Statistical correction of daily precipitation data from the climate models Jan Hnilica
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Statistical correction of daily precipitation data from the climate models Jan Hnilica Institute of Hydrodynamics of the Academy of Sciences of the Czech Republic Background A coarse spatial resolution of climate models leads to the loss of some physical phenomena, e.g. orographic precipitation, which results in the systematic deviation of model precipitation data from the observed values. Statistical approach to bias correction consist in finding an empirical relation between the simulated and observed data in the calibration period and applying of the relation to the model data from the period of interest. A validity of the derived relation in the future depends on the stationarity of the precipitation time series which brings an uncertainty within its application. Daily precipitation sum can be considered as gamma distributed random variable with pdf given by where α, β - the shape and scale parameters, respectively The dissimilar character of the observed and model data results in different course of pdfs: The aim of the paper is to find a linear transformation function, derived from an ) ( ) / exp( ) / ( ) ( 1 x x x pdf Solution Reduction of the high number of precipitation days To achieve a realistic number of precipitation days, the threshold value can be expressed in the form where Δ O , Δ M are the fractions of zero values in observed and model datasets, respectively. Then all values lower than or equal to x 0 are reduced to zero. Transformation of remaining nonzero values L1 method – scale modification Because X ~ Γ(α, β) implies AX ~ Γ(α, Aβ), such a value of A can be found, which minimizes the difference between pdf curves of the observed and modet data. This difference can be expressed as a sum of segments according to where y i are the values of the observed pdf, α, β are gamma parameters of model dataset. The gradient method was used to minimize D with respect to A. The subsequent transformation follows this scheme L2 method – simultaneous scale and shape modification To apply the more general function x T = Ax M + B, the coefficients A and B can be found by minimalization of the difference where parameters α´ and β´ depend on the actual values of A, B. The numerical derivatives were applied to minimize D with respect to A, B using the gradient method. The subsequent transformation follows this scheme ) ( M O 1 M 0 cdf x n ι i i i α β Α β Α x β Α x y D 1 1 ) Γ( )) ( / exp( ) ) ( / ( 0 M M 0 M T , , 0 x x x A x x x n i i i i x x y D 1 1 ) ´ ( ´ ) ´ / exp( ´ ) ´ / ( 0 M M 0 M T , , 0 x x B x A x x x We used the simulated daily precipitation sums from the regional climate model REMO (Jacob 2001) to test the methods described above. The model data in 10 × 10 km horizontal resolution were transformed to the observed data from 7 meteorological stations in the Malse River in the Czech Republic. We had at our disposal 38 years’ worth of time series observed daily precipitation sums from 1961 to 1998. For the calibration period we set the initial six years, 1961 – 1966. As the validation period we set the years, 1993 – 1998. Table 1 contains gamma parameters of the observed and transformed datasets from the validation period. Table 1 Results of the transformation Validation Station observed L1 L2 Benesov nad Cernou α 0.75 0.5 0.83 β 6.55 8.92 5.28 Dolni Dvoriste α 0.76 0.5 1.03 β 5.5 6.81 3.25 Kaplice α 0.81 0.49 0.74 β 5.62 8.14 5.37 Malonty α 0.74 0.58 0.83 β 5.92 8.32 5.74 Pohorska Ves α 0.74 0.48 0.94 β 6.72 9.47 4.73 Sobenov α 0.64 0.5 0.65 β 6.7 8.76 6.75 Stare Hute α 0.78 0.5 1 β 6.84 10.5 5.13 Results indicate, that L2 method usually leads to the more accurate transformation. Nevertheless, both methods bring the significant improvement. For more information and for comparison with nonlinear methods see Hnilica J., Pus V. (2012): Linear methods for the statistical transformation of daily precipitation sums from regional climate models. Theoretical and applied climatology. available online from: http://www.springerlink.com/content/101580/?Content+Status=Accep Contact: [email protected] Acknowledgements: This study was supported by the research grant IAA300600901 GA AS CR
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Statistical correction of daily precipitation data from the climate models Jan Hnilica Institute of Hydrodynamics of the Academy of Sciences of the Czech Republic. Background. Validation. Solution. R eduction of the high number of precipitation days - PowerPoint PPT Presentation
Transcript of Statistical correction of daily precipitation data from the climate models Jan Hnilica
Statistical correction of daily precipitation data from the climate modelsJan Hnilica
Institute of Hydrodynamics of the Academy of Sciences of the Czech Republic
Background A coarse spatial resolution of climate models leads to the loss of some physical phenomena, e.g. orographic precipitation, which results in the systematic deviation of model precipitation data from the observed values. Statistical approach to bias correction consist in finding an empirical relation between the simulated and observed data in the calibration period and applying of the relation to the model data from the period of interest. A validity of the derived relation in the future depends on the stationarity of the precipitation time series which brings an uncertainty within its application.
