1. EISCAT Radar Summer School 15th-26th August 2005 Kiruna
Calculation of the plasma-velocity vector Vikki Howells Rutherford Appleton Laboratory, UK
EISCAT Radar Summer School th-26th August 2005
Kiruna

2. Introduction Plasma velocity measurements using EISCAT
Calculation of plasma velocity vector, vp
Matrix Inversion
Least Squares Fit
CP4
Calculation of uncertainties
Strengths and weaknesses of each method
The RAL velcom program

3. Plasma velocity measurements

4. Line-of-sight velocity
e.g. for a northward-pointing beam
aspect angle, 
Vlos V
V| |
Vlos = V| | cos + VN sin 
VN B

5. Methods of measuring plasma velocity
Tristatic Method
Used to combine measurements from all three stations to give true estimates of the plasma velocity for a single scattering volume
Monostatic Method
Used to estimate a plasma velocity averaged over the three scattering volumes
Beamswinging Technique
Used to combine two velocity measurements
Mainly used for CP4-type modes.
Assumes V| | = 0

6. Tristatic Method
e.g. for CP1

7. Tristatic Method VK VS
This is not how to combine tristatic velocities…
The remote site do not measure line-of-site velocity…. VT
Tromsø
Sodankylä
Kiruna

8. Bistatic measurements of velocity
Scattering geometry of bistatic incoherent scatter radar.
Measure the “mirror velocity” Vm from the Doppler shift
Bragg wavelength
λ/(2cosΧ/2) Vp Χ
Incident signal
Scattered signal Vm

9. Tristatic measurement of plasma velocity
Velocities are measured simultaneously and have a common volume
Common volume is not fixed (i.e. you can point to where you like)

10. Monostatic Method
Tromsø
Total vector velocity is estimated by pointing the antenna in at least three different directions and measuring a component of velocity in each direction.
Commonly used technique at other monostatic IS radars

11. Beamswinging Method Used to combine CP4 velocities.
Geographic North
Magnetic North
Used to combine CP4 velocities.
Only have two measurements, so we assume V| | =0
We then have one measurement of VN and we can calculate VE

12. Methods of calculating the plasma velocity vector

13. Matrix Inversion Most common method
From the three components VT, VK, VS can be obtained the plasma velocity vector VP.
It’s components may be computed either in the geometric coordinate system (Geographic East, North and vertically upward) or, more usefully, in geomagnetic coordinates (VE, VN , V| | )

14. Local to geocentric
Convert radar positions to geocentric coordinates using the transformation matrix Rlg
Matrices for geocentric to local (Rgl) and local to geocentric (Rlg) transformations
θ = geographic latitude and Φ = longitude

15. Azimuth, elevation to geocentric
Convert az, el, height to geocentric coordinates
For a given scattering point Q, the vector
ε = elevation and α = azimuth

16. Geographic to Geomagnetic coordinates
Need to use a magnetic field model (IGRF 2005)
D = Dip angle
I = Inclination

17. IGRF Model

18. Matrix Inversion

19. Least Squares Fit
Instead of describing the set of simultaneous equations as a matrix, they can be written explicitly
For example:
Can be rewritten as
D = Dip angle
I = Inclination

20. Least Squares Fit
These set of equations may then be calculated by computing the minimum solution to a real linear least squares problem:
(|b-A*x|)
using the singular value decomposition (SVD) of A.
A is an M-by-N matrix which may be rank-deficient.
If A is a 3 x 3 array (like the matrix inversion), we will get exactly the same results using the least-squares fit and the matrix inversion method

21. CP2 CP2 pointing directions:
Find the common altitude at all three (or more than three) beams.
Assume that the plasma velocity varies little with time relative to the scan time of the radar
Overdetermined simultaneous equations
Can use all four pointing directions if we use a least-squares fit instead of a matrix inversion Up
Field aligned
North
East

22. Beam swinging Only have two beams
Geographic North
Magnetic North
Only have two beams
Work out the invariant latitude and calculate a common L-shell
IGRF model used to calculate the L-shells
At Tromsø, the west beam points BN, giving v┴N
Assume that v||=0
Can then calculate v┴E from the east beam
Van Eyken et. al JATP vol. 46, No. 6/7, 1984

23. Calculating Uncertainties

24. Calculation of uncertainties
For matrix inversion method (most commonly used):
Here every element of the matrix m is the square of the corresponding matrix M

25. Map of uncertainties

26. Map of uncertainties

27. Map of uncertainties

28. Problems with each method

29. Tristatic Method - Problems
At low elevations, the pointing positions become close to parallel
No longer have 3 orthogonal, independent measurements of Vp.
End up with singular matrix (which can’t be inverted)
Random errors can be large because they are a combination of random errors from all three sites

30. Monostatic Method – Problems
Systematic errors can be introduced due to horizontal gradients in the plasma velocity
Time resolution not as good as tristatic method
Assumes that the plasma velocity is constant over large distances and periods of tens of minutes
(Williams et. al 1984, JATP 47, 6/7 p521)

31. CP4 Beamswing Method – Problems
Assume V||=0
This is not always the case
Assuming that the plasma velocity does not change over 100s of km

32. The RAL velcom program

33. Velcom Calculates plasma velocity vectors using all the above methods
Can also be used for non-EISCAT data
Can be used for mainland and ESR data
But..
Uses RAL NCAR format data