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