In order to study the temporal evolution of the Matérn parameters, a more or less consecu-tive time period was chosen from the data, where each day contains more than a thousand pixel observations. For each selected day, a chain of ten thousand samples was run.
5.2.1 Ströömi
Figure 35 presents the temporal evolution of the smoothness parameterν in the Ströömi case from 13.10.2016 to 17.11.2016. The probability density estimates were calculated using the MATLAB functionksdensityand they were all normalised by dividing by their maximum values. A black curve connecting the maximum values of the density estimates was drawn as a visual aid.
Figure 35: Temporal evolution of the smoothness parameter at Ströömi.
Figure 36 illustrates the temporal evolution of the length-scale parameter`in the Ströömi case from 13.10.2016 to 17.11.2016. Note here that the`-axis is plotted in natural loga-rithmic scale because of the high variance in the length-scale parameter on certain days.
The computation took all in all almost 40 hours.
Figure 36: Temporal evolution of the length-scale parameter at Ströömi.
The smoothness parameter evolves continuously for the most part, but there are some more abrupt changes. Also there seems to be some kind of cyclicity. The length-scale parameter on the other hand behaves more chaotically with many abrupt changes and with no clear trends or patterns.
Figure 37 displays two cases of the turbidity at Ströömi, for which the parameter values are similar. The turbidity fields also look similar visually. There is a bit more uncertainty in the parameter values of the turbidity field in Figure 37b most likely partly because of the lower amount of observed pixels.
(a) 1.11.2016 (b) 3.11.2016
Figure 37: Turbidity at Stroomi on 1.11.2016 and 3.11.2016.
5.2.2 Porvoo
Figures 38 and 39 display the temporal evolution of the smoothnessνand the length-scale
`parameters in the Porvoo case from 29.08.2016 to 8.10.2016. Note that here the `-axis of the length-scale Figure is not in logarithmic scale, since there is no need for that. The computation took altogether around 27.5 hours.
Both the smoothness and the length-scale parameters behave quite chaotically in this case.
Compared to the Ströömi case, both parameters are now confined to smaller intervals. Im-portant thing to note about the Porvoo location is, that the bay alongside the city is shallow and is fed by the river Porvoonjoki to the north. This river brings with it suspended par-ticulate matter and nutrients from the extensive fields bordering the river. This surely has an effect on the turbidity fluctuations of the bay.
Figure 38: Temporal evolution of the smoothness parameter at Porvoo.
Figure 39: Temporal evolution of the length-scale parameter at Porvoo.
Figure 40 illustrates two cases of the turbidity at Porvoo, for which the value of the smoothness parameterνdiffers notably. It can also be visually verified, that the turbidity field in Figure 40b looks smoother than the turbidity field in Figure 40a.
(a) 4.10.2016 (b) 5.10.2016
Figure 40: Turbidity at Porvoo on 4.10.2016 and 5.10.2016.
5.2.3 Pakinainen
Figures 41 and 42 show the temporal evolution of the smoothnessν and the length-scale
`parameters in the Pakinainen case from 5.12.2016 to 14.01.2017. The computation took in total approximately three days and six hours.
Figure 41: Temporal evolution of the smoothness parameter at Pakinainen.
The smoothness parameter varies relatively continuously and there seems to be noticeable cyclicity.
Figure 42: Temporal evolution of the length-scale parameter at Pakinainen.
The length-scale parameter seems to evolve quite smoothly rising and falling in turn with two sharper changes.
Figure 43 exhibits two cases of the turbidity at Pakinainen with unalike length-scale pa-rameter ` values and comparable smoothness parameter ν values. Judging visually, the
smoothness does look alike and the scale of the turbidity field in Figure 43b does look greater compared to the other turbidity field; keeping in mind, that the uncertainty in the length-scale parameter`of the turbidity field in Figure 43b is rather huge.
(a) 11.12.2016 (b) 12.12.2016
Figure 43: Turbidity at Pakinainen on 11.12.2016 and 12.12.2016.
6 DISCUSSION AND CONCLUSIONS
We conclude that the method described in this thesis is able to characterise regions of turbidity based on the Matérn parameters and to investigate the temporal behaviour of these regions of turbidity. The same methods could perhaps also be extended to satellite measurements of algae related chlorophyll, vegetation related chlorophyll etc.
One possible way to extend what has been done in this thesis, would be to take into account the fact, that when there are barriers (for example, land) between two points, the distance in the Matérn covariance function should not be the shortest Euclidean dis-tance between the two points. It should rather be the shortest disdis-tance, that goes around the barrier residing between the two points. Another possibility would be to implement something akin to the method proposed in (Bakka et al., 2019).
The variogram is a traditional tool used in spatial statistics. It indicates the extent of spa-tial dependence of a spaspa-tial random field. The variogram (Cressie, 1993) is defined as 2γ(x1 −x2) ≡ Var(F(x1)−F(x2)). If the field is stationary and isotropic, then the variogram is only a function of the distanceh= kx1−x2k, 2γ(h)(Cressie, 1993). Es-timation of the variogram is usually done best by first estimating the empirical variogram and subsequently fitting a parametric model function to the empirical variogram in order to obtain 2γ as an explicit mathematical function (Cressie and Hawkins, 1980). Now there are various ways of estimating the empirical variogram; Particularly robust estima-tors are addressed in (Cressie and Hawkins, 1980). There are also several ways to fit a model to the empirical variogram, for example, restricted maximum likelihood (REML) and ordinary least-squares (OLS). A mathematical function, which is a valid variogram (the empirical variogram might not be valid), is needed for spatial prediction (kriging) (Cressie, 1993). Minasny and McBratney (2005) consider a Matérn variogram model for spatial variability in soil. The parameters of the Matérn model are determined by utilising REML and weighted nonlinear least squares (WNLS). Marchant and Lark (2007) also treat the use of the Matérn variogram model, but here its performance is, among other things, contrasted with spherical and exponential variogram models. This Matérn vari-ogram model approach to the parameter estimation should be compared with what has been done in this thesis in order to see how well the results of the different methods agree with each other.
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