
Citation 
 Permanent Link:
 http://digital.auraria.edu/AA00002140/00001
Material Information
 Title:
 Spatial filtering technique for remote measurement of wind velocity in turbulent atmosphere
 Creator:
 Vu, John Giang Truong
 Publication Date:
 1989
 Language:
 English
 Physical Description:
 62 leaves : illustrations ; 29 cm
Subjects
 Subjects / Keywords:
 Winds  Speed  Measurement ( lcsh )
Atmospheric turbulence  Measurement ( lcsh ) Atmospheric turbulence  Measurement ( fast ) Winds  Speed  Measurement ( fast )
 Genre:
 bibliography ( marcgt )
theses ( marcgt ) nonfiction ( marcgt )
Notes
 Bibliography:
 Includes bibliographical references (leaves 6162).
 General Note:
 Submitted in partial fulfillment of the requirements for the degree, Master of Science, Department of Electrical Engineering, Department of Computer Science and Engineering.
 Statement of Responsibility:
 by John Giang Truong Vu.
Record Information
 Source Institution:
 University of Colorado Denver
 Holding Location:
 Auraria Library
 Rights Management:
 All applicable rights reserved by the source institution and holding location.
 Resource Identifier:
 20960288 ( OCLC )
ocm20960288
 Classification:
 LD1190.E54 1989m .V8 ( lcc )

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Full Text 
SPATIAL FILTERING TECHNIQUE FOR REMOTE MEASUREMENT
OF WIND VELOCITY IN TURBULENT ATMOSPHERE
by
John Giang Truong Vu
B.S., University of Saigon, 1972
M.S., Roosevelt University, 1980
A thesis submitted to the
Faculty of the Graduate School of the
University of Colorado in partial fulfillment
of the requirements for the degree of
Master of Science
Department of
Electrical Engineering and Computer Science
1989
This thesis for the Master of Science degree by
John Giang Truong Vu
has been approved for the
Department of
Electrical Engineering and Computer Science
by
Arun K. Majumdar
'fL
John R. Clark
J.
Edward T. Wall
Date
ACKNOWLEDGEMENTS
I would like to thank Dr. Arun K. Majumdar for his
technical assistance and especially, for his support in
obtaining essential equipment for this work.
i v
Vu, John Giang Truong (M.S., Electrical Engineering)
Spatial Filtering Technique for Remote Measurement of Wind
Velocity in Turbulent Atmosphere
Thesis directed by Professor Arun K. Majumdar
Spatial filters applied on transmitting and receiving
apertures are used in lineofsight measurement of
transverse flow velocity of the wind in turbulent
transmission medium, exhibiting small random fluctuations in
its refractive index which is produced by irregularities
across the probe wave propagation path, provided that the
weak scattering approximation is appropriate.
The technique uses spatial filtering of scintillation
from a spatially filtered incoherent optical scource. A
simulated disturbance of the transmission medium was
generated in the laboratory, and data were recorded and
processed through a personal computer.
The form and content of this abstract are approved. I
recommend its publication.
/t >
Signed
Faculty member in charge of thesis
V
CONTENTS
CHAPTER
1. INTRODUCTION....................................1
2. THEORY OF SPATIAL FILTER OPERATION..............3
3. EFFECT OF SPATIALLY FILTERED APERTURE...........7
3.1 Spatial filter at the receiver terminal....7
3.2 Spatial filter at both termini............1 2
4. WEIGHTING FUNCTION.............................17
4.1 Path weighting function................... 17
4.2 Wave number weighting function............23
5. EXPERIMENT.....................................28
5.1 Spatial filters............................28
5.2 Transmitter source....................... 28
5.3 Turbulence source.........................3 0
5.4 Receiver system.......................... 31
5.5 Recording and Processing Instruments......31
5.6 Procedure.................................33
6. ANALYSIS AND CONCLUSIONS.......................4 5
6.1 Analysis.................................. 45
6.2 Conclusions................................60
BIBLIOGRAPHY.
