Table Of ContentNASA Contractor Report 4763
Evaluation of the MV (CAPON) Coherent
Doppler Lidar Velocity Estimator
B. Lottman and R. Frehlich
Cooperative Institute for Research in Environmental Sciences (CIRES)
University of Colorado ° Boulder, Colorado
National Aeronautics and Space Administration Prepared for Marshall Space Flight Center
Marshall Space Flight Center • MSFC, Alabama 35812 under Grant NAG8-253
January 1997
Contents
1 Introduction 1
2 Typical Parameter Regimes 1
3 Velocity Estimator Performance 3
4 Potential Uses 4
5 Computational Effort 4
6 Results M--10 5
7 Results M--16 9
8 Results M--32 10
9 Results M--64 12
10 Results M_-128 13
11 Results M----512 14
12 Acknowledgements 16
Appendix A, M----16 17
Appendix B_ M_-32 21
Appendix C, M--64 27
Appendix D, M----128 32
Appendix E, M--512 37
iii
List of Tables
Table 1. Parameter regime for a typical 2#m and 10#m LAWS type lidar. 2
Table 2. Parameter regime for a typical 2#m and 10#m boundary layer lidar. 2
Table 3. Parameter regime for typical 2#m and 10#m LAWS type lidar observing a
thick cloud. 3
Table 4. Parameter regimes evaluated in this paper. 3
Table 5. Computational effort for the CAPON Estimator based on estimator order
p, oversampling os, and the number of data points in the range cell M. 4
Table 6. Model Fit Parameters for M--10, i2--0.2 and 0.3. 8
Table 7. Model Fit Parameters for M--16, 1_--0.2, 0.3, 0.5 and 1.0. 10
Table 8. Model Fit Parameters for M----32, _----0.2, 0.3, 0.5, 1.0, 2.0, and 3.0. 11
Table 9. Model Fit Parameters for M--64, _--0.3, 0.5, 1.0, 2.0, and 3.0. 13
Table 10. Model Fit Parameters for M--128, _--0.5, 1.0, 2.0, 3.0, and 7.0. 14
Table 11. Model Fit Parameters for M--512, _----7.0 and 10.0. 15
List of Figures
Figure 1. Estimator performance as a function of oversampling os and order p for
M--32 and _--0.5. 3
Figure 2. Estimator performance vs. computational effort for M--512 and _=7.0. 5
Figure 3. Performance vs. oversampling os for M=10 and i2=0.2. 6
Figure 4. Performance vs. oversampling os for M=10 and Q=0.3. 6
Figure 5. Performance vs. order p for the CAPON estimator for M--10 and D----0.2. 6
Figure 6. Performance vs. order p for the CAPON Estimator for M--10 and _--0.3. 6
Figure 7. Single realizations of the CAPON estimates for M -- 10 and _------0.2 for
_=1,10,50,1000 with optimal estimator parameters. 7
Figure 8. Single realizations of the CAPON estimates for M----10 and Q=0.3 for
(I)=1,10,50,1000 with optimal estimator parameters. 7'
4
Figure 9. PDF of the Velocity Estimates using the CAPON estimator with M--10
and Q=0.2 for ff=1,10,50,1000. 7
iv
Figure 10. PDF of the Velocity Estimates using the CAPON estimator with M=10
and f_----0.3 for for q_--1,10,50,1000.
Figure 11. Performance of the CAPON estimator with M----10 and _----0.2 and 0.3
using optimal estimator parameters. Best fit model is (-). 8
Figure 12. Optimal parameters for the CAPON estimator with 64----10 and _--0.2
and 0.3.
Figure 13. Minimum estimator order p with M=10 and _----0.2 and 0.3 for no
greater than 5the good estimates.
