Review of various methods for interpolation of rainfall and their applications in hydrology

Transcription

1 Review of various methods for interpolation of rainfall and their applications in hydrology Bachelor Thesis Supervisor: Prof. Dr. Markus Disse, M.Sc. Kanwal Amin. Chair of Hydrology and River Basin Management, Technical University Munich Submitted by: Christopher Steimer Hogenbergstr München Submitted on: Munich, 5. September 2017

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5 Abstract The objective of the review is to provide an overview of applications of spatial interpolation techniques (Spline, KED, Copula based) for a flood and flash flood framework. Hydrological models require reliable estimations of precipitation fields to calculate the runoff. Focus lay on the interpolation performance of rainfall measurements in hourly to daily time intervals. The simulation techniques TBM and Random Mixing were also considered to interpolate values when a sufficient number of simulations were conducted. The flexible description of marginal distributions using copula based interpolation was found to outperform all other methods except for the RADOLAN precipitation estimate. Additionally, a description of a TBM based simulator for a flash flood forecast and warning system was found to already exist in literature. Keywords: spatial interpolation, hourly, Spline, KED, Copula, TBM, merging of data. Acknowledgments I want to thank Prof. Markus Disse and the chair of hydrology and river basin management for providing the interesting topic. Also, I want to thank Ms. Kanwal Amin for her support and help over the last few months and especially during the final stages when time ran short. I want to thank my parents for their various types of support over the last four years. Our long discussions helped me to structure and formulate my thoughts and ideas. Thank you for proof reading the thesis and helping me learn the subtleties of English scientific writing. Your advice and encouragement means the world to me.

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7 Table of contents 1 Introduction Deterministic Methods Arithmetic Mean Value Thiessen Polygon Inverse Distance Method Polynomial Interpolation Spline Interpolation Geostatistical Methods The Variogram Ordinary Kriging Non-stationary Methods Universal Kriging Kriging with external Drift Copula in Hydrology Basic Methodology Interpolation using Copulas Case Studies Multivariate frequency analysis Rainfall measurement with Radar RADOLAN KED based merging of data Copula based merging of data Simulation Techniques Turning Bands Method Monte Carlo Simulation Random Mixing Conclusion List of figures References... 52

8 Review of various methods for interpolation and their applications in hydrology 1 Introduction In recent years, there have been several dramatic flood events in central Europe with devastating loss to life and property. Private properties, public infrastructure and agricultural produce were severely damaged in the floods of 2002 in the Elbe catchment, 2013 in central Europe, and 2016 in the towns of Simbach and Braunsdorf. Hydrological models are used to assess the flood areas of climatological extreme events and assist in sizing possible protection measures. The models are fed with time series of precipitation both variable in time and space as well as time invariant parameters such as land use, soil type, etc. (depending on the complexity of the model). The most important input of a hydrological model is an accurate spatio-temporal precipitation distribution. Large precipitation inaccuracies will lead to inconclusive or unrepresentative results. Difficulties may arise when processing rainfall measurements as the observation stations only capture the rainfall of a very small area. Large fluctuations over a catchment area and a limited spatial density of rain gauges require methods to accurately estimate values between two observation stations. These estimation techniques are embraced by the term spatial interpolation. It is also possible to make forecasts from historic data sets. This process is called temporal interpolation and is often used for frequency analysis. The thesis focuses on spatial interpolation techniques and provides an overview of each method s basic theory and methodology. The main approaches presented are: Spline interpolation, copula based interpolation and merging of data, Kriging with external drift implementing radar data, the RADOLAN quantitative precipitation estimate, Turning Bands Method and Random Mixing. Also, the review focuses on case studies investigating the performance of methods on hourly to daily precipitation measurements. A short summary of each case study is presented. The goal was to find applications for any of these interpolation techniques in a flash flood framework to assist in developing a live forecasting system. The thesis is divided into six parts and the interpolation methods separated into three groups: deterministic (Chapter one), geostatistical (Chapter three) and copula based (Chapter four). Chapter five provides an overview of different data merging techniques to assimilate the point accurate rain gauge measurements with the spatial network from radar observations. In Chapter six, different simulation techniques for precipitation fields are summarised. The databases used for the thesis were from the University Library of the Technical University Munich and the Staatsbibliothek Bayern. 2

9 C.Steimer 2 Deterministic Methods Deterministic methods are applied because of their relatively simple structure and low computational effort. Consequently, they often serve as a univariate reference method for more complicated applications. 2.1 Arithmetic Mean Value The arithmetic mean value is, much like the median, an averaging tool of statistics to describe the observed data. It is the simplest method to estimate precipitation over an area and is best used if the data from the rain gauges is normally distributed and the terrain is relatively even. A rule of thumb is that the method can be applied if the standard deviation of any point is less than 10% of x avg [Grams, 2000]. The AMV is most useful when a tight mesh of rain gauges is available and when trying to estimate the precipitation over a larger period of time, such as monthly or yearly. x avg = 1 n x i = 1 n (x 1 + x x n ) ( 1 ) xavg: Arithmetic mean value xi: parameter value at point i n: sample size 2.2 Thiessen Polygon The Thiessen Polygon Method is a partially graphic method to assign point values from rain gauges to a certain area. Adjacent measuring stations are joined and mid-perpendicular lines are constructed. Polygons are created where the lines cross, marking the area of influence of each rain gauge. The rain gauge s value is assigned to the corresponding region. When the average precipitation of the entire region is of interest, the steps to follow are: 1. Construct the Thiessen Polygons 2. Determine the area proportions of each polygon and assign a weight 3. Calculate the average precipitation by using the weights [Thiessen, 1911] The advantages of this method are that it is very simple and these polygons only have to be constructed once, assuming the mesh of rain gauges does not change. 3

