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An adaptive spatial model for precipitation data from multiple satellites over large regions

Abstract

Satellite measurements have of late become an important source of information for climate features such as precipitation due to their near-global coverage. In this article, we look at a precipitation dataset during a 3-hour window over tropical South America that has information from two satellites. We develop a flexible hierarchical model to combine instantaneous rainrate measurements from those satellites while accounting for their potential heterogeneity. Conceptually, we envision an underlying precipitation surface that influences the observed rain as well as absence of it. The surface is specified using a mean function centered at a set of knot locations, to capture the local patterns in the rainrate, combined with a residual Gaussian process to account for global correlation across sites. To improve over the commonly used pre-fixed knot choices, an efficient reversible jump scheme is used to allow the number of such knots as well as the order and support of associated polynomial terms to be chosen adaptively. To facilitate computation over a large region, a reduced rank approximation for the parent Gaussian process is employed.

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References

  1. Agarwal, D.K., Gelfand, A.E., Citron-Pousty, S.: Zero-inflated models with application to spatial count data. Environ. Ecol. Stat. 9, 341–355 (2002)

  2. Albert, J.H., Chib, S.: Bayesian analysis of binary and polychotomous response data. J. Am. Stat. Assoc. 88(422), 669–679 (1993)

  3. Anderes, E.B., Stein, M.L.: Estimating deformations of isotropic Gaussian random fields on the plane. Ann. Stat. 36, 719–741 (2008)

  4. Austin, P.M., Houze, R.A.: Analysis of the structure of precipitation patterns in New England. J. Appl. Meteorol. 11, 926–935 (1972)

  5. Ba, M.B., Gruber, A.: Goes multispectral rainfall algorithm (gmsra). J. Appl. Meteorol. 40, 1500–1514 (2001)

  6. Banerjee, S.: On geodetic distance computations in spatial modeling. Biometrics 61(2), 617–625 (2005)

  7. Banerjee, S., Gelfand, A.E., Knight, J.R., Sirmans, C.F.: Spatial modeling of house prices using normalized distance-weighted sums of stationary processes. J. Bus. Econ. Stat. 22(2), 206–213 (2004)

  8. Banerjee, S., Gelfand, A., Finley, A., Sang, H.: Gaussian predictive process models for large spatial data sets. J. R. Stat. Soc. B 70(4), 825–848 (2008)

  9. Bardossy, A., Plate, E.J.: Space-time model for daily rainfall using atmospheric circulation patterns. Water Resour. Res. 28(5), 1247–1259 (1992)

  10. Bell, T.L., Kundu, P.K.: A study of the sampling error in satellite rainfall estimates using optimal averaging of data and a stochastic model. J. Climate 9, 1251–1268 (1996)

  11. Bell, T.L., Abdullah, A., Martin, R.L., North, G.R.: Sampling errors for satellite-derived tropical rainfall: Monte Carlo study using a space-time stochastic model. J. Geophys. Res. 95(D3), 2195–2205 (1990)

  12. Bell, T.L., Kundu, P.K., Kummerow, C.D.: Sampling errors of ssm/i and trmm rainfall averages: comparison with error estimates from surface data and a simple model. J. Appl. Meteorol. 40, 938–954 (2001)

  13. Chakraborty, A., Gelfand, A.E., Wilson, A.M., Latimer, A.M., Silander, J.A.: Modeling large scale species abundance with latent spatial processes. Ann. Appl. Stat. 4(3), 1403–1429 (2010)

  14. Chakraborty, A., Mallick, B.K., McClarren, R.G., Kuranz, C.C., Bingham, D.R., Grosskopf, M.J., Rutter, E., Stripling, H.F., Drake, R.P.: Spline-based emulators for radiative shock experiments with measurement error. J. Am. Stat. Assoc. 108, 411–428 (2013)

  15. Chib, S., Carlin, B.P.: On mcmc sampling in hierarchical longitudinal models. Stat. Comput. 9(1), 17–26 (1999)

  16. Cohen, A.C.: Truncated and Censored Samples, 1st edn. Marcel Dekker, New York (1991)

  17. Cooley, D., Nychka, D., Naveau, P.: Bayesian spatial modeling of extreme precipitation return levels. J. Am. Stat. Assoc. 102(479), 824–840 (2007)

  18. Cressie, N., Johannesson, G.: Fixed rank kriging for very large spatial data sets. J. R. Stat. Soc. B 70(1), 209–226 (2008)

