Multidimensional Hurst-Kolmogorov process for modelling … · Α=D- NH , N D, N D, Nithe number of...
1
European Geosciences Union General Assembly 2011 Vienna, Austria, 4-9 April 2011 Session HS7.5/NP6.7: Hydroclimatic stochastics Multidimensional Hurst-Kolmogorov process for modelling temperature and rainfall fields P. Dimitriadis 1 , D. Koutsoyiannis 1 , C. Onof 2 and K. Tzouka 1 1 Department of Water Resources and Environmental Engineering, National Technical University of Athens (www.itia.ntua.gr) 2 Department of Civil and Environmental Engineering, Imperial College of London 1. Abstract A multidimensional (MD) stochastic simulation model is presented, which is a direct extension Poly-line fits for 3D, 4D fields and lags 0 and 1, with the constrains ρD,r=1 for H=1 (ZD,r 1) and 5b. Discrete autocorrelation function of the MHK process 8. Case study on observed rainfall fields The application presented is based on an observed rainfall and temperature field South of the 10b. Dealing with anisotropy For this application, m is found 0 for the rainfall field and 2 for the temperature one, so that the A multidimensional (MD) stochastic simulation model is presented, which is a direct extension of the 1D simple scaling process, known as Hurst-Kolmogorov (HK) process following the analysis of the 2D extension of Koutsoyiannis et al. (2011). The MD HK (MHK) process can generate time-varying spatial geophysical fields (such as rainfall and temperature), consistent Poly-line fits for 3D, 4D fields and lags 0 and 1, with the constrains ρD,r=1 for H=1 (ZD,r 1) and ρD,0=0 for H=0.5 (CD0 more rapidly, so ZD,r0). Obviously, ρD,r=0 for H=0.5 but it seems that this happens for H coefficients very close to 0.5 where the Mathematica software could not converge. Hurst Coefficient D = 3 Hurst Coefficient D = 3 The application presented is based on an observed rainfall and temperature field South of the Indian Ocean (coordinates: 0N-30S, 55E-85E). The data were acquired from NASA satellite system (available on-line): http://disc2.nascom.nasa.gov/Giovanni/tovas/TRMM_V6.3B42.shtml (rainfall) For this application, m is found 0 for the rainfall field and 2 for the temperature one, so that the Hurst coefficients of 0.87 and 0.92 are approximately reached for all the sub-fields, respectively. generate time-varying spatial geophysical fields (such as rainfall and temperature), consistent with the observed long-term spatiotemporal persistence (slowly decaying autocorrelation over spatial or temporal displacement). The MHK process is formulated assuming anisotropy, so as to take into account possibly different autocorrelation decay rates (Hurst coefficients) in each 1.E+00 0.50 0.55 0.60 0.65 0.70 0.75 0.80 0.85 0.90 0.95 1.00 Hurst Coefficient D = 3 10 0.50 0.55 0.60 0.65 0.70 0.75 0.80 0.85 0.90 0.95 1.00 Hurst Coefficient Koutsoyiannis et al. (2010) Approximate Solution Mathematica Numerical Integration Poly. (Mathematica Numerical Integration) D = 3 http://disc2.nascom.nasa.gov/Giovanni/tovas/TRMM_V6.3B42.shtml (rainfall) http://gdata1.sci.gsfc.nasa.gov/daac-bin/G3/gui.cgi?instance_id=neespi_daily (temperature) The sample consists of a spatial grid 31 x 31 points (of a 1 o x 1 o spatial resolution, approx. 110 km 3D: Heq = 0.88 1D Time: H = 0.87 2D Space: H = 0.90 to take into account possibly different autocorrelation decay rates (Hurst coefficients) in each dimension of the field. The MHK process is also investigated through some applications based on observed temperature and rainfall fields. 2. Hurst phenomenon and the MHK process Z 3,0 = -1.204x 4 + 1.411x 3 + 3.600x 2 - 3.612x + 0.806 R 2 = 1 1.E-02 1.E-01 Z 3,0 Koutsoyiannis Approximated Solution Z 3 2 Z 3,1 x 110 km) and a temporal 270-days grid for September 2002 to 2010 (October was also needed for the calculation of the acrf). A “High tendency of high/low values to occur in natural events”: Hurst (1951) Slowly decaying autocorrelation over scale Power-law behaviour (Kolmogorov, 1940). 