Time Delay Estimation in Two-Phase Flow Investigation Using the γ-Ray Attenuation Technique

1 Faculty of Electrical and Computer Engineering, Rzeszow University of Technology, 35-959 Rzeszow, Poland 2 Faculty of Geology, Geophysics and Environmental Protection, AGH University of Science and Technology, 30-059 Krakow, Poland 3 Faculty of Physics and Applied Computer Science, AGH University of Science and Technology, 30-059 Krakow, Poland 4 Faculty of Electrical and Control Engineering, Gdansk University of Technology, 80-233 Gdansk, Poland


Introduction
Determination of the time delay between signals is a significant issue in many fields such as radar and sonar technologies, seismology, communications, and medical diagnostics.Over the past few decades numerous methods and algorithms have been developed ranging from cross-correlation to the advanced blind channel identification techniques [1][2][3][4][5][6].The scope of particular application method depends on the type of the analysed signals (random or determined, stationary or nonstationary) and their parameters (distribution of probability, as well as the signal-to-noise ratio).
Estimation of the time delay is also essential in contactless measurements of flow parameters, especially in the following two-phase mixtures transportation as liquid-gas, liquid-solids, or gas-solids.In these measurements, mutually delayed stochastic signals may be provided by sensors of such type as capacitance, optical, electrostatic, and thermometric or scintillation probes, situated on a wall of a pipeline or open channel [7][8][9].In such case, the measurement of time delay allows us the mean velocity determination of the flow dispersed phase.
Two-phase flow measurements utilizing radioactive isotopes have been used for more than 50 years.Those measurements can be divided into two types: the tracer method and the absorption based on an analysis of a mixture flowing through a beam of photons emitted by a closed gammaray source [10][11][12][13][14][15][16].The latter method is noninvasive and relatively safe in application.
The analysis of random signals provided by scintillation probes requires the usage of advanced statistical methods of signal processing in both time and frequency domain.Because these signals, after proper preprocessing, may be ergodic and Gaussian, such classical methods as the crosscorrelation function (CCF) and the phase of cross-spectral density may be applied [1, 3, 9-12, 14, 16-22].Recently, not so 2 Mathematical Problems in Engineering popular methods of the time delay estimation, as differential [3,[23][24][25], the cross-correlation with use of the Hilbert Transform [17,[26][27][28][29], and the method based on conditional averaging of the signals [30][31][32][33], have appeared.The usefulness of the last three in two-phase flow investigation by radioisotopes is not known in detail.They can be, in the first stage, examined using simulation.
The first part of the paper presents two mathematical models used in estimation of the time delay of random signals (Section 2).Then, the cross-correlation method is described together with the differential one as average magnitude difference function (AMDF), average square difference function (ASDF), and an original one combining the differential functions and the CCF (Section 3).Section 4 describes the basis of two-phase flow velocity evaluation by application of the gamma-ray absorption method.Consequently, the next section presents exemplary results of simulation studies of the metrological properties of the above functions, performed on the basis of mutually delayed stochastic signals simulation.The parameters of the above-mentioned models have been selected in a way which ensures the proper correspondence with real signals obtained in gamma-ray absorption measurements of a liquid-gas flow in a horizontal pipeline.The results of time delays and their standard uncertainties received in differential and combined methods have been compared with corresponding results of the classical cross-correlation procedure.Finally, Section 6 summarizes results and presents conclusions of the paper.

Signal Models in Time Delay Estimation of the Random Signals
In many time delay estimation issues, the relations between () and () signals received from two sensors can be expressed by the following formulas [17,20]: where () is a stationary random signal with Gaussian N(0,   ) distribution of probability, frequency band , and one-sided power spectral density: where  is a coefficient;  0 is the most probable transportation time delay; () and () are white noises with Gaussian N(0,   ), N(0,   ) distributions, not correlated with () and mutually not cross-correlated.The autocorrelation function of the () signal can be expressed as The following assumptions concerning the models of signals ( 1) can be stated: where   , and   are standard deviations of () and () signals and   (⋅),   (⋅) are their autocorrelation functions.
Depending on presence of the noise in one or both signals, three models can be considered [32], but in practice only two cases are worthwhile of deliberation: (i) Model I:   = 0,   =   ̸ = 0 and SNR  = SNR; then (ii) Model II: The disturbing signals in both channels have identical distributions N(0,   ) but have distinct, not cross-correlated realisations.

Cross-Correlation. A cross-correlation function of 𝑥(𝑡)
and () ergodic signals is defined by the following formula: where  is the averaging time and  is time delay [17].
The  0 transportation time delay is determined on position of the main maximum of the CCF.The typical waveform of the cross-correlation function is shown in Figure 1.
The normalized CCF value for  =  0 can be expressed in the following form [17]: When we substitute in (11) the formulas (5), as well as  = 1, then for  CCF ( 0 ) = (SNR) the following approach can be made.(i) Model I, for (6) signal: (ii) Model II, for (8) signals: The graphical representations of ( 12) and ( 13) are shown in Figure 2. Formulas ( 12) and ( 13) can be applied in the determination of SNR on the base of the normalised cross-correlation of the recorded signals.
The discrete estimator of CCF can be determined from the following equation: where  is the number of discrete values of () and () signals,  = /Δ,  = /Δ, and Δ is the sampling interval [34].

