spatial analysis - uncertainty

8/11/2019 Spatial Analysis - Uncertainty

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Modelling Slope Uncertainty in DEMs: Monte Carlo Simulation

Approaches for Error Propagation Work in R software

PART I

1.& 2.install.packages("sp")library(sp)control


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DEM30

8.

Random DEM30Mean 0.18St. Deviation 1.48

Question: Modul description

The modul RANDOM in Idrisi creates image with random values with a distribution that in ourcase was chosen as normal.

This modul is usefull when Montecarlo method is taken in consideration.

Question: What does it look like?The legend in the new image shows expected errors values of the original DEM30 withdistribution error from -4.91 to 5.54

Question: Can you identify any structure in the surface at all?

No, it is not possible identify any structure. This might depend that values are randomlydistributed on the new image without any spatial correlation.9.


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Mean = 0.18

sd(delta) =1.486844

Question: What does this random surface represent?The error distribution of the random data calculated by Random in Idrisi looks different from theone in Delta. Though, there are similar mean and standard deviation of the errors for both themodels. This perhaps is due to an absence of correlation between data in the Random errormodel. The random surface represents a simulated error surface based on the surface calledDelta.

10.

Created 50 errors random surfaces with mean equal to 0.18 and standard deviation equal to 1.48.

Histogram of delta

delta

Frequency

-5 0 5

0

50

100

150

200

250

300


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The two images are clearly different in quality. The disturbed Dem30dist_slope have lost a lot ofdetails about the slope at small resolution. For example, it is not visible anymore the line

where slope is between 0 and 0.01 degrees. However, it still possible to distinguish clusters indifferent regions of the raster and their contours.


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13.

50 realizations

14.

Slope on realizations


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15.

Mean of 50 realizations of slope


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Original slope from Dem30

Question: What is the essential difference in this case?

In this case for Mean of 50 realizations of slope the values are not correct. There are locationswhere the value looks very different from that one in the original slope of Dem30.

Standard deviation slope

16.


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Standard deviation/mean slope

Question: Where is the slope more reliable and least reliable?

Is there any relationship between relative error and elevation? Or relative error and slope

value?

RSE =

High values of RSE (Relative Standard Error) indicate a high variability in the dataset, thus thosevalues are less reliable and vice-versa there RSE values are low data are more reliable becausethere is less variance around the mean.Since we dont know the actual value then we use the measured value that in our case is theslope value. That means that there is a relationship between relative error and elevation. This canbe explained by saying that the true value will be represented by the measurement of its

relative error. The lower the relative error is the better the slope will match to the unknown truevalue.


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PART II

1. Delta.vg


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4.hist(delta.ok$var1.pred)

Question: Are there any major differences in the statistics between the two (delta.ok

VS delta histogram)?

Errors in Delta.ok are further from mean of zero and its distribution looks asymmetricalrespect to that one in Delta. However, the mean is close to that one in Delta equal to0.16908.

Histogram of delta.ok$var1.pred

delta.ok$var1.pred

Frequency

-1.0 -0.5 0.0 0.5 1.0 1.5 2.0

0

1000

2000

3000

4000

Histogram of delta

delta

Frequency

-5 0 5

0

50

100

150

200

250

30

0


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5. delta.sim=krige(delta~1,control,dem,delta.mvg,nmax=50,nsim=2)spplot(delta.sim[1],col.regions=bpy.colors(),at=seq(-5,5,0.2))

Figure 3 Conditional Gaussian Simulation

Figure 4 Random Simulation in Idrisi

The conditional Gaussian simulation Errors (figure 3) generated with gstat function looksvery similar with that done in Idrisi(figure 4). The only different is that in the Guassiansimulation is possible to note some cluster of errors that often represent those errorswhere standard deviation from mean if very high (blue and yellow zones in figure 3).

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-2

0

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4


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6.

hist(delta.sim$sim1) & hist(delta.sim$sim2)

Figure 6 Delta histogram errors of DEM

Figure 8 Conditional Gaussian Simulation errors Sim1

The histogram of the conditional Gaussian simulation gives us a better normaldistribution of the errors (figures 8, 9). Data are symmetrical with a mean 0.1159 fordelta.sim$sim1 (fig. 8). Standard deviation is 1.35 for delta.sim$sim1 that is lower than

that one in Delta equal to 1.48. This mean that data in the Gaussian simulation are closerto the mean of the errors. In other words, the values that are further spread out are thosevalues represented in yellow and blue colors in the map in figure 3.

Histogram of delta

delta

Frequency

-5 0 5

0

50

100

150

200

250

300

Histogram of delta.sim$sim1

delta.sim$sim1

Frequency

-4 -2 0 2 4

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500

1000

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Histogram of delta.ok$var1.pred

delta.ok$var1.pred

Frequency

-1.0 -0.5 0.0 0.5 1.0 1.5 2.0

0

1000

2000

3000

4000

Figure 5 Delta.ok histogram errors simulation

Histogram of delta.sim$sim2

delta.sim$sim2

Frequency

-4 -2 0 2 4

0

500

1000

1500

2000

Figure 7Figure 7 Conditional Gaussian

Simulation errors Sim2


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7. --

8.

50 realizations of Conditional Gaussian Simulation

9.

Figure 9 Developing 50 realizations of DEM surface by using

the errors generated by Gaussian conditional simulations.

Figure 10 Results of 50 Realizations produced by using Gaussian Simulation


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10.

Figure 11 Slope autocorrelation of all 50 disturbed DEMs

11.

Figure 12 Mean and SD of the disturbed slopes


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Figure 16 Standard Deviation of slope (Part I 15)

Figure 13 Mean and SD of the disturbed slopes

12.

Figure 14 Standard Deviation of autocorrelation slope (PartII 11 )

Figure 16 Mean Autocorrelation slope (PartII 11)

The mean and the standard deviation from both simulation look alike. It is very difficult to see any difference.

Figure 17 Mean slope (PartI 15)


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13.

The result of RSE looks similar with low values at the same positions and higher values almost at the same positiontoo in the center of the DEM. That means that both results equally describe if predicted data are reliable in ourmodel.

14.

With a stronger spatial autocorrelation in error sample, all errors will be strongly correlated and will not have amean of errors equal to 0. The elevations will have a systematic error that will lead to get some predictable model

that will depict as best as possible the true observed dataset.

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