2. Hurst phenomenon and the MHK process 1.E-03 Koutsoyiannis Approximated Solution Mathematica Numerical Integration Poly. (Mathematica Numerical Integration) Hurst Coefficient D = 4 Hurst Coefficient D = 4 Z 3,1 = 2.827 Η 3 - 1.284 Η 2 - 9.166 Η + 8.623 R 2 = 1 1 Spatial averaged rainfall and temperature fields. The rainfall field 3D: Heq = 0.95 1D Time: H = 0.92 2D Space: H = 0.97 ( ) ( ) μ μ = ( ) () - - , where A k k l Z Z v v d l Z μ = Ε[ ] , v Α = D(1 - H) , 1 D D ... = ∑ ∑ 1 D 1 1 1 D vk vk (k) v i ,...,i D i =(v -1)k+ i (v -1)k 1 Z= Z k • Z: random field of interest (assumed stationary and isotropic) 1.E+00 0.50 0.55 0.60 0.65 0.70 0.75 0.80 0.85 0.90 0.95 1.00 Hurst Coefficient D = 4 1.E+02 0.50 0.55 0.60 0.65 0.70 0.75 0.80 0.85 0.90 0.95 1.00 Hurst Coefficient Koutsoyiannis et al. (2010) Approximate Solution Mathematica Numerical Integration Poly. (Mathematica Numerical Integration) D = 4 The temperature field The rainfall field contained no missing values. • Z v : mean aggregated field (at a spatio-temporal scale) • v: vector index of random field indicating location in the field • k,l: any aggregation scales of the process • μ: mean of the process Z 4,0 = 1.267x 4 - 8.774x 3 + 17.626x 2 - 11.470x + 2.350 R 2 = 1 1.E-02 1.E-01 Z 4,0 Z 4,1 = -44.024 H 3 + 144.422 H 2 - 162.857 H + 63.460 R 2 = 1 1.E+01 Z 4,1 Poly. (Mathematica Numerical Integration) The temperature field contained a lot of missing values which were supplemented 1 = - eq H A/D The equivalent Hurst coefficient can be determined by the equation: • μ: mean of the process • = d : equal in distribution function • A: power law exponent of autocorrelation over scale • D: dimension of vector index space of random field (v) 1.E-03 1.E-02 Koutsoyiannis Approximated Solution Mathematica Numerical Integration Poly. (Mathematica Numerical Integration) 1.E+00 assuming linear regression. 1 1 = = ≤ = ∑ ∑ n n i i i i i i Α=D- NH , N D, N D, N i the number of dimensions of the sub-field i 1 = - eq H A/D The equivalent Hurst coefficient can be determined by the equation: 3. Hurst coefficient (H) of the MHK process 6. Simulation scheme for generating MHK process 9. Normalization of fields 11. Stochastic simulation model 3. Hurst coefficient (H) of the MHK process • HK process depends on the characteristic parameter 0<H<1. Here, the estimation of the H coefficient is done via the minimization of the square error (SE ) of the empirical (S (k) ) 2 and true 6. Simulation scheme for generating MHK process SMA stands for Symmetric Moving Average and it can be used to generate a stochastic process with any structure of autocorrelation or power spectrum (Koutsoyiannis, 2000). Here, the SMA 9. Normalization of fields The zero values of the natural field are replaced with the small value of 1e-5. 10 0 11. Stochastic simulation model Rainfall 10 1 Rainfall stochastic simulation at position coefficient is done via the minimization of the square error (SE H ) of the empirical (S (k) ) 2 and true (γ (k) ) variance over scale k of the process. A method of Tyralis and Koutsoyiannis (2010) for the estimation of H was extended to the MHK process (D dimensions). ~ D ~ 2 with any structure of autocorrelation or power spectrum (Koutsoyiannis, 2000). Here, the SMA scheme has been extended to D spatio-temporal dimensions (direct extension from 1D and 2D schemes). The simulated field should be converted to natural units by solving arithmetically the inverted transformation. -3 10 -2 10 -1 tocorrelation (-) 10 0 nce Simulated HK Process Unbiased 3D field H = 0.88 Rainfall stochastic simulation at position (5,5) on the grid for September 2002 0.30 0.40 m/hr) Synthetic Natural ( ) ( ) ( ) ~ 2 2 ( ) ( ) k k , E D 2H-1 D D N/k -1 S =RS , R(k;H)= N/k - N/k ( ) ~ ( ) 2 k k H ∑ 2 k' p k=1 SE = ln(S )-ln( γ ) / k , p = 2 D 1 ... ∑ ∑ q q v y =-q y =-q Z= αW y v-y 10 -5 10 -4 10 -3 Coefficient of Aut Natural Simulated 10 -1 Varian q = 100 0.00 0.10 0.20 0 5 10 15 20 25 30 Rainfall (mm → → 0< < 0.5 Anticorrelated <0 = 0.5 Independent ( = 0) H (ρ ) H ρ • The autocovariance γ (acvf) and autocorrelation function ρ (acrf) of the MHK are expressed as: where stands for continuous B=2D(1-H)=2A c ( ) and c r D L = (k) -B (1) (k) -B (0) (0) γ =k γ γ r •Z v : generated normalized random field of interest • W: discrete white noise (random field) with zero mean (μ w = 0) and unit standard deviation (σ w = 1) (since Z 10 0 10 1 10 2 10 -6 Lag (-) Temperature 10 0 10 1 10 2 10 -2 Scale (-) Note: All the negative values of the synthetic rainfall field are set equal to zero. 0 5 10 15 20 25 30 Days → → = 0.5 Independent ( = 0) 0.5 < <1 Correlated ( > 0) H ρ H ρ stands for continuous is a coefficient is the lag c L r D ( ) ( ) ( ) (0) ( ) (0) / / k cr r cr D L → = (k) (k) -B ρ =γ γ ρ r γ Note: The continuous acvf and acrf become infinite for scale 0 and lag 0, respectively. w w has been normalized). •α : field of coefficients that can be determined through the Fourier transform F of the autocovariance field • q: finite limit for the range of coefficients α y (for m, the desired number of autocorrelation coefficients that are to be preserved). 