10 Review of various methods for interpolation and their applications in hydrology One disadvantage of the method is that the orography is not considered and large jumps in value are shown at the boarder of the polygons. These jumps generally do not appear in reality. 2.3 Inverse Distance Method The Inverse Distance Method interpolates values by weighing measurements in close proximity higher than the values of points further away. The influence decreases quadratically with the distance. In his study [Caruso et al., 1998] defines a d min value to avoid infinite weight values for d = 0. A d max value is also defined to limit the range of influence. The following formula for f(x, y) states the analytical expression of the surface: N: number of data points di: Euclidean distance with point j w(di): weighting function N f(x, y) = i=1 w(d i)v i N w(d i ) i=1 ( 2 ) w(d j ) = { 1 d j 2 for d d min, 1 d 2 for d min < d < d max, 0 for d > d max. ( 3 ) Again, the Inverse Distance Method is very simple and easy to use which is one of its biggest advantages. It is applicable for a wide range of data as the method often delivers reasonable results and does not exceed the range of meaningful values [Caruso et al., 1998]. Caruso et al. [1998] analysed the results from four different spatial Interpolation Methods (Inverse Distance, Kriging, Hardy s Multiquadric Method and Tension Finite Difference). A quality criterion was provided for the performance of these methods on several data sets. Focus was on point value interpolation. Two viewpoints were defined: prediction and characterization. Following the viewpoint of prediction, the best interpolation method is the one minimizing the prediction error for an unmeasured location. Cross validation was used to avoid the problem of the unknown true values and thus the inability to calculate the prediction error directly. Finally, Theil decomposition [Theil, 1971] was used to compare the real values with the interpolation results in terms of R 2 p, bias, slope and residual. The second viewpoint, Characterization, defines the best method as the one delivering the most accurate global surface approximation. Statistical characteristics from a sampled set within the 4

11 C.Steimer actual surface were taken and associated to the entire surface. If the roughness of the surface from the experimental data r(d) (point data) is similar to the predefined roughness index from the calculated data R(d) (grid data), it seemed likely that the grid data will produce the same exact surface as would be received from the measured values. Caruso et al. [1998] found that methods depending on parameter value, such as IDM or Kriging (see Chapter 3), are very sensitive in their performance to parameter tweaking. IDM, in particular, depends on the chosen value of d max. It delivers best results for low values of d max,as it reduces the effect of smoothening. Another essential characteristic of IDM is the sensitivity to the density of the rain gauge network as confirmed for example by Noori et al. [2014]. A larger number of rain gauges increases the accuracy. They also confirmed the sensitivity of the IDM to choice of parameter d max in their study on interpolation accuracy on the Duhok Governorate, Iraq for annual rainfall. Their analysis shows that the optimization of the exponent through cross validation improves the interpolation results. 2.4 Polynomial Interpolation The goal of this method is to fit a global equation to the surface using algebraic or trigonometric functions. A general approach is [Ascher&Greif, 2011]: m h 0 = a k θ k (x 0, y 0 ) ( 4 ) k=1 h0: Value of interpolation at any point x0,y0 ak: kth polynomial coefficient θ k (x 0, y 0 ): kth monomial in terms of x 0, y 0 coordinates m: number of monomials determined by the degree of polynomial function Figure 1 Algebraic Monomials for a Maximum of Five degree Polynomial Function [Tabios et al., 1985] Since the interpolation function has a corresponding weight, it is easiest to describe by using a weighted linear combination of the observed values: 5

12 Review of various methods for interpolation and their applications in hydrology n h 0 = w j h j ( 5 ) j=1 Tabios et al. [1985] lists two approaches available to transform ( 4 ) into ( 5 ): The Least- Squares Approach and the Lagrange Approach. The Least-Squares Approach This approach requires the number of measuring stations j = 1,, n to be larger than the number of polynomials. It estimates the perimeter set a k, k = 1,, m by minimizing the difference F of the measured values at the sampling location h j and the values from a model ĥ j based on the same process. n F = [h j ĥ j ]² ( 6 ) j=1 The Lagrange Approach This method is also referred to as Lagrange Interpolation and is an exact method. The number of measuring stations must be equal to the number of monomials m = n. The coefficients a k are determined so that the process h fits all observed values. For a more detailed explanation of both approaches please refer to Tabios et al. [1985]. 2.5 Spline Interpolation A spline is a piecewise polynomial function that is fitted through the sampled points. The spline represents a two-dimensional curve on a three-dimensional surface [Hutchinson and Gessler, 1994]. Spline Interpolation is often preferred to Polynomial Interpolation as it tries to avoid the oscillating effect that may be observed with polynomial functions of high degrees. This method produces smooth and easily interpretable surfaces with low degree polynomials for the spline. The spline function is constrained at defined points (local technique). A specific number of neighbouring values is considered, therefore, the spline can adjust to local abnormalities without affecting the values of interpolation at other points on the global area [Burrough et al., 1986]. The constraints r are given by the degree m of the polynomial function: If r = 0 there are no restraints If r = 1 the function has to be continuous When r = m + 1 the mth derivative of the function has to be continuous for all points 6

