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Explore the principles and methods of interpolating hydrological variables. Learn about deterministic and stochastic interpolation, spatial correlation, geostatistical methods, and practical applications.
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1. Interpolation of Hydrological Variables Josef Frst
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
Bulls eye effect
controlled by exponent b
30. 30 Inverse Distance Weighting (IDW) contd. Bulls eye effect b = 2
31. 31 Inverse Distance Weighting (IDW) contd. grey: b = 0.1 red: b = 2
32. 32 Inverse Distance Weighting (IDW) contd. green: b = 10 red: 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