10 -1 10 0 on (-) Temperature 0 10 1 Simulated HK Process Unbiased Temperature stochastic simulation at position (5,5) on the grid for September 2002 20 4. Field Normalization MHK process generates random fields that follow the N(0,1). Here the following transformation Note: The continuous acvf and acrf become infinite for scale 0 and lag 0, respectively. •α y : field of coefficients that can be determined through the Fourier transform F γ of the autocovariance field γ Z (Koutsoyiannis, 2000, Koutsoyiannis et al. 2010). Rainfall: Temperature: 10 -3 10 -2 Coefficient of Autocorrelatio 10 -1 10 0 Variance 3D field H = 0.92 17 18 19 Temperature (oC) Synthetic Natural q = 100 MHK process generates random fields that follow the N(0,1). Here the following transformation (Papalexiou et al., 2007) is used, where its coefficients p i are estimated through the minimization of the square error of the transformed data and the N(0,1) distribution function. ( ) ( ) ( ) 2 - p 1 = 77.0, p 2 = 6,.0 p 3 = -1.0, p 4 = -2.2E-5, p 5 = 0.073, SE = 54.0 p 1 = 8.0E-7, p 2 = 1E4, p 3 = 7.4, p 4 = -35.2, p 5 = 0.00165, SE = 97.0 10 0 10 1 10 2 10 -5 10 -4 Lag (-) Co Natural Simulated 10 0 10 1 10 2 10 -2 Scale (-) 3D field H = 0.92 16 0 5 10 15 20 25 30 Days T Natural Note: The simulated autocorrelations seem to be smaller than the natural ones. This is due to the small q ( ) ( ) ( ) 5 2 1 2 3 4 4 3 1 1/ ln 1 p N Z pZ p p p p Z p - = + + + - + Note: The simulated autocorrelations seem to be smaller than the natural ones. This is due to the small q parameter that is chosen, as larger values would enormous increase the numerical simulation time. 5a. Discrete autocorrelation function of the MHK process 7. Spectral density and α y coefficients of SMA 10a. Dealing with anisotropy 12. Conclusions For the discrete acrf, one can adapt the Koutsoyiannis et al. (2010) approximate solution (KAS); this works well for D ≤ 2 and for D ≥ 3 and lags greater than 1; for lags 0 and 1, a poly-line fit for lags can give better results. ( ) ( ) B D discete D ρ j ,...,j I 2 j j j j dv dv r j jj 1 1 /2 2 2 2 2 1 ( ) γ ... ( ... 1 ... 1 ... , ... , - = = + + - - = + + ∈ ∫ ∫ The spectral density F γ of the stochastic field can be determined via the Fourier transform of the discrete form of autocovariance γ discrete (r). It can be shown that the Fourier transform F α , of the field α y , is related to F γ (for q=∞), thus the α y field can then be estimated. A separate Hurst coefficient should be assigned to the quantities that are non isotropic to each other. It is still not applicable to create a multi-dimensional model that can synthesize time-series assuming anisotropy (thus assuming a different autocorrelation behavior in each dimension) and that is why hydrologists tend to use multi-variate models. A proposed solution is to omit • A multi-dimensional (MD) stochastic simulation model is proposed, which is a direct extension of the 1D simple scaling process (HK or FGN). 2H 2H 2H 1,r , ρ =r+1 /2+r-1 /2-r ( ) { } discete , 1, 1 1, () , ρ CρC ρ ρ min = ≈ D Dr D r r r ( ) ( ) D D discete D Dr D D D D i v v D I 2 j j j j dv dv r j jj L 1 2 2 2 2 1 , (0) 1 1 1 1 0 0 γ ... ( ... 1 ... 1 ... , ... , = = = = + + - - = + + ∈ ∫ ∫ C field α y , is related to F γ (for q=∞), thus the α y field can then be estimated. ( ) ( ) /2 /2 /2 /2 1 /2 1 /2 1 /2 1 2 () () {2 } {2 } D D/2-B γ 0 0 2π π Fs= r γrJ πsr dr = r J πsr dr ∞ ∞ - - - - → ∫ ∫ D D D D D D D L s s that is why hydrologists tend to use multi-variate models. A proposed solution is to omit intermediate data of the field grids, so as the multi-acrfs decay at the same rate (at least for the first lags). So, omitting factor (omfc) m means that the (m*c)th cell is omitted in the model, where c is 0,1,…,maximum number of cells in each direction of the sub-field and in the diagonally