tempt to fit a distribution to the H0 data and de- velop dedicated detection。 Another is to borrow an

idea from robust statistics [28], where a nonlinear transformation, a Huber compression, is applied to data, rw  to give rwl

interval [180, 550]s。 The amplitude distributions before and after the fault are reasonably close to be- ing Gaussian。 The autocorrelation functions show considerable correlation, caused by the wave im- pact。  This is undesired for change detection since

the theory requires independent increments of the rw wl

residuals (whiteness)。 Signal whitening is therefore required。

2。3。Whitening of residuals

If the discrete time scalar residual ri(kTs) was generated through an auto regressive moving aver-

is an iterative procedure to determine the standard deviation of rwl from a window of antecedent data。 The factor of kσ is an engineering design choice, de- termining the level where the tail of the distribution should be compressed, here kσ = 3 was selected。

Results of the whitening and subsequent Huber compression are shown in Figs。 (5) and (6)。 It is noted that the accepted theoretical procedure is to first determine a nonlinear function that brings the distribution back to Gaussian and then to whiten the signals。 In this case, the raw residuals appeared to be Gaussian but not white, and the nonlinear transformation was applied after whitening。 The autocorrelation function of the compressed resid- uals were found to be reasonably independent, as shown in Fig。 (7)

Figure 6:  Time histories and cumulative probability  distri-

0 butions of residuals rw and rw before and after Huber Com-

pression in condition H0

Figure 5: Autocorrelation of whitened residuals r1 and r5 in a fault-free situation

In Fig。 (5), the correlation of residuals r1 and r5 in a fault-free situation reduces after the whiten- ing process,  compared with the residual  without a whitening process, shown in Fig。 (3) and (4)。 However, the whitened signals do not approximate the Gaussian distribution。  From  Fig。 (6), the  up-

Figure 7:  Autocorrelation of compressed residuals rw   and

per and lower tails of the residual CDF (blue point line) are not confirmed very well with the Gaussian distribution。

Fig。 (6) and (7) also shows the effect of applying the Huber compression。 In Fig。 (6), the whitened signals are shown as the blue line, while the res- ult of the compression is shown in green。  It is clear that the result of compression is a better fit to Gaussian distribution, which is a straight line in the probability plot。 Fig。 (7) shows the autocorrela- tion function of the signals with Huber compression, implying that signal correlation has not changed。 Even though the autocorrelation does not show in- dependent increments of the residual, this is the best result that could be achieved by using a  fixed

5   for condition H0

whitening filter on the the non-stationary signal at hand。 The change in residuals with loss of buoy- ancy element are by far the smaller compared to changes appearing when line breakage takes place。 Both phenomena are strongly detectable in the re- siduals Eqs。 (4)-(6)。 This is confirmed by the model tests, shown in Figs。 (3) and  (4)。

From the above observations, the following pro- cedure is adopted for analysis of residuals: residuals are first compressed, thereafter whitening is applied to each component and finally change detection is applied。   The residual can be assumed to have     a

Gaussian distribution before and after the change, and the detection aims to find a shift in mean value。