Spatial Interpolation
General
The Spatial Interpolation function offers the possibility to make spatial interpolations on a defined grid. Two spatial interpolation functions are available, the Kriging and the Inverse Distance methods.
The function interpolates point values to a user defined grid. The point series can be selected from the Series list box and the grid characteristics are entered in the Grid Characteristics frame. Interpolations can be made on a grid defined by geographic (latitude/longitude) co-ordinates or on a local meter co-ordinate system. Make sure that the station co-ordinates are entered in the same co-ordinate system as the selected system in this function.
...
Under normal use a grid is made for 1 timestep only, therefore the option "Make Interpolation for first timestep only" must be checked. In this case the user is asked for an Arc/Info ASCII filename to save the results to. In case the Kriging function is selected, two filenames must be entered, one for the estimation and one for the variance.
HYMOS also computes the average value of the grid, this value is stored in memory and can be viewed with the View button. If the option "Make Interpolation for first timestep only" is not checked, HYMOS computes the average value for all selected timesteps. This resulting series can be stored in the HYMOS database by pressing the <Save> button.
When an average value of a catchment must be computed, make sure the "Blank grid with catchment boundary data" checkbox is checked an a catchment is selected from the list. In this case HYMOS sets all interpolated values outside the catchment area missing and computes only the average value of the non-missing values.
Results of the interpolation function can be viewed and analyzed by the graph program, the viewer, the report, or by the Netter program.
Series Codes
The series to use in the spatial interpolation can be selected or de-selected by clicking the checkbox of the series. By clicking your right mouse button on top of the Series list box you can select or de-select all series.
Grid Characteristics
The characteristics of the grid on which the interpolation must be made has to be defined. First a selection must be made on the co-ordinate system to make the interpolation on. When a 'geographic' co-ordinate system is selected all the station co-ordinates of the selected series must have been entered in latitude/longitude co-ordinates in the 'station Characteristics' form. When a 'local co-ordinate' system is selected, the local X-co-ordinate and local Y-co-ordinate of the stations must have been entered in the 'station Characteristics' form. This local co-ordinate system is a co-ordinate system in meters.
...
- the location of the lower left cell, this is the centre of the cell,
- the Cell Size, equal in X and Y direction (entered in km),
- the number of grid cells in X and Y direction
Catchment Blanking
When an average value of a catchment must be computed, check the "Blank grid with catchment boundary data" checkbox to activate the catchment list box. The catchment list box contains the ID's and names of all catchments stored in the HYMOS database. Make sure that a closed polygon is entered as catchment boundary and that the co-ordinate system of the catchment boundary is equal to the selected co-ordinate system.
Extra Options
Two extra options are given:
...
When the first option is checked, HYMOS interpolates only for the first timestep of the selected processing period.
When the second option is selected, HYMOS divides the point values by their long time average value before making the spatial interpolations. This value must be entered for each point series in the "Series Characteristics" function.
Make Interpolation for first timestep only
Compute a interpolated surface for the first timestep only. When this button is not checked then for all timesteps of the selected period an interpolated surface will be computed
Kriging
By 'Kriging Technique' we denote a class of interpolators based on a stochastic approach.
There is a large array of Kriging techniques, sometimes obscured by cumbersome notations and mathematics. All these different Kriging techniques have a very simple and common line: they are all regression techniques and differ only by the particular set of functions of the data which are combined into an estimator. A unique advantage of the Kriging interpolation method is its ability to quantify the reliability of prediction, to provide an estimate plus a confidence interval.
Basic Idea of Kriging
The basic idea of the different Kriging methods is the same. If x denotes a point in space and Z(x) is a function of x which is known in the n observation points x1, x2, .., xn, we look for an estimate Z*(x0) of Z(x0) in an non-measured location of the form:
...
Each observation will contribute a different proportion to the total estimation variance of Z(x0). There are an infinite number of ways in which the weights can be allocated, and each will produce a different estimation variance. Among these at least one combination of weights must produce a minimum estimation variance. It is this combination which kriging seeks to find. If the covariance function; the semivariance or the covariance is known, the weights λi can be calculated.
Ordinary Kriging
Ordinary Kriging is the most utilised type of Kriging, it can be used when the variable is stationary and the covariance is known, in contrary the mean is unknown. Ordinary Kriging refers to the following model assumption:
...
The reliability of interpolation (e.g. error variance) is found with the solutions of the weights:
The estimation error variance σk2 can be regarded as depending exclusively on the number and the locations of the measurement locations. Therefore σk2 is an efficient tool for solving network optimisation problems such as the optimal choice of measurement locations. It must be emphasised that σk2 is not the variance of the actual real spatial estimation error but a modelled error. σk2 Provides a theoretical measure of the relative accuracy's of the various estimates.
Kriging in HYMOS
In HYMOS the Kriging interpolations are performed according to ordinary Kriging. Different spherical models can be selected among which the Spherical, Exponential, Gaussian and Power model. For more information on these models see the Spatial correlation function.
...
After the computation is done HYMOS asks the user to save the computed Grid files. For the Kriging interpolations the user must first enter a name for the interpolated values, the second file to save is the grid file with Kriging variances.
Inverse Distance
The basis of the inverse distance method is that the weights are inversely proportional to the distances. The weights may be raised to a power, not necessarily an integer value, to increase the effect of the weighting function. inverse distance squared is most commonly used:
...
The Kriging function is much more complex than the inverse distance function. However, when the phenomenon is studied and the semi-variogram is determined, this function is much more powerful than the inverse distance function. The advantages of Kriging are amongst others the declustering of measurement stations and a variance map with the accuracy's of the interpolation..
Application
The Spatial Interpolation function can be used with any kind of time series data, the only restriction is that no missing values are allowed in the time series. Computation procedure:
1. Select some series from the Select series list box.
2. Select the interpolation function to use and enter the function parameters (i.e. Kriging or inverse distance).
3. Enter the appropriate Grid Characteristics, explained in the Grid Characteristics section.
4. Press the <Execute> button to start the computation.
5. Enter a filename for the ASCII Grid file where the computed data must be stored to.
6. To show the results in Netter, select the "Options" - "Map Options" menu and load the ASCII file as a grid.
ASCII Raster File Format
The interpolated data are stored in an ASCII raster file format, also called the Arc/Info ASCII Grid file format. This is a simple and common format that can be used by many GIS, like ArcInfo and ArcView.
The format is very simple, basically a header followed by a list of cell values. The header that includes the following keywords and values:
| Section | ||||||||||
|---|---|---|---|---|---|---|---|---|---|---|
|
The first row of the data is the top of the data set, moving from left to right. Cell values should be limited by spaces. The number of cell values must be equal to the number of rows times the number of columns.
Example:
| Section | ||||||||||
|---|---|---|---|---|---|---|---|---|---|---|
|
377 372 366 362 357 353 348 344 340 336 ....
333 330 328 326 326 327 330 334 339 345 ....
......
Presentation of Grids in Netter
The Spatial Interpolation function produces two types of grid files, static and dynamic grid files. To produce static grid files, the option "Make Interpolation for first timestep only" must be checked. In this case Arc/Info ASCII files are saved containing interpolation results for only 1 timestep. To produce dynamic grid files the above mentioned option must be unchecked. In this case dynamic BIL files are saved, containing interpolated grids for more than 1 timestep.
...