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The spherical model of time and location calculation of the lightning discharge is given. The calculations are made by means of radio signals detection by sensors of the distributed network. The full solution of a problem of lightning discharge cloud-ground type location for three sensors is given. Based on this task the lightning location method for a network of sensors was developed. By means of computational experiments, the analysis of accuracy of the model depending on radio signals detection accuracy at observing stations was done.

Let

where,

The system of three Equations (1) is quite defined, as it has three unknown: coordinates

In a task we believe, that the lightning discharge can occur in any place on Earth surface, that is:

Because of physical sense, timepoint of discharge

On the other hand, the way passed by a radio signal to each sensor can’t be more than length of a semi-circle of a big circle of the Globe. From this follows that

where

Let is consider some features of the system (1). Using the geocentric coordinate system, which is connected with Earth, we will enter single vectors

Applying a scalar product of vectors, we can show distance from sensors to a lightning as

In geocentric coordinate system the scalar product disclose by the equation:

and, in spherical coordinate system:

Let’s put (5) into (1) and divide equations on

where,

We will consider that timepoints

Otherwise, we will change numbering of points

Let’s consider a difference of two equations of system (6)

For a spherical triangle with vertexes

From this inequation and (8) it follows that

The Inequation (9) represents a necessary condition for solvability of the system of Equation (1). Thus, if the difference of the lightning timepoints measured by each couple of sensors does not meet a requirement (9), to define location and time of a lightning discharge based on these sensors it is impossible.

The cause of disarrangement of an Inequation (9) can be in an error of identification of lightning discharges at sensors, or be a consequence of errors of measuring equipment. Further, we believe that the requirement (9) is met.

Geometrically each of the Equation (8) define a set of points

semi-transverse axis

Further, we assume that equation in (9) does not exist and the strict in equation we have

Thereby we exclude a case, when the hyperbole degenerates in an arch interval.

Unlike a flat case, hyperbole on the sphere is the limited closed curve and, moreover, it is coincides with a spherical ellipse. Really, using the identical equation:

let us find

where,

This equation defines a geometrical set of points

finition of an ellipse on the sphere. Focal length of an ellipse equally

plane passing through the center of Earth perpendicular to a vector

The visualization of the Equation (8) given by function graph:

which is represented in

Let us find the parametrical equation for a hyperbole (8). Let us add

Previously we will create orthogonal coordinate system with respect of which there will be a

Let us construct a single normal vector to the plane

where,

Thus, the

and direct

Thus, the

where,

For the finding dependence

After plugging (19) to (8) we will get an equation

which, after arccosines inversion changed to

Because of (12), the

from this we have

The Equation (21) give us unknown relationship between

if the

Therefore,

By using sine and cosine of the

Thus, the required parametrical equation for a hyperbole (8) is set by Equation (19) in which

By excluding 1st equation from 2nd and 3rd in (6) we will get equivalent system:

Two last equations describe two hyperboles, which have the general focus set by

Parametrical equation of the 2nd equation in the system (24):

where,

Parametrical equation of the 3rd equation in the system (24):

where,

At the same values

Let the vectors

Then, at the half-planes

If the scalar triple product of the vectors

By plugging (29) in (26), we will receive the parametrical equations of two hyperboles, which depend on one parameter

Vectors

By plugging cot values, we will get equation relative to

By identical transformations, the Equation (30) will be as

where,

Believe that

we get equation:

where,

Depending on value

1) If

2) If

3) If

After

To lightning discharge there can correspond only one point of intersection of hyperboles, which we will call actual. The second point is a consequence of crossing of two closed convex curves leaning at each other. This point we will call phantom.

Timepoints of both lightning discharges (actual and phantom) are defines by formula (32), which is develop from the first equation of system (28):

Thus, if the lightning detection network consists only of three sensors, then with conditions (12) keeping we have actual position of a lightning and, as a rule, there is a phantom point.

