Analysis and mapping of spatio-temporal hydrologic data: HYDROSTOCHASTICS

Transcription

1 Analysis and mapping of spatio-temporal hydrologic data: HYDROSTOCHASTICS Lars Gottschal & Irina Krasovsaia Gottschal & Krasovsaia

2 Mapping spatio-temporal hydrologic data. Runoff maps Huai He, China Consistent Water Balance Maps Huai He, China Time series: Gap filling and estimation at ungauged sites Flood & Low flow Magdalena, Colombia Moselle, France Gottschal & Krasovsaia

3 Methods of Analysing Variability Observations is the ey issue for analysing variability in space and time. Probability Theory and Statistical Methods give the necessary theoretical bacground to analyse and draw conclusions from these observations. The observations are seen as realisations of spatiotemporal random processes. Dependent on the problem and observations at hand it can be narrowed to a purely temporal process i.e. a time series or a purely spatial process. A final step is to consider observations as a random variable. Gottschal & Krasovsaia 3

4 Methods of Analysing Variability Characterisation of a random process Many important characteristics of random processes viz. homogeneity (stationarity), isotropy and ergodicity, permit a more effective use of the limited data amount available for estimation of probability distributions and their parameters. The strict definitions of these characteristics can be formulated with the help of the multivariate probability distribution function f X (x)=f X (x,x,,x M ). A random process is called homogeneous (stationary) if all multivariate distributions do not change with the movement in the parameter space (translation, not rotation). This implies that all probabilities depend on relative and not absolute positions of points u,u,..., u M in the parameter space. A process is called isotropic if the multivariate distribution functions remain the same even when the constellation of points u, u,..., u M is rotated in the parameter space. A random process is ergodic if all information about this multivariate distribution (and its parameters) is contained in a single realization of the random process. Gottschal & Krasovsaia 4

5 Methods of Analysing Variability Characterisation of a random process The parameter space of a random process usually includes an unlimited and infinite number of points. General characterisation by means of multivariate probability distributions is therefore only of a theoretical value. The alternatives are partial characterisation: i. Characterisation by distribution function (one dimensional). ii. Second moment characterisation. iii. Karhunen-Loève expansion i.e. a series representation in terms of random variables and deterministic functions of a random process. Gottschal & Krasovsaia 5

6 Mapping of spatio-temporal hydrologic data. Runoff maps Huai He, China Second moment characterisation. Consistent Water Balance Maps Second moment characterisation. Huai He, China Time series: Gap filling and estimation at ungauged sites Karhunen-Loève expansion Gottschal & Krasovsaia Magdalena, Colombia Flood & Low flow Characterization by distribution function (one dimensional). Moselle, France 6

7 Second moment characterisation Random process in space For a spatial random process the parameter space is defined in the plane u=(u,u ) over an area. For spatial processes first and second order moments are functions of space: E X u m X u var X u u ; X The mixed second order moment the spatial covariance is dependent on the absolute position between points u and u : cov X u X u EX u X u m u u X m The assumption of homogeneity does not apply to spatial random processes. The semivariogram may then replace the spatial covariance: u, u E X u X u Gottschal & Krasovsaia 7 X

8 Mapping of spatio-temporal hydrologic data. Can all descriptors be displayed on a map? The mean value does not change with scale and is therefore the only descriptor that correctly can be shown on a map. All other statistical descriptors show strong temporal and spatial scaling. Runoff can only be mapped along rivers. Gottschal & Krasovsaia Coefficient of variation. Gottschal et al. (979) 8

9 Mapping of spatio-temporal hydrologic data. Theoretical constraints along rivers Physical laws Continuity equation Spatio-temporal dynamics Statistical laws Unbiased expected values Sum of variancecovariance Sum of distributions Extreme value theory Gottschal & Krasovsaia 9