Figure 14. Performance of the CAPON estimator with M--16 and _:0.2, 0.3, 0.5,
and 1.0 using optimal estimator parameters. 9
Figure 15. Optimal parameters for the CAPON estimator with M:16 and _--0.2,
0.3, 0.5, and 1.0. 9
Figure 16. Minimum estimator order p with M--16 and f_--0.2, 0.3, 0.5, and 1.0 for
no greater than 5% and 10% increase in the standard deviation g of the good
estimates. 9
Figure 17. Performance of the CAPON estimator with M----32 and _2=0.2, 0.3, 0.5,
1.0, 2.0, and 3.0 using optimal estimator parameters. 10
Figure 18. Optimal parameters for the CAPON estimator with M--32 and 12=0.2,
0.3, 0.5, 1.0, 2.0, and 3.0. 11
Figure 19. Minimum estimator order p with M--32 and _----0.2, 0.3, 0.5, 1.0, 2.0
and
3.0 for no greater than 5% and 10% increase in the standard deviation g of
the good estimates. 11
Figure 20. Performance of the CAPON estimator with M_64 and 1_----0.3, 0.5, 1.0,
2.0, and 3.0 using optimal estimator parameters. 12
Figure 21. Optimal parameters for the CAPON estimator with M--64 and 1_--0.3,
0.5, 1.0, 2.0, and 3.0. 12
Figure 22. Minimum estimator order p with M_64 and _--0.3, 0.5, 1.0, 2.0 and
3.0 for no greater than 5% and 10% increase in the standard deviation .q of the
good estimates. 12
Figure 23. Performance of the CAPON estimator with M----128 and _----0.5, 1.0, 2.0,
3.0, and 7.0 using optimal estimator parameters. 13
Figure 24. Optimal parameters for the CAPON estimator with M----128 and _----0.5,
1.0, 2.0, 3.0, and 7.0. 14
Figure 25. Minimum estimator order p with M----128 and _=0.5, 1.0, 2.0, 3.0 and
7.0 for no greater than 5_0 and 10% increase in the standard deviation g of the
good estimates. 14
Figure 26. Performance of the CAPON estimator with M--512 and _=7.0 and 10.0
using optimal estimator parameters. 15
Figure 27. Optimal parameters for the CAPON estimator with M=512 and _--7.0
and 10.0. 15
Figure 28. Minimum estimator order p with M=512 and _--7.0 and 10.0 for no
greater than 5% and 10 % increase in the standard deviation g of the good
estimates. 15
vi
1 Introduction
space-based measurements of winds. Assuming
and ft are known, we evaluate the perfor-
mance of the CAPON estimator where the auto-
The first coherent Doppler lidars were based on
the C02 laser [1, 2, 3, 4]. More recently, solid regressive coefficients up to order p are deter-
mined from the biased covariance estimates and
state Doppler lidars have been successfully op-
erated [5, 6, 7]. Coherent Doppler lidar is un- the Yule-Walker solution [10, 12, 13]. The op-
der consideration for measurements of the global timal order p and the optimal sampling interval
for estimating the maximum of the spectral esti-
wind field from space [[3, 8, 9]]. A coherent
Doppler lidar operating at wavelength Acan es- mate S(f) for the CAPON estimator are deter-
timate the radial component of the velocity v -- mined by minimizing the standard deviation of
._f/2 as a function of range for every transmitted the good estimates. Various parameter regimes
will be considered.
pulse by estimating the Doppler frequency shift
f. The Probability Density Function (PDF) pro-
vides a complete statistical description of an es- 2 Typical Parameter Regimes
timator. For the better velocity estimators, the
PDF is characterized by a fraction bof uniformly A variety of parameter regimes relevant to space-
distributed bad estimates or random outliers and based measurements are evaluated in this paper.
a distribution of good estimates centered on the The basic space-based parameters were deter-
true mean velocity. Frehlich and Yadlowsky[10] mined by the following criteria;
investigated the performance of mean-frequency 1)A velocity search space vs of 50 rn/s was se-
estimators by modeling the distribution of good lected. The sampling interval T_ for complex
estimates as a Gaussian PDF with standard de- data is given by T_ = ,k/(2vs). The sampling
viation g. Approximately universal curves of interval for 2 and 10 #m lidar is 20 and 100 ns
performance of velocity estimates (and mean- respectively.
