Ionosphere Research and its applications

 

1. Methods ionosphere model identification.

Ionosphere physical phenomenon is very complicated. Therefore ionosphere models are heuristic and  are created by experimental data by interpolation.

1.1 Identification by table.

There are a set of different tables those contains experimental data. These tables looks like:

L

B, Gs

Density of protons

0.1

0.4

1.0

4.0

10.0

30.0

50.0

100.0

200.0

400.0

       

1

2

3

4

5

6

7

8

9

10

11

12

       

       

       

1.20

0.18035

2.50E+03

2.50E+03

2.46E+03

2.40E+03

2.30E+03

2.30E+03

2.00E+03

1.40E+03

7.00E+02

1.40E+02

0.19000

1.20E+03

1.20E+03

1.12E+03

1.07E+03

1.03E+03

9.60E+02

9.49E+02

7.00E+02

3.70E+02

7.00E+01

0.20000

4.00E+02

3.79E+02

3.67E+02

3.56E+02

3.44E+02

3.27E+02

3.10E+02

2.40E+02

1.40E+02

2.20E+01

0.20500

2.05E+02

1.91E+02

1.82E+02

1.74E+02

1.70E+02

1.63E+02

1.48E+02

1.26E+02

7.71E+01

1.16E+01

0.21000

1.00E+02

8.98E+01

8.69E+01

7.84E+01

7.28E+01

6.58E+01

5.93E+01

4.50E+01

2.80E+01

4.70E+00

0.21500

4.20E+01

3.57E+01

3.29E+01

2.69E+01

2.42E+01

1.85E+01

1.62E+01

1.02E+01

2.50E+00

5.37E-01

0.22000

1.00E+01

8.00E+00

7.15E+00

5.46E+00

4.58E+00

3.00E+00

2.00E+00

8.63E-01

1.60E-01

1.00E-01

0.22400

1.00E+00

1.00E+00

8.04E-01

2.00E-01

1.00E-01

1.00E-01

1.00E-01

1.00E-01

1.00E-01

1.00E-01

       

       

       

       

1.30

       

0.14185

3.48E+04

3.44E+04

3.40E+04

3.10E+04

2.67E+04

1.60E+04

1.30E+04

7.82E+03

2.97E+03

4.99E+02

0.16000

1.62E+04

1.58E+04

1.54E+04

1.46E+04

1.29E+04

9.00E+03

7.30E+03

4.40E+03

1.67E+03

2.57E+02

0.18000

6.61E+03

6.52E+03

6.43E+03

6.04E+03

5.47E+03

4.10E+03

3.35E+03

2.14E+03

8.06E+02

1.15E+02

0.20000

1.72E+03

1.77E+03

1.74E+03

1.60E+03

1.50E+03

1.20E+03

1.02E+03

7.82E+02

2.97E+02

3.69E+01

0.20500

1.15E+03

1.09E+03

1.13E+03

9.69E+02

9.89E+02

8.04E+02

6.65E+02

5.02E+02

2.12E+02

2.47E+01

0.21000

5.65E+02

5.37E+02

6.14E+02

5.32E+02

5.20E+02

4.04E+02

3.39E+02

2.60E+02

1.19E+02

1.31E+01

0.21500

1.91E+02

1.94E+02

2.03E+02

2.04E+02

1.82E+02

1.45E+02

1.36E+02

9.64E+01

6.17E+01

5.50E+00

0.22000

3.48E+01

3.44E+01

3.40E+01

3.10E+01

3.01E+01

2.67E+01

2.45E+01

2.25E+01

1.67E+01

1.58E+00

0.22500

2.90E+00

1.00E+00

1.00E+00

9.62E-01

9.49E-01

9.06E-01

8.75E-01

8.49E-01

7.69E-01

2.83E-01

0.22800

6.20E-01

1.00E-01

1.00E-01

1.00E-01

1.00E-01

1.00E-01

1.00E-01

1.00E-01

1.00E-01

1.00E-01

       

