Interpolation of Hydrological Variables

1. Interpolation of Hydrological Variables Josef F�rst

2. 2 Learning objectives In this section you will learn: Overview of most common interpolation methods To understand the principles of deterministic and stochastic interpolation methods Ability to select the appropriate interpolation method for a hydrological problem Overview of practical problems

3. 3 Outline Introduction Regionalisation and Interpolation Principle of Interpolation Deterministic and statistical interpolation methods Global and local Interpolation Choice of interpolation method Deterministic interpolation Stochastic interpolation Spatial correlation Geostatistical interpolation Practical problems

4. 4 Problem A fundamental problem of hydrology is that our models of hydrological variables assume continuity in space (and time), while observations are done at points. The elementary task is to estimate a value at a given location, using the existing observations

5. 5 Introduction Hydrological data have variability in space and time Spatial variability is observed by a sufficient number of stations Time variability is observed by recording time series Spatial variability can be in different range of values or in different temporal behaviour A continuous field v = v(x,y,z,t) is to be estimated from discrete values vi = v(xi,yi,zi,ti)

6. 6 Introduction contd. Global estimation: characteristic value for area Point estimation: estimation at a point P = P(x,y) We need data AND a conceptual model, how these data are related, (i.e. a conceptual model of the process) If the process is well defined, only few data are needed to construct the model

7. 7 Example A groundwater table in a confined, homogeneous, isotropic aquifer under steady state discharge from a well is described by the Thiem well formula. Theoretically, the observation of 2 groundwater heads in different distance from the well is sufficient to reconstruct the complete g.w. surface

8. 8 Introduction contd. Hydrological variables are random and uncertain ? geostatistical methods Mostly 2D consideration ? v = v(x,y,t)

9. 9 Regionalisation and Interpolation Regionalisation: identification of the spatial distribution of a function g, depending on local information as well as by transfer of information from other regions by transfer functions. Regionalisation therefore means to describe spatial variability (or homogeneity) of Model parameters Input variables Boundary conditions and coefficients

10. 10 Regionalisation and Interpolation contd. Regionalisation includes the following tasks (and more): Representation of fields of hydrological parameters and data (contour maps) Smoothing spatial fields Identification of homogeneous zones Interpolation from point data Transfer of point information from one region to others Adaptation of model parameters for the transfer from point to area

11. 11 Principles of interpolation Given z = z(x,y) at some points we want to estimate z0 at (x0, y0)

12. 12 Principles of interpolation contd. Weighted linear combination The methods differ in the way how they establish the weights z can be a transformed variable, if, e.g., certain statistical properties must be maintained

13. 13 Deterministic or statistical interpolation Deterministic methods attempt to fit a surface of given or assumed type to the given data points Exact Smoothing Statistical (stochastic) methods treat a set of observations as an arbitrary realisation of a 2D stochastic process

14. 14 Example: Precipitation data zi(t) of station I out of N stations contain P independent events. We can interpret them as P different scalar fields. The spatial distribution of precipitation in a single event is a random realisation of one 2D stochastic process.

15. 15 Deterministic or statistical interpolation contd. Stochastic processes have a deterministic (or structural) and a random component. The random component can have spatial autocorrelation which is used in interpolation.

16. 16 Global and local interpolation an interpolation method is working globally, if all data points are evaluated in the interpolation. Local interpolation techniques use only data points in a certain neighbourhood of the estimated point 2-step procedure:densification

17. 17 Choice of interpolation method depends primarily on the nature of the variable and its spatial variation Examples: Rainfall, groundwater, soil physical properties, topography

18. 18 Example: Interpolation of rainfall spatial correlation depends on time aggregation

19. 19 Example: Groundwater data groundwater tables have smooth surface, but trend! Hydrogeological information is highly random, has faults, few points with �good� data

20. 20 Example: soil physical properties Highly random: infiltration rate, soil water content, hydraulic conductivity geostatistical methods few points with �good� data ? use of additional �soft� information: soil maps, correlation with other data (elevation, slope)

21. 21 Example: topography Elevation of a ground point can be measured at any time, repeated measures, etc... Exact interpolation properties of a terrain surface ? see DEM

22. 22 Deterministic interpolation methods Polynomials Spatial join (point in polygon) Thiessen polygons TIN and linear interpolation Bi-linear interpolation Spline Inverse Distance Weighting (IDW) Radial basis functions