ones. • The HK process and the SMA generation algorithm are extended for any dimension D of the field (the autocorrelation function is extended for D≤4 and a methodology is proposed for greater dimensions). Hurst Coefficient r m = 1/√D 0.50 0.55 0.60 0.65 0.70 0.75 0.80 0.85 0.90 0.95 1.00 Hurst Coefficient r = 2√D ,0 ,0 (0) ρ 1 ( 1) D D D D D I L D γ = ≈ + (2H-1)[D(2H-1)+1] , C = C , B C ρ r Note that for great lags: ≈ D Dr 0 0 s s ( ) [ ] 2 2 1 1 () 1 1 2 2 2 B-D γ Γ D-B F LEs , Eπ , for <B<D <H< - , Γ where - + → = = → ∈ D B D s s B D Temperature 1.0 is 0,1,…,maximum number of cells in each direction of the sub-field and in the diagonally ones. Rainfall 1.0 Spatial Temporal m = 2, so that the spatial H=0.97 a methodology is proposed for greater dimensions). • The MHK process is formulated assuming anisotropy through a methodology of changing the m omitting factor of the grids until the minimum Hurst 1.E+00 0.50 0.55 0.60 0.65 0.70 0.75 0.80 0.85 0.90 0.95 1.00 ρ2,r ρ3,r ρ4,r 1.E+00 0.50 0.55 0.60 0.65 0.70 0.75 0.80 0.85 0.90 0.95 1.00 [ ] 2 2 2 Γ B D great lags discete 0 discrete 1 1 Thus it can be shown that for 1 2 2 B-D γ , , F CγEs , <H< - , s ≈ ∈ D discrete D 0.6 0.8 elation 0.6 0.8 relation m = 2, so that the spatial H=0.97 becomes approximately equal to the temporal H=0.92. m ≈ 0, as the temporal and of changing the m omitting factor of the grids until the minimum Hurst coefficient of the sub-fields is reached. • A 3D spatio-temporal model is applied based on an observed rainfall and 1.E-01 ρ D,r ρ2,r ρ3,r ρ4,r ρ2,r m ρ3,r m ρ4,r m ρ 1,r 1.E-02 1.E-01 γ/C D = 2 D = 3 ΖD,2 discrete discrete Also it can be assumed that for 0< <1and a coefficient B-D γ , ,F s , H K ≈ D D K From the above equations and assumptions it can be derived that: 0.2 0.4 Autocorre Spatial Temporal 0.2 0.4 Autocorr m ≈ 0, as the temporal and spatial H coefficients are very close to each other. temperature field. References 1.E-03 1.E-02 1.E-03 1.E-02 Koutsoyiannis et al. (2010) Approximate Solution Mathematica Numerical Integration D = 4 ( ) ( ) 1/( 1) 1, m 1 1, m 1, m 1 D D D D r r r D - → = Cρ C =ρ ρ CC discrete extension of 1 SMA γ y 0 F F α ρ( y ;H'), where = → ≈ D α α ' ( ) / 2 ' ( 0.5) / 2 - = - → = + B D B D H H From the above equations and assumptions it can be derived that: 0.0 0 5 10 15 20 25 30 lag Temporal Spatial for m=2 0.0 0 5 10 15 20 25 30 lag Since, the autocorrelagram is more sensible than the climacogram (as the first is the second Hurst, H.E. (1951) Long term storage capacities of reservoirs. Trans. ASCE 116, 776-808. Kolmogorov, A.N., WienerscheSpiralenund einigeandereinteressanteKurvenin HilbertschenRaum, Dokl. Akad. NaukURSS, 26, 115–118, 1940. Koutsoyiannis, D. (2000) A generalized mathematical framework for stochastic simulation and forecast of hydrologic time series. Wat. Resour. Re. 36(6), 1519-1534. 1.E-03 Comparison of equation ρD,r with the KAS for lag 1/√D for a 2D, 3D and 4D field. It can be observed that the min function equals the first term when rm ≥ 1/√ D. For lags r greater than 1 (thus r≥ 2), the KAS can be used. , , , Z / ρ 1/ Dr D Dr Dr I ≈ =C 0 ( ) ( ) ( ') ( ') D D C HEH C H EH α ∞ and 0,q= γ = 1 D D 1 ,..., ... → ∑ ∑ ∑ D q q q,q,...,q 2 2 2 0 y y 0 0 y =-q y =-q =-q,-q,...,-q γ= α α =γ / ρ( ;H') y y Since, the autocorrelagram is more sensible than the climacogram (as the first is the second derivative of the second, Koutsoyiannis, 2010), it is rather more appropriate to work with the second. So, one should change the m until the minimum Hurst coefficient is reached. Koutsoyiannis, D. (2010), A random walk on water, Hydrol. Earth Syst. Sci., 14, 585–601, 2010. Koutsoyiannis et al., (2011), Two-dimensional Hurst-Kolmogorov dynamics and its application to the study of rainfall fields (in preparation). Papalexiou, S.M., A. Montanari, and D. Koutsoyiannis, Scaling properties of fine resolution point rainfall and inferences for its stochastic modelling, European Geosciences Union General Assembly 2007, Geophysical Research Abstracts, Vol. 9, Vienna, 11253, European Geosciences Union, 2007. be used. D Union General Assembly 2007, Geophysical Research Abstracts, Vol. 9, Vienna, 11253, European Geosciences Union, 2007.
Transcript of Multidimensional Hurst-Kolmogorov process for modelling … · Α=D- NH , N D, N D, Nithe number of...