For system of the Equation (24) both of its solutions are equal. Therefore, to allocate a lightning actual point, additional information is necessary. For example, for this purpose it is possible to use a vector of induction of a magnetic component of the accepted radio signal of a lightning which coordinates can be received by means of the bidirectional magnetic antenna [

In a demonstration example, three measurement points which are conditionally placed near the cities of Almaty, Taldykorgan and Kapshagay (Republic of Kazakhstan) were selected. The corresponding timepoints are calculated in the assumption that the lightning discharge occurred near Astana city. Zero was taken for initial counting of time. Given data for calculation are presented in

Speed of radio waves―299,792.458 km/s,

Average Earth radius―6371.308 km.

As a result of the task solution, two points of the lightning discharge with various timepoints of a discharge are defined. Results of calculation are given in

Parameters of the actual and phantom points of the lightning discharge are mutually reversible. That is, if according to a phantom point, provided in

Lightning discharge location and timepoint in a phantom point are displaced with respect to actual.

Let’s consider a set of N sensors located randomly on the Earth’s surface in the points

We will put in compliance to each point

Sensor area | Longitude of the sensor, degree | Latitude of the sensor, degree | Lightning discharge detection timepoint, ns |
---|---|---|---|

Almaty | 76.92848 | 43.25654 | 3,236,010 |

Taldykorgan | 78.36667 | 45.01667 | 2,872,390 |

Kapshagay | 77.06304 | 43.86681 | 3,049,610 |

Solution # | Longitude, degree | Latitude, degree | Timepoint of lightning discharge, ns |
---|---|---|---|

1 (Astana region) | 77.0000030 | 50.999975 | 0.0 |

2 (Phantom point) | 78.2293052 | 44.4711265 | 0.002666825 |

equations

We will make subsystems from the equations of system (33), including three equations to each subsystem. We will call such subsystems as triads. In total, it is possible to make M triads from N equations, where

Let each triad of the Equation (33) have two solutions. We will construct the set of solutions

If the functional

If

For an assessment of the received solution of system (33) we will allocate the subset including the M actual solutions from a set of all solutions of triads. For this purpose we use the natural assumption, that on the actual solution the functional

We will determine an arithmetic average value on a subset of actual solutions

For the accuracy ranking of the approximation

and at time-difference

The grade of dispersion of intersection points of hyperboles is characterized by an average square deviation from the position of the solution

For the analysis of the given technique a number of numerical experiments was executed. In all experiments the same group of 6 sensors with coordinates are given in

Numerical experiments are given for three points of a lightning placed in various regions of Kazakhstan, removed from each other―in the North, West and East. For each lightning coordinates are set and conditionally exact time of detection within 1ps accuracy is calculated. The zero moment of each lightning discharge in each experiment is taking. Lightning parameters are given in

The problem of each experiment consisted in lightning position and initial moment calculation with triangles technique depending on the accuracy of lightning detection on stations. Time of lightning detection was set by rounding of exact values of the moments of detection given in

Station # | Longitude, degree | Latitude, degree | Region of the Station |
---|---|---|---|

1 | 76.92848 | 43.25654 | Almaty |

2 | 78.36667 | 45.01667 | Taldykorgan |

3 | 77.06304 | 43.86681 | Kapshagay |

4 | 71.36667 | 42.9 | Taraz |

5 | 74.995 | 46.8481 | Balkhash |

6 | 73.76139 | 43.59833 | Shu |

Station # | Lightning coordinates: -Longitude 71˚, -Latitude 51˚, -Astana city | Lightning coordinates: -Longitude 51˚, -Latitude 44˚, -Aktau city | Lightning coordinates: -Longitude 85˚, -Latitude 47˚, -Zaisan city | |||
---|---|---|---|---|---|---|

Lightning detection timepoint, s | Distance from station to lightning, km | Lightning detection timepoint, s | Distance from station to lightning, km | Lightning detection timepoint, s | Distance from station to lightning, km | |

1 | 0.003236008550 | 970.1310 | 0.006938195622 | 2080.0186 | 0.002525797123 | 757.2149 |

