This section presents the H
s
estimations made by the proposed MLP-based method for the two platforms (Ekofisk and FINO 1) considered in the study. They are compared with the measurements made by in-situ sensors (buoy). To validate the proposed method, these results are compared with the ones obtained by the standard method. The comparisons are made in the designing and testing stages. The aim of comparing the results obtained in both stages is to realize whether the performances obtained during the designing stage are maintained for a data set never processed before (testing data set) or not. In other words, we want to know, once the MLP-based estimator is designed, how the proposed method works from a point of view of performance and time of designing/execution.
5.1. Comparison of the standard method and MLP-based Hs estimators for the Ekofisk platform
The H
s
estimations made by the standard and MLP-based methods considering the data of the Ekofisk platform are presented first for the designing stage, and second for the testing stage.
5.1.1. Designing stage of the MLP-based Hs estimator for the Ekofisk platform
In this stage, the data of the training and validation data sets of the Ekofisk platform (see Figure 7a) are considered. Considering the data measured by the buoy for both data sets and applying the learning algorithm described in Section 4.2, a time plot of the estimates made by the proposed MLP-based method is depicted in Figure 8. As can be subjectively observed, these estimates approximate with high accuracy the measurements made by the buoy.
From Figure 7a and Figure 8, some limitations are observed in both methods. These limitations are clearer observed in the scatter plots (estimates Vs measurements) presented in Figure 9. Focusing on the results obtained by the standard method (see Figure 9a), a general H
s
underestimate (negative bias of the H
s
estimate error) is observed for the whole range of H
s
. An opposite effect is particularly observed for sea states mainly conditioned by swell (λp ≳ 200 m). Focusing on the results obtained by the MLP-based method (see Figure 9b), not only better performance is observed in general (null bias of the H
s
estimate error), but also better performance is observed particularly for sea states mainly conditioned by swell (λp ≳ 200 m). Apart from these general and particular improvements, its general performance is also improved because more accurate estimates are obtained, being it denoted by the reduction of standard deviation (SD) of the H
s
estimate error (from 0.44 m to 0.35 m) and the increase of the correlation coefficient between estimates and measurements (from 0.93 to 0.96). But, there are still two limitations in the proposed approach. First, there are still some outliers in the H
s
estimates. These outliers are observed for bimodal sea states (swell and wind-generated waves are strong) with very high λ
p
values, being observed for H
s
∈ [4.5, 5.0] m. These outliers concern only to a few data of the designing data sets, being it the reason why the MLP is not properly learning from the environmental conditions of these data. In other words, the MLP is learning from the environmental conditions of the majority of the data. And second, there are still some H
s
underestimations, but they are lower in number than for the standard method and not predominant because the mean error of the H
s
estimate is close to 0 m.
5.1.2. Testing stage of the MLP-based Hs estimator for the Ekofisk platform
Once the standard method and MLP-based H
s
estimators are designed, and they are autonomously working, we analyze whether the performances and limitations discussed above continue being present or not using a new data set, the testing data set. A time plot of the H
s
measured by the buoy and estimated by the standard method for the testing data set was presented in Figure 7a, whereas for the H
s
estimated by the MLP-based approach is depicted in Figure 10.
As occurred in the designing stage, some limitations are observed from Figure 7a and Figure 10, which can be better observed in the scatter plots of Figure 11. Focusing on the estimates obtained by the standard method (see Figure 11a), the limitations of this method previously observed for the training and validation data sets are endorsed. In this way, an H
s
overestimate is observed for swell-dominated sea states (λp ≳ 200 m) and an H
s
underestimate is observed in general (negative bias of the H
s
estimate error). Focusing on the estimates obtained by the MLP-based method (see Figure 11b), most of the conclusions obtained in the designing stage are endorsed in this stage, but with some differences. In this case, a low overestimate of H
s
(+0.14 m) is obtained in general, what did not happen in the designing stage. But, the problem of overestimating H
s
for swell-dominated sea states (λp ≳ 200 m) continues being solved, being clearly observed in the region of H
s
∈ [0.5, 1.0] m. The second difference concerns to the high reduction rate of outliers. The third difference concerns to the high concentration of H
s
estimates close to the line of null error and between the lines denoting an H
s
error of ±0.5 m. This high concentration of estimates denotes high accuracy in the estimates, improving the results obtained by the standard method. This improvement can be observed by the decrease of the SD of the H
s
estimate error (from 0.27 m for the standard method to 0.22 m for the MLP-based method) and the increase of the correlation coefficient (from 0.95 for the standard method to 0.97 for the MLP-based method). The last difference concerns to the presence of H
s
underestimations, which presence is practically negligible in the test case.
5.2. Comparison of the standard method and MLP-based Hs estimators for the FINO 1 platform
As done for the case of study of the Ekofisk platform, a study of the performances of the H
s
estimators based on the standard method and MLPs is made in the designing and testing stages.