13 C.Steimer The spline for m = 1 is called linear, for m = 2 quadratic, and for m = 3 cubic. [Hartkamp et al., 1999]. The three-dimensional case for interpolating surfaces instead of lines may also be referred to as bicubic [Burrough & MacDonnell, 1998]. A common application for splines, apart from the exact interpolation, are the thin plate smoothening splines [Hartkamp et al., 1999]. This application is used to recover spatial coherent signals for spatial interpolation and to remove the noise/error from a set of data [Hutchinson & Gessler, 1999]. Monitoring the variance of the residuals over the data set is possible with smoothening spline interpolation [Naoum&Tsanis, 2002]. When there is a measurement error ε so that: y(x i ) = z(x a ) + ε(x a ) ( 7 ) a thin plate smoothening spline f is derived under the following conditions [Myers, 1994]: n F = [f(h j ) f (h j ))]² = 0 j=1 ( 8 ) and [Df (x)]²dx D: second order derivative R: Region of interest R is minimized ( 9 ) The problem is easily solved for the single dimension analysis. Adding additional dimensions and extending to a multivariate analysis increases complexity. This is because the degree of anisotropy and the individual weights of the partial derivatives in D must be chosen [Myers, 1994]. To allow a compromise between smoothness and exactness, a constant smoothening parameter λ is introduced into the function, which is related to the variance of the error term [Myers, 1994]: n F = [f(h j ) f (h j ))]² + λ [Df (x)]²dx is minimized ( 10 ) j=1 R The smoothening parameter λ is found by minimizing the generalized cross validation function (GCV). The GCV is a method to determine the overall predictive error of the fitted function by removing each value one by one. The sum of the square difference of each omitted data value is calculated and the spline function fitted to the remaining values [Hutchinson, 1998a]. Once the order of the derivative has been chosen, the GCV can be automated [Tait et al., 2006]. The thin plate smoothening spline estimates the spatial correlation as one and is calibrated by a single parameter. Compared to geostatistical methods that require prior estimation of several 7

14 Review of various methods for interpolation and their applications in hydrology correlation parameters, e.g. Kriging (Nugget, Sill and Range of the variogram), this is an advantage when interpolation results are urgent [Hutchinson, 1998a]. Splines and Kriging retrieve results of similar accuracy [Hutchinson&Gessler, 1995], so the key advantage of spline interpolation is the operational simplicity [Hutchinson, 1994]. There have been several remarks to the formal connection between splines and different types of Kriging, e.g. Dubrule [1984], Matheron [1981], Hutchinson&Gessler [1994], or Myers [1994]. An extensive introduction to the theoretical aspects of spline interpolation is given in Wahba [1990]. Case Studies A comparison of the performance of thin plate smoothening splines to more simplistic methods, such as Thiessen Polygon or the isohyetal method can be found in Taesombat&Sriwongsitanon [2009]. Their analysis showed that the splines approach clearly outperformed the other methods in terms of ME, MAE and RMSE. In the study conducted by Hutchinson [1998a,b], splines were applied to a two dimensional smoothening of data with short range correlation. A criterion was developed to assess the number of necessary station removals to erase the correlation. This criterion was validated in the second part of the study [Hutchinson, 1998b] for a three dimensional spline interpolation incorporating position vectors and elevation from a DEM. A physical explanation was found for the short range correlation, which was associated with topographic effects. Prior to all processes, a square root transformation was applied to the raw data to reduce the skew in the data set. The results from the interpolation of 100 daily precipitation values indicated a reliable calibration of topographic dependencies and a flexible use of thin plate smoothening splines. The fast computation of this method suggests a possible application in emergency situations and live rainfall estimation. One example for the application of a second order derivative trivariate thin plate smoothening spline interpolation [ANUSPLIN package] is given in Tait et al. [2006]. The country under investigation, New Zealand, has unique climatic conditions so the results should not be generalised. The information was taken from a data set containing daily precipitation measurements over 44 years. It was shown that the error in interpolation of daily rainfall reduces when a hand drawn annual rainfall map is included as additional information instead of elevation. The benefit of the square root transformation to daily rainfall values as described in Hutchinson [1998a] was confirmed. Another use of Spline interpolation in the field of hydrology is demonstrated by Guenni&Hutchinson [1998]. The parameters of a rainfall model were fit to periodic functions 8

15 C.Steimer to account for the seasonal change. The periodic functions were then interpolated using thin plate smoothening splines. Thus, the model parameters represented the spatial variability of rainfall patterns over a region and over time. The dependency of precipitation on elevation was included in the interpolation process. A precise reconstruction of historic values was shown by comparing interpolation results with measuring stations excluded during the fitting process. In the study conducted by Jeffrey et al. [2001], a climatological data set covering the entire Australian territory was constructed by thin plate smoothening spline interpolation of daily ground observations. Kriging was performed on monthly aggregations. Records of daily precipitation values in Australia date back to the year Jeffrey et al. [2001] developed a practical, extensive and timesaving data base for researchers in the field of hydrology. In order to better interpret the interpolation surfaces, error estimations were included (R 2, ME, MAE, RMSE obtained by comparison with cross validation results). 9

