1. Introduction

Information of a thunderstorm activity collected by a SPMS can be used by secondary processing algorithms to identify a real Predicting the occurrence and development of a thunderstorm near the insured object requires the development of appropriate singlepoint systems for monitoring thunderstorm activity. For mobile insured objects, such as aircraft carriers, oil tankers, these systems must be small and included in the equipment of the objects [1]. Primary processing algorithms which used to determine location of the radiation source can lead to errors in calculation the bearing of discharge. It inevitably occurs in a single-point monitoring systems (SPMS). The error depends on the orientation of the equivalent dipole of the radiation source. If the dipole moment is vertical then the error is minimal, otherwise the direction finding error is completely determined by the horizontal projection of the equivalent dipole and can be equal to π radians.

location of the thunderstorm foci. It is particularly critical when processing pre-storm radiation which consist of intra- and inter-cloud discharges. Secondary processing algorithms can reduce the uncertainty of determining the coordinates of the thunderstorm location by analyzing the location parameters of radiation sources that found for a time interval. This approach has been announced at ”2019 Russian Open Conference on Radio Wave Propagation (RWP), Kazan, Russia” [2] in the field of study that deals with electromagnetic methods of lightning detection. This paper is an original extension of this work.

Each cloud and ground lightning flash consist of many individual physical processes. These processes can be associated with characteristic electric and magnetic fields. Lightning emits significant electromagnetic energy in the radio-frequency range from below 1 Hz to near 300 MHz. This electromagnetic radiation can be detected by lightning locating systems. Lightning data have utility in different areas of importance meteorological service organizations. Typically, meteorological organizations use this data to protect life and property. Commercial organizations use lightning data to provide weather warnings, forecasts to aviation, agriculture, energy, mass media.

There are plenty of lightning locating systems that uses different radio-frequency ranges, for example, the very low-frequency (VLF) range from 3 to 30 kHz, the low-frequency (LF) range from 30 to 300 kHz. Accurate lightning locating systems necessarily employ multiple sensors. But single-point passive monitoring system are also popular as they are inexpensive, mobile and do not require communication. Many companies are engaged in the production of such devices [3]–[6].

A challenging problem which arises in this domain is processing pre-thunderstorm radiation (that is before the first lightning flash). During the project ISTC 1822 [7] were recorded 2.5 million atmospherics and less than 10% were defined as a lightning discharge, the rest radiation were classified as pre-thunderstorm radiation. Algorithms used in multiple sensors systems require registration of the atmospherics by several sensors, it is not always possible. Otherwise, single-point systems generally filtered out this data or it processed incorrectly.

Algorithms to determine location of the radiation source uses the magnetic direction finding (MDF) technique to calculate the bearing of discharge. It works really well for multistation systems especially with combination with time of arrival (TOA) technique. But using MDF technique in SPMS leads to significant errors for the discharges whose equivalent dipole has an orientation different from vertical.

In general, the results of the project showed possibility of creating a new generation of passive thunderstorm detection systems to extend their functionality. One way to overcome these problems is not only revision of the technical characteristics of sensors, but also development of new mathematical models and algorithms for analyzing thunderstorm activity. The problem of identifying the position of thunderstorm foci is practically important to forecasting the development of thunderstorm. The registration of the cloud radiation by a single-point lightning protection system, for the purposes of forecasting thunderstorm development, can be reached by expansion of receiving hardware dynamic range and development of mathematical and software.

The standard model for the lightning discharge is based on a vertical dipole that cannot adequately represent all discharges. Modern methods of analyzing the electromagnetic radiation source (EMRS) field allow to determine the parameters of the EMRS location by many algorithms based on the dipole as a model of the EMRS [1], [8]–[10]. A more correct approach is to simulate a lightning discharge as a arbitrarily oriented dipole and it possible by using special a group of E − H methods that allow arbitrary orientation of dipoles [11, 12]. These algorithms allow us to estimate the position parameters of this dipole: the distance to the radiation source and the pseudo bearing, and can be used for processing prethunderstorm radiation. In this case, signals from various sources of electromagnetic radiation are registered, the appropriate mathematical model of which is an arbitrarily oriented electric dipole. Processing signals from dipoles with a predominant horizontal projection leads to uncertainty in the estimation of the bearing on the [2, 12] discharge, which significantly reduces the quality of dipoles location detection. One way to reduce this uncertainty is to analyze the information received from all dipole sources.

The aim of the work is describing approach for the secondary processing of monitoring results of thunderstorm activity in VLF range, allowing to get plausible estimates of the location of a thunderstorm center based on its pre-storm radiation.

A review of the status of passive storm monitoring systems and the demonstration of the use of lightning-position detection systems for the passive radar of hazardous meteorological phenomena are presented in [8]. Modern methods for analyzing the field, allowing to determine the parameters of the EMRS, characterizing its location and orientation are presented in the works [12].