10 Mapping of spatio-temporal hydrologic data. Point of departure - Mean values The long term annual mean value is the statistical characteristic that can be expected to have the highest accuracy. It contains the main reliable signal in data. In a second step we can study the relative anomalies from this average pattern in other descriptors. It is the first order moment and as such the basic characteristic of the distribution function. It is the only characteristic that does not depend on the dynamics of the runoff process. For higher order moments time-space interpolators are needed. It does not change with scale and is therefore the only descriptor that correctly can be shown on a map. If it is combined with mapping of long-term means of precipitation and actual evapotranspiration to be able to close the water balance the accuracy is improved. Gottschal & Krasovsaia 0

11 Mapping of spatio-temporal hydrologic data. Second order moment The spatio-temporal variance-covariance function is the fundamental tool for consistent mapping (spatio-temporal interpolator) of second order moments along rivers An important achievement is the possibility of separating scaling effects (variance reduction) and non-homogeneity in variance A proper model needs to be able to account for: - the role of temporal support for explaining the changes in the autocorrelation function with changing duration of averaging. - the role of spatial support (basin area) for explaining differences in variance reduction. - the role of the degree of intermittency i.e. proportion between quic flow and baseflow Gottschal & Krasovsaia

12 Karhunen-Loève expansion Expansion of random processes into orthogonal functions Karhunen-Loève expansion: a series representation of a random process in terms of random variables and deterministic functions. The deterministic functions can either be postulated: - Harmonic Analysis; and Wavelet Analysis, or can be determined from the data by analysis: - Empirical orthogonal functions and Principal components. Gottschal & Krasovsaia

13 Karhunen-Loève expansion Random process Random variable X ( u) u Parameter space u (,,3 dimensions) Alternative formulations: X Normalizing factor (square root of variance, eigenvalue) Orthogonal function - postulated: sin & cos; wavelets -calculated from data: PCA; EOF ( u) u X ( u) u Gottschal & Krasovsaia 3

14 X Karhunen-Loève expansion Empirical orthogonal functions The Empirical orthogonal function (EOF) approach was developed in meteorology for dimensionality reduction, i.e. getting rid of the large amount of redundant information contained in meteorological data. Principal components (Amplitude functions) t, u t u Eigenfunctions (weights) Generalisation of the Karhunen-Loève expansion: The eigenfunctions (the eof s) (u) are functions in the two dimensional space u and the random variables are now times series (t) named amplitude functions or principal components. Gottschal & Krasovsaia 4

15 Karhunen-Loève expansion Empirical orthogonal functions X t, u t u Gottschal & Krasovsaia 5

16 Characterization by distribution function (one dimensional). Your data might be modelled as a random variable if the order in which data have been collected is of no importance (i.i.d. condition) Low flow sample and theoretical (Weibull) distributions of discharge series in the Moselle River, France. If your data are not i.i.d. then a random process is the proper model for your data and you need to turn to a second order moment approach including the covariance function. Standard statistical tests are only valid for random variables. Gottschal & Krasovsaia 6

17 F F DISTRIBUTION Lognormal: F x Gamma: x x 0 x x e Pearson type III: x Characterization by distribution function (one dimensional). Standard distributions x ln x m exp x 0 n n x 0 dx x x e n dx dx m MOMENTS m exp m Cs exp exp n m Cs Cs n n / mn n / 3 n 3exp n 3 exp m Gottschal & Krasovsaia 7

18 Gottschal & Krasovsaia DISTRIBUTION MOMENTS Gumbel (EV): u x x F exp exp Cs u m Generalised Extreme Value (GEV): u x x F exp Cs u m Generalised Pareto (GPD): u x x F Cs u m 3 Characterization by distribution function (one dimensional). Extreme value distributions 8

19 Mapping of spatio-temporal hydrologic data. Runoff maps Huai He, China Second moment characterisation. Yan Ziqi Time series: Gap filling and estimation at ungauged sites Karhunen-Loève expansion Li Linqi Gottschal & Krasovsaia Magdalena, Colombia Consistent Water Balance Maps Second moment characterisation. Yan Ziqi Flood & Low flow Huai He, China Characterization by distribution function (one dimensional). Yu Kun-xia Moselle, France 9