16 Review of various methods for interpolation and their applications in hydrology 3 Geostatistical Methods Knowledge about the deterministic processes causing and influencing precipitation events is limited. A stochastic concept can help to estimate the uncertainty of precipitation and other parameters found in nature. The term Kriging, as it is known today, was introduced by the French mathematician Georges Matheron, who in 1963 developed the theoretical background to the practical methods of South African Mining Engineer Danie Krige. In the early 50s, Krige developed his method to optimize the mining process by making estimates on natural resources still left in the soil. Improved computational resources introduces these methods to other fields of science such as hydrology. 3.1 The Variogram This Chapter is a short revision of the theory used for Kriging. For a more detailed explanation, please refer to [Oliver&Webster, 2015]. In the theory of regionalized variables, a regionalized variable is defined as the realisation of a random variable at a certain location u. The change of value from one point to another may be either discontinuous and erratic or continuous. The only definite observation is that points in close proximity to each other are more likely to correspond and share a similar value than those at a larger distance. This explains why, unlike the approach in the previous Chapter, it is not possible to use deterministic methods alone to find out the parameters of interest [Grams, 2000]. Continuing this approach, a set of values at point z(u) can also be interpreted as the realisation of a set of random variables Z(u) in the limiting area D {Z(u), u D}. This set of random variables defines the random function Z(u), which allows to estimate parameter values for points without measured values [Akin & Siemens, 1988].Several assumptions must be made concerning the structure of the random function Z(u) and its degree of stationarity. The stationarity provides information about the spatial homogeneity of the random variables and is determined by the first moment (Expected value of a random variable) and the second moment (the variance, the covariance and the semi variogram). The following is then defined: the variance function Var[Z(u)], the covariance function C(h) and the function γ (h), which is called semi variogram: 10

17 C.Steimer m = E[Z(u i )] ( 11 ) Var[Z(u)] = E{[Z(u) m] 2 } ( 12 ) C(h) = E[(Z(u + h) m) (Z(u) m)] ( 13 ) γ(h) = 1 2 Var[Z(u + h) Z(u)] = 1 2 E [(Z(u + h) Z(u))2 ] ( 14 ) A general hypothesis to simplify the problem is the strong stationarity. This implies that the distribution of the random function depends only on the configuration of the points and not on their global position. To simplify the problem further, two assumptions must be made: The first assumption is the second order stationarity. It implies [Mantoglou&Wilson, 1982]: The expected value of the random function Z(u) is constant over the domain D The covariance of two random variables corresponding to two locations depends only on the vector h separating these two points According to [Goovaerts, 1997], if the conditions of the second order stationarity are met, the covariance function C(h) and the Semi variogram γ(h) are linked through the following equation: γ(h) = C(0) C(h) ( 15 ) Also for h = 0: C(0) = Var[Z(u)] ( 16 ) If the random variables do not fulfil this requirement of same finite variance, a second assumption, the intrinsic hypothesis, is formulated which is slightly weaker than the first. It still implies the first point of the second order stationarity but adapts the other to the requirements. It now states that [Elfeki, 2000]: The expected value of the random function Z(u) is constant all over the domain D The variance of the increment corresponding to two different locations depends only on the vector separating them It can be shown that the second order stationarity implies the intrinsic hypothesis however the contrary does not apply [M.Stein, 1999]. The intrinsic hypothesis is sufficient for most applications in geostatistics. 11

18 Review of various methods for interpolation and their applications in hydrology Modelling the variogram The most common form in practice to describe spatial variance of regionalized variables is the experimental variogram [Grams, 2000]. It is built on observed data, is limited by the amount of data at hand and limited for the estimation of the variogram for the vector h i. For Kriging, an accurate experimental variogram is essential as inaccuracies will distort any subsequent results [Oliver & Webster, 2015]. The experimental variogram is described by the formula: γ (h) = 1 2 N(h) [Z(u i) Z(u j )]² u i u j =h ( 17 ) Here N(h) is the number of pairs of locations separated by the vector h. To make a statement for every vector h i a theoretical model must be fit to the experimental variogram. There are several common models in practice, such as the nugget model, the spherical model, exponential variogram, Gaussian variogram, Linear Variogram or complex models. Another advantage of fitting a model to the experimental variogram is the consequential smoothening of fluctuation in the data. The data must also be checked for anisotropy. Anisotropy is detected by comparing Semi variograms of one point in different directions and indicated by a change in the shape of the curve. For a more detailed explanation of the different models and the isotropy, please refer to [Oliver&Webster, 2015], [Stein, 1999] or [Hartkamp et al., 1999]. Figure 2 1Examples of most commonly used variogram models a) spherical, b) exponential, c) Gaussian, and d) linear [Hartkamp et al., 1999] 12

19 C.Steimer 3.2 Ordinary Kriging Assumptions are the same for Ordinary Kriging and the variogram: variation is random and spatially dependant and the intrinsic hypothesis applies to the underlying random process. Kriging is often referred to as Best Linear Unbiased Estimator (B.L.U.E.). It estimates the errors of the interpolation, reduces it to a minimum and predicts unbiased values [Grams, 2000]. Another advantage of this method is that the spatial structure of the variables is included. The result from the Kriging method is a linear sum of data which can be calculated for one, two or three dimensions. In practice, the 2D approach is most common [Oliver&Webster, 2015]. The following notation has been taken from Oliver&Webster [2015]. Point Kriging Point Kriging can deliver the value at any unmeasured location u 0 by using a linear estimator λ i for the measured values at locations u 1, u 2,, u n. N Z(x 0 ) = λ i z(u i ) i=1 N λ i = 1 i=1 ( 18 ) As the sum of all linear estimators equals one, the unbiased prediction is ensured. Using the Intrinsic Hypothesis, the prediction variance is calculated by the formula: σ 2 (u) = Var[Z(x 0 )] = E[{Z(u 0 ) z(u 0 )} 2 ] N N N = 2 λ i γ(u i u 0 ) λ i λ j γ(x i x j ) i=1 i=1 j=1 ( 19 ) The semivariances (u j u 0 ) and (x i x j ) are taken from the variogram model. The goal is to minimize σ²(u) under the unbiased conditions and to find the corresponding weights. The weights are chosen to fulfil: E[Z(x 0 ) z(x 0 )] = 0. To solve this, the Lagrange multiplier µ is introduced: N Z(x 0 ) = λ i γ(u i u j ) + μ(u 0 ) = γ(u j u 0 ) ( 20 ) i=1 The solutions to this linear equation system are the values of the linear estimators. This system is called Kriging system. In parallel, the Kriging variance is determined by: 13

