_{1}

We show a quantitative technique characterized by low numerical mediation for the reconstruction of temporal sequences of geophysical data of length
*L* interrupted for a time Δ
*T* where
. The aim is to protect the information acquired before and after the interruption by means of a numerical protocol with the lowest possible calculation weight. The signal reconstruction process is based on the synthesis of the low frequency signal extracted for subsampling (subsampling
∇
_{Dirac} = Δ
*T* in phase with Δ
*T*) with the high frequency signal recorded before the crash. The SYRec (SYnthetic REConstruction) method for simplicity and speed of calculation and for spectral response stability is particularly effective in the studies of high speed transient phenomena that develop in very perturbed fields. This operative condition is found a mental when almost immediate informational responses are required to the observation system. In this example we are dealing with geomagnetic data coming from an uw counter intrusion magnetic system. The system produces (on time) information about the transit of local magnetic singularities (magnetic perturbations with low spatial extension), originated by quasi-point form and kinematic sources (divers), in harbors magnetic underwater fields. The performances of stability of the SYRec system make it usable also in long and medium period of observation (activity of geomagnetic observatories).

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 ^{−1} [sec]). The time variation of the geomagnetic field vertical component ΔZ is studied in the present work. The magnetovariogram (^{−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 (

- 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 [

- 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).

We use a standard numerical technique [_{measure} = 1.0 [sec] (Ñ_{measure} = sampling rate = 0.5 [sec]). The cutoff wavelength applied for our LP filter is Ʌ_{cut} = 1.5 [sec] after an oversampling operation performed to stabilize the action of the Fourier procedure for the frequency band of our interest [

The series of raw data there is an artificial singularity generated by the change in the power supply of the measurement network from which the data of the paper is coming from. Between samples 291 and 292 there is a signal drop of about 20 [nT] (

These problems can be referred also to the well-known phenomenon of aliasing. To remove this problem (after elimination of the spikes) we extract a subset of 110 samples centered on step 291 - 292 (

In this series the discontinuity is between the sample 51 and the sample 52. To obtain a value of discontinuity representative of the series and not only of the two samples 51 - 52, we assign to the sample 51 the value of the linear regression of the interval 1 - 51 (series 1) and to the sample 52 that of field 52 - 110 (series 2). The difference between these two values is adopted as compensation “k” factor (

The equalization of the total series is obtained by the relationship Series (1 - 51) + k = Series (1 - 51) + (−20 [nT]) (

As is well known, the discontinuities of data in the numerical series generates falls (loses) of direct and indirect information [

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 [_{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

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]) (

where

Δ t crash = t / 10 (3)

and

t = recorded period

The LFR procedure starts with the extraction of a continuous series of data from

the interrupted series (observed Fobs function) for a subsampling action execute with a sampling rate equal to the length of the signal interruption

∇ u s = Δ t crsah = t f crash − t i crsah (4)

where t f crash signal interruption end time

and t i crsah signal interruption start time

and subsampling action b in phase with the interruption (

This action produces a continuous subsampled function (F_{us}). Then we proceed to a resampling of F_{us} by polynomial approximation of 5 order with a sampling rate Ñ_{pol} equal to the F_{obs} one.

∇ p o l = ∇ o b s s (5)

The result function (called the F_{pol} polynomial function) contains the low-frequency information of F_{obs} (except for computational approximations) and it has sampling rate and length equal to F_{obs}.

L p o l = L o b s ; ∇ p o l = ∇ o b s (6)

le due funzioni hanno uguale densità numerica ρ e fase φ e sono quindi confrontabili in TD (time domain)

ρ F p o l = ρ F o b s ; φ F p o l = φ F o b s (7)

Obviously the lower the length of the signal interruption the better the reconstruction in accordance with the general rule

lim ∇ u s → ∇ o b s F u s = F o b s (8)

The resampling action has a cost in terms of computation weight. This cost is directly proportional to the length of the sector subjected to resampling for this reason it is necessary to make a compromise in the choice of this length. In the present case we propose a calculation segment of length L_{pol}

L p o l = BcW + Δ t c r a s h + BcA = 4 Δ t c r a s h (9)

where BcW (Before crash Window)

BcW = 2 Δ t crash (10)

and AcW (After crash Window)

AcW = Δ crash (11)

This choice, in our opinion, is the best compromise between the weight of data processing and effectiveness in limiting the numerical pollution due to the suture of the crash.