  19. Denison, D.G.T., Mallick, B.K., Smith, A.F.M.: Bayesian mars. Stat. Comput. 8(4), 337–346 (1998)

  20. Felgate, D.G., Read, D.G.: Correlation analysis of the cellular structure of storms observed by raingauges. J. Hydrol. 24, 191–200 (1975)

  21. Finley, A., Sang, H., Banerjee, S., Gelfand, A.: Improving the performance of predictive process modeling for large datasets. Comput. Stat. Data Anal. 53(8), 2873–2884 (2009)

  22. Finley, A.O., Banerjee, S., MacFarlane, D.W.: A hierarchical model for quantifying forest variables over large heterogeneous landscapes with uncertain forest areas. J. Am. Stat. Assoc. 106(493), 31–48 (2011)

  23. Friedman, J.: Multivariate adaptive regression splines. Ann. Stat. 19(1), 1–67 (1991)

  24. Fuentes, M.: Spectral methods for nonstationary spatial processes. Biometrika 89, 197–210 (2002)

  25. Fuentes, M., Reich, B., Lee, G.: Spatial-temporal mesoscale modelling of rainfall intensity using gauge and radar data. Ann. Appl. Stat. 2, 1148–1169 (2008)

  26. Furrer, R., Genton, M.G., Nychka, D.: Covariance tapering for interpolation of large spatial datasets. J. Comput. Graph. Stat. 15(3), 502–523 (2006)

  27. Gelfand, A.E., Kim, H.J., Sirmans, C.F., Banerjee, S.: Spatial modeling with spatially varying coefficient processes. J. Am. Stat. Assoc. 98(462), 387–396 (2003)

  28. Gelfand, A.E., Banerjee, S., Finley, A.O.: Spatial design for knot selection in knot-based dimension reduction models. In: Mateu, J., Müller, W.G. (eds.) Spatio-Temporal Design: Advances in Efficient Data Acquisition, pp. 142–169. Wiley, Chichester (2012)

  29. Guhaniyogi, R., Finley, A.O., Banerjee, S., Gelfand, A.E.: Adaptive Gaussian predictive process models for large spatial datasets. Environmetrics 22(8), 997–1007 (2011)

  30. Higdon, D.: A process-convolution approach to modelling temperatures in the North Atlantic Ocean. Environ. Ecol. Stat. 5(2), 173–190 (1998)

  31. Higdon, D.: Space and space-time modeling using process convolutions. In: Anderson, C., Barnett, V., Chatwin, P.C., El-Shaarawi, A.H. (eds.) Quantitative Methods for Current Environmental Issues, pp. 37–56. Springer, London (2002)

  32. Higdon, D., Swall, J., Kern, J.: Non-stationary spatial modeling. In: Bernardo, J.M., Berger, J.O., Dawid, A.P., Smith, A.F.M. (eds.) Bayesian Statistics, vol. 7, pp. 181–197. Oxford University Press, Oxford (1999)

  33. Huffman, G.J., Adler, R.F., Stocker, E.F., Bolvin, D.T., Nelkin, E.J.: A trmm-based system for real-time quasi-global merged precipitation estimates. In: TRMM International Science Conference, Honolulu, pp. 22–26 (2002)

  34. Huffman, G.J., Adler, R.F., Bolvin, D.T., Gu, G., Nelkin, E.J., Bowman, K.P., Hong, Y., Stocker, E.F., Wolef, D.B.: The trmm multisatellite precipitation analysis (tmpa): quasi-global, multiyear, combined-sensor precipitation estimates at fine scales. J. Hydrometeorol. 8, 38–55 (2007)

  35. Joyce, R.J., Janowiak, J.E., Arkin, P.A., Xie, P.: Cmorph: a method that produces global precipitation estimates from passive microwave and infrared data at high spatial and temporal resolution. J. Hydrometeorol. 5, 487–503 (2004)

  36. Jun, M.: Non-stationary cross-covariance models for multivariate processes on a globe. Scand. J. Stat. 38, 726–747 (2011)

  37. Jun, M., Stein, M.L.: Nonstationary covariance models for global data. Ann. Appl. Stat. 2(4), 1271–1289 (2008)

  38. Kammann, E.E., Wand, M.P.: Geoadditive models. J. R. Stat. Soc., Ser. C, Appl. Stat. 52(1), 1–18 (2003)

  39. Kaufman, C.G., Schervish, M.J., Nychka, D.W.: Covariance tapering for likelihood-based estimation in large spatial data sets. J. Am. Stat. Assoc. 103(484), 1545–1555 (2008)