European Geosciences Union General Assembly 2011 Vienna, Austria, 4-9 April 2011 Session HS7.5/NP6.7: Hydroclimatic stochasticsEuropean Geosciences Union General Assembly 2011 Vienna, Austria, 4-9 April 2011 Session HS7.5/NP6.7: Hydroclimatic stochastics
Multidimensional Hurst-Kolmogorov process for modelling temperature and rainfall fieldsMultidimensional Hurst-Kolmogorov process for modelling temperature and rainfall fieldsP. Dimitriadis1, D. Koutsoyiannis1, C. Onof2 and K. Tzouka1
1Department of Water Resources and Environmental Engineering, National Technical University of Athens (www.itia.ntua.gr)Department of Water Resources and Environmental Engineering, National Technical University of Athens (www.itia.ntua.gr)2Department of Civil and Environmental Engineering, Imperial College of London
1. AbstractA multidimensional (MD) stochastic simulation model is presented, which is a direct extension Poly-line fits for 3D, 4D fields and lags 0 and 1, with the constrains ρD,r=1 for H=1 (ZD,r �1) and
5b. Discrete autocorrelation function of the MHK process 8. Case study on observed rainfall fieldsThe application presented is based on an observed rainfall and temperature field South of the
10b. Dealing with anisotropyFor this application, m is found 0 for the rainfall field and 2 for the temperature one, so that the A multidimensional (MD) stochastic simulation model is presented, which is a direct extension
of the 1D simple scaling process, known as Hurst-Kolmogorov (HK) process following the analysis of the 2D extension of Koutsoyiannis et al. (2011). The MD HK (MHK) process can generate time-varying spatial geophysical fields (such as rainfall and temperature), consistent
Poly-line fits for 3D, 4D fields and lags 0 and 1, with the constrains ρD,r=1 for H=1 (ZD,r �1) and ρD,0=0 for H=0.5 (CD�0 more rapidly, so ZD,r�0). Obviously, ρD,r=0 for H=0.5 but it seems that this happens for H coefficients very close to 0.5 where the Mathematica software could not converge.
Hurst CoefficientD = 3 Hurst CoefficientD = 3
The application presented is based on an observed rainfall and temperature field South of the Indian Ocean (coordinates: 0N-30S, 55E-85E). The data were acquired from NASA satellite system (available on-line):
http://disc2.nascom.nasa.gov/Giovanni/tovas/TRMM_V6.3B42.shtml (rainfall)
For this application, m is found 0 for the rainfall field and 2 for the temperature one, so that the Hurst coefficients of 0.87 and 0.92 are approximately reached for all the sub-fields, respectively.
generate time-varying spatial geophysical fields (such as rainfall and temperature), consistent with the observed long-term spatiotemporal persistence (slowly decaying autocorrelation over spatial or temporal displacement). The MHK process is formulated assuming anisotropy, so as to take into account possibly different autocorrelation decay rates (Hurst coefficients) in each
1.E+00
0.50 0.55 0.60 0.65 0.70 0.75 0.80 0.85 0.90 0.95 1.00
Hurst CoefficientD = 3
10
0.50 0.55 0.60 0.65 0.70 0.75 0.80 0.85 0.90 0.95 1.00Hurst Coefficient
Koutsoyiannis et al. (2010) Approximate SolutionMathematica Numerical IntegrationPoly. (Mathematica Numerical Integration)
D = 3 http://disc2.nascom.nasa.gov/Giovanni/tovas/TRMM_V6.3B42.shtml (rainfall) http://gdata1.sci.gsfc.nasa.gov/daac-bin/G3/gui.cgi?instance_id=neespi_daily (temperature)
The sample consists of a spatial grid 31 x 31 points (of a 1o x 1o spatial resolution, approx. 110 km
3D: Heq = 0.88 1D Time: H = 0.87 2D Space: H = 0.90
to take into account possibly different autocorrelation decay rates (Hurst coefficients) in each dimension of the field. The MHK process is also investigated through some applications based on observed temperature and rainfall fields.
2. Hurst phenomenon and the MHK process
Z3,0 = -1.204x4 + 1.411x3 + 3.600x2 - 3.612x + 0.806
R2 = 11.E-02
1.E-01
Z3,
0
Koutsoyiannis Approximated Solution Z 3 2
Z3,
1
Poly. (Mathematica Numerical Integration) The sample consists of a spatial grid 31 x 31 points (of a 1 x 1 spatial resolution, approx. 110 km x 110 km) and a temporal 270-days grid for September 2002 to 2010 (October was also needed for the calculation of the acrf).
A
“High tendency of high/low values to occur in natural events”: Hurst (1951) � Slowly decaying autocorrelation over scale� Power-law behaviour (Kolmogorov, 1940).
2. Hurst phenomenon and the MHK process1.E-03
Koutsoyiannis Approximated SolutionMathematica Numerical IntegrationPoly. (Mathematica Numerical Integration)
Hurst CoefficientD = 4Hurst CoefficientD = 4
Z3,1 = 2.827 Η3 - 1.284 Η2 - 9.166 Η + 8.623
R2 = 1
1
Spatial averaged rainfall and temperature fields. The rainfall field
3D: Heq = 0.95 1D Time: H = 0.92 2D Space: H = 0.97
( ) ( )µ µ
=
( ) ( )- - , whereA
kk lZ Zv vd lZμ = Ε[ ] , v Α = D(1 - H) ,
1
D
D
...=
∑ ∑1
D
1
1 1 D
v k v k(k)
v i ,...,iDi =(v -1)k + i (v -1)k
1Z = Z
k
• Z: random field of interest (assumed stationary and isotropic)
1.E+00
0.50 0.55 0.60 0.65 0.70 0.75 0.80 0.85 0.90 0.95 1.00Hurst CoefficientD = 4
1.E+02
0.50 0.55 0.60 0.65 0.70 0.75 0.80 0.85 0.90 0.95 1.00
Hurst Coefficient
Koutsoyiannis et al. (2010) Approximate Solution
Mathematica Numerical Integration
Poly. (Mathematica Numerical Integration)
D = 4
The temperature field
The rainfall field contained no missing values.
