VAV_05

 

 

 

The main algorithm of VAV is tidal analysis according to the Method of the Least Squares (MLS). The application of MLS is based on a model of the tidal signal, which consists in the following.

Every tidal phenomena has a corresponding theoretical tidal signal whose expression  at time t is

 

                                          (1)

 

Here  is frequency of a tide or tidal wave, taking m known discrete values . In this expression  is expressed in radians per unit of time. In practice, we deal with frequencies  in deg/hr (degrees of arc per hour), in cpd (cycles per day) and in cph (cycles per hour).

VAV uses  provided by the development of Tamura (1987) of the tide-generating potential with  tides with  and 2 constant terms with . A list of the tides is stored in the file tamura.inp (see Section C).

The quantities  in (1) are called theoretical amplitude and phase at time  of the tide with frequency . All parameters of (1), i.e.   are precisely known quantities.

In the observed data of the tidal phenomena we have an observed tidal signal whose expression  at time t is

 

                                         (2)

 

where  takes the same known discrete values as in (1).

The quantities  and  are called observed amplitude and observed phase at time t.  Unlike the theoretical , the observed  and  are unknown quantities, which are subject to estimation by the tidal analysis.

The relation between (2) and (1) can be described by using the so-called amplitude factor  and phase shift or phase lag  (denoted also by the Greek ). Namely, we have

 

 where                       (3)

 

In the classical methods of analysis in the Earth tide domain the directly estimated unknowns used to be a set of the observed  (actually  and ). This kind of unknowns does not allow a correct application of MLS, because it is impossible to take into account all tides. The obstacle is that the tides are concentration in frequency bands or tidal groups with very close , which cannot be separated. I.e., if we include all tides in the observational equations of MLS, they become linearly dependent. If we include a restricted number of main tides, the equations are not correct and we shall get biased estimates of  and incorrect estimates of the precision.

VAV uses the unknowns proposed and applied by Venedikov (1961, 1966). They are a set of values of the , more concretely a set of the quantities  and . Under the assumption that  is a constant for groups of tides with very close  we can create correct linearly independent equations, including the model (3) with all tides and a moderate number of unknown coefficients . Through the estimates of the unknowns, respectively of  we can get the estimates of all observed , not only of a small number of biased amplitudes and phases of some main tides.

The use of this kind of unknowns for the Earth tide data is based on the theory of the Earth deformation, as well as on an abundant experience.

It is easy to show that (3) is equivalent to the model of the ocean tidal signal, proposed by Munk and Cartwright (1966). The admittance function  in their model is a continuous function of , theoretically defined for all .

In our model  is a stepwise function, remaining constant for some short intervals of , covering a tidal group, which is not defined for intervals of , empty of tidal energy. We do not see a contradiction with the model of Munk and Cartwright which makes us believe that VAV can also be efficient for the ocean tidal data with a careful selection of the tidal groups.

VAV uses an approximation of the drift by independent polynomials in short time intervals of length  hours. The coefficients of the polynomials, which are different in different intervals, are treated as unknowns by MLS. In our examples we shall use  and  but other values of the same order are also available.

The application of MLS uses a separation of the drift unknowns. This operation appeared to be equivalent to a filtration of the intervals, which separates the main tidal frequencies and eliminates the drift. Through the filtration we get the data transformed from the time domain in the time/frequency domain, i.e. in a set of filtered numbers. In this sense we can consider  as a time window, through which pass the data. Further MLS is applied namely in the time/frequency domain. This means that we create and solve the observational equations about the filtered numbers.

The use of the transformed or filtered data allows getting frequency dependent residuals and thus - frequency dependent estimates of the precision. In such a way VAV takes into account the colored character of the noise (red noise).

VAV allows the data to have gaps, jumps and perturbations of any kind between the intervals, as well as small gaps within the intervals. The principle to deal with the gaps is the most natural one: we create observational equations for the existing data and we do not create them for not existing data, as well as for perturbed data, which we want to ignore. This allows avoiding the interpolations and the reparations of doubtful data, operations, introducing anomalies and noise with very unpleasant properties.

            An important task, followed by VAV is to study carefully the data. The purpose is, on the one side, to clean the data from anomalies and thus to get better analysis results. On the other side, we take into account, in particular for the Earth tide data, that every anomaly may be a geophysical signal.

In principle, the Earth tide observations, at the moment, may reveal with the highest possible sensibility and precision the slightest motions at the Earth surface. Hence, if any kind of deformation can serve as an earthquake and volcano precursor, it can be caught by our observations. One of the deplorable consequences of the interpolation/reparation of the data is that we cannot distinguish when an anomaly is a signal and when it is artificially introduced.

In the creation of VAV we have tried to include in one and the same program many different options. Nevertheless, the practical use of VAV is very simple, but the explanation of how to work is not so simple. Due to this in the following text we have chosen to give the explanations by using examples of the most commonly used applications of VAV. We hope that it is enough to start with a restricted volume of options, after which, through consultations with the authors of VAV, we can go further.

It may be encouraging for the users of VAV to know that this program is the product of the work of experienced specialists during many years. This work has started by the creation of the first method and computer program for tidal analysis (Venedikov, 1966), which has used successfully MLS by taking into account all tides, providing a frequency dependent estimation of the precision and dealing with data having gaps and arbitrary length.