaling) where the regularization term is selected to impose the desired smoothness in the solution while allowing for some steep gradients (called ℓ1-norm or total variation regularization). We demonstrate the duality between this geometrically inspired solution and its Bayesian statistical interpretation, which is equivalent to assuming a Laplace prior distribution for the precipitation intensities in the derivative (wavelet) space. When the observation operator is not known, we discuss the effect of its misspecification and explore a previously proposed dictionary-based sparse inverse downscaling methodology to indirectly learn the observation operator from a data base of coincidental high- and low-resolution observations. The proposed method and ideas are illustrated in case studies featuring the downscaling of a hurricane precipitation field.

Keywords

Sparsity Inverse problems 1-norm regularization Non-smooth convex optimization Generalized Gaussian density Extremes Hurricanes 

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References

  1. Badas MG, Deidda R, Piga E (2006) Modulation of homogeneous space-time rainfall cascades to account for orographic influences. Nat Hazard Earth Syst 6(3):427–437. doi: 10.5194/nhess-6-427-2006 CrossRefGoogle Scholar
  2. Bateni SM, Entekhabi D (2012) Surface heat flux estimation with the ensemble Kalman smoother: joint estimation of state and parameters. Water Resour Res 48(3). doi: 10.1029/2011WR011542
  3. Beck A, Teboulle M (2009) A fast iterative shrinkage-thresholding algorithm for linear inverse problems. SIAM J Imaging Sci 2(1):183–202. doi: 10.1137/080716542 CrossRefGoogle Scholar
  4. Bertsekas DP (1999) Nonlinear programming, 2nd edn. Athena Scientific, Belmont, MA, p 794Google Scholar
  5. Chen S, Donoho D, Saunders M (2001) Atomic decomposition by basis pursuit. SIAM Rev 43(1):129–159CrossRefGoogle Scholar
  6. Chen SS, Donoho DL, Saunders MA (1998) Atomic decomposition by basis pursuit. SIAM J Sci Comput 20:33–61CrossRefGoogle Scholar
  7. Deidda R (2000) Rainfall downscaling in a space-time multifractal framework. Water Resour Res 36(7):1779–1794CrossRefGoogle Scholar
  8. Ebtehaj AM, Foufoula-Georgiou E (2011) Statistics of precipitation reflectivity images and cascade of Gaussian-scale mixtures in the wavelet domain: a formalism for reproducing extremes and coherent multiscale structures. J Geophys Res 116:D14110. doi: 10.1029/2010JD015177 CrossRefGoogle Scholar
  9. Ebtehaj AM, Foufoula-Georgiou E, Lerman G (2012) Sparse regularization for precipitation downscaling. J Geophys Res 116:D22110. doi: 10.1029/2011JD017057 CrossRefGoogle Scholar
  10. Ebtehaj AM, Foufoula-Georgiou E (2013) Variationl downscaling, fusion and assimilation of hydrometeorological states: a unified framework via regularization. Water Resour Res. doi: 10.1002/wrcr.20424 CrossRefGoogle Scholar
  11. Figueiredo M, Nowak R, Wright S (2007) Gradient projection for sparse reconstruction: application to compressed sensing and other inverse problems. IEEE J Sel Topics Signal Process 1(4):586–597. doi: 10.1109/JSTSP.2007.910281 CrossRefGoogle Scholar
  12. Flaming GM (2004) Measurement of global precipitation. In: Geoscience and remote sensing symposium, 2004. IGARSS’04. Proceedings. 2004 IEEE international, vol 2, p 918–920Google Scholar
  13. Freitag MA, Nichols NK, Budd CJ (2012) Resolution of sharp fronts in the presence of model error in variational data assimilation. Q J Roy Meteor Soc. doi: 10.1002/qj.2002 CrossRefGoogle Scholar
  14. Hansen P (2010) Discrete inverse problems: insight and algorithms, vol. 7. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PAGoogle Scholar
  15. Harris D, Foufoula-Georgiou E, Droegemeier KK, Levit JJ (2001) Multiscale statistical properties of a high-resolution precipitation forecast. J Hydrometeor 2(4):406–418CrossRefGoogle Scholar