• Z: random field of interest (assumed stationary and isotropic)• Zv: mean aggregated field (at a spatio-temporal scale)• v: vector index of random field indicating location in the field• k,l: any aggregation scales of the process• μ: mean of the process
Z4,0 = 1.267x4 - 8.774x3 + 17.626x2 - 11.470x + 2.350
R2 = 1
1.E-02
1.E-01
Z4,
0 Z4,1 = -44.024 H3 + 144.422 H2 - 162.857 H + 63.460
R2 = 11.E+01
Z4
,1
Poly. (Mathematica Numerical Integration) The temperature field contained a lot of missing values which were supplemented 1= −
eqH A / DThe equivalent Hurst coefficient can be determined by the equation:• μ: mean of the process
• =d: equal in distribution function• A: power law exponent of autocorrelation over scale• D: dimension of vector index space of random field (v) 1.E-03
1.E-02
Koutsoyiannis Approximated SolutionMathematica Numerical IntegrationPoly. (Mathematica Numerical Integration)
1.E+00
were supplemented assuming linear regression.
1 1= =
≤ =∑ ∑n n
i i i ii i
Α = D - N H , N D, N D, Ni the number of dimensions of the sub-field i
1= −eq
H A / DThe equivalent Hurst coefficient can be determined by the equation:
3. Hurst coefficient (H) of the MHK process 6. Simulation scheme for generating MHK process 9. Normalization of fields 11. Stochastic simulation model3. Hurst coefficient (H) of the MHK process• HK process depends on the characteristic parameter 0<H<1. Here, the estimation of the Hcoefficient is done via the minimization of the square error (SE ) of the empirical (S(k))2 and true
6. Simulation scheme for generating MHK processSMA stands for Symmetric Moving Average and it can be used to generate a stochastic process with any structure of autocorrelation or power spectrum (Koutsoyiannis, 2000). Here, the SMA
9. Normalization of fieldsThe zero values of the natural field are replaced with the small value of 1e-5. 10
0
11. Stochastic simulation modelRainfall
101
Rainfall stochastic simulation at position coefficient is done via the minimization of the square error (SEH) of the empirical (S(k))2 and true (γ(k)) variance over scale k of the process. A method of Tyralis and Koutsoyiannis (2010) for the estimation of H was extended to the MHK process (D dimensions).
~DN / k - 1
~ 2
with any structure of autocorrelation or power spectrum (Koutsoyiannis, 2000). Here, the SMA scheme has been extended to D spatio-temporal dimensions (direct extension from 1D and 2D schemes).
The simulated field should be converted to natural units by solving arithmetically the inverted transformation.
-3
10-2
10-1
Coeff
icie
nt
of
Auto
corr
ela
tion (
-)
100
Vari
an
ce
Simulated
HK Process
Unbiased3D field H = 0.88
Rainfall stochastic simulation at position (5,5) on the grid for September 2002
0.30
0.40
Rai
nfal
l (m
m/h
r)
Synthetic
Natural
( ) ( )( )
~
2 2( ) ( )k k
, E
D
2H -1D D
N / k - 1S = R S , R(k; H) =
N / k - N / k
( )~
( )2k
k
H
∑2
k'p
k =1
SE = ln(S )- ln(γ ) / k , p = 2
D 1
...∑ ∑q q
vy = -q y = -q
Z = α Wy v-y
10-5
10-4
10-3
Coeff
icie
nt
of
Auto
corr
ela
tion (
-)
Natural
Simulated
10-1
Vari
an
ce
q = 100
0.00
0.10
0.20
0 5 10 15 20 25 30
Rai
nfal
l (m
m/h
r)
→
→
0 < < 0.5 Anticorrelated < 0
= 0.5 Independent ( = 0)
H (ρ )
H ρ
• The autocovariance γ (acvf) and autocorrelation function ρ (acrf) of the MHK are expressed as:where
stands for continuous
B = 2D(1 - H) = 2A
c ( ) and
c r DL=(k) -B (1) (k) -B
(0) (0)γ = k γ γ r•Zv: generated normalized random field of interest
• W: discrete white noise (random field) with zero mean (μw = 0) and unit standard deviation (σw = 1) (since Z
100
101
102
10-6
Lag (-)
Simulated
Temperature
100
101
102
10-2
Scale (-)
Note: All the negative values of the synthetic rainfall field are set equal to zero.
0 5 10 15 20 25 30Days
→
→
= 0.5 Independent ( = 0)
0.5 < < 1 Correlated ( > 0)
H ρ
H ρ
stands for continuous
is a coefficient
is the lag
c
L
r D
( )
( ) ( ) (0) ( ) (0)/ /k
c r r c r DL→ =(k) (k) -Bρ = γ γ ρ r γ
Note: The continuous acvf and acrf become infinite for scale 0 and lag 0, respectively.
• W: discrete white noise (random field) with zero mean (μw = 0) and unit standard deviation (σw = 1) (since Zhas been normalized).