       

       

       

       

       

       

1.40

0.11358

1.40E+05

1.40E+05

1.40E+05

1.20E+05

7.89E+04

3.60E+04

2.50E+04

1.40E+04

4.58E+03

7.00E+02

0.12000

1.10E+05

1.10E+05

1.00E+05

9.49E+04

6.58E+04

3.10E+04

2.20E+04

1.20E+04

4.00E+03

5.89E+02

0.14000

5.30E+04

4.80E+04

4.30E+04

3.79E+04

3.10E+04

1.70E+04

1.30E+04

7.50E+03

2.40E+03

3.50E+02

0.16000

2.59E+04

2.20E+04

2.18E+04

1.86E+04

1.60E+04

1.00E+04

8.00E+03

4.70E+03

1.40E+03

2.10E+02

0.18000

9.69E+03

9.10E+03

8.30E+03

8.00E+03

7.20E+03

5.40E+03

4.20E+03

2.50E+03

8.30E+02

1.10E+02

0.20000

3.00E+03

3.00E+03

3.00E+03

2.70E+03

2.59E+03

2.20E+03

2.00E+03

1.00E+03

3.79E+02

3.79E+01

0.20500

2.04E+03

2.12E+03

2.16E+03

1.79E+03

1.83E+03

1.56E+03

1.43E+03

7.08E+02

3.03E+02

2.46E+01

0.21000

1.24E+03

1.33E+03

1.36E+03

1.12E+03

1.15E+03

9.42E+02

9.12E+02

4.45E+02

2.19E+02

1.32E+01

0.21500

7.30E+02

7.80E+02

7.95E+02

6.59E+02

6.76E+02

5.51E+02

5.07E+02

2.69E+02

1.36E+02

6.40E+00

0.22000

4.00E+02

4.00E+02

4.00E+02

3.75E+02

3.30E+02

2.80E+02

2.49E+02

1.50E+02

6.00E+01

2.59E+00

0.22500

1.96E+02

1.72E+02

1.67E+02

1.86E+02

1.31E+02

9.60E+01

9.71E+01

7.31E+01

1.39E+01

8.89E-01

0.23000

8.22E+01

6.47E+01

6.07E+01

8.02E+01

4.32E+01

3.08E+01

2.75E+01

2.81E+01

2.94E+00

3.41E-01

0.23600

2.07E+01

1.57E+01

1.43E+01

1.74E+01

9.02E+00

6.64E+00

4.99E+00

5.25E+00

5.45E-01

1.47E-01

0.24000

3.50E+00

3.00E+00

2.70E+00

2.00E+00

1.60E+00

1.00E+00

8.42E-01

5.32E-01

1.60E-01

1.00E-01

0.24300

1.00E+00

1.00E+00

9.04E-01

4.02E-01

1.00E-01

1.00E-01

1.00E-01

1.00E-01

1.00E-01

1.00E-01

       

1.50

       

0.09234

5.31E+05

5.20E+05

5.00E+05

3.78E+05

1.97E+05

3.77E+04

2.37E+04

1.33E+04

4.34E+03

5.65E+02

0.10000

3.46E+05

3.37E+05

3.26E+05

2.48E+05

1.30E+05

3.11E+04

2.04E+04

1.15E+04

3.76E+03

4.84E+02

0.12000

1.29E+05

1.27E+05

1.23E+05

9.49E+04

5.31E+04

1.99E+04

1.45E+04

8.28E+03

2.70E+03

3.42E+02

 

We need approximation of these models by interpolation law. This task is solved by usage of nonlinear regression. Nonlinear regression in statistics is the problem of fitting a model.

y = f(x,\theta) + \varepsilon

to multidimensional x,y data, where f is a nonlinear function of x, with regression parameter θ.