61
VI
FIGURES
FIGURE
3.1 Sinusoidal turbulence at the receiving plane.......8
3.2 Two transmitters configuration......................10
3.3 Zerosum receiving aperture.........................11
3.4 Zerosum spatial filter function and the
corresponding approximation as a function
of the coordinate x for a circular aperture
of radius r........................................1 3
3.5 Geometry of two filters at both termini.............14
3.6 Nonzerosum spatial filter function and the
corresponding approximation as a function
of the coordinate x for a circular aperture
of radius r........................................1 6
4.1 Normalized path weighting as a function of
the normalized path position for 6 cycles
transmitter and 2 cycles receiver filters..21
4.2 Normalized path weighting as a function of
the normalized path position for transmitter
and receiver with 6 cycles filters.................22
4.3 Normalized path weighting as a function of
the normalized path position for 6 cycles
transmitter and 18 Cycles receiver.........22
4.4 Power spectral density of the receiver filter
VII
as function of wave number (2pi/2.54 cm)...........26
4.5 Power spectral density of the receiver filter
as function of wave number (2pi/1.52 cm)...........27
4.6 Power spectral density of the receiver filter
as function of wave number (2pi/15^2 cm).........27
5.1 Diagram of the experiment configuration.............29
5.2 4 mm spatial filter.................................30
5.3 Temperature and velocity profile of the
heat source.................... ...................32
5.4 Output signal of the 2mm filter system...........35
5.5 Output signal of the 4mm filter system:..........36
5.6 Output signal of the 6mm filter system...........37
5.7 Output signal of the 8mm filter system...........38
5.8 Output signal of the unmodulated aperture...........39
5.9 Power spectral density of the unmodulated
aperture...........................................4 0
5.10 Noise of the system................................41
5.11 Power spectral density of the system noise........42
5.12 "Normalized" power spectral density of the
unmodulated aperture...............................43
5.13 "Normalized" power spectral density of the
system noise........................................44
5.14 Signal power spectral density of the 2mm
filter system.................. ...................46
5.15 "Normalized" signal power spectral density
VIII
of the 2mm filter system...............................47
5.16 Signal power spectral density of the 4mm
filter system..........................................4 8
5.17 "Normalized" signal power spectral density
of the 4mm filter system................................49
5.18 Signal power spectral density of the 6mm
filter system..........................................5 0
5.19 "Normalized" signal power spectral density
of the 6mm filter system...............................51
5.20 Signal power spectral density of the 8mm
filter system...........................................52
5.21 "Normalized" signal power spectral density
of the 8mm filter system.............................. 53
5.22 Power spectral density of the 4mm filter
system with heat source at 1/3 of
propagation path length.................................56
5.23 Power spectral density of the 4mm filter
system with heat source at 2/3 of
propagation path length.................................57
5.24 Power spectral density of the 8mm filter
system with heat source at 1/3 of
propagation path length.................................58
5.25 Power spectral density of the 8mm filter
system with heat source at 2/3 of
propagation path length.................................59
CHAPTER 1
INTRODUCTION
The performance of optical systems that operate in the
atmosphere is affected by the presence of turbulent
refractive index fluctuations. Accurate prediction of system
performance depends on knowledge of the strength of those
fluctuations along the propagation path. Since the index of
refraction fluctuations in the optical portion of the spectrum
are dominated by temperature fluctuations, one can measures
the temperature fluctuations with an array of small, fast
thermometers and infer the refractive index fluctuations and
other parameter, such as the wind velocity transverse to the
propagation path of the optical system. However, this method
is inconvenient where measurements are carried out at
different locations or when the location at which
measurements are desired is inaccessible.
For this reason, acoustic, radar, and optical remote
sensors have been used to measure profiles of refractive
turbulence. The method of acoustic and radar for refractive
index measurements continue to be a useful technique.
However, it has a number of serious limitations. For acoustic
wave, it is generally useful at ranges of a few hundred meters
or less because of the strong absorption of acoustic energy by
2
the atmosphere. It is also limited to obtaining vertical
profiles of turbulence because of ground reflection problem.
For radar, the range can be much greater than that of
acoustic sensor. But it shares the ground clutter problem of
the acoustic sensor. Therefore, radar sensors have been used
exclusively for obtaining vertical profiles of turbulence.
Another drawback of radar sensors are poorer spatial
resolution and influence of humidity effects. The magnitude
of the humidity contribution must be known to make
predictions of optical effects based on radar Cn2, strength of
the turbulence parameter measurements.
The final class of remote sensing techniques that has
been used is based on lineofsight optical propagation
through turbulence. It used the relationship between the
position of a turbulent scattering layer and the spatial scale
of the scintillation produced by that scattering layer at a
receiver some distance away. By installing the proper spatial
filters at both termini, one can extract information to
measure the desired quantities of the transmission medium
such as its refractive index and related parameters. For this
thesis, the desired quantity to be measured is the wind
velocity.
CHAPTER 2
THEORY OF SPATIAL FILTER OPERATION
Local measurements of turbulence along a lineof sight
path could be made by using an incoherent receiver with
spatially modulated aperture response (a spatial filter
receiver) to detect the scintillation pattern produced by an
incoherent transmitter across which the intensity
distribution had been modulated (a spatial filter transmitter).
