In the westside of the La Spezia port—ITA (φ = 44 03 59.58N, λ = 09 50 49.22E, elevation e = −4.5 [m], date 14.09.2012, time Δt 09.10am: 09.20am (GMT)) the magnetogram of Figure 1 has been observed by means a magnetovariograph (sampling rate s = 5 × 10⁻¹ [sec]). The time variation of the geomagnetic field vertical component ΔZ is studied in the present work. The magnetovariogram (Figure 1) consists of 1200 samples for a length L = 10 [min]; ΔZ has an accuracy of 1 × 10⁻¹ [nT] and is returned in the graphs (and in the calculation) with precision of 1 [nT] (approximation by truncation). In theory, the measured function solves a spectral window Λmin = 2 [s] → Λmax = 300 [s]. The planetary index K of geomagnetic field activity during the experiment was = 1 (standard K → H comp.).

Preliminary observation of raw data shows a typical harbour high-noise coastal proximity field. It is characterized by a very large spectral window dominated by a high frequency noise band close to the Nyquist frequency (which probably carries important aliasing components foreign to the object of the present study) interfered by a lower frequency noise band. The magnetovariogram qualitative observation (observed field) highlights some preliminary physical informations on the origin of the local electromagnetic noise. For description of the parameters of our interest, we divide this magnetogram in 4 windows (W1, 2, 3, 4) plus a continuous subset (W5) of the window 4. The magnetogram is characterized by a very large spectral window dominated by a high frequency noise band close to the Nyquist frequency (which probably carries important aliasing components foreign to the object of the present study) interfered by a lower frequency noise band (Figure 1).

- W1 (samples 1 → 102): particularly disturbed field due to the unfolding of the chain of magnetometers. One of these devices provided the magnetogram di.

- W2 (samples 122 → 149): noise generated by the power supply test. In correspondence with the shift AC → DC there is a intensity drop of the ΔZ signal background of about 20 [nT]. The phenomenon is related to the electromagnetic activity of the generator and its geometric position with respect to the sensor considered.

- W3 (samples 150 → 298): transition to AC power supply with increased intensity of the induced magnetic background (ΔZ) of about 20 [nT]. Time variation dZ_t/dt

d Z t / d t ≃ 1.3 × 10 − 1 [ nT / sec ] (1)

Probably dZ_t/dt is not related of the magnetic noise produced by the electrical supply system but it is coming from an unknown artificial source [1].

- W4 (samples 298 → 1201): definitive return to the power supply by generator and inversion of the phenomenon of magnetic interference (see W2) on the background of the ΔZ signal.

- W5 (samples 750 → 1050, sub-window W4); decrease of the high frequency noise (noise W5 ≃ 20% noise W4). The phenomenon is probably dues to the temporary shutdown of a high frequency noise source close to the sensors. In the qualitative discussion of the characteristics of the total magnetogram, this fact requires that the subset W5 is defined individually and not as a part of W4 that contains it. On the other hand, the previous and next data set W4 is more homogeneous and for this reason it is preferable to define W5 as a subset of W4 rather than dividing the data into three subsets.

Furthermore, the magnetogram is characterized by the presence of probably instrumental spikes (same times classifieds as “electronic disturbance”). We observe also several electronic spikes and an artificial edge imposing two numerical interventions to stabilize the series (cleaning data action): 1—to delete spikes and 2—to delete edge induced by the AC/DC power passage and vice-versa. The standard information processing protocols on data whit high time transient would require the application of numerical techniques for signal strengthening (i.e. FFT filtering) to make the signal information more usable. In the present case we do not apply these procedures to subject the synthetic signal reconstruction protocol to maximum operative stress.

The data cleaning action is developed in two distinct steps: first one elimination of electronic noise (spikes), second one elimination of the edge (man made noise, data 291 - 292).

As is well known, the discontinuities of data in the numerical series generates falls (loses) of direct and indirect information [5]. We define information directly lost the ones contained in the time of unregistration and indirectly lost (sometimes called the second order information fall) those are due to the spectral pollution coming from the interruption. Longer is the crash window (interruption of the data sequence) higher is this pollution level. The main cause of this action is the practical impossibility of the Fourier calculation protocols to define the geometric conditions of non-derivability (sharp edges).