•α : field of coefficients that can be determined through the Fourier transform F of the autocovariance field
• q: finite limit for the range of coefficients αy (for m, the desired number of autocorrelation coefficients that are to be preserved). 10
-1
100
Coeff
icie
nt
of
Auto
corr
ela
tion (
-)
Temperature
0
101
Simulated
HK Process
Unbiased
Temperature stochastic simulation at position (5,5) on the grid for September 2002
20
4. Field NormalizationMHK process generates random fields that follow the N(0,1). Here the following transformation
Note: The continuous acvf and acrf become infinite for scale 0 and lag 0, respectively. •αy: field of coefficients that can be determined through the Fourier transform Fγ of the autocovariance field γZ (Koutsoyiannis, 2000, Koutsoyiannis et al. 2010).
Rainfall: p = 77.0, p = 6,.0 p = -1.0, p = -2.2E-5,
Temperature: 10
-3
10-2
Coeff
icie
nt
of
Auto
corr
ela
tion (
-)
10-1
100
Variance
3D field H = 0.9217
18
19
Tem
per
atu
re (
oC)
Synthetic
Natural
q = 100
MHK process generates random fields that follow the N(0,1). Here the following transformation (Papalexiou et al., 2007) is used, where its coefficients pi are estimated through the minimization of the square error of the transformed data and the N(0,1) distribution function.
( ) ( ) ( )2−
Rainfall: p1 = 77.0, p2 = 6,.0 p3 = -1.0, p4 = -2.2E-5, p5 = 0.073, SE = 54.0
Temperature: p1 = 8.0E-7, p2 = 1E4, p3 = 7.4, p4 = -35.2, p5
= 0.00165, SE = 97.010
010
110
210
-5
10-4
Lag (-)
Coeff
icie
nt
of
Auto
corr
ela
tion (
-)
Natural
Simulated
100
101
102
10-2
Scale (-)
3D field H = 0.9216
0 5 10 15 20 25 30Days
Tem
per
atu
re (
oC)
Natural
Note: The simulated autocorrelations seem to be smaller than the natural ones. This is due to the small q( ) ( ) ( )52
1 2 3 4 4 31 1 / ln 1p
NZ p Z p p p p Z p− = + + + − +
Note: The simulated autocorrelations seem to be smaller than the natural ones. This is due to the small qparameter that is chosen, as larger values would enormous increase the numerical simulation time.
5a. Discrete autocorrelation function of the MHK process 7. Spectral density and αy coefficients of SMA 10a. Dealing with anisotropy 12. Conclusions5a. Discrete autocorrelation function of the MHK processFor the discrete acrf, one can adapt the Koutsoyiannis et al. (2010) approximate solution (KAS); this works well for D ≤ 2 and for D ≥ 3 and lags greater than 1; for lags 0 and 1, a poly-line fit for lags can give better results.
( ) ( )BDdiscete D
ρ j ,..., jI 2 j j j j dv dv r j j j
1 1/22 2 2 21( )
γ ... ( ... 1 ... 1 ... , ... ,−
= = + + − − = + + ∈∫ ∫ �
y
The spectral density Fγ of the stochastic field can be determined via the Fourier transform of the discrete form of autocovariance γdiscrete(r). It can be shown that the Fourier transform Fα, of the field αy, is related to Fγ (for q=∞), thus the αy field can then be estimated.
A separate Hurst coefficient should be assigned to the quantities that are non isotropic to each other. It is still not applicable to create a multi-dimensional model that can synthesize time-series assuming anisotropy (thus assuming a different autocorrelation behavior in each dimension) and that is why hydrologists tend to use multi-variate models. A proposed solution is to omit
12. Conclusions• A multi-dimensional (MD) stochastic simulation model is proposed, which is
a direct extension of the 1D simple scaling process (HK or FGN).
2H 2H 2H
1,r, ρ = r + 1 / 2 + r - 1 / 2 - r( ){ }discete , 1, 1 1,( ) ,ρ C ρ C ρρ min= ≈D
D r D r rr
( ) ( )D
BDdiscete D
D r D D D D i
v vD
ρ j ,..., jI 2 j j j j dv dv r j j j
L1
/22 2 2 21, (0) 1 1 1 1
0 0
( )γ ... ( ... 1 ... 1 ... , ... ,
−
= =
= = + + − − = + + ∈ ∫ ∫ �
C
field αy, is related to Fγ (for q=∞), thus the αy field can then be estimated.
( ) ( )/ 2 / 2
/2
/ 2 1 / 2 1/ 2 1 / 2 1
2( ) ( ) {2 } {2 }
D D / 2- B
γ
0 0
2π πF s = r γ r J πsr dr = r J πsr dr
∞ ∞
− −− −→∫ ∫
D D
D D DD DL
s s
that is why hydrologists tend to use multi-variate models. A proposed solution is to omit intermediate data of the field grids, so as the multi-acrfs decay at the same rate (at least for the first lags). So, omitting factor (omfc) m means that the (m*c)th cell is omitted in the model, where cis 0,1,…,maximum number of cells in each direction of the sub-field and in the diagonally ones.
• The HK process and the SMA generation algorithm are extended for any dimension D of the field (the autocorrelation function is extended for D≤4 and a methodology is proposed for greater dimensions).