20 Analysis and mapping of spatio-temporal hydrologic data References I. Methods of Analyzing Variability. Gottschal, L. Methods of analysing variability. In Anderson, M.G. & McDonnell, J.J. (005) Encyclopaedia of Hydrological Sciences, John Wiley & Sons Vol Ch. 6 pp Gottschal, L. Ch. 7 Low flow and drought time series modelling. In Hydrological Drought L. Tallasen and H.A.J. van Lanen (Eds.), Elsevier, Runoff maps Gottschal, L. 993: Correlation and covariance of runoff. Stochastic Hydrology and Hydraulics, 7:85-0 Gottschal, L. 993: Interpolation of runoff applying objective methods. Stochastic Hydrology and Hydraulics, 7:69-8 Sauquet, E., Gottschal, L. & Leblois, E. (000) Mapping average annual runoff: A hierarchical approach applying a stochastic interpolation scheme. Hydrological Sciences Journal 45(6): Gottschal, L., Krasovsaia, I., Leblois, E. & Sauquet, E. (006) Mapping mean and variance of runoff in a river basin. Hydrol. Earth Syst. Sci. 0, -6. Gottschal, L., Leblois, E and Søien, J. (0) Correlation and covariance of runoff revisited. Journal of Hydrology, doi:0.06/j.jhydrol Gottschal, L., et al Leblois, E and Søien, J. (0). Distance measures for hydrological data having a support. Journal of Hydrology. doi:0.06/j.jhydrol Mapping water balance components. Gómez, F., Krasovsaia, I., Gottschal, L. & Leblois, E. (006) Interpolation of water balance components for Costa Rica. IAHS Publ. 308, IAHS Press, Wallingford, UK. Yan, Z., Gottschal, L., Krasovsaia, I. and Xia, J. (0) To the problem of uncertainty in interpolation of annual runoff. Hydrology Research 43(6): Yan, Z., Gottschal, L., Leblois, E. and Xia, J. (0) Joint mapping of water balance components in a large Chinese basin. Journal of Hydrology.doi:0.06/j.jhydrol Yan, Z., Gottschal, L., Wang, J. (06) Signal to noise ratio in water balance maps with different resolution. Journal of Hydrology 543: 8 9. Gottschal & Krasovsaia 0

21 Analysis and mapping of spatio-temporal hydrologic data: References II 4. Time Series: Gap filling and estimation at ungauged sites Krasovsaia,I. & Gottschal,L. (995) Analysis of regional drought characteristics with empirical orthogonal functions, In: Z.Kundzewicz (ed.) New Uncertainty Concepts in Hydrology. Cambridge Univ. Press Krasovsaia, I., Gottschal, L. & Kundzewicz, Z.W. (999) Dimensionality of Scandinavian river flow regimes. Hydrological Science Journal 45(5): Sauquet E., Krasovsaia I. & Leblois E. (000) Mapping mean monthly runoff pattern using EOF analysis. HESS, 4 (): Krasovsaia I. and Gottschal L. (00) River flow regimes in a changing climate. Hydrological Science Journal. 47(4), Krasovsaia I., Gottschal L., Leblois E. and Sauquet E. (003) Dynamics of River Flow Regimes viewed through Attractors. Nordic Hydrology 34(5) Sauquet, E., Gottschal, L. and Krasovsaia, I (008) Estimating mean monthly runoff at ungauged locations: an application to France. Hydrology Research 39(5-6): Gottschal, L., Krasovsaia, I, Dominguez, E., Caicedo, F., Velasco, A. (05) Interpolation of monthly runoff along rivers applying empirical orthogonal functions: Application to the Upper Magdalena River, Colombia. Journal of Hydrology 58 : Li, L., Krasovsaia, I., Xiong, L., Yan, L. (07) Analysis and projection of runoff variation in three Chinese rivers. (07) Hydrology Research 48(5): Li, L., Gottschal, L., Krasovsaia, I., Xiong, L. (08) Conditioned empirical orthogonal functions for interpolation of runoff time series along rivers: Application to reconstruction of missing monthly records. Journal of Hydrology 556: Floods and Low flow Sauquet E., Gottschal L., Krasovsaia I & Leblois E. (006). Predicting river flow statistics at ungauged locations a hydrostochastic approach. IAHS Publication, 307, IAHS Press, Wallingford, UK Krasovsaia, I., Gottschal, L, Leblois, E. & Pacheco, A. (006) Regionalization of flow duration curves. IAHS Publ. 308, IAHS Press, Wallingford, UK. Pacheco, A., Gottschal, L. & Krasovsaia, I. (006) Regionalization of low flow in Costa Rica.. IAHS Publ. 308, -6. IAHS Press, Wallingford, UK Gottschal, L., Krasovsaia, I., Yu, K.-X., Leblois, E. and Xiong, L. (03) Joint mapping of statistical streamflow descriptors. Journal of Hydrology 478: 5 8 Gottschal, L., Yu, K.-X. Leblois, E. and Xiong, L. (03) Theoretical derivation of the distribution of minimum streamflow series. Journal of Hydrology 48: 04 9 Yu, K-X., Xiong, L. and Gottschal, L (04) Derivation of low flow distribution functions using copulas. Journal of Hydrology 508: Xiong, L., Yu, K-X., and Gottschal. L (04) Estimation of the distribution of annual runoff from climatic variables using copulas. Water Resources Research 50: Yu, K-X., Gottschal, L., Xiong, L., Li, Z., Li P. (05) Estimation of the annual runoff distribution from moments of climatic variables Journal of Hydrology 53: Yu, K-X., Gottschal. L, Zhang, X., Li, P., Li,Z., Xiong, L., Sun, Q, (08) Analysis of non-stationarity in low flow in the Loess Plateau of China. Hydrological Processes In Press.

