Uncertainty analysis in spatial environmental modelling. Kasia Sawicka Athens, 29 Mar 2017

Transcription

1 Uncertainty analysis in spatial environmental modelling Kasia Sawicka Athens, 29 Mar 2017

2 Uncertainty propagation overview Input ± U Model (± U) Output ± U Input data Model parameters Model structure Solution method Model output Observ. data Data ± U 2

3 Environmental data vary in space E.g. Rainfall Categories like soil type Environmental quality indicators like soil ph 3

4 Deriving maps is encumbered with error E.g. interpolation error or error associated with maps derived from other maps Pb predictions [ppm] input map A 1 input map A input map A m Output map 4

5 Fact of life: maps stored in the GIS database are rarely if ever error-free Causes: generalisation, digitisation, measurement, classification and interpolation errors Consequence: errors will propagate through GIS operations and spatial models Key research question: given the errors in the inputs to the GIS operation, how large are the errors in the output? 5

6 Uncertainty propagation analysis involves three steps: 1. DEFINITION of a (statistical) uncertainty model for spatial objects and attributes 2. IDENTIFICATION of the uncertainty model (estimate its parameters) 3. Perform the actual UNCERTAINTY PROPAGATION ANALYSIS 6

7 DEFINE and IDENTIFY a (statistical) uncertainty model DEFINITION A i x = b i x + V i (x) IDENTIFICATION Marginal pdf at each location (both its shape and parameters) b i (x) is our (deterministic) representation of the variable (i.e. the map as stored in the GIS) V i (x) is the (stochastic) error about it (typically zero mean, but non-zero variance and spatially correlated) V i x = A i x b i (x) Spatial correlation (correlogram or semivariogram) Cross-correlation with other uncertain inputs

8 Performing the actual UNCERTAINTY PROPAGATION ANALYSIS 1. Taylor series approximation method 2. Monte Carlo method Advantages Monte Carlo method: Yields the full output pdf, not only the mean and variance Works with any model Easy to implement 8

9 Monte Carlo method Introduce by means of an example 9

10 Example: computing slope from DEM for a 2 by 2.5 area in the Austrian Alps 10

11 Slope map computed from the DEM (percent): 11

12 Now let the uncertainty about the elevation be 10 meter 12

13 Realisations of uncertain DEM: 13

14 Corresponding slope maps: 14

15 Histograms capture uncertainty in slope: 15

16 Monte Carlo algorithm: 1. Repeat N times (N 100): 1. Simulate a realisation from the probability distribution of the uncertain inputs using a pseudo-random number generator 2. Run the model with these inputs and store the result 2. Analyse the N model outputs by computing summary statistics such as the mean and standard deviation (the latter is a measure of the output uncertainty) 16

17 Beware: the world is deterministic and hence there are only pseudo random number generators 17

18 Monte Carlo method with spatial inputs Requires that we simulate from the probability distribution of a spatially distributed variable Can be done with spatial stochastic simulation: instead of kriging, which produces the most likely value (conditional expectation), we generate a possible reality, by simulating from the probability distribution of the spatial variable, using a pseudo-random number generator Spatial correlations are taken into account Most popular technique is sequential Gaussian simulation

19 Monte Carlo approach as before, e.g. for uncertainty analysis in deriving stream networks from elevation data

20 Potential applications in industry involved with hydrology Urban drainage models Wastewater treatment plant models

21 ksawicka spup repository This project has received funding from the European Union s Seventh Framework Programme for research, technological development and demonstration under grant agreement no

22 Sequential Gaussian simulation, how does it work 1. Visit a location that was not measured 2. Krige to the location using the available data, this yields a probability distribution of the target variable 3. Draw a value from the probability distribution using a random number generator and assign this value to the location 4. Add the simulated value to the data set, and move to another location 5. Repeat the procedure above until there are no locations left

23 Simulations, with conditioning data points