Finally, the 451-→ 513 data window is extracted from the F_{pol} and it is inserted in phase in the crash window of the F_{obs} series.

Where the sample 451 is time of the start of the crash t_{i} and 513 is the time of the end of the crash t_{f}.

The graphic performance of the LFR reconstruction is shown in

This signal reconstruction technique is excellent for protecting the physical information of medium-low frequency signals less for those of high frequency

signals (particularly if L_{crash} is long). It is not a heavy numerical technique but it is too penalizing for high frequency observations.

HFR is a very simple and fast calculation option. It is based on the action of a counter that detects on quasi-real time the absence of the data in the measured serie and replaces them with the corresponding series immediately preceding the metrological crash (in reverse sequence). For example if the values n + 1, n + 2, n + 3, n + 4 are lost they are substituted with the data sequence is n, n − 1, n − 2, n − 3 (

The operation stops when the counter detects a new measured data. In a qualitative way we can affirm HFR technique is reliable if the series in question is stationary (or with a low temporal increase) and if its spectral window is not too large [_{crash} = t/10 by means the HFR technique.

The percentage error “e%” of HFR is defined in the graph of

The percentage error produced by HFR in a stationary numerical environment can be considered acceptable while that for non-stationary numerical environment is not acceptable.

If the original numerical series is not stationary, the HFR has no acceptable capacity to protect spectral information.

As seen reconstructions of the LFR and HFR signal show advantages in execution speed but also heavy disadvantages in information protection performance. In particular, LFR has a calculation speed compatible with an environmental

control-reaction system but does not sufficiently protect the high frequency band (our maximum interest). While HFR is very fast but very dangerous in the case of non-stationarity of the measured function (F_{obs}). Reconstruction (SYRec) procedure is the merge of the numerical actions of LFR and HFR. It is built to merge the best qualities of the two standard techniques of reconstructions. SYRec sums in Time Domain (in phase) the series of data of length Δt_{crash} coming from LFR and the correspondents ones of HFR. In this way SYRec integrates the control of low frequencies (from LRF capability) to that of high frequencies (from HFR capability).

According to

- Definition of the number and position of lost samples (the subset of Fobs of length L_{crash} = t_{crash}) by means a sequential counter.

- Lost data reconstruction by means the merge of the low and high frequency information according to LFR and HFR procedures.

- and then to insert on the in the L_{crash} window of F_{obs} this composed data series.

The sub-sampled series F_{us} is extracted (in phase) from F_{obs} crashed by means of a Dirac subsampling function (

∇ = Δ t crash = L crash (12)

The undersampled series (F_{us}) is continuous and transparent to the interruption Δt_{crash} window and it has

L = L o b s (13)

From F_{us} the polynomial function F_{pol} of length 4L_{crash} is extracted (in the present case we decide 4L_{crash} to be best length for information protection effectiveness ratio/calculation weight) and sample step Δt_{pol} = Δt_{obs}

F ( p o l ) { F ( u s ) x = − a x n + ( − b x n − 1 + c x n − 2 + ⋯ ) + m x L ( p o l ) (14)

where L_{pol}

L p o l = 4 L crash = 2 L befoare crash + L crash + L after crash (15)

and sampling rate

Δ t p o l = Δ t o b s (16)

from the F_{pol} we extract the LFR subset data of length L_{crash} in phase with Δt_{crash} (_{LFR }

W LFR { t i L p o l → t i − 1 L crash ⇒ 0 t i L crash → t f L crash ⇒ 1 t f + 1 L crash → t f L p o l ⇒ 0 (17)

where t_{i} is the crash window start time and

where t_{f} is the crash window end time (

The high frequency component is added to the low frequency over-sampled serie with reverse phase. The final result of the action is shown in

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)

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” (

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) [

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)

_{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 (

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 [

With reference to

According to

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

We compare at last the efficacy of SYRec action with LFRs suture protocol in use in our reference magnetometric stations (

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 [

(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