Error propagation in environmental modelling with gis download

Introduction [1] The authors of Wood et al. Their comment brings focus to the discussion and shows that the proposed. Read Article. Download PDF. Share Full Text for Free beta 4 pages. Complex models can be subdivided into those in which spatial interactions play no role type I , and those in which spatial interactions are an integral part of the modelling type 2.

Heuvelink et al. Spatial variations in crop growth can be estimated by measuring or estimating the values of site-specific parameters at several locations for which the model results are obtained. The spatial variation of calculated crop growth can then be depicted by interpolating from the sample sites over the area of interest van Diepen et al.

The quality of the maps of the model results is then dependent on the quality of the data about the site-specific parameters, the quality of the crop growth model itself, the kind of spatial variation over the area and the methods used for interpolating the results. Models of groundwater, pesticide leaching, surface runoff and atmospheric circulation often require data about the spatial variation ofthe properties ofthe volume or surface through or over which they work.

Because sampling is expensive and data points are sparse, the attribute values for each spatial element a grid cell or cube must be interpolated and are consequently never known exactly. These models often contain elements of both types of GIS operations.

Models and errors This paper concentrates on the problem of error propagation in raster GIS when quantitative data from various overlays are combined by a continuous differentiable mathematical function of type 1 to yield an overlay or map of an output variable. The problem that is addressed here is: given values and standard deviations of input attributes, find the level of errors bP associated with the model outputs P.

If this can be done on a cell-by-cell basis then maps of both P and bP can be made. Note that this paper does not cover error propagation through the type 2 operation as defined above nor through qualitative models and logical models. Neither does it extend to complex models such as those used for simulating crop growth, groundwater movement or erosion. The sources of error in P By definition a model is an approximation of reality.

Some models describe reality better than others, and thus may be considered to be more appropriate. But more complex models often require more and better data.

The model bias will be large when an inappropriate or incompletely defined model is chosen. The choice of the model thus plays an important role in the error bP.

We assume here than an appropriate model has been selected i. As an example, consider a multiple regression model of the form given as equation 3 ,in which the Ai values are grid maps of input variables and the IXj values are coefficients. These are I the errors associated with the model coefficients IX i and 2 the errors associated with the spatial variation of the input attributes Ai.

The errors associated with the model parameters For any given model the values ofits parameters need to be chosen, estimated or adjusted to fit the area in which it is to be used.

The values of model parameters can rarely be known exactly; usually their values must be estimated and there is always an associated error of estimation. It is not always easy to determine the errors associated with the model parameters without having a thorough knowledge of the model and the real situation where it is applied. However, when the model is a multiple regression equation like equation 3 , the model error is given by the standard errors of the coefficients lXi and their correlations.

The errors associated with the spatial variation of the input data Because the values of each input attribute Ai can vary spatially, the model will encounter a different set of input values at each location x.

The spatial variation of the Ai and the associated errors can be described in two ways. The first and most commonly encountered method divides the area of interest into polygons on the basis of other information such as land use or soil type. If observations of the Ai are available for points within each delineated area then average values and standard deviations can be calculated for each Ai for each polygon or mapping unit Marsman and de Gruijter The polygons are converted to raster format yielding two overlays per attribute, the first overlay contains the average value of Ai and the second overlay contains the variance, both calculated for each polygon separately.

The second method involves interpolating the Ai from point observations directly to the grid. This could yield better predictions than simple polygon averages providing sufficient point observations are available.

If optimal interpolation techniques based on regionalized variable theory kriging are used then each prediction of the value of Ai at a point or cell x is accompanied by an estimate of the variance of that prediction known as the kriging variance , so the problem of determining the errors associated with each value of Ai at each cell x is solved directly.

Because some readers may not be familiar with regionalized variable theory a short review follows. Further details have been given by Burgess and Webster a, b , Burrough , Davis , Journel and Huijbregts , and Webster Note that we are using kriging only as one way to obtain estimates of the values of the Ai and their kriging variances for every cell x; for this paper it is not necessary to consider the reliability of these estimates and how they may vary with the size of the data set or the exact method used.

Regionalized variable theory assumes that the spatial variation of any variable can be expressed as the sum of three major components.

These are a a structural component, associated with a constant mean value or a polynomial trend, b a spatially-correlated random component, and c a white noise or residual error term that is spatially uncorrelated. Let x be a position in one, two or three dimensions.

The variation of the noise terms over space is summarized by a function known as the variogram. This function, which describes how one-half the variance of differences! Var [Alx -Alx-h ] known as the semivariance varies with sample spacing or lag , h, provides estimates of the weights used to predict the value of the Ai at unsampled points X o as a linear weighted sum of measurements made at sites Xi nearby.

The weights are chosen to. The kriging variances can also be derived. The kriging method yields predictions for sample supports areas that are the same size as the original field samples.