Daily precipitation sum can be considered as gamma distributed random variable with pdf given by
where α, β - the shape and scale parameters, respectively
The dissimilar character of the observed and model data results in different course of pdfs:
The aim of the paper is to find a linear transformation function, derived from an analysis of the probability distribution, for which pdf of transformed data tightly approximates pdf of the observed data.
)()/exp()/()(
1
xxxpdf
SolutionReduction of the high number of precipitation daysTo achieve a realistic number of precipitation days, the threshold value can be expressed in the form
where ΔO, ΔM are the fractions of zero values in observed and model datasets, respectively. Then all values lower than or equal to x0 are reduced to zero.
Transformation of remaining nonzero valuesL1 method – scale modificationBecause X ~ Γ(α, β) implies AX ~ Γ(α, Aβ), such a value of A can be found, which minimizes the difference between pdf curves of the observed and modet data. This difference can be expressed as a sum of segments according to
where yi are the values of the observed pdf, α, β are gamma parameters of model dataset. The gradient method was used to minimize D with respect to A. The subsequent transformation follows this scheme
L2 method – simultaneous scale and shape modificationTo apply the more general function xT = AxM + B, the coefficients A and B can be found by minimalization of the difference
where parameters α´ and β´ depend on the actual values of A, B. The numerical derivatives were applied to minimize D with respect to A, B using the gradient method. The subsequent transformation follows this scheme
)( MO1
M0 cdfx
n
ι
iii αβΑ
βΑxβΑxyD1
1
)Γ())( / exp())( / (
0MM
0M
T ,,0
xxxAxx
x
n
i
iii
xxyD1
1
)´(´)´/exp(´)´/(
0MM
0M
T ,,0
xxBxAxx
x
We used the simulated daily precipitation sums from the regional climate model REMO (Jacob 2001) to test the methods described above. The model data in 10 × 10 km horizontal resolution were transformed to the observed data from 7 meteorological stations in the Malse River in the Czech Republic. We had at our disposal 38 years’ worth of time series observed daily precipitation sums from 1961 to 1998. For the calibration period we set the initial six years, 1961 – 1966. As the validation period we set the years, 1993 – 1998. Table 1 contains gamma parameters of the observed and transformed datasets from the validation period.
Table 1 Results of the transformation
Validation
Station observed L1 L2
Benesov nad Cernouα 0.75 0.5 0.83β 6.55 8.92 5.28
Dolni Dvoristeα 0.76 0.5 1.03β 5.5 6.81 3.25
Kapliceα 0.81 0.49 0.74β 5.62 8.14 5.37
Malontyα 0.74 0.58 0.83β 5.92 8.32 5.74
Pohorska Vesα 0.74 0.48 0.94β 6.72 9.47 4.73
Sobenovα 0.64 0.5 0.65β 6.7 8.76 6.75
Stare Huteα 0.78 0.5 1β 6.84 10.5 5.13
Results indicate, that L2 method usually leads to the more accurate transformation. Nevertheless, both methods bring the significant improvement.
For more information and for comparison with nonlinear methods see Hnilica J., Pus V. (2012): Linear methods for the statistical transformation of daily precipitation sums from regional climate models. Theoretical and applied climatology.
available online from: http://www.springerlink.com/content/101580/?Content+Status=Accepted
Contact: [email protected]: This study was supported by the research grant IAA300600901 GA AS CR