In general the aim of the reconstruction of missing data is not to retrieve the information carried by the lost data but to protect the information carried by the recorded data. Numerical reconstruction techniques are many, their spectral effects are often related to the ratio transience of the sampled function-sampling density [6]. The most satisfactory reconstructive solutions are heavy, require not short calculation time and therefore are not compatible with control systems that base their effectiveness on the immediacy of the response [4]. Moreover, if the systems are made up of many peripherals (e.g. a magnetometer array that observes the magnetic singularity variations over a large area), it is essential to delegate part of the measurement intelligence to each peripheral so as not to compromise the effectiveness of the system in any eventual single instrumental crash For this reason, heavy computational protocols are not operative in the case, for example, of environmental magnetic control-reaction systems. But on the other hand it is fundamental to equip the system with a numerical suture protocol for unregistered periods. In fact, the crash condition of duration T can generate a much longer system blindness condition. In the present case a fall of the system of duration T = 120 [s] samples = 60 [sec] blinds the system for a period T = 3 [min]. More, operational practice shows that pollution diffusion is greater and more extensive on harmonics of our interest (higher frequency harmonics) [7]. If the system must act on these harmonics it is necessary to proceed to a suture as effective and fast as possible. Working towards the high frequencies (relative to this type of signals essentially expressed in the low frequency band) we have the following problem: an effective suture is necessary to contain the pollution of the information and at the same time it is necessary to use light and fast calculations for not compromise the response speed of the system. The solution adopted in the magnetic control system of the HaPS Harbor Protection System (EDA) project is the “SYRec” algorithm (SYNTHETIC REConstruction). SYRec calculates a sub-sampled function (LFf Low Frequency function) with sampling rate = Ñ_DIRAC = Δt_crash = L_crash and phase equal Δt_crash. From LFf we extract a series of lengths Δt_crash and phase Φ = ΦΔt_crash and then insert it in the observed series crash window. This step is the reconstruction of the low frequency component. To this component is added (in inverse progression) the signal of length Δt_crash measured immediately before the crash (HFf High Frequency function). The response of the merge is a numerical series of length L = Δt_crash in phase with Δt_crash containing the low frequencies of the entire recording and the high frequencies closer to the crash period.

SIREC L crash = LFf L crash + HFf L crash (2)

We discuss and compare the results obtained by the SYRec standard with those of two standard fast suture actions:

- SYRec standard (signal SYnthetic REConstruction). Enough fast and highly effective in containing information pollution.

- LFR low frequency reconstruction. Enough fast but unsatisfactory for high frequencies.

- HFR high frequency reconstruction. Fast but unsatisfactory (in some spectral conditions harmful).

To start the study of the effectiveness of the suture protocols we generate the interrupted magnetogram of Figure 7 from the observed magnetogram of Figure 6 (noise condition = hard) and proceed to the synthetic reconstructions. The information contents of the reconstructed magnetograms coming from LFR, HFR, SYRec techniques will be compared with those of the continuous (and clean) observed magnetogram (Figure 6).

We start generating the broken magnetogram with an artificial interruption of length Δt = 30 [sec] between the sample 452 and the sample 512 (border amplitudes values of the crash period = 48 [nT], 54 [nT]) (Figure 7). The artificial interruption is long ΔT

where

Δ t crash = t / 10 (3)

and

t = recorded period

To quantify the efficacy of LFR, HFR and SYRec in information protection we compare the amplitude spectrum of the original causal function F_obs cleaned and continuous with the spectra of the three continue reconstructed causal functions F(LFR), F(HFR), F(SYRec) Figure 17.

To protect this action from edge effects all the series considered are previously subjected to smoothing by means the smoothing function so called “cosine bell” (Figure 18).

The action of the cosine bell function is defined in (18) where F_n is the result of the cosine bell smoothing

F n = { 1 2 ( cos π ( n + L ) M ) | − ( L + M ) < n < L 1 | − L ≤ n ≤ L 1 2 ( cos π ( n − L ) M ) | L < n < L + M (18)

This datum manipulation produces spectral stability benefits unrelated to the type of suture technique adopted. This increase of spectral stability is about the same for the three techniques of reconstruction (LFR, HFR, SYRec) and therefore their spectral comparative analysis does not lose validity.

We observe that in the frequency domain the cosine bell produces heavier distortive effects especially when applied to short numerical series.

The quantification of these distortions is easily computable from (19) (FFT of cosine bell) [5] :

W ω = 2 L sin ω L ω L + 2 M sin ω M ω M cos ω ( L + M )     + M [ cos ( ω + ω 0 ) M + ω L ] sin ( ω + ω 0 ) M ( ω + ω 0 ) M     + cos [ ( ω − ω 0 ) M + ω L ] sin ( ω − ω 0 ) M ( ω − ω 0 ) M (19)

where

ω 0 = π 2 M (20)

ω = 2 π f (21)

but also these effects are distributed in approximately the same way on the Fourier Transforms of the Fobs, FLFR, FHFR, FSYRec and therefore do not intervene in the effectiveness of their comparison.

This condition justifies the validity of the differential spectral comparison of W_LFR, W_HFR, W_SYRec with W_obs where

W function = F F T   F function (22)

Figure 19 shows the spectral difference between W_obs and repetitively W_LFR, W_HFR, W_SYRec.

