Mostly the overall estimates seem reasonable in BoB and in GoB, but in some cases there seem to occur abrupt changes in ice thickness. These are probably due to changes in SAR backscattering level. To avoid occasional sudden jumps in the ice thickness values, we have made some preliminary tests with a simple method restricting the changes in the ice thickness distributions between temporally adjacent ITC’s. In our simple first model the distributions are described by two parameters mean (μ) and standard deviation (σ) and the distributions are assumed to be Gaussian. The change in these values is restricted by applying a Kalman filter [3] to the two-valued time series, x=[μ σ] with the assumption that the distribution parameters remain the same. First, we write the system (or evolution or prediction) equation and then the observation equation for the Gaussian noise:
ˆ
x−k =Axˆk−1+vk, vk∼N(O,Q)
zk=Hxˆ−k +wk, wk∼N(O,R) (3) In our case we assume that both the the system operatorAand observation operationHare just identity matrices. In the Kalman filtering we have the following updating steps. First the state vector and its error covariance are predicted according to:
ˆ
x−k =Axˆk−1+vk
Pk−=APk−1AT+Q (4)
the indexkrefers to time. Next, the Kalman gain (matrix)Kis computed, and the predicted state estimate and its error covariance are analyzed (corrected) on the basis of observationsz:
Kk=Pk−HT(HPk−HT+R)−1 ˆ
xk=xˆ−k +Kk(zk−Hxˆ−k) Pk= (I−KkH)Pk−
(5)
In the case of Gaussian distributions the scaling of thickness valuesHto new onesHnewis obtained as:
Hnew=μe+H−μ
σ σe, (6)
whereμandσare the mean and standard deviation for the ice thicknessH of previous ice level chart.
Currently the parameters for the Kalman filter are experimental, i.e. the process noise covariance and
Figure 7: ITC on March 25th 2006 with the restricted change in distribution (left), the ITC without any restrictions (middle), and the operational ITC (right).
observation noise covariance matrices,QandRwere not estimated from data. An example of one test is shown in Fig. 7. It can be seen that the result is closer to the operational ITC. Also other smoothing methods based on Kalman or Kalman type filtering will be studied. One of the basic ideas in the Kalman filtering is to correct the predicted state according to known (or estimated) error statistics. Due to the complicated nature of error statistics in our case we will study also other dynamic updating schemes where the correction can based on the physical arguments. The assumption of Gaussianity is not true for ice thickness distributions. For example log-normal distributions could be closer to the ice thickness distributions.
Wet snow on the sea ice may change the SAR signal drastically, e.g. by increasing theσ0 in the level ice case. Thus wet snow conditions and melting/refreezing cycles may cause rapid oscillations in the ice thickness values given by the algorithm although the ice cover thickness remains practically the same. We are also going to study including weather information in the algorithm, either directly from temperature information or snow information produced by an updated version HIGHTSI. We expect that the use of this information will reduce the abrupt changes in estimates.
We have also made some preliminary studies of longer-term ice motion statistics based on the motion estimated from SAR data. We have computed the monthly distributions of the estimated ice motion for the winter 2006 for six Baltic Sea areas: eastern Gulf of Finland, western Gulf of Finland, Gulf of Riga, Bay of Bothnia, southern Gulf of Bothnia, and Archipelago Sea. We are going to study could of the ice motion history statistics be used to further improve the algorithm performance.
6 Conclusion
The results with the current algorithm were clearly better than the results achieved by our earlier algo-rithm [4], the meanL1error for the earlier algorithm compared to the measurements was 11.8 cm. The results clearly show that this new ice thickness algorithm has potential for operational use in the area of the Baltic Sea. Also the usability in other sea areas will be studied.
There seems to be clear differences between the algorithm performance in different areas of the Baltic Sea. This suggests that the training should be performed separately for different sea areas, at least for the Gulf of Bothnia and Gulf of Finland, possibly also for the archipelago sea and Gulf of Riga. The
for melting period will also be performed.
The use and computation of the features derived from the ice motion will be studied more carefully, and also including other additional features (e.g. segment size and shape) into the classification will be studied. We also aim to study how to utilize the divergence (computed from ice motion) in open water detection. We also study the possible use of areal drift statistics derived from SAR data.
It was also noticed that sometimes large thin ice areas suddenly turn into much thicker ice regions.
Also abrupt changes to other directions for large areas were observed. Physically this kind large scale thickness transitions are unrealistic. We will try to avoid these jumps by adoptinga Kalman filter or Kalman filter type approach. The critical steps are the definition of the state vector and the determination of the allowed state evolution.
In some cases there also exist obvious disagreements between HIGHTSI model output and the SAR data and we are also studying the ways to handle these disagreements efficiently. Also the parametriza-tion of the HIGHTSI model will be studied to find optimal parametrizaparametriza-tions for different sea areas and conditions.
With the current parameter setting the linear and nonlinear models produced rather similar results.
However, the linear modulation model assumes a rather simple linear relationship between the inputs and the output, and with more input variables a more refined model will probably be required, e.g. a generalized additive model. Another alternative for modeling the input-output relationships is to use the MLP training for learning the relationships. The optimization of the nonlinear estimation algorithm parameters will be studied and also the use of hybrid model combining the results of linear and nonlinear model will be studied. The training and test data sets will be extended to cover several years. We have SAR images and operational ITC’s for this purpose available, and we are able to make the model runs, using the also archived forcing data, for all the ice seasons with available ITC’s and SAR data.
We also have interests to apply these algorithms in other sea areas than Baltic Sea only. Our main interest are in the Arctic areas.
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