Hurst Coefficientrm = 1/√D0.50 0.55 0.60 0.65 0.70 0.75 0.80 0.85 0.90 0.95 1.00
Hurst Coefficientr = 2√D
,0
,0 (0)
ρ 1
( 1)
D
D D
D D
IL D
γ= ≈+
(2H - 1)[D(2H - 1) + 1], C =
C, B
Cρ
rNote that for great lags: ≈ D
D r
0 0∫ ∫s s
( )[ ]
22 1 1
( ) 1 12 2 2
B- D
γ
Γ D - BF L E s , E π , for < B < D < H < - ,
Γwhere
− + → = = → ∈�D B
Ds s
B D
Temperature1.0
is 0,1,…,maximum number of cells in each direction of the sub-field and in the diagonally ones.
R a in fa ll1 .0
S pa tia lTempora l m = 2, so that the spatial H=0.97
a methodology is proposed for greater dimensions).
• The MHK process is formulated assuming anisotropy through a methodology of changing the m omitting factor of the grids until the minimum Hurst
1.E+00
0.50 0.55 0.60 0.65 0.70 0.75 0.80 0.85 0.90 0.95 1.00
ρ2,r ρ3,r ρ4,r
1.E+00
0.50 0.55 0.60 0.65 0.70 0.75 0.80 0.85 0.90 0.95 1.00 [ ]2 2 2Γ B D
great lags
discete 0 discrete
1 1Thus it can be shown that for 1
2 2
B- D
γ, , F C γ E s , < H < - , s≈ ∈�D discreteD
0.6
0.8
Au
toco
rrel
atio
n0 .6
0 .8
Au
toco
rrel
atio
nm = 2, so that the spatial H=0.97 becomes approximately equal to the temporal H=0.92.
m ≈ 0, as the temporal and
of changing the m omitting factor of the grids until the minimum Hurst coefficient of the sub-fields is reached.
• A 3D spatio-temporal model is applied based on an observed rainfall and temperature field.
1.E-02
1.E-01
ρ D,r
ρ2,r ρ3,r ρ4,rρ2,r m ρ3,r m ρ4,r m
ρ1,r
1.E-02
1.E-01
γ/C
D = 2
D = 3
ΖD
,2
discrete discreteAlso it can be assumed that for 0< <1 and a coefficientB- D
γ, , F s , H K ≈ D DK
From the above equations and assumptions it can be derived that:0.2
0.4
Au
toco
rrel
atio
n
Spatial
Temporal
0 .2
0 .4
Au
toco
rrel
atio
n
m ≈ 0, as the temporal and spatial H coefficients are very close to each other.
• A 3D spatio-temporal model is applied based on an observed rainfall and temperature field.
References
1.E-03
1.E-02
1.E-03
1.E-02
Koutsoyiannis et al. (2010) ApproximateSolutionMathematica Numerical IntegrationD = 4 ( ) ( )1/ ( 1)
1, m 1 1, m 1, m 1
DD D
D r r r D
−→ =C ρ C = ρ ρ C C
discrete
extension
of 1 SMAγ y 0F F α ρ( y ; H'), where= → ≈
Dα α ' ( ) / 2 ' ( 0.5) / 2− = − → = +B D B D H H
From the above equations and assumptions it can be derived that:0.0
0 5 10 15 20 25 30lag
Temporal
Spatial for m=20 .0
0 5 1 0 1 5 2 0 2 5 3 0la g
Since, the autocorrelagram is more sensible than the climacogram (as the first is the second
Hurst, H.E. (1951) Long term storage capacities of reservoirs. Trans. ASCE 116, 776-808.
Kolmogorov, A.N., WienerscheSpiralenund einigeandereinteressanteKurvenin HilbertschenRaum, Dokl. Akad. NaukURSS, 26, 115–118, 1940.
Koutsoyiannis, D. (2000) A generalized mathematical framework for stochastic simulation and forecast of hydrologic time series. Wat. Resour. Re. 36(6), 1519-1534. 1.E-03
Comparison of equation ρD,r with the KAS for lag 1/√D for a 2D, 3D and 4D field. It can be observed that the min function equals the first term when rm ≥ 1/√ D.
For lags r greater than 1 (thus r≥ 2), the KAS can be used.
, , ,Z / ρ 1 /D r D D r D rI≈= C 0 ( ) ( )
( ') ( ')
D
D
C H E H
C H E Hα ∞and 0,q=
γ=
1 D
D 1
,...,... →∑ ∑ ∑�����
D
q q q,q,...,q2 2 2
0 y y 0 0y = -q y = -q = -q,-q,...,-q
γ = α α = γ / ρ ( ; H')y
ySince, the autocorrelagram is more sensible than the climacogram (as the first is the second derivative of the second, Koutsoyiannis, 2010), it is rather more appropriate to work with the second. So, one should change the m until the minimum Hurst coefficient is reached.
Koutsoyiannis, D. (2010), A random walk on water, Hydrol. Earth Syst. Sci., 14, 585–601, 2010.
Koutsoyiannis et al., (2011), Two-dimensional Hurst-Kolmogorov dynamics and its application to the study of rainfall fields (in preparation).
Papalexiou, S.M., A. Montanari, and D. Koutsoyiannis, Scaling properties of fine resolution point rainfall and inferences for its stochastic modelling, European Geosciences Union General Assembly 2007, Geophysical Research Abstracts, Vol. 9, Vienna, 11253, European Geosciences Union, 2007. min function equals the first term when rm ≥ 1/√ D.be used. D Union General Assembly 2007, Geophysical Research Abstracts, Vol. 9, Vienna, 11253, European Geosciences Union, 2007.