frequency estimates) are produced by plotting 2)For numerical weather prediction, current
the standard deviation g normalized by the sig- models use a 1 km height resolution. This im-
nal spectral width in velocity space wv = Aw/2 plies a 1.5 km range cell (line of sight) for typical
versus the parameter (I) -- SNR. M for fixed satellite scanning geometries. The range resolu-
.Q= wMTs where w is the signal spectral width, tion is Ap+Ar where Ap isthe distance the pulse
Ts is the sampling interval of the complex data, moves per velocity estimate, i.e., Ap = MT_c/2
SNR is the Signal-to-Noise Ratio. and M is the where c [m/s] is the speed of light. Ar [m] is
number of complex data points per observation. the full width half max (FWHM) spatial extent
is fixed for fixed range resolution and fixed of the pulse, i.e. Ar = Atc/2 where At Is] is
transmitted pulse length, the typical comparison the FWHM temporal extent of the pulse. A Ap
case for lidar performance. For shot-noise domi- of 1500m was selected. Since M = 2Ap/(Tsc),
nated operation, the parameter • = riHN ,,where M = 500 and 100 for 2 and 10 pm lidars re-
7JHis the heterodyne efficiency (rlu _ 0.4 in the spectively. For a large Ap the spectral width
far field. The far field condition is usually valid w of the returned signal is usually dominated
for space-based platforms.) and N, is the aver- by wind turbulence. The lidar signal is approx-
age number of photo-electrons per estimate, q_is imated as a Gaussian with a spectral width in
the average number of coherent photo-electrons velocity space w,, of 1m/s and since w = 2wv/A,
per estimate. The parameter ft is proportional w is 1 and 0.2 MHz for a 2 and 10 pm lidar re-
to the number of independent samples of the sig- spectively, ft = wMTs where _t is proportional
nal per range gate, sometimes called the %peckle to the speckle count. _ = 10.0 and 2.0 for a 2
count". and 10 t_rn lidar respectively.
Anderson [11] proposed the Minimum Vari- Examples of parameter regimes for a 2pro and
ance (MV) or CAPON spectral estimator for 10#m LAWS type lidar are shown in Table 1.
spectral width is determined by the transmitted
2 20 1.0 1536 512 10.2
pulse.
2 20 1.0 768 256 5.1
The basic parameter set for thick cloud obser-
2 20 1.0 384 128 2.5
vation using a LAWS type system is determined
10 100 0.2 1920 128 2.5
as follows;
10 100 0.2 960 64 1.3
1)Sampling interval is selected based on a veloc-
10 100 0.2 480 32 0.6
ity search space of 50 rn/s and the lidar wave-
Table 1. Parameter regime for a typical 2#m length A. The sampling interval 7'8 for 2 and 10
and 10#m LAWS type lidar. Ais in #m. Ts is in /_rn lidar is 20 and 100 ns respectively.
ns. w is in MHz. Ap is in m. 2)The observation time MTs is determined as a
fraction of the FWHM of the transmitted pulse
Typical lidar parameters for boundary layer
At. MTs = kAt; where k is a user defined con-
measurements are also of interest. The basic pa-
stant, which is related to _t by _ = k_/_c
rameters for the typical boundary layer lidar are
or fl=.1873 k. At determines M.
based on the following;
1)50 m/s was selected for the velocity search
_At
space. 2 and 10/_rn was selected for lidar wave-
20 0.30 48 16 0.2
length A. The sampling time Ts for complex data 2
2 20 0.20 48 16 0.3
for a 2 and 10 #m lidar is 20 and 100 ns respec-
2 20 0.12 48 16 0.5
tively. 2 20 0.06 48 16 1.0
2)The boundary layer is typically 1000 m in
20 0.60 96 32 0.2
depth. To characterize activity within the layer, 2
2 20 0.40 96 32 0.3
measurements are usually taken at least every
2 20 0.24 96 32 0.5
100 m. A range cell length Ap of 48m was se-
2 20 0.12 96 32 1.0
lected and M = --2_c, or M = 16. 2 20 0.06 96 32 2.0
For a small Ap the spectral width w of the 2 20 0.04 96 32 3.0
returned signal is determined by the transmit- 2 20 0.80 192 64 0.3
t(;d pulse (pulse dominated) and not wind tur- 2 20 0.48 192 64 0.5
bulence. For a 2 #m example, the FWHM 2 20 0.24 192 64 1.0
pulse width At is 0.3 ps. The spectral width 2 20 0.12 192 64 2.0
w (pulse dominated) is w = v/-_2/2/rAt or 2 20 0.08 192 64 3.0
w = 0.1873/At and f_ = wMTs or l] = 0.2. 10 100 0.94 150 10 0.2
100 0.62 150 10 0.3
For very large spectral width w, the velocity 10
100 1.50 240 16 0.2
search space becomes comparible with w. In this 10
100 1.00 240 16 0.3
regime the PDF's do not have a uniform layer 10
of outliers and the two parameter model is not 10 100 0.60 240 16 0.5
100 0.30 240 16 1.0
valid. The upper bound oil f_ is when the 16a 10
spectral width of the signal approachs the full Table 2. Parameter regime for a typical 2#m
velocity search space. For example, for M = and 10#m boundary layer lidar. _ is in m. T, is
16, 16w < 1/Ts, or wTs <_ 1/16_ therefore f_ = in ns. At is in/_s. Ap is in m.