Clifford and Lataitis (1) predict that for a weak
homogeneous turbulence, the temporal auto covariance of the
detected signal normalized to its mean unfiltered value is
given by
C(t) = 4 C%(x) (1)
Where
CxW = 2*k2L J dsj d2K $ (K,0,s) sin2[K2Ls(1s)/2k]
* exp(iKv(s)t) Ft[K(1s)]2 Fr(Ks)2 (2)
is the spatially filtered, temporal, spherical wave,
logamplitude covariance,
t : time delay
k : optical wave number
L : path length
4
s : path position of the turbulence
v : transverse turbulent velocity
K : wave number of the turbulent refractive index
(K,0,s) path dependence refractiveindex spectrum
fr(p),ft(p): transmitter and receiver intensity
modulation
Fr2, Ft2 : power spectra of the transmitter and
receiver intensity modulation and are defined by
Fu(r) = Id2 P exp(ir.p) ft r(p) (3)
Equation (2) is valid only if the refractive index irregularities
can be treated as conservative passive additives that are
advected by the flow without changes in their spatial
characteristics.
Considering the ideal case of a spatial filter
transmitter and spatial filter receiver of infinite transverse
extent with sinusoidal intensity modulation function ft(p) a
cos(Kt.p) and fr(p) a cos(Kr.p), respectively. The filter are
identified by their spatial wave vectors Kt = at 2rcM,t and Kr
= ar2n/Xr, where at and ar are unit vector defining the
orientation of the spatial filters, and X.r are the
transmitter and receiver spatial wavelengths. The modulated
5
power spectra are given by
Fu(r)2a6(rKu) + S(r+Ku) (4)
for 5 is the delta function, Equation (1) reduces to
C(t) = 87ck2LI
where
I = Ids Jd2K Ft[K(1s)]2 Fr(Ks)2 (6)
K : wave vector of the turbulence and defined as
K = Kt+Kr (7)
s : position of the turbulence along the propagation
path and defined as s = Kr/(Kt + Kr) (8)
With an assumption that the refractive turbulence is
isotropic and that Kt and Kr lie along the same direction (i.e.,
the rotational orientation of the filters is the same). Equation
(5) indicates that the intergral over the frequency spectrum
yields the signal variance, which is proportional to the
refractive index spectrum evaluated at path position s and
single turbulence wave vector K.
If only the level of the spectrum as measured by Cn2
changes along the path, the spectrum can be written as the
product of a pathdependent term Cn2(s), and a term '(K) that
describes the shape of the spectrum. Thus, the sigal variance
6
can then be used to extract an estimate of Cn2 at path
position s and wave vector K. Further, the temporal frequency
w produced by each such K(s) is simply
w(s) = K(s). v(s) (9)
where
v(s) : linear velocity of the transverse turbulence at
path position s
By varying Kt and Kr so that s changes, one can profile
Cn2 along the path by observing the variations in a2 and
measure the turbulence velocity from the temporal frequency
w.
CHAPTER 3
EFFECT OF A SPATIALLY FILTERED APERTURE
3.1 Spatial Filter at the Receiver Terminal
Aperture filtering has generally been considered in a
limited sense, as the low pass filter action of a large
aperture, in removing the effects of small scatterers from
the receiver output. In fact, large apertures can be used to
emphasize such small scatterers when the aperture
illumination function is properly designed to provide a band
pass spatial filter.
To illustrate the effects of a spatially filtered
aperture, assume the turbulence wavenumber K(s) at position
s between the transmitter and the receiver, contains a single
component K(s) as shown in Figure (3.1) and acts as a weak
sinusoidal lens, imposing a sinusoidal phase modulation upon
the wavefront of the probing beam. It is obvious that the
effect of this component at the receiving plane (L), is a
similar sinusoidal modulation of the received field, which
spatial period is the projection of turbulence from the source
to the receiving plane. If the turbulance has a velocity v(s),
the disturbance at the receiving plane has a velocity greater
by the factor of L/s.
8
Fig.(3.1) Sinusoidal turbulence at the receiving
plane, produced by a sinusoidal refractive
turbulence at s, of wave number K(s). The relative
velocity and scale of the turbulence at the receiving
plane are the projections of the turbulence at s
from the transmitter
A particular path has been set up with a point
source at L=0, and a receiving aperture at L of such a nature
that only a disturbance with a certain spatial wavenumber K
causes variance at the receiver output. If the refractivity
disturbance of all wavenumbers exist at all points along the
path, only certain of these will produce an output at the
receiver. Specifically, only those wave numbers K(s) that
satisfy
Kr=K(s)s/L, (10)
because only these wavenumbers have K as their projection.