  40. Kidd, C.: Satellite rainfall climatology: a review. Int. J. Climatol. 21, 1041–1066 (2001)

  41. Lee, G.W., Zawadzki, I.: Variability of drop size distributions: time-scale dependence of the variability and its effects on rain estimation. J. Appl. Meteorol. 44, 241–255 (2005)

  42. Lethbridge, M.: Precipitation probability and satellite radiation data. Mon. Weather Rev. 95(7), 487–490 (1967)

  43. Marchenko, Y.V., Genton, M.G.: Multivariate log-skew-elliptical distributions with applications to precipitation data. Environmetrics 21(3–4), 318–340 (2010)

  44. McConnell, A., North, G.R.: Sampling errors in satellite estimates of tropical rain. J. Geophys. Res. 92(D8), 9567–9570 (1987)

  45. Negri, A.J., Xu, L., Adler, R.F.: A trmm-calibrated infrared rainfall algorithm applied over Brazil. J. Geophys. Res. 107(D20), 8048–8062 (2002)

  46. Paciorek, C., Schervish, M.: Spatial modelling using a new class of nonstationary covariance functions. Environmetrics 17, 483–506 (2006)

  47. Richardson, S., Green, P.J.: On Bayesian analysis of mixtures with an unknown number of components. J. R. Stat. Soc. B 59(4), 731–792 (1997)

  48. Rodríguez-Iturbe, I., Mejía, J.M.: The design of rainfall networks in time and space. Water Resour. Res. 10, 713–728 (1974)

  49. Sampson, P.D., Guttorp, P.: Nonparametric estimation on nonstationary spatial covariance structure. J. Am. Stat. Assoc. 87, 108–119 (1992)

  50. Sang, H., Gelfand, A.E.: Hierarchical modeling for extreme values observed over space and time. Environ. Ecol. Stat. 16(3), 407–426 (2009)

  51. Sang, H., Huang, J.Z.: A full scale approximation of covariance functions for large spatial data sets. J. R. Stat. Soc. B 74(1), 111–132 (2012)

  52. Schmidt, A.M., O’Hagan, A.: Bayesian inference for non-stationary spatial covariance structure via spatial deformations. J. R. Stat. Soc. B 65, 743–758 (2003)

  53. Simpson, J., Adler, R.F., North, G.R.: A proposed Tropical Rainfall Measuring Mission (TRMM) satellite. Bull. Am. Meteorol. Soc. 69(3), 278–295 (1988)

  54. Sorooshian, S., Hsu, K.L., Gao, X., Gupta, H., Imam, B., Braithwaite, D.: Evaluation of Persiann system satellite-based estimates of tropical rainfall. Bull. Am. Meteorol. Soc. 81(9), 2035–2046 (2000)

  55. Stein, M., Chi, Z., Welty, L.: Approximating likelihoods for large spatial data sets. J. R. Stat. Soc. B 66, 275–296 (2004)

  56. Sun, Y., Li, B., Genton, M.G.: Geostatistics for large datasets. In: Porcu, E., Montero, J.M., Schlather, M. (eds.) Advances and Challenges in Space-Time Modelling of Natural Events, vol. 207, pp. 55–77. Springer, Berlin (2012)

  57. Tanner, T.A., Wong, W.H.: The calculation of posterior distributions by data augmentation. J. Am. Stat. Assoc. 82, 528–549 (1987)

  58. Vicente, G.A., Scofield, R.A., Menzel, W.P.: The operational goes infrared rainfall estimation technique. Bull. Am. Meteorol. Soc. 79(9), 1883–1898 (1998)

  59. Weng, F.W., Zhao, L., Ferraro, R., Pre, G., Li, X., Grody, N.C.: Advanced microwave sounding unit (amsu) cloud and precipitation algorithms. Radio Sci. 38(4), 8068–8079 (2003)

  60. Wilheit, T.T.: A satellite technique for quantitatively mapping rainfall rates over the ocean. J. Appl. Meteorol. 16, 551–560 (1977)

  61. Wilheit, T.T., Chang, A.T.C., Rao, M.S.V., Rodgers, E.B., Theon, J.S.: A satellite technique for quantitatively mapping rainfall rates over the oceans. J. Appl. Meteorol. 16(5), 551–560 (1977)

  62. Xie, P., Arkin, P.A.: Global monthly precipitation estimates from satellite-observed outgoing longwave radiation. J. Climate 11, 137–164 (1998)

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Acknowledgements

The authors acknowledge the Texas A&M University Brazos HPC cluster that contributed to the research reported here (http://brazos.tamu.edu).

Author information

Correspondence to Avishek Chakraborty.