5.2.1. Designing stage of the MLP-based Hs estimator for the FINO 1 platform
In this stage, the data of the training and validation data sets of the FINO 1 platform are considered (see Figure 7b). The H
s
measured by the buoy for these data sets is used in the MLP learning algorithm described in Section 4.2. A time plot of the estimates made by the proposed MLP-based method for both data sets is plotted in Figure 12. Comparing both figures, we observe that the estimates made by the proposed method approximate the measurements made by the buoy with an accuracy higher than the one obtained by the standard method.
The accuracy mentioned above can be clearer observed in the scatter plots of Figure 13. Focusing on the estimates obtained by the standard method, we observe that, even when the bias of the H
s
estimate error is close to null, there are still some over and underestimates present. So, comparing these results with the results obtained for the Ekofisk platform (see Figure 9a and Figure 13a), it is observed that poorer estimations are made. It can be objectively observed by the decrease of the correlation coefficient of the temporal series (0.93 for Ekofisk and 0.89 for FINO 1), while the SD of the H
s
estimate error is maintained (0.44 m for both platforms). It is important to note that, as occurred for the Ekofisk platform, an overestimation of H
s
is still made for swell-dominated sea states (λp ≳ 200 m), as observed in Figure 13a. On the other hand, and focusing on the results obtained by the MLP-based method (see Figure 13b), a performance improvement, with respect to the results achieved by the standard method, is observed in general, with a negligible bias in the H
s
error estimate. This performance improvement is subjectively observed in Figure 13b because the estimates are more concentrated between the curves of H
s
error ±0.5 m. This performance improvement can be objectively observed by the reduction of the SD of the H
s
estimate error (from 0.44 m for the standard method to 0.24 m for the MLP-based method) and the increase of the correlation coefficient (from 0.89 for the standard method to 0.97 for the MLP-based method). Finally, it is also important to note that the particular problem of overestimating H
s
for swell-dominated sea states is solved by the proposed method. As an example, see how the overestimates made by the standard method in the range H
s
∈ [1.0, 2.5] m are corrected by the MLP-based method. But, there are still some H
s
underestimates present.
5.2.2. Testing stage of the MLP-based Hs estimator for the FINO 1 platform
This section shows the results obtained when processing a new data set of the FINO 1 platform, the testing data set. A time plot of the H
s
estimates and measurements made by the standard method and the buoy for this data set, respectively, are plotted in Figure 7b, whereas Figure 14 presents the estimates made by the MLP-based method. As can be subjectively observed from these figures, better performance is obtained by the proposed method, achieving more accurate estimates of H
s
.
The above mentioned accuracy can be better observed by the scatter plots presented in Figure 15. Making an analysis as the one presented for the designing stage, similar conclusions can be obtained in the testing stage for both methods. So, considering the estimates achieved by the standard method, the statistical results given in Figure 15a for the testing data set are maintained with respect to the ones given in Figure 13a for the designing data sets. Moreover, the problem of overestimating H
s
for swell-dominated sea states is still present in the standard method. On the other hand, and considering the results achieved for the MLP-based method, a small decrease of the performance obtained in the testing stage (see Figure 15b) with respect to the one obtained in the designing stage (see Figure 13b) is observed. Moreover, no problem is observed when H
s
is estimated for swell-dominated sea states (λ
p
≳ 200 m). But, the problem of having a few underestimates is still present in the proposed method. As a conclusion, the advantages and limitations of both methods are endorsed when processing a new data set in the testing stage.
5.3. Comparison of the standard and MLP-based Hs estimators for both platforms
This section presents a comparison of the performances achieved by the standard method and MLP-based H
s
estimators when working with data from the Ekofisk or FINO 1 platforms. Since the most important aspect of the methods is to observe how they work once designed, i.e., when they are autonomously working, this comparison is made using the performances obtained for the testing data set. In this way, Table 2 summarizes the statistical results of the H
s
estimate error and the correlation coefficients obtained by both methods in the testing stage and for both platforms. Moreover, the performance improvements achieved by the proposed method with respect to the standard one are also given for comparison purposes.
Table 2 Comparative of the statistics of the H
s
estimates made by the standard method and proposed MLP-based estimators once designed, i.e., when processing the testing data set As can be observed in Table 2, the proposed method always outperforms the standard one, regardless of the platform. Moreover, it is observed that the achieved improvement is even higher for the FINO 1 platform. But, why does it happen? As described in Section 2, this platform is located in an area of the North Sea where swell-dominated sea states are commonly present. In this way, it is observed that the proposed method works better than the standard one in this kind of sea states. Finally, it is important to note that, comparing the results obtained by the proposed MLP-based method for both platforms, the performances are similar. It denotes that the proposed method presents a great robustness against sea state changes and maintains its performance regardless of the sea state conditions where the marine radar images are obtained. It is important to note that since each non-coherent X-band marine radar is calibrated in each site, obtaining different calibration parameters in each one during their calibration campaigns, different estimates of sea state parameters are made, such as the SNR parameter. So, the MLP-based estimator must be designed (tuned) for each radar site, as done for tuning the constants c0 and c1 of Equation (9) in the standard method.
