1/38

Imaging complex structures

in the ionosphere

Cathryn Mitchell,

Department of Electronic and Electrical Engineering

2/38

Introduce tomography as a concept

Outline some simple algorithms

A real system - Earth’s upper atmosphere (the ionosphere)

Other systems and applications

Outline

3/38

What is tomographic imaging?

?

?

?

?

e.g. Take a grid containing four numbers

Each number represents the magnitude of the property of the system

For example the density of an object

It could be the density of human body tissue

X-ray absorption is proportional to density

(left. Cranial blood vessels imaged by xray tomography)

4/38

How does tomography work?

Take measurements that pass through and are affected by an object

By taking many measurements from different angles you can determine the spatial distribution of the integrated quantity

?

?

?

?

For example if each of the above measurements (integrated quantities) are all equal to 10, find the density in each pixel

5/38

Were there enough measurements?

Four equations, four unknowns

… but there are many possible answers

?

?

?

?

5

5

5

5

9

9

1

1

7

7

3

3 … etc

Equations not all independent

6/38

If I can’t change the geometry?

Need some prior information

For example, if you know that the second pixel is four times the value of the first then you can solve the equations with these measurements

?

?

?

?

2

2

8

8

7/38

Formulate as a set of equations, e.g.

A11x1+A12x2 =10

A23x3+A24x4 =10

A31x1+A33x3 =10

A13x3 = 5 * 1.41

...

?

?

?

?

M1

M2

M3

M4

If I can choose the geometry? (1)

The influence of each measurement, M, is weighted by its length, A, through each density, x

8/38

Put in the path lengths and the unknowns are now only the densities

1 x1+ 1 x2 =10

1 x3 + 1 x4 =10

1 x1+1 x3 =10

1.41 x3= 5 * 1.41

...

5

5

5

5

If I can choose the geometry? (2)

9/38

• What is the fundamental thing that is measured? Attenuation of an x-ray? Change in a radio signal phase? How does this relate to my unknown quantity?

• How accurately did I measure it? How does this error relate through to my solution? (real data can be strange)

• Have I measurements over many angles? If not then how can I compensate for missing information?

•Do I even know the spatial path of the integral? Is the signal refracted? Is the problem linear?

Questions about imaging

10/38

Simple algorithms

11/38

1. Set all the image pixels along the ray pointing to the sample to the same value

2. The final backprojected image is then taken as the mean of all the backprojected views.

Backprojection

Diagram from http://www.dspguide.com

12/38

1. All the pixels in the image array are set to some arbitrary value.

2. An iterative procedure is then used to gradually change the image array to correspond to the profiles.

3. The algorithms attempts to address: how can the pixel values intersected be changed to make them consistent with this particular measurement?

4. An iteration cycle consists of looping through each of the measured data points.

Iterative algorithms

13/38

x x +

y x

k 1jk

k

i ij jk

j 1

n

ij ijj 1

n ijj

The Algebraic Reconstruction Technique

x is the density, j is the pixel index, k is the iteration index, y is the measurement of ray index, i, and delta the path length.

ART was used in the first commercial medical CT scanner

14/38

The ionosphere

15/38

What is the ionosphere?ionosphere

Earth

The ionosphere can be considered to be basically a thick shell of free electrons surrounding the Earth, starting at about 90 km altitude and extending to well beyond 700 km altitude

16/38

What is the ionosphere?

It is where our atmosphere meets space - starting at 100 km above the Earth

The thin atmosphere is bombarded with solar radiation and becomes ionised – hence the name the ionosphere

The ionisation consists of a dynamic sea of atoms or molecules and free electrons

The number of free electrons in a given volume is referred to as the electron density

17/38

What is complex about it?

Physics - the ionosphere is connected to the Sun (through radiation and magnetic fields) and to the Earth (through dynamical coupling to the lower atmosphere) – there are many inputs to the ionosphere system

18/38

Solar bombardment of the Earth

Acknowledgements for NASA movie

19/38

What is complex about it?