Nonlinear regression operates with selections. This software implements two methods of manipulation with selections. The first method loads selection iteratively, i.e., step by step. The second one loads selection at once.

The architecture of nonlinear regression software that uses iterative regression is presented in the following scheme:

Architecture

Let us describe the components of this scheme. The iterator provides data-in selections x and y. The y is the Left part of fitting the equations. The Transformation corresponds to the nonlinear function f, and generates the Left part of the fitting model. The Processor coordinates all the actions and corrects the Regression parameters. Let us consider an example of approximation of particles' density by following function:

Quadratic approximation 

where B, E are induction and energy and a, b, c, d, f, d are coefficients those we should define. Following picture shows as this problem is solved by our technology.

Program implementation 

Let us desribe elements of this picture. The SQL element performs an SQL query to database. It looks like:

SQL query 

The Function element contains approximation function.

Approximation function 

Right part of the window shows that variable x is linked to parameter B of end y is linked to patameter Energy of Selection. The Processor solves task of regression.

1.2 Visualization of results.

Scientific results requires visualization. Our technology enables us different kinds of 3D and 2D visualization. For example 3D visualization of obtained function looks like:

3D Visualization 

The 2D grayscale visualizaion is presented on the following picture:

Grayscale visualization 

Light places correspond to maximal values of function. Another kind of 2D visualization is rainbow one. It is presented at the following picture:

Rainbow visualization 

Red color correspond to maximal vaues and violet correspond to minimal ones.

1.3 Identification with motion model.

Let us consider following situation. We have a spacecraft. Its initial condition are known. We have measurements of parameters of ionosphere and we wish to define regression dependence of ionosphere parameter from Earth's magnetic field and height. The task is decomposed by two steps.

 

1.3.1 Step 1. Simulation.

During this step we shall obtain time dependencies of magnetic induction and height of time. Following picture present scheme of simulation

Physical picture 

 

We have Earth and Earth's magnetic field. Model of the field is rigidly linked to Earth's reference frame. However we should simulate magnetic field near spacecraft. So we install virtual sensor of the field at spacecraft. Our technology performs this operations by the following way:

Model of simulation 

This picture contains Motion model of spacecraft. This model is used for positioning of Spacecraft's frame. The F1 link means that 6D position of Spacecraft's frame is considered relatively Eearth's frame. The L 2 link means that Magnetic Field is considered relatively Eearth's frame. Link L 3 means that virtual Sensor is installed at Spacecraft's frame. The Output calculates module of magnetic field by the following formula:

|B| 

We can notice that this formula calculates square of vector product of induction vector by itself. During this step we've calculated dependences of magnetic field module on time.

1.3.2 Information processing.

Now we already have dependences of magnetic field, and height. Processing of this data is presented in the following picture:

Processing 

So the Height, B and Experimental data components contains data of spacecraft's height, magnetic induction and ionospheric experimental data. The term Regression formula does not requires comments.

 

2. Example of the complicated model application.

Models of ionosphere are used for determination of damage by high energy particles. To define this damage we should have comprehensive motion model of spacecraft. Quantatively this model is presented at following picure:

Controlled flight

Currents of its equipment interacts with Earth's magnetic field. Spacecraft has a photovoltaic that is an elastic vibrations body. It is used a flywheel for angular stabilization of the spacecraft. Now let us construct this situation:

Controlled flight simulation 

Let us describe components of this picture. The 3D motion model is a motion model of spacecraft as material point. It is used in 6D motion model of Spacecraft Body. The models of Photovoltaic and Flywheel are connected to model of Spacecraft Body. The Mechanical connection linklink means mechanical connection. The parameters of Spacecraft Body are used in Spacecraft frame. The virtual Sensor is used for determination of parameters of Magnetic Field and Damage field near spacecraft. The Damage field is the field of ionosphere particles. Parameters of fields those are defined by Sensor are used in Transformation. Transformation calculates damage effect and control laws of Spacecraft Body and Flywheel.