too large to invalidate the method if D is kept at a constant value. Moreover, a single individual is expected to stay within a signicantly smaller area than the one spanned by the entire domain thereby supporting the choice of a con-stant value. The value ofhis set to 10.88 km, the average value of the extremes.
The database contains the bathymetry of the region as well as seven tidal con-stituents: M2, S2, N2, K2, O1, K1, M4. It was created in 1994 and is based on a storm surge model and meteorological data. Little is known of the uncertainties of the database other than qualied guesses from people experienced with the database.
It is certain that the maximal uncertainty is found at the shores where the tide is extreme and that, at open sea and at a distance from amphidromic points, the database should be reliable to within 10 cm on the tidal prediction.
6.2.2 Tidal prediction
The tidal variation for a single constituent at a given position is calculated in the simple way
z=f A(x) cos[ωt−θ(x) +G],
whereA(x)andθ(x)are amplitude and phase respectively that depends on po-sition,x. The constantsf andGare calibration parameters that are calculated separately and depends on the tidal constituent and the t= 0 denition. The prediction of the seven constituents given by the database are summed to yield the complete tidal prediction at the given position.
6.3 Analysis of stationary tags
For the observational likelihood to be calculated, a model that describes the possible random variations in a tidal pattern, must be formulated. When a tidal pattern is observed in the depth record, the sh is assumed to be station-ary at the sea oor. It is therefore relevant to study stationstation-ary tags to assess the uncertainties that are non sh related e.g. tidal prediction uncertainty, tag measurement uncertainty, inuences of weather etc. Tags are kept stationary by so called minipods.
6.3.1 Analysis of tidal noise
A change in weather conditions will impose uctuations on the depth record even at the sea bed level. This is conrmed in an examination of the depth measured by the stationary tag #1536, that shows a period of increased noise, Figure6.5.
08/08/01 09/08/01 10/08/01
Figure 6.5: Measurements of depth and temperature from Tag #1536 at the 9th of August, 2001. The depth has increased uctuations and the temperature drops approximately a degree at the time.
According to the Danish Meteorological Institute2 low pressure, 995 hPa, was observed in western Jutland at the 9th of August 2001, leading to increased wind speeds of >17 m/s on the main land. The harsh weather conditions seem to aect the data recorded by the stationary tag, see Figure6.5. The depth is measured with signicantly increased noise due to waves and the temperature is seen to drop about 1◦C possibly because of mixing with surrounding colder water.
From Figure 6.5 there are obviously two types of variation in the observed depth. Variation following the tide which has the approximate period of 12.4 hours and the superposed white noise type variation from the waves that varies at a frequency higher than the tag sample rate of 1 minute. The observations are assumed to follow the stochastic process
Zi=Di+Ei,
2www.dmi.dk
6.3 Analysis of stationary tags 59 where Ei is white noise i.e.Ei ∼ N(0, σ2E), andDi is a slowly varying process that comprises the mean depth, the tidal variation and storm surge. A one-dierencing of the process is performed to remove the slow process leaving the superposed noise
Vi=Zi+1−Zi= (Di+Ei)−(Di−1+Ei−1)'Ei−Ei−1.
The white noise variance, σ2E, is estimated as half of the empirical variance of Vi
σ2E= 1 2V(Vi).
The green intervals in Figure 6.5, consists of 800 data points and cover the period where the storm is at its highest. The standard deviation of the white noise is found to
b σE=√
0.5·0.072763 = 0.19074 m.
This type of variation in depth cannot be predicted by the tidal model and must therefore be incorporated in the error model.
The assumption of Gaussianity of Vi and therebyEi, can be checked by a Q-Q plot of the quantiles of the Gaussian distribution to quantiles of the empirical distribution of data. A Q-Q plot forViis displayed in Figure6.6along with the autocorrelation function (acf ) forVi.
0 5 10 15 20
Figure 6.6: Statistical analysis of Vi. Left pane: Q-Q plot forVi. Right pane:
acf for Vi. The process shows apparent Gaussianity as assumed.
The Q-Q plot shows that the quantiles of Vi have strong agreement with the quantiles of a standard Gaussian distribution. The theoretical autocovariance function for a dierenced white noise process is given by
Cov(Vi, Vi+∆) =E(ViVi+∆)−E(Vi)E(Vi+∆).
The process has zero expectancy hence the last term can be omitted and after some computation it is found that
E(ViVi+∆) =
Normalisation of this autocovariance function by 2σ2E gives an acf similar to the estimated shown in Figure6.6right pane.
Finally, a test for distribution is performed. The one, commonly recognised as the most powerful, is the Anderson-Darling test (D'Agostino and Stephens, 1986). This is similar in methodology to the Kolmogorov-Smirnov test (Conover, 1971) but allows for the parameters of the test distribution to be estimated from the data. The hypotheses are
H0: The data comes from a Gaussian distribution.
H1: The data does not come from a Gaussian distribution.
The result was that the H0 hypothesis is rejected at a signicance level of α = 0.05, but not at the α = 0.025 level with a test statistic of 0.75731. It should be noted that the Anderson-Darling test assumptions of independent observations in Vi were violated because of the correlation. Even so, the test passed at an acceptable signicance level to allow for practical implementation.
It is concluded based on the above analysis that changes in weather conditions can lead to a white noise eect on the depth measurements with a standard deviation of at least σbE = 0.19074m. This noise covers also the uncertainty inherent in the resolution of the tag and other unknown non-sh related white noise sources. Inspection of the observations tells that the variance of the white noise is time varying but is for simplicity modelled here with this constant value.
6.3.2 Tidal prediction uncertainty
The POL database has a resolution of approximately12×12km on the bathy-metry as well as on the amplitude and phase of the tide. An arbitrarily ne grid of the tide can be obtained by interpolation but this is not appropriate for the bathymetry. Therefore it has no meaning to rene the resolution with the intention to get a more precise geolocation.
It may be worthwhile, though, to interpolate the phase and amplitude to check
6.4 Uncertainty due to database resolution 61