2. Primary processing of EMRS signal

Figure 1 shows the geometry of the problem: the earth’s surface is plane Oxy, the observation point is origin O, the polar coordinates of the radiation source are r, ψ, θ , and the angular coordinates of the radiation source dipole moment are ψ₀, θ₀.

Figure 1: Geometry of the problem

There are a dipole P₀ and an imaginary dipole P₁ which is the mirror image of the dipole P₀. P₀ results a field (E,H) consisting of the field (E(P₀), H(P₀)) of the source P₀, and the reflected field represented by the field (E(P₁), H(P₁)) of the P₁.

The electric E and magnetic H components of the field in terms of the Cartesian system of coordinates with origin at the observation point O and the O_z axis being the normal of the bounding plane have representation [9]

c is speed of light in vacuum, ₀ is permittivity of vacuum.

The values of the parameters r, ϕ, u, v are measured as a result of primary processing, for each pulse signal received from a radiation source [9]. The quality of estimating the location of an individual source is quite low. Really, a vertical dipole (that is when

It’s easy to see that for s < 1, that is for θ₀ approaching the vertical,

in other words, the maximum likelihood estimates are the same as for the vertical dipole: ψ = ϕ and θ = arcsinu; and for s ≥ 1, that is for orientation with a significant horizontal component, poor angular resolution is gained.

An adequate mathematical model of the pre-thunderstorm EMRS is an inclined electric dipole with a predominance of the horizontal component. Processing signals from dipoles with a predominant horizontal projection leads to uncertainty in the estimation of the angle coordinates of the [2, 9] discharge, which significantly reduces the quality of dipoles location detection.

The multipoint system of passive monitoring of thunderstorm activity not only removes the uncertainty of location [9], but also allows to determine the orientation of the dipole [13, 14]. Nevertheless, the improvement of forecasting methods using single-point systems remains relevant [15, 16].

One way to reduce the uncertainty of thunderstorm location is to analyze the information received from all dipole sources for time T.

The purpose of secondary processing is to estimate the location of the thunderstorm center by information about all dipoles location parameters for time T.

3. Secondary processing of EMRS signals from thunderstorm foci

It is known that the Cartesian coordinates x, y of the possible position of the EMRS satisfy the condition [9]:

which determines the OXY projection of the EMRS possible positions set. The coordinates of the possible location EMRS belong to the set of solutions of the system of equations

and OZ axis of the coordinate system OXYZ to be divided into segments

                 C(k) = {z : k∆ ≤ z ≤ (k + 1)∆},            k = 0, 1, …, M.

In the future we will use the following notation:

C_OXYZ(r,u,ϕ) = {(i, j,k) | C(i, j,k) ∩ S , ∅}

is set of cells in which the EMRS can be locate with a non zero probability;

C_OXY(r,u,ϕ) = {(i, j) | ∃k : C[i, j,k] ∩ S , ∅} is OXY projection of set C_OXYZ(r,u,ϕ);

C_OZ(r,u,ϕ) = {k | ∃C(i, j) : C(i, j,k) ∩ S , ∅}

is OZ projection of set C_OXYZ(r,u,ϕ).

For each cell containing solutions of the (15) system, we find the probability that the EMRS belongs to this cell when some hypothesis is fulfilled. In the future, we will use the found probabilities to build a three-dimensional map of the membership level of cells that are part of the thunderstorm center.

3.1      Hypothesis H₁:

Uniform probability distribution on C_OXY(r,u,ϕ)

It is following from (15) that on the OXY plane, the valid points farthest from the origin of the coordinates are (x₁, y₁) and (x₂, y₂)

The coordinates x, y for all possible ERMS positions belong to segment [(x₁,y₁), (x₂,y₂)] ⊂ OXY between the points (x₁,y₁) and (x₂,y₂). The all possible radiation source locations can be represented in a parametric form as the set of points:

Then, the probability of EMRS in cell C[i, j,k] is equal to

3.2     Hypothesis H₂:

Uniform probability distribution on C_OZ(r,u,ϕ)

It follows from (15) that the set of points of all possible locations of EMRS can be represented in a parametric form other than (20) :

3.3     Hypothesis H₃:

Uniform probability distribution on C_OXYZ(r,u,ϕ)

Since (20) and (21) give different representations of set L(r,u,ϕ), there are also two ways to represent a set of cells with a non-zero probability of having the EMRS:

3.4    Three-dimensional map of the level belonging of thunderstorm foci to space cells

So, for each cell containing solutions of the (15) system, we found the probability that the EMRS belongs to this cell when hypothesis H ∈ {H₁, H₂, H₃} is fulfilled. Let us use the found probabilities to build a three-dimensional map of the membership level of cells that are part of the thunderstorm center when hypothesis H is fulfilled.