  16. Krajewski WF, Smith JA (2002) Radar hydrology: rainfall estimation. Adv Water Resour 25(8–12):1387–1394. doi: 10.1016/S0309-1708(02)00062-3 CrossRefGoogle Scholar
  17. Kim S-J, Koh K, Lustig M, Boyd S, Gorinevsky D (2007) An interior-point method for large-scale ℓ1-regularized least squares. IEEE J Sel Topics Signal Process. 1(4):606–617. doi: 10.1109/JSTSP.2007.910971 CrossRefGoogle Scholar
  18. Kumar P, Foufoula-Georgiou E (1993) A multicomponent decomposition of spatial rainfall fields. 2. Self-similarity in fluctuations. Water Resour Res 29(8):2533–2544CrossRefGoogle Scholar
  19. Kumar P, Foufoula-Georgiou E (1993) A multicomponent decomposition of spatial rainfall fields. 1. Segregation of large- and small-scale features using wavelet transforms. Water Resour Res 29(8):2515–2532CrossRefGoogle Scholar
  20. Lewicki M, Sejnowski T (2000) Learning overcomplete representations. Neural Comput 12(2):337–365CrossRefGoogle Scholar
  21. Lovejoy S, Mandelbrot B (1985) Fractal properties of rain, and a fractal model. Tellus A 37(3):209–232CrossRefGoogle Scholar
  22. Lovejoy S, Schertzer D (1990) Multifractals, universality classes and satellite and radar. J Geophys Res 95(D3):2021–2034CrossRefGoogle Scholar
  23. Mallat S, Zhang Z (1993) Matching pursuits with time-frequency dictionaries. IEEE Trans Signal Proces 41(12):3397–3415. doi: 10.1109/78.258082 CrossRefGoogle Scholar
  24. Mallat S (1989) A theory for multiresolution signal decomposition: the wavelet representation. IEEE Trans Pattern Anal Mach Intell 11(7):674–693. doi: 10.1109/34.192463 CrossRefGoogle Scholar
  25. Nykanen DK, Foufoula-Georgiou E, Lapenta WM (2001) Impact of small-scale rainfall variability on larger-scale spatial organization of land-atmosphere fluxes. J Hydrometeor 2(2):105–121CrossRefGoogle Scholar
  26. Perica S, Foufoula-Georgiou E (1996) Model for multiscale disaggregation of spatial rainfall based on coupling meteorological and scaling. J Geophys Res 101(D21):26–347CrossRefGoogle Scholar
  27. Rebora N, Ferraris L, Von Hardenberg J, Provenzale A et al (2006) Rainfall downscaling and flood forecasting: a case study in the Mediterranean area. Nat Hazard Earth Syst 6(4):611–619CrossRefGoogle Scholar
  28. Rebora N, Ferraris L, Von Hardenberg J, Provenzale A (2006) RainFARM: rainfall downscaling by a filtered autoregressive model. J Hydrometeor 7:724–738CrossRefGoogle Scholar
  29. Sapozhnikov VB, Foufoula-Georgiou E (2007) An exponential Langevin-type model for rainfall exhibiting spatial and temporal scaling. Nonlinear Dyn Geosci :87–100Google Scholar
  30. Tibshirani R (1996) Regression shrinkage and selection via the Lasso. J R Stat Soc Ser B Stat Methodol 58(1):267–288Google Scholar
  31. Venugopal V, Roux SG, Foufoula-Georgiou E, Arneodo A (2006) Revisiting multifractality of high-resolution temporal rainfall using a wavelet-based formalism. Water Resour Res 42(6):6. doi: 10.1029/2005WR004489 CrossRefGoogle Scholar
  32. Venugopal V, Roux SG, Foufoula-Georgiou E, Arnéodo A (2006) Scaling behavior of high resolution temporal rainfall: new insights from a wavelet-based cumulant analysis. Phys Lett A 348(3):335–345Google Scholar

Copyright information

© Springer Science+Business Media Dordrecht (outside the USA) 2013

Authors and Affiliations

  • E. Foufoula-Georgiou
    • 1
  • A. M. Ebtehaj
    • 2
  • S. Q. Zhang
    • 3
  • A. Y. Hou
    • 3
  1. 1.Saint Anthony Falls Laboratory, Department of Civil EngineeringUniversity of MinnesotaMinneapolisUSA
  2. 2.Saint Anthony Falls Laboratory, Department of Civil Engineering, School of MathematicsUniversity of MinnesotaMinneapolisUSA
  3. 3.NASA Goddard Space Flight CenterGreenbeltUSA

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