The temporal frequency w produced by each K(s) is
simply
9
w(s)= K(s) v(s) (11)
From the reasoning above, it is clear that the spatial
filter allows only one wave number K at each position to
contribute to the receiver output; this wave number K is
different for each position. If the velocity is uniform, each of
these wavenumbers may be associated with a unique temporal
frequency in the receiver output. Thus, the position variable
along the path is translated into a frequency variable.
In order to calculate the disturbance position and its
velocity, further information is needed and can be obtained by
placing a second source in the same plane but at a distance d
away from the first source as shown in Figure (3.2).
The fluctuations from these two sources are recorded
separately, and the relative phase (9) of these fluctuation is
determined for each frequency. From Figure (3.2), the
separation of the two paths at position s is
d = d (Ls) / L (12)
Therefore, the magnitude of the phase difference is

Using equation (10), (11), and (12) to solve for the path
position s, the turbulence wavenumber K(s), and the
turbulence velovity v(s)
s= L/(1+(p(w)/Krd) (14)
K(s)= K(1+q>(w)/Krd (15)
10
Fig.(3.2) Two Transmitters configuration
v(s)= w/(K+cp(w)/d) (16)
From chapter 2, it showed how the spatial filter
interrogates the turbulent medium. In an actual
implementation of a single spatially filtered aperture (at the
receiver terminal), the ideal filter that was discussed can
only be approximated. Consequently, the receiver will
respond not only to a single wavenumber at a single path
position, but to some finite resolution cell in both wave
number and in path position. Also, a certain amount of
measurement noise is unavoidable. This will limit the
precision to which the measurement can be made.
The receiver that most closely approximates the ideal
receiver is probably the zero sum receiver shown in
11
Fig.(3.3) Zerosum receiver configuration
Figure (3.3). The incident light is collected by an aperture of
radius r Behind the aperture is a spatial filter consisting of
reflective stripes alternating with transmissive stripes.
These stripes are of equal width, given by jt/Kr so that the
filter has a fundamental frequency of Kr .
The light transmitted by the filter is collected by a
photodetector followed by a logarithmic amplifier.
The reflected light is similarly detected by a second
detector. The difference between these two detector outputs
is then fed into a spectrum analyzer to provide turbulence
12
strength as a function of height. To sample different regions
of the turbulence spectrum, the effective frequency of the
spatial filter would be changed.
The filter just described in zerosum receiver is
extremely difficult to treat analytically and is quite hard to
implement. Thus, it is approximated by the normalized
function
fr(x,y) = (7cr2)"1exp[(x2+y2)/2r2]cos(Krx) (17)
An example of this function is presented in Figure (3.4).
3.2 Spatial Filter at Both Termini
The scheme described in 3.1 has one drawback in the
use of a second source at a known distance away from the
first source, and a special configuration for the receiver.
This arrangement may prove to be difficult and impossible for
some applications.
The most satisfactory solution, however, is obtained by
using spatial filters at both the receiver and transmitter
terminals. This configuration is called a nonzerosum
receiver. From the general geometry of Figure (3.5), the two
filters act independently to weigh the effects of turbulence
on the receiveing aperture, providing the apertures to be
considered incoherent, so that powers may be summed
without regard to carrier phase.
13
Figure (3.4) Zerosum spatial filter function(dashed line) and
the corresponding approximation (solid line) as a function of
the horizontal coordinate x for a circular aperture of radius r
The receiving spatial filter acts on the projection Ks/L
of the turbulence wave number and the transmitting acts on
the projection in the reverse direction, K(Ls)/L, reflecting
the reciprocity of the situation; the disturbance of energy
from each point on each aperture must be weighted by the
entire opposite filter.
14
If the two filters are resonated at spatial wave
number Kt and Kr then
Kt = K(s)(Ls)/L (17)
and
Kr = K(s) s/L (18)
From equations (17) and (18)
s = Ly[1+(Kt/Kr)] (19)
K(s) = Kt+Kr
(20)
15
From equations (19) and (20), the resonant wave
number and the location of the turbulence are determined by
simple relationships that involve only the resonant wave
numbers of the filters; the location of the turbulence can be
found along the propagation path by simply varying the filter
wavenumber Kt or Kr, or both Kt and Kr may be varied
proportionally to vary K(s) while s remains constant.
For ideal filters, all variance observed at the receiver
output must originate at the known location s and resonant
wave number K(s) of the turbulence. Also, since the temporal
frequency appropriate to the resonant wavenumber K(s) of the
turbulence is simply K(s)v(s), measurement of the fluctuation
frequency leads immediately to the determination of the
turbulence velocity.
The configuration of spatial filters at both termini
(nonzerosum receiver) has several advantages over the
zerosum receiver (single spatially filtered aperture)
configuration. In this configuration, the receiver consists of
a single detector and a spatial filter that, transmits alternate
strips. The approximate normalized function for this receiver
is given by
fr(x,y) = (2rcr2)1exp[(x2+y2)/2r2][1+cosKrx] (21)
Which is plotted along with the actual function in Figure (3.6)
16
Fig.(3.6) nonzerosum receiver filter function (dashed line)
and its approximation (solid line).