The set of discharges registered for the time period T is defined as R(T). The cell grade membership to thunderstorm of the radiation source defined as the sum of all of radiation during this period.

Further we demonstrate the possibility to estimate the thunderstorm center location by (23) formula applying simulation.

4. Simulation

Let’s define two artificial thunderstorms R₁(T) and R₂(T) for modeling purposes. All discharges of the thunderstorms will have the same location R=60 km, ∆=10km, ψ = π/4, θ = π/3.

Figures 3–8 demonstrate the observation point (x = 0, y =

0, z = 0) and the thunderstorm location cell is C(i = 3, j = 3, k = 3) (marked with a blue square). Cell color shows the cell grade membership level to the thunderstorm (see figure 2).

Figure 2: Scale of the level of belonging of a cell to a thunderstorm center

The R₁(T) emits 256 pulse signals from dipole radiation sources with orientation ψ₀ ∈ {πk/16 : k = 1,2,…,32}, θ₀ ∈ {πk/8 : k = 0,1,…,8}, that is EMRS are different incline dipoles, which is typical for a thunderstorm center. Location parameters values r, u, v, ϕ are calculated by (5)-(6) formulas.

Figure 3: Map of R₁(T) under hypothesis H₁

Figure 4: Map of R₁(T) under hypothesis H₂

Figures 3–8 demonstrate three projections of the threedimensional map of thunderstorm foci R_l(T), l ∈ {1, 2} level belonging to space cells under hypothesis H ∈ {H₁, H₂, H₃}

Figure 5: Map of R₁(T) under hypothesis H₃

Figures 3, 5 make it possible to identify the desired cell

C(i = 3, j = 3, k = 3). Figure 4 does not allow to identify the desired cell at all.

Table 1 shows cells with grade membership level to the thunderstorm R₁(T) no less 0.9 and similar to visual analysis of the projections hypothesis H₁ and H₃ found the desired cell.

Table 1: Cells with grade membership level to the thunderstorm R₁(T) no less 0.9

Hypothesis	COXYZ(r,u,ϕ) ∩ {(i, j,k) : gradeRH1(T)(i, j,k) > 0.9}
H1	{(3,3,3), (3,2,3), (4,3,3)}
H2	{(1,5,0), (1,5,1), (3,3,4), (5,1,0), (5,1,1)}
H3	{(3,3,3)}

But the best hypothesis is hypothesis H₃, because it will allow to identify the cell with a thunderstorm center uniquely and accurately.

The R₂(T) has the same location as R₁(T) but emits only 100 pulse signals from dipole radiation sources with orientation ψ₀ ∈ {2πk/100 : k = 1,2,…,100} , θ₀ = π/2, that is EMRS are different horizontal dipoles, which is typical for a pre-storm center. Location parameters values r, u, v, ϕ are calculated by (5)-(6) formulas.

Figures 6–8 demonstrate three projections of the threedimensional map of thunderstorm foci R₂(T) level belonging to space cells under hypothesis H ∈ {H₁, H₂, H₃}.

Figure 6 demonstrates that the desired cell does not belong to cells with maximum level of membership.

Figure 7: Map of R₂(T) under hypothesis H₂

Figure 8 demonstrates that the desired cell belong to cells with maximum level of membership only.

Figure 8: Map of R₂(T) under hypothesis H₃

Table 2 also shows the best hypothesis is H₃. Hypotheses H₁ and H₂ maps do not contains the desired cell and does not allow to unambiguously establish the location of the foci.

Table 2: Cells with grade membership level no less 0.9 to the thunderstorm R₂(T)

Hypothesis	COXYZ(r,u,ϕ) ∩ {(i, j,k) : gradeRH2(T)(i, j,k) > 0.9
H1	{(3,3,4)}
H2	{(3,3,4)}
H3	{(3,3,3), (3,3,4)}

5. Conclusion

These findings provide a potential new technique to processing of the results of thunderstorm activity monitoring in the VLF range. The processing of all data from the SPMS during thunderstorm activity introduces a lot of uncertainty and makes it difficult to determine the real location of the thunderstorm center. The methods of analysis and visualization of the entire set of received signals presented in this paper reduce the uncertainty in determining the coordinates of the thunderstorm location. Processing of pre-thunderstorm radiation is significantly important to forecasting the development of thunderstorm.

The artificial single-point thunderstorm model described in the paper is a very simple, which is very convenient for research, however, a real thunderstorm can have a more complex structure, therefore it is important to carry out modelling with more complex models of thunderstorms, for example, check the proposed approaches for their ability to identification of multiple foci. This is an interesting topic for future work.

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