The parameter estimation in the case of a homogeneous region in Section 4.1.4 was not very sensitive to added noise; The estimated parameter values corresponded nicely with the actual underlying parameter values. The standard deviation of the added zero-mean Gaussian noise was 0.1 or roughly 3 % of the maximum absolute value of the obstructed non-noisy realisation, which is a high amount of noise.
In the case of the inhomogeneous region with a boundary in Section 4.2.3, the parameter estimation was a lot more sensitive to added noise. The standard deviation of the added noise was 0.001 or roughly 0.04 % of the maximum absolute value of the obstructed non-noisy realisation. With higher values of the standard deviation (for example, 0.005 or 0.01), the estimated ν and` parameter values were not close to the true values, but the estimated parameters of the border were highly accurate even with heavy added noise.
5 EXPERIMENTS WITH REAL-WORLD DATA
As a real-world data analysis example, we use the turbidity data from the Baltic Sea (SYKE data, contains modified Copernicus data 2016-2017) provided by the Finnish En-vironment Institute (SYKE). The data is openly available for download at the server1. Using Web Coverage Service, the data can be found under the coverage identification EO_HR_WQ_S2_TURB. The data is based on the MultiSpectral Instrument (MSI) ob-servations of the Sentinel-2 mission of the European Union’s Copernicus program from 2016 onwards. The estimation of the turbidity data from the satellite observations has been done using a neural network model called Case-2 Regional CoastColour (C2RCC) (Brockmann et al., 2016).
Three locations were selected for analysis. The first one is Ströömi, Kustavi, which is a strait in the Archipelago Sea off the southwestern coast of Finland. The second one is the bay beside the city of Porvoo on the northern coast of the Gulf of Finland. The third one is near Pakinainen in the Archipelago Sea. Figure 25 displays the three chosen locations on the map of the southern part of Finland.
Figure 25: The chosen locations highlighted on the map of the southern part of Finland (Contains modified Copernicus data, SYKE 2020).
1https://geoserver2.ymparisto.fi/geoserver/eo/ows
5.1 Spatial analysis
Estimation of the Matérn parameterθcan be done as in Section 4.1, since we assume that the turbidity field is homogeneous. Ten thousand samples was chosen as a sufficiently large size for the sample chains.
5.1.1 Ströömi
Figure 26 shows the turbidity at Ströömi on 23.10.2016. The location is a very narrow but long strait. In the Figure, the grey pixels represent land and the white pixels represent unobserved areas of water due to obstructions like cloud cover. The units on both the x- and y-axis are in kilometers. The size of the field in Figure 26 is 119 pixels in the x-direction and 159 pixels in the y-direction.
Figure 26: Turbidity at Ströömi on 23.10.2016.
Figure 27 illustrates the samples drawn from the posterior ofθ using AM in the case of Ströömi on 23.10.2016. The computation of the samples took nearly 80 minutes. Figure 28 presents the chain of samples drawn from the posterior of θ and the marginal his-tograms of the parameters. The chain looks great here. The conditional sample mean of θ, computed after removing the first 100 samples, is [0.582, 0.393].
0.45 0.5 0.55 0.6 0.65 0.7
Figure 27: Samples drawn from the posterior ofθ.
101 2000 4000 6000 8000 10000 0.45
101 2000 4000 6000 8000 10000 0.3 Figure 28: Sampled chain and marginal histograms.
5.1.2 Porvoo
Figure 29 shows the turbidity at Porvoo on 29.08.2016. The location is a shallow bay fed by the small river Porvoonjoki. The river flows into the bay in the top part in Figure 29.
The grey pixels once again stand for land and the white pixels represent unobserved areas of water. The units are in kilometers. The size of the field in Figure 29 is 172 pixels in the x-direction and 133 pixels in the y-direction.
Figure 29: Turbidity at Porvoo on 29.08.2016.
Figure 30 presents the samples drawn from the posterior of θ using AM in the case of Porvoo on 29.08.2016. The computation of the samples took virtually 48 minutes. Note the ranges of the axes here. The variance inν is actually a lot less than the variance in`.
Figure 31 depicts the chain of samples drawn from the posterior ofθand the marginal his-tograms of the parameters. The conditional sample mean ofθ, computed after removing the first 100 samples, is [0.102, 0.796].
0.098 0.1 0.102 0.104 0.106 0.108 0.75
0.8 0.85
Figure 30: Samples drawn from the posterior ofθ.
101 2000 4000 6000 8000 10000 0.098
101 2000 4000 6000 8000 10000 0.75
0.8 0.85
(b)
0.098 0.1 0.102 0.104 0.106 0 Figure 31: Sampled chain and marginal histograms.
5.1.3 Pakinainen
Figure 32 shows the turbidity at Pakinainen on 05.12.2016. The location is a much wider strait compared to Ströömi. The grey pixels still represent land and the white pixels still stand for areas we do not have measurements from for some reason. The units are once more in kilometers. The size of the field in Figure 32 is 101 pixels both in the x- and the y-direction.
Figure 32: Turbidity at Pakinainen on 05.12.2016.
Figure 33 illustrates the samples drawn from the posterior ofθ using AM in the case of Pakinainen on 05.12.2016. The computation of the samples took almost 160 minutes.
Figure 34 depicts the chain of samples drawn from the posterior of θ. The chain looks great here also. The conditional sample mean ofθ, computed after removing the first 100 samples, is [0.108, 0.323].
0.095 0.1 0.105 0.11 0.115 0.12
Figure 33: Samples drawn from the posterior ofθ.
101 2000 4000 6000 8000 10000 0.095
101 2000 4000 6000 8000 10000 0.28
0.1 0.105 0.11 0.115 0 Figure 34: Sampled chain and marginal histograms.