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The Mars Lidar Simulation Model (MLSM) Multi-Paired Algorithm (MPA) |
Multi-Paired Algorithm (MPA)
The MPA matches forward and aft lidar line-of-sight wind velocities to compute weighted horizontal wind components. The MPA weights the winds by angular separation from orthogonality and by the distance between the lidar shots ( Emmitt and Wood, 1988) as shown below.
awt = 1.0 - ((p/2 - (ßa - ßf)) * (2/p))4
Dwt = 1-(((fa × Qc × COS(Qa) - ff × Qc × COS(Qf))2
+ ((Qa + 90) × Qc - (Qf + 90) × Qc)2)½)/D
where
awt - angle shot pair weight,
Dwt - distance shot pair weight,
Qc - degrees to kilometers conversion factor at equator (km/deg),
D - diagonal distance across the grid area (km),
fa - longitude of the aft shot (deg),
ff - longitude of the forward shot (deg),
Qa - latitude of the aft shot (deg),
Qf - latitude of the forward shot (deg).
The U and V horizontal wind components are computed, respectively, as follows:
U = ((VLOSa / (COS(Q) × COS(ßa)) - (VLOSf × TAN(ßa))) /
(COS(Q) × SIN(ßf))/(1.0 - TAN(ßa)/TAN(ßf))
V = (VLOSa - U × COS(Q) × COS(ßa))/(COS(Q) × SIN(ßa))
where
U - U horizontal wind component (m/s)
V - V horizontal wind component (m/s)
VLOSa - line-of-sight lidar wind for aft shot (m/s)
VLOSf - line-of-sight lidar wind for forward shot (m/s)
Q
- elevation angle (deg)ßa - scanner angle for aft shot (deg)
ßf - scanner angle for forward shot (deg).
An analytical expression for the MPA errors dependent upon shot separation was derived. The line-of-sight (LOS) velocities for two shots from different perspectives can be expressed as follows:
VLOS1 = (U1 cosF1 + V1 sinF1) × sinQ + W1 cosQ + N1
VLOS2 = (U2 cosF2 + V2 sinQ2) × sinQ + W2 cosQ + N2
where:
VLOSi - line-of-sight velocity for shot i (m/s)
Ui - u component of the wind at location i (m/s)
Vi - v component of the wind at location i (m/s)
Wi - w component of the wind at location i (m/s)
Ni - random noise
Fi - azimuth for shot i from mathematic +x (deg)
Q - elevation angle from the nadir (deg)
Given the scarcity of shots, the following assumptions are made to solve for the horizontal wind components:
u1 = u2; v1 = v2; w1 = w2 = 0.0; F2 = -F1; N1 = N2 = 0
Thus:
VLOS1 = (U × cosF1 + V sinF2) × sinQ
VLOS2 = (U cosF2 + V sinF2) × sinQ
To get two equations in two unknowns (u,v) we substitute Q1 for Q2:
VLOS1 = (u cosF1 + v sinF1) × sinQ
VLOS2 = (u cosF1 - v sinF1) × sinQ
Solving for u:
VLOS1 + VLOS2 = 2 u cosF1 × sinQ
u = (VLOS1 + VLOS)/(2 cosF1 sinQ)
Solving for v:
VLOS1 - VLOS2 = 2 v sinF1 × sinQ
v = (VLOS1 - VLOS2)/(2 sinF1 × sinQ)
However, if U2,U1 and V2, V1 then we make the following substitution:
U2 = U1 + u
V2 = V1 + v
VLOS2 = (u cosF2 + v sinF2) × sinQ + (u cosQ2 + v sinF2) × sinQ
= (u cosF - v sinF) × sinQ + (u cosF - v sinF) sinQ
Solving for u and v:
u = (VLOS1 + VLOS2)/(2 cosF sinQ) + (du/2 - (dv tanF)/2
v = (VLOS1 - VLOS2)/(2 sinF sinQ) - (du)/(2tanF) + (du)/2
The correct u and v values would be u + ½ u and v + ½ v. Therefore, the errors due to having different horizontal velocities at shot locations 1 and 2 are:
Uerror = - dv tanF/2
Verror = - du/2 tanF
If u and v are statistically non-zero (i.e., related to some coherent structure) then there will be a residual error regardless of the number of shots used in the velocity estimates. We can compute the average u and v errors for a given shot pair spacing.
For example (F1 = 301o):
du/dx = 10-5 s-1
u = du/dx ×x
where x is x(u2) - x(u1))
Let x = 50 km and u = .5 m/s
uerror = 0
verror = .43 m s-1
Last Updated: 02/07/2007