20 Review of various methods for interpolation and their applications in hydrology N σ 2 k = λ i γ(u i u 0 ) + μ(u 0 ) ( 21 ) i=1 Block Kriging When predicting values for areas larger than the dataset, Block Kriging is an option. The approach from Point Kriging is adapted to have averaged values of the weighted data z(u 1 ), z(u 2 ),, z(u n ) for each Block B. The following includes the Lagrange Multiplier for the Kriging system: N Z(B) = λ i γ(u i u j ) + μ(u j, B) i=1 The sum of the weights is still 1. The prediction variance is calculated by: ( 22 ) Var[Z(B)] = E[{Z(B) z(b)} 2 ] N N N = 2 λ i γ (u i, B) λ i λ j γ( u i u j ) γ (B, B) i=1 i=1 j=1 ( 23 ) γ'(ui,b): average semi-variance between data point ui and Block B γ'(ui,b): average semi-variance within B (within Block variance) and the Block Kriging Variance is: N σ 2 (B) = λ i γ (u i, B) + μ(b) γ (B, B) i=1 ( 24 ) 3.3 Non-stationary Methods Unfortunately, in some real life applications, the stationary Kriging methods do not deliver adequate results due to a known change in the natural parameter value. This systematic change is called drift, in which case the intrinsic hypothesis is no longer met. The change of parameter undermines the meaningfulness of the experimental variogram. The underlying problem of nonstationary situations is that either an estimation of the drift or knowledge of the variogram is required to determine the other, but often both are unknown [Kumar, 2010]. Two common examples of non-stationary methods are introduced below Universal Kriging Universal Kriging was the first geostatistical method dealing with non-stationary random functions. It splits the random function Z into a deterministic component m(u) and a stochastic component S(u), as [Kumar, 2010]: 14

21 C.Steimer Z(u) = m(u) + S(u) ( 25 ) m(u) = E[Z(u)]: expected value of the regionalized variable Z at location u, also called drift Unlike stationary methods, the mean varies over the area of interest and shows a functional dependency on the spatial location. The mean can be approximated locally by the following model and can be a function of linear, quadratic or higher form [Kiš, 2016]: k m(u) = a i f i (u) ( 26 ) i=1 ai: ith coefficient to be estimated from the data fi: ith basic function of spatial coordinates that describes the drift k: the number of functions used in modelling the drift The Random function is assumed to be intrinsic, i.e. the expected value of the increment, for every vector h, is zero: E[R(u + h) R(u)] = 0 (Unbiasedness) and has a finite variance independent of the point x [Journel & Huijbregts, 1978]. The experimental variogram is determined in the same way as in (16) and the weights are also chosen to satisfy the statistical conditions of: Unbiasedness: E[Z (u 0 ) Z(u 0 )] = 0 and Minimum of Variance: Var[Z (u 0 ) Z(u 0 )] = minimum The solution to minimizing the variance while under the constraint of unbiasedness and use of the Lagrange multiplier μ leads to the following universal Kriging system [Journel & Huijbregts, 1978]: N k λ j γ(u i, u j ) + μ l f l (u i ) = γ(u i, u 0 ) i = 1,2,, N j=1 l 1 N λ i f l (u i ) = f l (u 0 ) { i=1 ( 27 ) j = 1,2,, k By solving these equations simultaneously, the optimum values for λ i can be obtained [Kumar, 2010]. If determining a variogram from the data is not possible, there are iterative methods to finding a solution. Firstly, the type of drift is determined (usually order of polynomial). Then, a theoretical variogram γ is used to calculate the drift coefficients. After the experimental variogram has been calculated from the residuals, the theoretical and the experimental 15

22 Review of various methods for interpolation and their applications in hydrology variogram are compared to see if the correspondence between the two curves is sufficient or not. If not sufficient, step 2 is repeated with a different theoretical model. There is, however, no guaranty that the results will converge [Bredehoeft et al., 1983] Kriging with external Drift External, deterministic data may be incorporated into the model of Kriging with external Drift (KED) [Ahmed&de Marsily, 1987]. Examples are a digital elevation model (DEM), effective elevation of a larger area, or smoothened and original radar data [Rabiei&Haberlandt, 2015]. To include the information, m additional variables Y k (u), k = 1,, m are introduced and the assumption of constant expected value is replaced by: m E[Z(u) Y k (u)] = b 0 + b k Y k (u) ( 28 ) For any value of the unknown variables b 0/k=1,,m the assumption of unbiasedness for the linear estimator still holds but the parameters b 0/k vary in space. The variance of the increment of two points is only dependant on their distance to each other. The interpolation value at location u 0 is a weighted combination of n surrounding observations u i. The values for the weights λ i are found by solving the following linear equation system and minimizing the estimate standard k=1 deviation: n m λ j γ(u i u j ) + μ 0 + μ k Y k (u i ) = γ(u i u 0 ) j=1 k=1 n j=1 n λ j = 1 λ j Y k (u j ) = Y(u 0 ) j=1 ( 29 ) i = 1,, n and k = 1,, m n: number of considered neighbours m: number of additional variables μ k : m + 1 Lagrange Multipliers The resulting Kriging system (29) consists of n + m + 1 equations. When dealing with an entire time series of precipitation, steps (28), (29) and the construction of the variogram are performed independently for each time step of the series. The coefficients bk will not only change in space, but also in time. The variable Y must be known at all locations u for an 16

