EISCAT Radar Summer School 15 th -26 th August 2005 Kiruna Calculation of the plasma-velocity vector Vikki Howells Rutherford Appleton Laboratory, UK
EISCAT Radar Summer School 15th-26th August 2005
Kiruna
Calculation of the plasma-velocity vector
Vikki Howells
Rutherford Appleton Laboratory, UK
• 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
Introduction
Plasma velocity measurements
Line-of-sight velocitye.g. for a northward-pointing beam
B
Vlos
VN
V| |
aspect angle,
V
Vlos = V| | cos + VN sin
c
f2VΔf los
• 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
Methods of measuring plasma velocity
Tristatic Methode.g. for CP1
Sodankylä
Tromsø
Kiruna
VT
VKVS
Tristatic Method
This is not how to combine tristatic velocities…
The remote site do not measure line-of-site velocity….
Bistatic measurements of velocity
Scattering geometry of bistatic incoherent scatter radar.
Measure the “mirror velocity” Vm from the Doppler shiftVp
Vm
Bragg wavelength λ/(2cosΧ/2)
Χ
Incident signal Scattered signal
c
2χfcos2VΔf m
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)
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
Beamswinging Method
Used to combine CP4 velocities.
Only have two measurements, so we assume V| | =0
We then have one measurement of VN and we can calculate VE
Geographic North
Magnetic North
Methods of calculating the plasma velocity vector
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 (VE, VN , V| | )
Z
N
E
S
K
T
V
V
V
C
V
V
V
c
fVf m2
Local to geocentric
Convert radar positions to geocentric coordinates using the transformation matrix Rlg
sin 0 cos
sincos cos sinsin
coscos sin- cossin
R
sin sincos coscos
0 cos sin-
cos- sinsin cossin
R
lg
gl
Matrices for geocentric to local (Rgl) and local to geocentric (Rlg) transformations
θ = geographic latitude and Φ = longitude
sinrQT
sincosrQT
coscosrQT
z
y
x
Azimuth, elevation to geocentric
ε = elevation and α = azimuth
Convert az, el, height to geocentric coordinates
For a given scattering point Q, the vector
Z
N
E
S
K
T
V
V
V
C
V
V
V
c
fVf m2
Geographic to Geomagnetic coordinates
z
N
E
| |
N
E
z
N
E
| |
N
E
V
V
V
sinI cosDcosI- sinDcosI-
cosI cosDsinI sinDsinI
0 sinD- cosD
V
V
V
V
V
V
B
V
V
V
Need to use a magnetic field model (IGRF 2005)
D = Dip angle
I = Inclination
IGRF Model
z
N
E
| |
N
E
V
V
V
B
V
V
V
Z
N
E
S
K
T
V
V
V
C
V
V
V
1
z
N
E
| |
N
E
CBM
where
V
V
V
M
V
V
V
Matrix Inversion
S
K
T
S||K||T||
NSNKNT
ESEKET
| |
N
E
V
V
V
M M M
M M M
M M M
V
V
V
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
z
N
E
| |
N
E
V
V
V
sinI cosDcosI- sinDcosI-
cosI cosDsinI sinDsinI
0 sinD- cosD
V
V
V
IsinVIcosDcosVIcosDsinVV
IcosVIsinDcosVIsinDsinVV
0DsinVDcosVV
zNE||
zNEN
NEE
D = Dip angle
I = Inclination
c
fVf m2
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
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
North
East
Field aligned
Beam swinging
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
Geographic North
Magnetic North
Calculating Uncertainties
• For matrix inversion method (most commonly used):
2S
2K
2T
2||
2N
2E
ΔV
ΔV
ΔV
m
ΔV
ΔV
ΔV
Calculation of uncertainties
Here every element of the matrix m is the square of the corresponding matrix M
2S
2K
2T
2S||
2K||
2T||
2NS
2NK
2NT
2ES
2EK
2ET
2||
2N
2E
ΔV
ΔV
ΔV
M M M
M M M
M M M
ΔV
ΔV
ΔV
c
fVf m2
Map of uncertainties
c
fVf m2
Map of uncertainties
c
fVf m2
Map of uncertainties
Problems with each 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
Tristatic 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)
Monostatic Method – Problems
• Assume V||=0
• This is not always the case• Assuming that the plasma velocity does not
change over 100s of km
CP4 Beamswing Method – Problems
The RAL velcom program
• 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
Velcom