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 15th-26th August 2005 Kiruna

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

Plasma velocity measurements

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

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

Tristatic Method e.g. for CP1

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

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

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. 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

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 (VE, VN , V| | )

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

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

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

IGRF Model

Matrix Inversion

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

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 Field aligned North East

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

Calculating Uncertainties

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

Map of uncertainties

Map of uncertainties

Map of uncertainties

Problems with each method

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

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)

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

The RAL velcom program

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