23 C.Steimer estimation. Similarly to the iterative process to determine the variogram in Universal Kriging, there are several non-trivial solutions for KED when neither the variogram nor the drift are known. In theory, the variogram is calculated from the residual: r(u) = Z(u) m(u). One way of estimating the unknown drift is to derive the residual variogram from the residuals of a slightly tampered KED system (29). The iterative process is applied to an approximate variogram, based for example on Z(u) and improved step by step. Another approach is to only consider locations unaffected by the drift m(u) = 0. This allows using Z(u) = r(u) directly for the inference of the residual variogram [Haberlandt, 2007]. KED performance is optimal if the additional information Y(u) is provided in a high spatial resolution such as a regular grid. Additional information on the performance of KED is given in Chapter 4.1 in the study of Bàrdossy&Pegram [2013]. The common use for merging raingauge and radar data is illustrated in Chapter 5.2 with several case studies on the topic. 17

24 Review of various methods for interpolation and their applications in hydrology 4 Copula in Hydrology At the beginning of the new millennium, rapid progress was made in the application of copula theory to hydrology. Using copulas, common problems of spatial interpolation methods were avoided, e.g. their dependency on the density of the observation network and the complexity of the underlying terrain [Vogl et al., 2012]. Although frequently used in the financial sector to evaluate investment risk patterns for extreme events, copula was a new tool for describing spatial dependency in the Hydrological community. Other than traditional interpolation methods such as Kriging, copulas recognize that the spatial dependency structure can vary with different percentile values [Bárdossy, 2006]. Also, the marginal distributions are not limited to the Gaussian assumption. The copula model permits each variable to be characterized by its own parametric family of univariate distributions. These properties are very useful when investigating conditional return periods in extreme value statistics, see Salvadori et al. [2007]. A short summary of the basic copula methodology is given in this Chapter using the notation from Bàrdossy [2006] and Bàrdossy&Li [2008]. For a detailed introduction to copula theory please refer to Joe [2001], or Nelsen [1999]. Genest&Favre [2007] describe copula theory with a practical example of a hydrological model for a bivariate case. 4.1 Basic Methodology A copula is a multivariate distribution function on the n-dimensional unit cube with uniform marginals on [0,1]. The multivariate copula in n dimensions is C: [0,1] n [0,1], with uniform marginals C(u i ) = u i if u i = (1,,1, u i, 1,,1), which must be zero if any of the arguments are zero C(u) = 0 if u = (u i,, u n ) and there is i so that u i = 0. The corresponding probability within each unit of the n-dimensional hypercube must be nonnegative, as: 2 n 1 n ( 1) n i=1 j ic(u 1 + j 1 1,, u n + j n n ) 0 i=0 n 1 If 0 u i u i + i 1 and i = j k 2 k k=0 ( 30 ) Sklar [1959] proved that every multivariate distribution F(u i,, u n ) can be linked by a copula function C and its marginal: F Ui (u i ): F(u i,, u n ) = C (F Ui (u i ),, F Un (u n )), ( 31 ) 18

25 C.Steimer where F Ui (u i ) is the ith one dimensional marginal distribution of the multivariate distribution. This simplifies the problem of specifying a probability model for multivariate observations. It was also shown that if the distribution function is continuous, F Ui (u i ) and C(u i ) are uniquely determined. Copulas can be constructed from the distribution functions with the following [Nelsen, 1999]: C(u) = C(u 1,, u n ) = H(F 1 1 (u 1 ),, F 1 n (u n )) ( 32 ) H: multivariate distribution function with margins F 1 1 (u 1 ),, F 1 n (u n ) F 1 n (u n ): inverse distribution functions of the marginals Figure 3 Marginal distributions [Vogl et al., 2012] It is simple to transform the experimental data, as the copula is invariant to monotonic transformations of the marginals [Genest&Favre, 2007] (e.g. taking the natural logarithm, applying a Box-Cox transformation or a normal score transformation). Copulas may be interpreted as the pure expression of the dependency without the influence of the marginal distributions. In the bivariate case, the copula of their joint distribution function is: C(u 1 u 2 ) = u 1 u 2. In full dependency, the copula becomes: C(u 1 u 2 ) = min (u 1 u 2 ). It is also important to consider whether the copula density is symmetrical to the minor axis of the unit square and, thus, shows the same corresponding dependency for low or high values. This applies for a bivariate copula if the following is fulfilled: The survival copula C : C (u, v) = C(1 u, 1 v) 1 + u + v and, consequently, the copula density: c(u, v) = c(1 u, 1 v). Assuming that C is absolutely continuous, the density is calculated as follows: c(u 1,, u n ) = n C(u 1,,u n ) u 1 u n. ( 33 ) 19