Finally, the time needed for designing (training with external validation) and testing an MLP is reported for both platforms. The time values presented below are obtained implementing the designing and testing stages of the MLP-based approach in Matlab 2007a and using a standard personal computer with a 2.4 GHz Intel Core2 Duo CPU, 4 GB of DDR2 PC2-5300 RAM and running Linux. The measured average time values are:
-
Designing time of an MLP for the Ekofisk platform using the training and validation data sets of Figure 7a: ≈ 30 s in average, considering a total of ≈ 30000 measurements.
-
Designing time of an MLP for the FINO 1 platform using the training and validation data sets of Figure 7b: ≈ 55 s in average, considering a total of ≈ 47500 measuremensurements.
-
Time for processing a given measurement (vector composed of: , λ
p
and T
m
) once the MLP is designed: ≈ 100 μs in average, regardless of the platform.
From an operational point of view, the design (train) of the MLP is proposed to be performed during the calibration campaign of the radar, when the data from the buoy are available.
5.4. Influence of the dimensioning and composition of the designing data sets
In the previous sections, we observed how the proposed method based on MLPs outperform the standard method when estimating H
s
. For doing so, we considered large data sets for designing the MLP and high values of H
s
in them. But, what does it happen when neither the designing data sets are so large nor it incorporates high values of H
s
? For finding an answer to this question, we reduce the number of measurements (dimensioning) considered in the designing data sets of the experiments made for each platform, and vary their composition by selecting the time instants for which the measurements do not present high values of H
s
.
Starting with the measurements of the Ekofisk platform, we divide the database as presented in Figure 16. Comparing this database division with the one used originally (see Figure 7a), the following differences are found:
-
Reduction of the number of measurements used in the designing data sets in approximately 70%: from ≈ 40000 to ≈ 12600.
-
Reduction of the maximum H
s
considered in the designing data sets in approximately 17%: from ≈ 7.8 m to ≈ 6.5 m.
Considering this new division of the Ekofisk database, we design the MLP (tuning of its parameters), as done for the original case of study, and we test it. The estimates obtained by the standard and proposed methods when processing the new testing data set are depicted in the scatter plots of Figure 17. Comparing these results with the ones obtained in the original case (see Figure 11), we observe several important aspects. First, the bias of the H
s
estimate error is very similar each other, being still very low. Second, the SD of this error is increased with respect to the original ones in both methods. Third, the correlation coefficient is maintained in both cases and methods, being very high again. Fourth, the problem of overestimating the H
s
for high values of λ
p
is solved again by the proposed method. Fifth, there are still some underestimations of H
s
in the proposed method, but its number is much lower than the one obtained in the standard method. Sixth, the dispersion of the measurements is greater for high values than for low values of H
s
in the MLP-based method. It happens because there were not data available of these heights in the designing data sets, but the H
s
estimation still maintains high accuracy. And seventh, since the number of measurements used now for designing MLPs is lower than in the original case, the time needed for training an MLP is reduced in ≈ 60%: from ≈ 30 s in the original case to ≈ 12 s with this new data set dimensioning. Since the size of the MLP does not vary in the experiments, the time needed for obtaining an estimate of the H
s
is the same as in the original case of study, i.e., ≈ 100 μs.
Finally, we perform a similar experiment (design and test) as made above for the Ekofisk platform data, but with the data of the FINO 1 platform. In this way, we divide the database as presented in Figure 18, where we apply deeper modifications in the composition of the designing data sets with respect to the original case (see Figure 7b). These modifications are:
-
Reduction of the number of measurements used in the design data sets in approximately 50%: from ≈ 47500 to ≈ 23000.
-
Reduction of the maximum H
s
considered in the designing data sets in approximately 50%: from ≈ 10.0 m to ≈ 5.0 m.
The H
s
estimates obtained by the standard and proposed methods when processing the new testing data set of the FINO 1 platform data are depicted in the scatter plots of Figure 19. Comparing these results with the ones obtained in the original case (see Figure 15), we observe similarities and differences with respect to the aspects observed in the previous analysis performed for the Ekofisk platform. Focusing on the differences, we observe four main aspects. First, the SD of the H
s
error estimate is reduced in this case. Second, there are some underestimations for the whole range of H
s
(see Figure 19b), but they are not so strong as in the original case (see Figure 15b). Even when they exist, they are less in number and lower in error than the ones obtained by the standard method (compare Figure 19a and 19b). Third, the levels of dispersion and underestimation are higher in the highest range of H
s
values (H
s
> 7 m). It happens because there were not available data of this kind when designing (training) the MLP. But even with that, these levels are not very high and the proposed approach is still working properly. And fourth, the time needed for training an MLP is reduced in ≈ 45%:from ≈ 55 s in the original case to ≈ 30 s with this new data set dimensioning. The time for obtaining an estimate of the H
s
is the same as in the original case and as for the other platform, i.e., ≈ 100 μs.