Since there is only one detector to contribute to the
noise of the nonzerosum receiver, the signaltonoise ratio
could be improved and the actual implementation of the
system is much simpler than the zerosum receiver
configuration.
CHAPTER 4
WEIGHTING FUNCTION
To investigate the expected performance of a given
spatial filter system theoretically, it is necessary to
calculate the corresponding path and wave number weighting
functions that describe the sensitivity of the system to
difference path positions and spatial wave numbers of the
refractive turbulence spectrum.
4.1 Path Weighting Function
With an assumption that the path dependence of the
refractive index spectrum is due solely to Cn2 variations
along the path, then
4>(K,0,S) = Cn2(s) '(K,0) (22)
By rewriting Eq.(1), a path weighting function P(s) for C_2
profiling is defined as
o2 4C/(0) = J ds Cn2(s).P(s) (23)
Where
P(s) = 8ick2 L / d2K
* Ft[K(1s)]2 Fr[Ks]2
(24)
18
Calculation for the modulation power spectra of the
transmitter Ft2 and receiver Fr2 is needed to further
evaluate of Eq.(24).
A simple way to generate spatial filters is to mask the
transmitter and receiver apertures with equally spaced
parallel strips of tape. The resulting modulation function f(p)
is zero outside the aperture and wherever there is a strip of
masking tape, and is one elsewhere. The spatial wavelength A,
of the filter is the distance between the centers of the
stripes of tape.
For striped cicular apertures, which are the most likely
to be encountered in practice, the spatial power spectrum is
found to be
F(T)2 = F(rx,ry)2 = I Isinc(n7t/2) exp(inKfA)
*J1c(r[(rxnK)2+ry2]1/22 (25)
Where
r : radius of the circular aperture
A : spatial shift of the stripes
Kf : spatial wave number of the filter
n : number of stripes across the aperture
J1c(X) = J1(X)/X
Equation (25) can be used in equations (24) and (33) to
generate the path weighting and wave number weighting
19
function for both the zerosum and nonzerosum filter
system. The resulting expressions are complex and require
numerical analysis technique for evaluation.
Analytical tractable expressions that yield some
physical insight can be derived by using an approximate form
for the aperture modulation functions as mentioned in Chapter
3. Consider the case of a nonzerosum circular Gaussian
aperture with a perfectly sinusoidal modulation and arbitrary
rotational orientation with modulation filter function as
shown by Fig.(3.6).
F(r)2 = 1/4(exp(r2r2/2) + 1/4(exp[(r+K)2r2/2]
+ exp[(rK)2r2/2])) (26)
Equation (26) is an approximation to the modulation power
spectrum of the nonzerosum vertically striped spatial filter
described by Eq.(25). It is reasonably accurate, provided there
are many spatial cycles acrsoss the aperture.
With the assumption that the Cn2 distribution does not
favor the ends of the path and there is no rotational mismatch
of the spatial filters, the path weighting function can be
approximated by
P(s) = 8.48*10'2L3r'7/3Cn2(s){s2(1s)2/(s2+(1s)2)7/6
+ C'(s) exp [(ss')V(s)]} (27)
Where
20
Y(s) = {2[s2+(1s)2]}1/2/[( Kt + Kr ) r] (28)
C(s) = 0.48[r/(l_/k)1/2]173 [(Kt+Kr)(L7k)1/2]'11/3
* [(1s)2+s2]8/3 {1+R/[1+s(R1)]}11/3
* sin2{1/2[(Kt+Kr)(Uk)1/2]2
*[1 +s(R1 )/(1 +R)]2s(1s)/[s2+(1s)2]2} (29)
R = K/Kt
s0' = [* +(Kt/Kr)]1
(30)
(31)
The relative contribution of each term in Equation (27)
to the signal variance is depended on the size of C'(s) and the
width y(s). C'(s) is a strong function of the quantity
which reflects the refractive index spectral weight
associated with the sampled wave number K=Kt+Kr For
smaller transmitter and receiver spatial wave lengths, the
magnitude of the second term relative to the first term
decreases because the spatial filter component of the signal
comes from smaller scales in the turbulent flow.
The width 2y(s) = 2[(Kt+Kr)r]"1 of the exponential also
decreases, reflecting the reduction in size of the^ scattering
volume that contributes to the spatially filtered component
of the signal.