The observation of the spectral difference indicates:

- the reconstruction of the signal performed in HFR pollutes both the high frequency band and the low band and therefore it is not suitable for an accurate reconstruction respecting the information capacity in the signal itself;

- the LFR reconstruction solves, in large part, the HFR problems in low frequency band but is more or less transparent to high frequency band and therefore its performance is not sufficient (especially for high frequency studies);

- SYRec contrasts in a very effective manner the spectral distortions both for the low and high frequency components. The price paid is increased computation time. But this cost does not exceed (in general) about 10% of the computational time of the other two data reconstruction procedures. It is acceptable.

To complete the comparative analysis of SYRec spectral efficacy, we compare its results with those of the low-frequency reconstructed “borders smoothed” LFR_bs (Figure 20).

This suture technique is able to contain spectral instabilities of both high frequency and low frequency but is very heavy as calculation time. It is therefore valid for “data collection” and observatory studies [9] but it is not suitable for “quick reading-response” activities of environmental control systems. This comparison allows us to test the efficacy of SYRec with a synthetic high stability signal reconstruction procedure.

With reference to Figure 20 (general view) and more to Figure 21(a), Figure 21(b) (particular view) we consider subsets 1 - 26 (a) and 26 - 42 (b).

According to Figure 22 in order to obtain the LFRs suture we proceed to the calculation sub-sets A and B

Subset A

F A x = cos x [ 0 , π ] 1 → 0 (23)

where

0 ≡ t i crash − 1 ,     π ≡ t i crash − 26 (24)

Sub-set B

F B x = cos x [ π , 2 π ] 0 → 1 (25)

where

0 ≡ t f crash + 1 ,     π ≡ t f crash + 26 (26)

The results of the smoothing process are shown in Figure 23.

We compare at last the efficacy of SYRec action with LFRs suture protocol in use in our reference magnetometric stations (Figure 24). The gain in spectral stability obtained by LFRs (Figure 24(e_LFRs)) is qualitatively compared to that of SYRec (Figure 24(SYREC)).

The difference in deviation between the two reconstruction methods spectra with respect to the original continuous function spectrum is evidently greater for

the LFRs technique. This fact is fundamental for the study of the real energy distribution between the elementary harmonics of the reconstituted function and therefore fundamental for magnetic measurements of singularity [10] [11]. SYRec is preferable to LFRs both in terms of calculation speed and performance stability of results. Following to this performance, the SYRec protocol has been included (dec. 01 2018) also in medium-term magnetic observation stations

(MTM) (4 - 12 [days]) (magnetoimetric reference stations for detection of low energy magnetic signals, from quasi-point sources and kinetics active in high electromagnetic noise environment).

The purpose of SYRec protocol is the production of a suture system of interrupted magnetic recordings not very demanding for the automatic calculation area and highly effective in protecting the physical information of the recording. This target is vital in operative evaluation of detection systems of low energy, quasi-punctiform and kinetic sources in high noise magnetic environment. The performance of the SYRec numerical protocol was tested in over 80 trial actions performed in critical high noise conditions both in an underwater environment (protection of ports) and in terrestrial environments (protection of critical and urban structures), both military and civil. The SYRec response was compared with the most commonly used fast techniques to suture of interrupted numerical series (LFR, HFR). This operational has shown a decisive improvement in the protection of information. This result is obtained by paying an increase of the calculation weight not exceeding 10% respect to LFR techniques (more effective than the HDR technique but also more demanding from the computational point of view). This information security protocol worked for over 240 consecutive hours (Isola Rossa observatory) without showing instability. In this context, a very clear gain in terms of computation time and weight was also highlighted. The SYRec computational procedure was therefore included in the medium-term-magnetometric observatories (MTM). Today, this procedure validated and its difficulty were structuring started.

This study is part of EU and Italian Harbor/Base Protection research action. It was supported by SEGREDIFESA of the Italian Ministry of Defence under the National Military Research Plan R & T, projects C.A.I.Ma.N., La.Ma.1.0_2.0 and European Defence Agency by Project Ha.P.S. (SWE Lead Nation, ITA, GER, NOR). Field trials were managed by MARI.CO.DRAG Marina Militare in La Spezia Harbour (ITA) and by WTD71 Bundesmarine in Eckerfoerde Horbour (GER). This study was coordinated by CSSN-MM. Isola Rossa Geomagnetic Base Station (ITA) was managed in logistic collaboration with the S&T CMRE SP (ITA) (Capo Teulada Experiment-2007).

Thanks to all of you.

The author declares no conflicts of interest regarding the publication of this paper.

Faggioni, O. (2019) The Information Protection in Automatic Reconstruction of Not Continuous Geophysical Data Series. Journal of Data Analysis and Information Processing, 7, 208-227. https://doi.org/10.4236/jdaip.2019.74013