Physics - the ionosphere is connected to the Sun (through radiation and magnetic fields) and to the Earth (through dynamical coupling to the lower atmosphere) – there are many inputs to the ionosphere system

Measurements - the ionosphere slows and refracts radio signals – hence ray bending means that some measurement paths are known and some are not. There are many measurements that give different information

Structure – the electron density can form structures on many spatial scales – from metres up to thousands of kilometres. Knowledge of all of these scales is useful for different science and applications

20/38

S a te lliteP o s itio n 1

S a te lliteP o s itio n 2

R e fe re n c eR e c e iv e r A

0 °

1 0 °

2 0 °

E a rth

Io n o sp h e re

5 ° 1 5 °

S a te lliteP o s itio n 3

R e fe re n c eR e c e iv e r B R e fe re n c e

R e c e iv e r C

1 0 0 k m

4 0 0 k m7 0 0 k m

1 0 0 0 k m

R

S

ds ),N(r, = TEC

Tomographic imaging

21/38

Path integrals (A) are determined through a 3-D voxel representation of the ionosphere of electron concentration (x) for TEC observations (b).

bAx

Tomographic inversion (1)

22/38

A relatively short period is chosen for the time-dependent inversion, for example one hour, and data collected at typically 30 second intervals are considered.

The mapping matrix, X, is used to transform the problem to one for which the unknowns are the linear changes in coefficients (X) of a set basis functions

Tomographic inversion (2)

bAWX

23/38

Tomographic inversion (3)

b(AX)W 1

).b/w).U(V.(diag(1W T

WXx

24/38

To put this into words.

First we formed a set of mathematical models of the ionosphere – these were spherical harmonics and orthogonal functions

Next the corresponding values along each measurements path was found by the appropriate integration through each model

A least-squares fit is made between the measurements and all the corresponding paths through the models to find an appropriate weighting (W) for each model

These weightings are applied back to the model electron densities

Tomographic inversion (4)

25/38

Horizontal Variation

Spherical Harmonics

Model (eg IRI)

Height profile (to create EOFS)

Thin Shell (variable height) Chapman profiles Epstein profiles Models (eg IRI)

TIME:

None Zonal/Meridional Zonal/Meridional & Radial

Co-ordinate frame

Geographic Geomagnetic

Inversion type

2-D (latitude-height or thin shell) 3-D (2-D with time evolution or latitude-longitude-altitude)4-D (latitude-longitude- altitude-time)

Graphics options

Vertical profiles of Ne

Horizontal profiles of Ne

TEC maps

Electron concentration images (latitude vs height) at one longitude.

Electron concentration images (longitude vs height) at one latitude.

TEC movies

Electron concentration movies

Diagram showing options in ionospheric part of MIDAS program.

MIDAS imaging

26/38

Acknowledgements: IGS network

Tomographic imaging – MIDAS – northern hemisphere

27/38

28/38

Tomographic imaging – MIDAS – northern hemisphere

29/38

30/38

31/38

32/38

33/38

Users of images of the ionosphere

GPS navigators – the ionosphere induces an error on the position and sometimes a fading of GPS signals – aircraft landing systems and surveyors

Radio communications planners – the ionosphere refracts lower frequency signals and can distort signals at higher frequencies

Scientists – investigating the interactions between the Earth and the Sun

34/38

The ionosphere causes two problems for GPS navigation:

Group delay to the signal propagation time that is proportional to the total electron content. This can change the apparent position by tens of metres.

Scintillations of the signal are related to small-scale irregularities in electron density. These can cause temporary loss of the signals.

Research applications

35/38

These effects are not an issue for most GPS users but they are important for safety-critical applications.

A clear example of a GPS signal broken up in an auroral arc has already been identified

Relating the physics to the impacts on systems is important, especially in new technologies like GPS and the new European Galileo.

Research applications

36/38

More examples of tomography

• Oil exploration

• Mapping sea temperatures

• Imaging of human brain

• Bone structure in race horses

• Atmospheric water vapour

37/38

“From oil pipelines to pork pies, electromagnetic tomography is providing new ways to look inside industrial processes”http://physicsweb.org/articles/world/16/6/8

“oil, gas and water flowing in a pipeline from the North Sea can be imaged”

“A uniform and consistent product is often essential for commercial success, and images from the inside of mixing vessels can determine when the contents are of even quality. On food production lines, for example, defects in fruit can be automatically detected”

Other areas of tomography

38/38

Questions?