For many purposes it is preferable to obtain predictions of average values of the attribute being mapped for blocks of a given size, for example, when combining data from optimal interpolation with data that have been transformed to a raster overlay at a given level of resolution, or that have already been collected for pixels of a given size e.

In this case the kriging equations can be modified to predict an average or smoothed value of Ai over a block or cell. This procedure is known as block kriging Burgess and Webster b. In many cases the kriging variances obtained by either point or block kriging are lower than those obtained by computing general averages for each polygon Burrough , Stein et aI.

The balance of errors The errors jP in the output maps of P accrue both from the errors in the model coefficients and from spatial variation in the input data. Knowledge of the relative balance of errors that accrue from the model and the separate inputs allows rational decisions to be made about whether extra sampling effort is needed a to determine the model parameters more precisely, or b to map one or more of the inputs more exactly.

Comparing the error analyses for several different models e. If the relative contributions to the overall error are known, then the allocation of survey resources between the tasks of model calibration and mapping can be optimized.

Equation 5 is a generalized form of equations 1 and 3 ,more suited for the mathematical analysis that will follow. The arguments z, of 9 may consist of both the input attributes Ai at some grid cell and the model coefficients IX;, but input attributes at neighbouring cells are excluded here.

Therefore models represented by equation 2 are not covered here and will be the subject of a subsequent paper. In order to study the propagation of errors we assume that the inputs, Zi' are not exactly known, but are contaminated by error model coefficient errors and spatial variation of the inputs.

The problem is to determine the magnitude of the error in y caused by the errors in the Zi' Therefore we interpret y as a realization of a random variable, Y. This random variable is fully characterized by its probability density function p. The standard deviation e of Y, defined as the square root of the variance, may be interpreted as a measure for the absolute error within y.

The relative error, also termed the coefficient of variation, is represented by the quotient of a and u. Downloaded By: [Wageningen UR] At: 5 October Propagation of errors in spatial modelling The problem of error propagation involves determining the variance a 2 of Y, given the means Jli and variances a; of the Z, values, their correlation coefficients TU and the type of operation, g.

In the case of operations on classified data the corresponding random variables would be discretely distributed.

In that case a full probabilistic description would require the specification of all class-membership probabilities Goodchild and Min-hua , Drummond Unfortunately the desirable properties of the continuous case that we describe below are not directly transferable to the discrete case.

If the joint p. However, this quickly leads to problems with numerical complexity because numerical integration is required unless 9 is linear in its arguments.

Even for the relatively simple case of the product of two normally- distributed random variables much computing time is required Meeker et al. An alternative approach to estimating error propagation is to run a model many times using Monte Carlo simulations of all inputs and parameter values; but apart from the considerable computing problems, these methods do not satisfy an analytical form and the results cannot be transferred to new situations Dettinger and Wilson Because we are interested only in the mean and variance of Y, rather than its p.

Fortunately, in the neighbourhood p, the higher-order terms of the Taylor polynomial are small in comparison with lower-order terms. Therefore, neglecting these higher order terms leads to the approximation of 9 by a Taylor series of finite order, which is a computable polynomial.

It can be easily verified that equations 8 and 9 are equivalent to the results presented by Burrough , p. We note an interesting feature resulting from equation 9. If the correlation between the arguments Z, is zero, then the variance of Y is simply, 10 This shows that the variance of Yis the summation of parts, each to be attributed to one of the inputs Zj.

This partitioning property allows one to analyse how much each input contributes to the final error. Being able to do this is of great importance when deciding which of the inputs' errors should be diminished to reduce the final error. In practice, it is not always known if the various inputs are correlated or are truly independent. Applications of the theory 5. A prototype software package for error propagation In order to obtain a map showing how both the results of the model and the errors are distributed over space it is necessary to carry out the above procedure for each cell in the raster overlay.

A map of predicted cell means and a map ofthe associated variances is required for each input attribute. Integrating environmental component models. Development of a software framework. A software framework for process flow execution of stochastic multi-scale integrated models. Dynamic numerical models are powerful tools for representing and studying environmental processes through time. Usually they are constructed with environmental modelling languages, which are … Expand.

Assessing uncertainty dimensions in land-use change models: using swap and multiplicative error models for injecting attribute and positional errors in spatial data. View 1 excerpt, cites background. Uncertainty and sensitivity analysis: tools for GIS-based model implementation.

Highly Influential. View 2 excerpts, references methods and background. Integration of simulation modeling and error propagation for the buffer operation in GIS. Error propagation modeling in layer-based GIS is based on explicit mathematical models representing the mechanisms whereby errors in source layers are modified by GIS data transformation functions.

View 1 excerpt, references methods. Development and test of an error model for categorical data. Geography, Computer Science. Modelling soil map-unit inclusions by Monte Carlo simulation. A research paradigm for propagating error in layer-based GIS.