Figures (4.1), (4.2), and (4.3) show the approximated
[(Kt+Kr)(L/k)'^]
1/2H1/3
21
normalized path weighting function P(s) with 6 spatial cycles
across the transmitter and 2, 6, and 18 spatial cycles across
the receiver.
The approximated path weighting functions exhibit an
enhancement at the expected path position of 1/4, 1/2, and
3/4 path length.
Fig.(4.1) Normalized path weighting as a function of the
normalized path position for 6 cycles transmitter
and 2 cycles receiver spatial filters
22
Fig.(4.2) Normalized path weighting as a function of the
normalized path position for transmitter and
receiver with 6 cycles spatial filters
normalized path position
Fig.(4.3) Normalized path weighting as a function of the
normalized path position for 6 cycles transmitter
and eighteen cycles receiver
23
4.2 Wave number weighting function
The wave number weighting function can be defined as
o2=Jd2KW(K) _ (32)
Where ..
W(K) = 87ck2LO(K,0) J ds Cn2(s) sin2[K2Ls(1 s)/2k]
*Ft[K(1s)]2Fr[Ks]2 (33)
A problem with this definition for W(K) is that it is a
function of the distribution of Cn2 along the path. If the
distribution is known, then the resulting expression for W(K)
will accurately describe the contribution of different wave
numbers to the spatially filtered signal. However, the
distribution of Cn2 is not known. To avoid the problem,
assume that Cn2 is constant across the resolution cell of the
system, then replace it with its value at the expected
sampling position s, and remove it from the intergrand.
Subject to the same approximations used in 4.1 to
obtain Equation (27), the wave number weighting function
i s
W(K) = S.18*10'2k2LCn2K'11/3{D(K)exp(K2/A2)
+H(K)(exp[[K(K,+Kr)]2/A2)
+exp([K+(Kt+Kr)]2/A2))} (34)
Where
24
D(K)=7c1/2/Kr[erf(Kr/2)]sin2(K2L/8k)
H(K) = 1 /32{(tc) 1 /2/Kr[erf[Kr(1 s')]+erf (Krs']}
* sin2[K2Ls'(1 s)/2k]
*exp[r2(KtKr)2sin20/4], 0
0 < s' < 1
0, otherwise
(35)
A = 2/r
s' = K[K(KtKr)]/2K2
(36)
0 = angle between K and (KtKr)
K : magnetude of K
erf : error function
The first term in Equation (30) describes the
unmodulated whole aperture response and indicates that only
scales larger than the aperture (i.e; K ^ 2/r) contribute to this
component of the signal. These larger spatial scales
correspond to the background level in Figures (4.1), (4.2), and
(4.3).
The second term describes the effect of the aperture
modulation, which samples the flow at spatial scales
corresponding to K=Kt+Kr These scales are much smaller
than those that contribute to the first term, and correspond
to the spatial filter enhancement of the path weighting
functions in Figures (4.1), (4.2), and (4.3). The spectral
resolution associated with the spatial filter component of the
25
signal is A = 2/r. The location of the peak response is at
K=Kt+Kr so that if rK(s)+Kr 1, the low wave number and
spatial filter components will be well separated.
For a vertically oriented and match spatial filters
system, Equation (34) is reduced to
W(Kx,Ky) 5.8*10'2k2LCn2(Kx2+Ky2)'11/6
*{D(Kx,Ky)exp[(Kx2+Ky2)/A2]
+H(Kx,Ky)exp(Ky2/A2)(exp[[Kx(Kt+Kr)]2/A2]
+exp[[Kx+(Kt+Kr)2/A2])} (37)
Where
D(Kx,Ky)=1/1[jt/(Kx2+K 2>]1/2erf[r/2(Kx2+K2)1/2]
*sin2[(Kx2+K2)L/8k]
H(Kx,Ky) = (32r)1[,i/(Kx2+Ky2)]1/2
(erf[r(Kx2+Ky2)1/2(1s')]
+erf[r(Kx2+Ky2)1/2s']}
*exp{r2(KtKr)2Ky2/[4(Kx2+Ky2)]}
*sin2[(Kx2+Ky2)Ls'(1s')/2k], 0 Â£ s Â£ 1
0, otherwise (38)
With
s' = [1(KtKr)Kx/(Kx2+Ky2)]/2
(39)
26
Kx>Ky describe the horizontal and vertical wave number.
To obtain a clearer view of the effect of the filter
wave number upon a spatial filter system, the output power
spectrum as a function of the filter wave number should be
observed. This is shown by Figures (4.4), (4.5), and (4.6),
providing the transmission function of the aperture contains
only the horizontal component and the transmitter
fundamental wavelength of 2.54cm, which corresponds to the
receiver fundamental wavelength of 2.54cm, 1.51cm, and
15.2cm respectly.