26 Review of various methods for interpolation and their applications in hydrology Figure 4 (left) Normal copula with the correlation r=0.85 and (right) the corresponding copula density (symmetrical) [Bàrdossy, 2006] Depending on the application, a more appropriate copula function is chosen to respect the different dependencies for high, medium or low values. Copulas are sorted into families. The most frequently used in hydrology include elliptical (Normal, Gaussian and Student-t ), Archimedean (Clayton, Gumbel-Hougaard, Frank, Joe, and Ali-Mikhail-Haq ), extreme value (Hüsler-Reiss, Galambos, Tawn, and t-ev ) and other families (Plackett and Farlie-Gumbel- Morgenstern ). Some Copulas are constructed for specific purposes e.g. corresponding to a different multivariate dependency (e.g. v-transformed Normal Copula, as explained in [Bàrdossy&Li, 2008]). The distribution families are described in more detail in the following publications: Favre et al. [2004], Balakrishnan [2009] and Qiu [2012]. Measuring dependency As described previously, the bivariate distribution H(x, y) determines the joint behaviour of the random sample (X n Y n ) given from pairs (X, Y) of continuous variables. The random samples (X n Y n ) are converted into rank space (x i, y i ) according to the value. Genest&Favre [2007] demonstrate how the copula remains invariant to monotone increasing transformation. They also derive an empirical copula Cn from the ranked values. C n (u 1 u 2 ) is a rank based estimator for the unknown quantity C(u 1 u 2 ): n R i C n (u 1 u 2 ) = 1 n 1 ( n + 1 u 1, n + 1 u 2) i=1 R i : Rank of X i among X 1,, X n S i : Rank of Y i among Y 1,, Y n S i ( 34 ) 20

27 C.Steimer One standard way of measuring nonparametric dependency of multivariate variables is Spearman s Rho ρ n : ρ n = n i=1 (R i R )(S i S ) n (R i R ) 2 n i=1 (S is ) 2 i=1 n with R = 1 n R i = i=1 n [ 1; 1] n = 1 n S i i=1 ( 35 ) and ρ n = 12 uvdc θ (u, v) 3 [0,1]² Another well-known rank based correlation coefficient is Kendall s τ: ( 36 ) τ n = P n Q n 4 ( n 2 ) = n(n 1) P n 1 ( 37 ) And τ n = 4 C θ (u, v)dc θ (u, v) 1 [0,1]² ( 38 ) P n /Q n : number of concordant and discordant pairs If (X i X j )(Y i Y j ) > 0 then two pairs (X i Y i ), (X j Y j ) are concordant, otherwise discordant The rank (or monotone) association refers to positive or negative correlation of two variables [Hao&Sing, 2016]. Both coefficients can also be expressed solely by the copula function as proven by [Schweizer&Wolff, 1981]. Pearson s correlation coefficient r is less suitable for this application as it depends on the marginal distributions and is not invariant to monotonic transformations, unlike Kendall s and Spearman s measures. Results may be missleading for nonlinear dependency measures of joint distributions. A very informative and extensive summary of methods measuring multivariate dependency is given by Hao&Sing [2016]. Figure 5 Probability function, parameter space, generating function and relationship of non-parametric dependency measure with association parameter for the most frequently used Archimedean copulas in hydrology (flood risk analysis)[mitková et al., [2014] 21

28 Review of various methods for interpolation and their applications in hydrology 4.2 Interpolation using Copulas Most of the frequently used high dimensional copulas do not fulfil certain requirements when dealing with random functions dependency in space. The necessary assumptions for the bivariate copula are [Bàrdossy&Li, 2008]: 1. Stationarity: This implies that the copula C corresponding to the assigned random values of the points is a function solely of the vector h separating them and independent from the location x. The second-order stationarity (or intrinsic hypothesis) is weaker than the assumption that a random function has translation invariant bivariate marginal copulas. 2. Fréchet upper bound: For a set of very close points in space, there should be a parametrization of the copula which is close to full dependency. 3. The bivariate copula should enable the multivariate copula to reflect the spatial configuration of any point n. The procedure for spatial interpolation using copulas was described by Bàrdossy&Li [2008]. Kazianka [2012] developed a toolbox for Matlab describing a multivariate Copula for the set of spatial data D = z(x 1 ),, z(x n ) as follows: F(u i,, u n ) = C θ,λ (F η (z(x 1 ),, F η (z(x n )) ( 39 ) η: parameters of the univariate distribution function, do not vary over location λ: copula specific parameters with control of the shape of the multivariate distribution θ: parameter to describe the correlation of data points with distance The steps for the interpolation are [Kazianka, 2012]: 1. A univariate distribution is fit to the spatial data set. The observed values z i are transformed to z i using the univariate distribution specific to the multivariate distribution as described in (32) to generate the applied copula. This transformation is rank preserving: z i = H Z (F 1 Z (z i )) ( 40 ) 2. As described previously, a correlation model is fit to the rank correlations which describes how the dependency varies in distance and parameterizes the copula. This may be repeated with different theoretical copula families. 3. The parameters of the univariate distribution and copula are optimized using maximum likelihood: 22