2 | 0.002872390603 | 861.1210 | 0.007214464807 | 2162.8421 | 0.001859757582 | 557.5413 |

3 | 0.003049609604 | 914.2500 | 0.006932949958 | 2078.4460 | 0.002368726661 | 710.1264 |

4 | 0.003005907395 | 901.1484 | 0.005485512347 | 1644.5152 | 0.003881926243 | 1163.7722 |

5 | 0.001821388535 | 546.0385 | 0.006309817053 | 1891.6355 | 0.002533458619 | 759.5118 |

6 | 0.002831421721 | 848.8389 | 0.006076166255 | 1821.5887 | 0.003188416712 | 955.8633 |

Lightning timepoint measurement accuracy | Lightning longitude, degree | Lightning latitude, degree | Time of lightning, s | Accuracy | The mean time of lightning, s | Mean square deviation | Deviation from the exact position, km |
---|---|---|---|---|---|---|---|

1 ns | 70.999986 | 51.000013 | −0.000000005 | 0.001680 | 0.000000000 | 0.007895 | 0.001740 |

10 ns | 71.000011 | 51.000016 | −0.000000003 | 0.019296 | 0.000000061 | 0.130168 | 0.001950 |

100 ns | 71.000296 | 50.999367 | 0.000000229 | 0.257204 | 0.000001077 | 1.595166 | 0.073341 |

1 μs | 71.790813 | 50.205197 | 0.000348181 | 46.011732 | 0.000193392 | 112.521931 | 58.525675 |

Lightning timepoint measurement accuracy | Lightning longitude, degree | Lightning latitude, degree | Time of lightning, s | Accuracy | The mean time of lightning, s | Mean square deviation | Deviation from the exact position, km |
---|---|---|---|---|---|---|---|

1 ns | 50.999841 | 43.999976 | −0.000000043 | 0.121148 | −0.000000446 | 0.673305 | 0.012965 |

10 ns | 50.999862 | 43.999927 | −0.000000040 | 0.347070 | −0.000001193 | 1.180598 | 0.013691 |

100 ns | 50.977009 | 43.996995 | −0.000006195 | 5.769364 | 0.000013007 | 33.453457 | 1.869205 |

1 μs | 51.312113 | 44.043791 | 0.000084374 | 1.269249 | 0.000081988 | 60.537080 | 25.427550 |

Lightning timepointmeasurment accuracy | Lightning longitude, degree | Lightning latitude, degree | Time of lightning, s | Accuracy | The mean time of lightning, s | Meansquaredeviation | Deviation from the exact position, km |
---|---|---|---|---|---|---|---|

1 ns | 85.000026 | 47.000002 | −0.000000006 | 0.029547 | −0.000000104 | 0.074809 | 0.001987 |

10 ns | 85.000094 | 47.000023 | −0.000000022 | 0.024272 | 0.000000058 | 0.076753 | 0.007576 |

100 ns | 84.996640 | 46.999239 | 0.000000879 | 0.525445 | −0.000000782 | 6.067994 | 0.268508 |

1 μs | 84.974926 | 46.994068 | 0.000006787 | 53.924626 | −0.000172967 | 238.321958 | 2.012793 |

Comparing change of result of lightning discharge time and location calculation on ratio of increase of an error of lightning detection time, it is possible to make a conclusion that the presented technique of calculation is effective and reliable if the detection accuracy does not exceed 100 nanoseconds. Since the accuracy of 1 microsecond range of hyperboles intersection points dispersion for different triads which is characterized by increase in distance between them and accident of their mutual position considerably increases.

The work is supported by the Grant 0100/GF4 of Ministry of Education and Science of the Republic of Kazakhstan.

Anatoliy Lozbin,Yuriy Shpadi,Alexander Inchin, (2016) Triangles Technique for Time and Location Finding of the Lightning Discharge in Spherical Model of the Earth. Journal of Geoscience and Environment Protection,04,125-135. doi: 10.4236/gep.2016.44016