Fig.(4.4) Power spectral density of the receiver filter as
function of wave number (2 pi/2.54 cm)
Spectrum
27
.5) Power spectral density of the receiver filter as
function of wave number (2 pi/1.52 cm)
Fig.(4.6) Power spectral density of the receiver filter as
function of wave number (2 pi/15.2 cm)
CHAPTER 5
EXPERIMENT!
An experiment was performed in the Laboratory to
verify the theory described in previous chapters. A diagram of
the experimental configuration is given in Figure (5.1).
5.1 Spatial Filters
Several attempts to use masking tape on a piece of
glass proved to be futile due to the size constraint. As an
alternative, a Me Draw program was used to generate the
needed stripes for different wavelengths. The hard copies
were then transferred to the transparencies. This method was
quite good for spatial wavelengths greater than 4mm, but
proved to be less than satisfactory for smaller wavelengths.
Filters with wavelengths of 2mm, 4mm, 6mm, 8mm,
12mm, and 16mm were generated using the described method,
Figure (5.2) showed 4mm spatial filter.
5.2 Transmitter Source
Any incoherent light (incandescent, L.e.d, etc..) could be
used for a transmitter source, but for possibility future
longer range experiment, a Laser source was chosen.
29
Fig.(5.1) Diagram of experiment configuration
30
Fig.(5.2) 4mm spatial filter
In this experiment, a HeNe Laser ( Spectra Physic
Stabilite, model 124A ) was used. This Laser put out 15mW of
optical power at a wavelength of 632.8nm.
To convert a coherent Laser beam into a incoherent
light source, a grounded glass portion of a microscope slide
was put at the Laser output and placed at the focus of a 25cm
focallength lens. The clear aperture of the lens was a round
of 10cm in diameter. An adjustable aperture was used to
reduce the collimated incoherent beam from 10cm to 5cm
cross section diameter. To avoid saturation and to improve
signal to noise ratio, a neutral density filter (16%
transmission) was placed in front of the grounded glass.
5.3 Turbulence Source
31
The turbulence source selected was a heat gun that
blows hot air out through a 5 cm diameter barrel. This was
placed at the same height as the optical path at a horizontal
distance of 6 inches. The jet from the heater was
characterized roughly with an thermometer and a 6 cm
diameter impeller anemometer as shown in Figure (5.2).
5.4 Bacskeisystsm
The receiver system consisted of a 6 cm diameter lens
with a focal length of 25 cm to focus the optical signal onto
the photodetector. The detector was a photodiode with a
JFET operational amplifier builtin (EG&G HUV1100B), the
opamp was configured as a transimpedance amplifier with
feed back resistor of 2 Mohm. The output of the amplified
detector was coupled to a voltage amplifier to increase the
output signal amplitude.
To block background noise, the collected optical signal
from the collecting lens was passed through an interference
filter with a passband of 20nm centered at an optical
wavelength of 632.8 nm.
5.5 Recording and Processing Instruments
Electrical output of the signal was then connected to an
Osciloscope, a Spectrum analyzer, and a PC with builtin A/D
board (DT 2814).
32
Fig. (5.3): Temperature and velocity profile of the heat source
33
The Osciloscope was used to align the system, primary
evaluation of the system feasibility was done by the
Spectrum analyzer. The PC was then used to record and
process the signal.
5.6 Erossdura
Data was recorded for the Spatial Filter of 2mm, 4mm,
6mm, and 8mm wavelength which shown by Figures (5.4),
(5.5), (5.6), and (5.7), respectively. Each data run consisted of
1000 sample points at the sampling rate of 20KHz.
Before the recorded data got processed further, a base
line of the system response was established to observe the
response of the unmodulated aperture on the turbulence (no
filter, heat gun on) in the time domain, shown in Figure (5.8),
and its power spectral density, which is shown in Figure (5.9).
To check the noise level of the system, a dry run (no filter,
heat gun off) was performed and is shown in Figures (5.10)
and (5.11).
To enhance the peak response visually, a "normalized"
power spectral density function U(f) was defined as:
U(f) = W(f)*f/ J W(f) df
Where
f : frequency in Hertz
W(f) : power sectral density of the signal
Integration upper bound = 10Khz
34
Integration lower bound = 0Hz
Figures (5.12) and (5.13) showed the graphs of U(f) v.s
frequency for the unmodulated aperture and the system noise.
Figure (5.9) shows that the unmodulated aperture acts
as a low pass filter with a cut off frequency at about 500Hz.
Figure (5.11) shows that the system noise is a periodic
function riched in harmonics with the fundamental frequency
is at about 60Hz and its harmonics are distributed to the
full spectrum of the system with bias in the low (<500Hz) and
high end (>2500Hz).