Differential Methods.
The discrete estimators for AMDF and ASDF can be represented by the following formulas: In both of the above differential methods the determination of transportation time delay consists of localization of the position of main minimum of the corresponding function [23].
Examples of graphical representations of AMDF and ASDF are presented in Figure 3.
where  is a small positive value.
The  has been introduced in order to prevent the division by zero, because the AMDF and ASDF functions in an ideal case of a lack of noise can assume a null for  =  0 .For   ̸ = 0, which usually takes place in measurements, the addition of  in the denominator of ( 16) is not necessary.Exemplary graphical representations of CCF/AMDF and CCF/ASDF functions and their comparison with CCF are shown in Figure 4.
The combination of CCF and AMDF functions has already been employed in the paper [25] for an acoustic signal processing.However the combination of CCF and ASDF is an original proposal of the authors.

Gamma-Ray Absorption Method in Two-Phase Flow Investigation
Nuclear methods have been in use for many years in measurement of two-phase flows in pipelines and open channels [10-16, 30, 35, 36].The gamma-ray absorption is relatively simple in application but requires conformance with strict safety requirements for staff and the environment protection from effects of ionizing radiation and radioactive pollution.
The principle of measuring a flow velocity of gas phase transported by liquid in a horizontal pipeline is presented in Figure 5. Two beams of gamma ray emitted by sealed radioactive sources and formed by collimators are partially absorbed by the flow of a medium.Impulses   () and   () are received on the outputs of scintillation probes, situated at the  distance on the opposite side of the pipeline, and then counted down within Δ sampling intervals, giving the mutually delayed discrete stochastic signals () and ().These signals describe the instantaneous state of flow in the studied section of the pipe [10][11][12][13][14].
Based on the  0 time delay of these signals one can calculate the average velocity of the gas phase   = / 0 .
Signals from the probes, after conditioning (centring, filtration), are ergodic and have normal probability distributions [30].Figure 6 presents a normalized cross-correlation function obtained for signals received in the BUB005, BUB006, and BUB010 experiments for water-air flow with the following parameters:  = 300000, Δ = 1 ms, and  = 97mm.The velocity of gas flow through the tested pipeline section of 30 mm inner diameter has been equal to 0.90 m/s (BUB005), 0.76 m/s (BUB010), and 0.71 m/s (BUB006), respectively.
A closed 241 Am source of 59.5 keV energy gamma ray has been used in the experiments mentioned above together with scintillation detectors based on NaI(Tl) crystals.The laboratory stand and geometry of a requisite absorption set are described in detail in papers [14,30,35].

Exemplary Results of Simulation Studies
During flow measurements signals provided by scintillation probes contain not only statistical information about the studied mixture but also disturbing signals caused by the gamma radiation background, electronic noise, and fluctuations of the gamma ray decay.The proposed modelling of such signals can be performed with use of I and II models described in Section 2, with properly selected parameters.Model II, defined by ( 8), has been used in the reported study.In result of the simulation, white noise signal was processed by a low-pass Butterworth filter with parameters  given in Table 1 which gives () and () signals and ensures similarity of the shape and amplitude of the normalized cross-correlation functions to the  CCF () functions shown in Figure 6.The  1 () and  2 () disturbing signals have been Gaussian white noises with N(0,   ) distributions, not correlated mutually and with the recorded signals.Figure 7 presents the normalized cross-correlation functions obtained by modelling with parameters given in Table 1.
A high level of similarity is observed between CCF functions from Figure 6 and the one in Figure 7 in vicinity of the maximum point, which is essential in estimation of the time delay.Due to the above procedure, the models of () and () signals allow determination of the functions ( 14)-( 16) and estimation of the transportation time delay and its standard uncertainty.Figures 8 and 9 present exemplary of the ASDF and AMDF differential functions (Figure 8) as well as CCF/ASDF and CCF/AMDF combined functions (Figure 9) obtained for models corresponding to the BUB006 experiment.Furthermore Figure 9 additionally shows the CCF function.For easier comparison of all functions, their normalization in relation to the maximum value was applied.
Determination of the transportation time delay for all examined functions consists in estimation of the extreme point position.In the paper, this step was achieved by interpolation of the selected part of a given characteristic by the Gauss function: where  0 is a normalization level of the Gauss function and  is a standard deviation of its distribution (Figure 10).Then the τ0 transportation time delay estimator is determined as the first moment of the fitted normal distribution [10,30], while (τ 0 ) is its standard uncertainty [37], and in this case it is equal to the standard deviation of the mean value: where  is a number of points used in the interpolation procedure.For all functions described above the similar fitting procedure has been used.The results obtained in this way are listed in Table 2.
For example, fitting a Gaussian curve to CCF, AMDF, and CCF/ASDF functions for models corresponding to the BUB006 experiment is presented in Figure 11.

Conclusions
This paper proposes the use of the AMDF and ASDF differential functions as well as CCF/ASDF and CCF/ASMF combined methods for time delay estimation in a radioisotope investigation of two-phase flows.Intentionally selected examples allow one to compare the proposed methods with the CCF cross-correlation distribution of signals delivered by measurements.In this task simulations have been performed for computer-generated models of stochastic signals,

Figure 1 :Figure 2 :
Figure 1: Typical graphical representation of a CCF function.

Figure 5 :
Figure 5: The principle of gamma-absorption measurement of two-phase flow.

Figure 10 :
Figure 10: Explanation of fitting the Gauss function (-) to the estimated time delay distribution (•).

Table 1 :
Parameters of simulation approach.

Table 2 :
Exemplary results of simulation for selected experiments.Method τ0 [ms] (τ 0 ) [ms] (τ 0 )/(τ 0 ) CCF [-] Experiment corresponding to one received from the scintillation probes in gamma-ray absorption measurements of liquid-gas flows in a horizontal pipeline.The advantage of the simulation method