23. 23 Polynomials

24. 24 Spatial join (point in polygon) assign spatial properties by spatial join

25. 25 Thiessen polygons Thiessen polygons, Voronoi Tesselation a point in the domain receives the value of the closest data point step-wise function

26. 26 TIN and linear interpolation Surface is approximated by facets of plane triangles Continuous surface, but discontinuous 1st derivative

27. 27 Bi-linear interpolation Simple and fast refinement in a 2-step interpolation Resampling of continuous raster fields

28. 28 Splines Spline estimates values using a mathematical function that minimizes overall surface curvature, resulting in a smooth surface that passes exactly through the input points. Conceptually, it is like bending a sheet of rubber to pass through the points while minimizing the total curvature of the surface.

29. 29 Inverse Distance Weighting (IDW) Default method in many software packages b = 2 Bull�s eye effect controlled by exponent b

30. 30 Inverse Distance Weighting (IDW) contd. Bull�s eye effect b = 2

31. 31 Inverse Distance Weighting (IDW) contd. grey: b = 0.1red:b = 2

32. 32 Inverse Distance Weighting (IDW) contd. green: b = 10red:b = 2

33. 33 Inverse Distance Weighting (IDW) contd. Interpolated values are always between Min and Max of data Sensitive to clustering and outliers

34. 34 Radial Basis Functions (RBF) �rubber membranes� supported at data points for smooth surfaces if many data points available

35. 35 Stochastic (geostatistical) Interpolation Analysis of the spatial correlation in the random component of a variable Optimum determination of weights for interpolation

36. 36 Stochastic (geostatistical) Interpolation contd. Experimental semivariogram things nearby tend to be more similar than things that are farther apart

37. 37 Stochastic (geostatistical) Interpolation contd. Theoretical semivariogram: fit function through empirical s.v.

38. 38 Stochastic (geostatistical) Interpolation contd. Ordinary Kriging

39. 39 Stochastic (geostatistical) Interpolation contd. Kriging goes through a two-step process: variograms and covariance functions are created to estimate the statistical dependence (called spatial autocorrelation) values, which depends on the model of autocorrelation (fitting a model), prediction of unknown values

40. 40 Stochastic (geostatistical) Interpolation contd. Kriging yields the estimated value AND the estimation variance

41. 41 Stochastic (geostatistical) Interpolation contd. problems of kriging Assumption of stationarity is not justified in many hydrological variables Spatial trends enhancements of kriging Universal Kriging (spatial trends) Indicator Kriging (inhomogeneities) Probabilistic Kriging (data with errors) Co-kriging (using correlation to other variables) External drift kriging

42. 42 Example: comparison of methods for interpolation of precipitation (month)

43. 43 Interpolation of elevation surface using different methods available in GIS: Mitas, L., Mitasova, H., 1999

44. 44 Interpolation of elevation surface using different methods available in GIS: Mitas, L., Mitasova, H., 1999

45. 45 Interpolation of elevation surface using different methods available in GIS: Mitas, L., Mitasova, H., 1999

46. 46 Interpolation of elevation surface using different methods available in GIS: Mitas, L., Mitasova, H., 1999

47. 47 Interpolation of elevation surface using different methods available in GIS: Mitas, L., Mitasova, H., 1999

48. 48 Interpolation of elevation surface using different methods available in GIS: Mitas, L., Mitasova, H., 1999

49. 49 Practical problems Inhomogeneous density of points Search radius

50. 50 Practical problems contd. Over- and undershoots: 2 close points define a steep gradient which has long range influence if distance to next points is large

51. 51 Practical problems contd. Special configurations of points (contour lines, profiles, raster) Points along contour lines ? add points

52. 52 Practical problems contd. Points along profile lines

53. 53 Practical problems contd. Points along profile lines

54. 54 Practical problems contd. Points on regular grid

55. Akkala et al. (2010) Interpolation techniques and associated software for environmental data. Env. Progr. & Sust. Energy (29/2) 134-141. 55

56. 56

57. 57 Summary and conclusions Interpolation is a matter of weighting the data points The nature of the variable determines the method of interpolation Deterministic methods Stochastic (geostatistical) methods Analysis of spatial correlation Optimum interpolation (BLUE) Reliability of interpolation (variance) GIS interpolation often simplistic, �smooth maps