29 C.Steimer L(θ, η, D) = c θ (F η (z(x 1 ),, F η (z(x n ))) f η (z(x i )) ( 41 ) A goodness-of-fit test is conducted after determining the copula parameters. There are several methods available comparing the fit of a theoretical copula C θ,λ to the empirical copula C n, which was determined from an n-tuple of random variables within a data set. Methods to find the best fit are listed by Vogl et al. [2012]. 4. The expected value at the unknown location x 0 is assessed through integration of the conditional copula density. The conditional copula density is calculated from n transformed values of the closest observations points to the interpolation point x. c x,n (u u 1 = H Z (z 1 ),, u n = H Z (z n )) n i=1 = h n+1 (z, z 1,, z n ) h 1 (z ) h 1 (z 1 ) h n (z n ) 1 c n (u 1, u 2, u n ) ( 42 ) The second term is often neglected as c n is constant over the range of u [0, 1] and, therefore, does not influence the estimation results. Finally, the prediction is calculated by: 1 Z (x 0 ) = F 1 η (u)c θ (u D)du 0 ( 43 ) To summarize, copula based spatial interpolation describes dependency of a set of variables as a function of distance between individual points and expresses results in form of ranked correlations. The interpolated value is calculated as an expected value of the copula density function at the unknown location [Wasko et al., 2013]. Parameter estimation Bàrdossy&Pegram [2013] suggest an alternative method to Kazianka`s model to determine the parameters of the marginal distribution of precipitation. Considering the high variability for each individual time step, the proposed method is a compromise between a parametric approach and an empirical distribution. The parametric approach is not able to model the broad variety of marginal distributions whereas an empirical distribution will lead to problems when inter- and extrapolating the observed values. Thus, a nonparametric density estimation method with gamma kernels is used. For small time scales (e.g. hourly or daily), some measured values will likely be zero. Therefore, the distribution is partially discrete and continuous. The separation in wet and dry values requires a special treatment and complicates the parameter estimation as 23

30 Review of various methods for interpolation and their applications in hydrology well as the interpolation process. This censored copula approach improves the interpolation for days with a high number of dry stations. To estimate the parameters of the copula, again there is a differentiation between dry and wet stations. For the parameter estimation of the dry stations, Bàrdossy&Pegram. [2013] suggests the incomplete integral method as described in Bàrdossy [2011]. For wet stations, the maximum likelihood method is applied as described in the previous subchapter. If a Gaussian copula is applied, there are no copula specific parameters to be determined [Wasko et al., 2013]. For the exact procedures, please refer to Bàrdossy&Pegram [2013] and Bàrdossy [2011]. Incorporating elevation One simplified way to incorporate elevation into copula based interpolation is suggested by Bàrdossy&Pegram [2013]. They construct a joint copula of topography and precipitation in the form of: C(u 1,, u k, v 1,, v l ) where u represents the precipitation and v the topography. The data for the topography (smoothened or original) is taken from a DEM and thus provided in a much higher resolution than the precipitation data. This implies that possibly k l. The topography is also invariant over time. To solve this problem, only the elevation at the interpolation site is considered and l set to 1. The joint copula is assumed to be of the same family as the spatial copula of the precipitation corresponding to a selected time t j. The cross correlation of a selected site with measured precipitation to any other location is directly calculated with the high resolution of topography data from the DEM. To connect the precipitation at several control points to the elevation at a target point, a correlation model is needed: r(h(x), Z(x i, t j ) = r e (t j )r tj (x i x) ( 44 ) h(x): elevation at point x Z(x i, t j ): rainfall at control location x i at time t j r e (t j ): correlation parameter of bivariate copula at time t j r tj (x i x): spatial correlation corresponding to the separation vector (x i x) 4.3 Case Studies The study conducted by Bárdossy&Pegram. [2013] is probably the most extensive investigation on copula based interpolation of rainfall on different time scales (daily to annual). They compared copula based interpolation to other Kriging methods, mainly KED with 24

31 C.Steimer different forms of un-/smoothened topography serving as external input on over 7 million implementations. The methods examined were: 1. v-transformed normal copulas with directionally smoothened topography: Due to the higher numerical demand only applied to a few selected cases of daily precipitation 2. Gauss Copula with directionally smoothened topography 3. Ordinary Kriging 4. KED with original topography 5. KED with directionally smoothened topography 6. KED with gauss-transformed directionally smoothened topography All interpolations were carried out with 12 points in the local neighbourhood of the point of interest. Only a very small improvement in estimate was observed for larger numbers of points due to the Shielding effect. Extensive simulations investigating the computational cost of the integrals showed that the one-point approximation gave more satisfying results compared to the full integration or the Rosenblueth approximation. The final results were then compared using a split sample technique. A random half of the data set is used for parameter estimation and interpolation, and the other half is used to validate the results from said interpolation. The quality of the interpolation was determined using four different measuring functions: The overall Bias B 1, the temporal Bias B 2, the spatial Bias B 3, and the squared error B 4. The coefficient of determination D between measured and interpolated precipitation was also calculated. Each formula to measure the bias and the determination can be found in Bárdossy&Pegram, [2013]. The goal of every interpolation is to minimize the error, thus the optimal results show Bias of 0 and D = 1. Some of the preliminary results in the evaluation of the individual performance were: Implementing topography does not improve KED in terms of B 4 on short timescales (daily) but the temporal Bias B 2 is significantly reduced. Directionally smoothened elevation outperforms the topographical elevation in every case. The estimation standard deviation was also calculated to determine a method of providing an error estimate, e.g. for ground truthing. A model must be checked in terms of point error estimate to be considered for plausible larger scale error estimation. Further results were: The copula based models are more accurate in estimating uncertainty than the Kriging based approaches. 25