22 Doubt is the beginning of wisdom Aristotle 384 BCE - 3 BCE Plato (left) and Aristotle in Raphael's 509 fresco, The School of Athens. Aristotle holds his Nicomachean Ethics and gestures to the earth, representing empirical observation, whilst Plato gestures to the heavens, representing The Forms, and holds Gottschal & Krasovsaia his Timaeus.

23 Doubt - Uncertainty Probability theory is the deductive science of uncertainty. Natural variability Ris analyses Knowledgeuncertainty Decision model uncertainty Statistics is the inductive science of uncertainty in time in space data model parameters aim formulation values time horizon Gottschal & Krasovsaia 3

24 Than you for your attention! Gottschal & Krasovsaia 4

25 Advices: Observations We must differ between the possible scale of variability of processes in nature may have and what our observations can reveal. The optimal solution is that the nowledge of the natural process scale guides us in how to observe with respect to extent, spacing and support. It is common that it is rather the data collection technology and economy that determines how to observe data. When comparing data from different observation platforms the support and its influence must be considered. Gottschal & Krasovsaia 5

26 Advices: Observations A motto is let the data spea for themselves i.e. do not impose at an early stage in the analysis a predetermined model formulation for how data should behave. Allow data to demonstrate its pattern of variability across time and space and see whether this pattern reveals any structure/signal and with what accuracy it can be determined. This structure/signal (or lac of such) will then indicate what model to be used, its complexity and expected precision for prediction. Gottschal & Krasovsaia 6

27 Advices: Models All models are based on some assumptions e.g. normality, homogeneity, stationarity, isotropy etc. in case of stochastic methods. Always chec to what extent these assumptions are satisfied and what the consequence will be if this is not the case. A model is called robust if the performance of it is little sensitive to these underlying assumptions. Gottschal & Krasovsaia 7

28 Advices: Models The river structure is a ey factor to understand the specific character of streamflow statistics. Only longterm mean values can be represented correctly on a map. All other descriptors need to be mapped along rivers to correctly account for scaling effects. Is there any significant information beyond second order moments to loo for? the same information is repeated in different statistics probability theory provides consistent parametric relations Loo for joint approaches that reflect how the different statistics are interrelated. Gottschal & Krasovsaia 8

29 Than you for your attention! Gottschal & Krasovsaia 9