Additional information

The research of Bani K. Mallick and Avishek Chakraborty was supported by National Science Foundation grant DMS 0914951. Research of Marc G. Genton was partially supported by NSF grants DMS-1007504 and DMS-1100492. The research in this article was also partially supported by Award No. KUSC1-016-04 made by King Abdullah University of Science and Technology (KAUST).

Appendix:  Marginalizing out ν y and \(\sigma^{2}_{y}\) for estimation of spline parameters in μ y (s)

Appendix:  Marginalizing out ν y and \(\sigma^{2}_{y}\) for estimation of spline parameters in μ y (s)

Denote by … all parameters except \(\nu,\sigma^{2}_{y}\). Let P=[ϕ 1[x(s)],ϕ 2[x(s)],…,ϕ k [x(s)]], S=y(s)− y . We have,

$$\begin{aligned} &p\bigl(y(\mathbf{s})|\ldots \bigr) \\ &\quad \propto \int_{\nu_y} \int _{\sigma_y^2} p\bigl(y(\mathbf{s}) | \nu_y, \sigma_y^2,\ldots\bigr) p\bigl(\nu_y | \sigma^2_y\bigr) p\bigl(\sigma^2_y \bigr) d\sigma_y^2 d\nu_y, \\ &\quad \propto \bigl(2\pi\tau^2_y\bigr)^{-k/2} \int _{\nu_y} \int_{\sigma^2_y} \bigl( \sigma^2_y\bigr)^{-\frac{n+k}{2} - a_\sigma-1 } \\ &\qquad {}\times\exp \biggl[ - \frac{1}{2 \sigma^2_y} \bigl( S^T D^{-1}S + \nu_y^T \nu_y/\tau^2_y + 2 b_\sigma\bigr) \biggr] d\sigma^2_y d \nu_y, \\ &\quad \propto \bigl(2\pi\tau^2_y\bigr)^{-k/2} \varGamma\biggl( \frac{n}{2} + a_\sigma\biggr) \\ &\qquad {} \int _{\nu_y} \biggl( \frac{S^T D^{-1}S + \nu_y^T \nu_y /\tau^2_y}{2} + b_\sigma \biggr)^{-\frac{n+m+k}{2} - a_\sigma} d\nu_y. \end{aligned}$$

Now write \(S^{T} D^{-1}S + \nu_{y}^{T} \nu_{y} = \nu_{y}^{T} A \nu_{y} - 2 \nu_{y}^{T} B + C\), where \(A = P^{T}D^{-1}P + \frac{ I_{k}}{\tau^{2}_{y}}\), B=P T D −1 S y , \(C = S_{y}^{T} D^{-1}S_{y}\). Then we have, \(S^{T} D^{-1}S + \nu_{y}^{T} \nu_{y} + 2 b_{\sigma}= (\nu_{y} - \mu_{k})^{T} \varSigma ^{-1}_{k} (\nu_{y} - \mu_{k}) + c_{0k}\), where μ k =A −1 B,Σ k =A −1,c 0k =Cb T A −1 b+2b σ . Denote d=n+2a σ . Then

$$\begin{aligned} &p\bigl(y(\mathbf{s})|\ldots\bigr) \\ &\quad \propto \bigl(\pi\tau^2_y \bigr)^{-k/2} c_{0k}^{-\frac {d+k}{2} }\varGamma\biggl( \frac{d+k}{2} \biggr) \int_{\nu_y} \biggl[ \frac{1}{d} (\nu_y - \mu_k)^T \\ &\qquad {}\times \biggl( \frac{ c_{0k} \varSigma_k}{d} \biggr)^{-1} (\nu_y-\mu _k) + 1 \biggr]^{-\frac{d+k}{2} } d\nu_y. \end{aligned}$$

The integrand is the pdf (up to a constant) for the k-variate t distribution with mean μ k , dispersion \(\frac{ c_{0k} \varSigma_{k}}{d}\) and degrees of freedom d. Hence, we obtain the closed form expression for \(p(y(\mathbf{s}) |\ldots) \propto(\tau^{2}_{y})^{-k/2} c_{0k}^{-\frac{d}{2} } |\varSigma_{k}|^{1/2}\).

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Chakraborty, A., De, S., Bowman, K.P. et al. An adaptive spatial model for precipitation data from multiple satellites over large regions. Stat Comput 25, 389–405 (2015). https://doi.org/10.1007/s11222-013-9439-8

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Keywords

  • Large data computation
  • Nonstationary spatial model
  • Precipitation modeling
  • Predictive process
  • Random knots
  • Reversible jump Markov chain Monte Carlo
  • Satellite measurements