wMT_ < M/16, or f_ _<1.
This regime is similar to the high SNR small
Examples of typical parameter regimes for the range gate scenario for boundary layer measure-
2#m and 10#m boundary layer lidar are shown ments. The parameter sets for the thick cloud
ill Table 2. target with various transmitted pulse widths:
For a LAWS system, observation of thick for 2#m (T._=2Ons) and 10pro (Ts=lOOns) ar_
clouds isof interest. Thick clouds generate a sig- shown in Table 3.
nal sinfilar to a hard target return and the signal Tables 1 through 3 give the general regimes of
interest. The parameter sets evaluated in this timator accuracy. In velocity space, the sam-
paper are shown in Table 4. pling interval Av = v_/(M os) where os is the
oversampling, os = 1 means that the spectral
2 prn 10 #m function is sampled with 2_4 data points. The
At k M _ At k M gt maximum value of the spectral estimate and the
0.20 1.0 10 0.2 1.00 1.0 10 0.2 two closest neighbors are fit to a parabola. The
0.13 1.6 10 0.3 0.63 1.6 l0 0.3 velocity that produces the peak of the parabola
0.32 1.0 16 0.2 1.60 1.0 16 0.2 is selected as the estimate. The value of x that
0.20 1.6 16 0.3 1.00 1.6 16 0.3 maximizes f(z) for mfiformly sampled Xk is
(}.11 2.7 16 0.5 0.59 2.7 16 0.5 f(x2) - f(xo)
0.64 _1.0 32 0.2 3.20 1.0 32 0.2 Xpeak = Xl + 2(2f(xl) -- f(Xo) -- f(x2))
0.40 1.6 32 0.3 2.00 1.6 32 0.3 where x0 is the index of the spectral function to
(}.24 2.7 32 0.5 1.18 2.7 32 0.5 the left of the peak, Xl is the index of the spec-
O.8O 1.6 64 0.3 2.37 2.7 64 0.5 tral function at the peak, and x2 is the index of
{).47 2.7 64 0.5 the spectral function to the right of the peak.
O.95 2.7 128 0.5 Generally, performance is less sensitive to
Table 3. Parameter regime for typical 2#m and oversampling than estimator order. Examples of
10/m_, LAWS type lidar observing a thick cloud. oversampling and order sensitivity for M = 32
At is in p._. and f_ = 0.5 are shown in Fig. 1. Performance is
normalized by the minimum standard deviation
of the good estimates for each regime gm,n.
M o2 0.3 05 10 2.0 3.0 7.0 100
l l(J x X 1.50
I I I I I
16 X X X X
1.40 o _=1 p=16
32 X X X X X X o @=10 p=12
1.30 = _ _=50 p=6
64 X X X X X A _=1000 p=2
j1.20
128 X X X X X X
512 X X 1.10
Table 4. Parameter regimes evaluated in this 1.00
l)al)er. 0.90 I I I I I
1 2 3 4 5
OS
1.50
3 Velocity Estimator Perfor-
1.40 o @=1 OS=2
mance
o @=10 OS=2
1.30 * @=50 OS=2
.E
Sinmlations of complex coherent Doppler lidar E1.20
data were l)roduced by the method described 1.10
in Frehlich and Yadlowsky[10]. This produces
1.00
data with a specified auto-covariance function
I i I i i ] I i I i i
which is chosen as a Gaussian function defined 0.90(_ 5 10 15 20
P
by the parameters ¢P, [_, and M [see Eq. (22) of
Figure 1. Estimator i)erformance as a function
Ref.[10]]. This is a good approximation for 2#m
coherent lidars[14]. of oversampling os and order p for M=32 an(l
ft=0.5. Curves are offset by 0.10. The la error
For the CAPON estimator, performance is a
function of both estimator order and oversam- bars are less than the symbol size.
piing. Oversampling refers to saint)ling the sl)ec- The technique for determining the optimal es-
tral fimction at smaller intervals to improw,, es- timator parameters (p, o._') for given basic pa-
rameters(M. fL _) 5 Computational Effort
is to first determine opti-
mal oversampling os with an approximate opti-
mal order, then determine optimal order p us- Computational effort for complex data is shown
in Table 5. Flops are complex floating point
ing the oversampling result os. Parameters that
operations[15].
give the best performance within the statistical
limitations of the simulation are considered the
I Multiplications (flops)
optimal parameters.
Autocorrelation (2M - p)(1 + p)/2
For large M with large 12, estimator accuracy AR coefficients
does not improve with oversampling (os = 1pro-
Yule Walker p2 + p
duces the best performance). For the M = 512
CMV coefficients p(p + 1)/2
fl = 7.0 and 10.0 cases order sensitivity is com-
FFT M osLog2(M os)/2
puted but oversampling sensitivity is not com-
Total(Approx) (M + p)(1 + p)
puted.
+M os Log2(M os)/2
The results of each parameter set include: per-
formance as a function of oversampling and or- Additions (flops)
der, single realizations of the CAPON spectral
Autocorrelation (2M - p)(1 + p)/2 + M
estimates, and PDF's for the velocity estimators
AR coefficients
for a subset of cases using optimal estimator pa-
Yule Walker p2 + p
rameters.
CMV coefficients p(p + 1)/2
FFT M osLog2(M os)
Total(Approx) (M + p)(1 + p) + M
+M os Log2(M os)
4 Potential Uses
Table 5. Computational effort based on estima-
tor order p, oversampling os, and the number of
The standard deviation of the good estimates data points in the range cell M. flops are com-
plex floating point operations.
g and the fraction b of uniformly distributed
bad estimates can be used to generate simu- For a typical 2pro LAWS regime (M = 512,
lated velocity data for a given parameter regime. -- 10.0), the maximum optimal order p is
To generate velocity data, velocity values are about 12 and optimal oversampling o_ is 1. Thus
drawn from one of two random number gener- the dominant terms for computational effort are
ators. The good estimates are represented by a (2M - p)(1 + p)/2 + MLog2(M)/2 for complex
Gaussian random variable with standard devia-
multiplications and (2M - p)(1 + p)/2 + M +
tion g, while the bad estimates are represented
MLog2(M ) for complex additions.
by a uniform random variable over the velocity Computational effort (total of complex mul-
search space vs. Selection of a good or bad es- tiplications and additions) for a typical LAWS
timate is based on the parameter b, the fraction
regime (M=512, _t=7.0) versus estimator per-
of uniformly distributed bad estimates. formance is shown in Fig. 2. The solid lines rep-
For a LAWS type mission, the final data prod- resent the total computational effort whereas the
uct will be generated from multiple shots in a dashed lines are the computational effort for the
resolution cell with a specified scanning configu- FFT portion of the estimation routine. For this
ration e.g. a conical scan[9] . The desired data case, computational effort is not dominated by
product is the average wind vector over the res- the FFT portion of the estimation routine.
olution cell in 3D. The PDF of any multishot Since computational effort for the CAPON es-
velocity algorithm can be derived from the sin- timator depends oil order p, a reduced order re-
gle shot PDF's using a Monte Carlo analysis. duced performance (reduced from optimal) esti-