For this reason, the experiment was limited to spatial
filter with a wavelength of 2mm, 4mm, 6mm, and 8mm. A
signal of the system that has wavelength longer than 8mm or
shorter than 2mm would be buried under the noise floor.
Amplitude (Volt)
Figure (5.4): Output signal of the 2mm filter system.
Amplitude (Volt)
Figure (5.5): Output signal of the 4mm filter system.
CO
CD
Amplitude (Volt)
Figure (5.6): Output signal of the 6mm filter system.
CO
Amplitude (Volt)
Figure (5.7): Output signal of the 8mm filter system.
CO
CO
2
5
1
5
0
5
1
5
2
5
T
T
T
T
T
Figure (5.8): Output signal of the unmodulated aperture.
Spectrum
xlO4
Figure (5.9): Power spectral density of the unmodulated aperture.
o
Amplitude (Volt)
Figure (5.10): Noise of the system.
100
90
80
70
60
50
40
30
SO
10
0
500 1000 1500 2000 2500 3000 3500 4000 4500
Frequency (Hz)
Figure (5.11): Power spectral density of the system noise.
U(f)
1 1r111111r
0.9 
0.8 
0.7 
0.6
0.5 
0.4 
0.3 
0.2
Mn**i* 'ftiAfA.nM *lfl \ i.ilift lli ftfc i II >if illmi t M n ill ifli i I i i L niiirM i n U ift
0 1000 2000 3000 4000 5000 6000 7000 8000 9000
Frequency(Hz)
Figure (5.12): "Normalized" power spectral density of the unmodulated
10000
apperture.
Â£
CO
U(f)
Figure (5.13): "Normalized" power spectral density of the system noise.
CHAPTER 6
ANALYSIS AND CONCLUSION
6.1 Analysis
For the described spatial filter system (double
spatially filtered apertures, non zerosum receiver), with the
wind velocity of 5m/s. The theoritically calculated peak
frequency responses are:
Filter wave length(mm) Frequency(KHz)
2 5
4 2.5
6 1.67
8 1.25
Figures (5.14) and (5.15), (5.16) and (5.17), (5.18) and (5.19),
(5.20) and (5.21) showed the power spectral density and its
"normalized" for the 2mm, 4mm, 6mm, and 8mm filter,
respectively.
For the 2mm wavelength spatial filter, a peak at 5KHz
was expected. Inspecting Figures (5.14) and (5.15) shows no
such peak. In Chapter 3, it was proved that a spatial filter
allowed only one wave number of the turbulence at each
position s to contribute to the receiver output, and the
turbulence wave number must satisfy the relation
100
go
80
70
60
50
40
30
20
10
0
t1111111r
500 1000 1500 2000 2500
n/Af
3000 3500 4000
fi rAnbi"ixAi
4500
Frequency (Hz)
Figure (5.14): Signal power spectral density of the 2mm filter system.
>
5000
05
U(f)
Figure (5.15): "Normalized" signal power spectral density of the 2mm
filter system.
^i
Spectrum
Figure (5.16): Signal power spectral density of the 4mm filter system.
00
U(f)
Figure (5.17): "Normalized" signal power spectral density of the 4mm
filter system.
Spectrum
Figure (5.18): Signal power spectral density of the 6mm filter system.
cn
o
U(f)
Figure (5.19): "Normalized" signal power spectral density of the 6mm
filter system. cn
Spectrum
Figure (5.20): Signal power spectral density of the 8mm filter system.
cn
ro
U(f)
Figure (5.21): "Normalized" signal power spectral density of the 8mm
filter system.
cn
co
54
K(s) = Kt + Kr
In the case of a 2mm filter, K(s) must equal 4ti/.002 m"1, and
from the above observation, it is known that the turbulence
did not contain such wave number.
Visual inspection of Figure (5.17) shows a hump
starting at about 1.5 KHz, peaking at about 2.3 KHz and then
gradually falling off at higher frequencies.
Inspection of Figure (5.18) shows a fast rise at about
800Hz, peaking at about 1.3 KHz and falling off slowly at
higher frequencies.
Inspection of Figure (5.20) shows a fast rise at about
500 Hz, peaking at about 1KHz and tapering off at higher
frequencies.
To understand these discrepencies, recall that a
uniform wind velocity was assumed and with this
assumption the high frequency cut off is expected to be sharp.
The turbulence generator (heat gun) has some maximum
velocity in the center of the stream and its velocity
decreased at the edges as shown in Figure (5.3). Also, the
velocity of the heat gun decreased when the heating element
kickedin, and increased when the heating element
kickingoff. Therefore, the measured values showed a
slightly shift in the peak frequency responses and slowly
rolled off at higher frequencies .
Although the output frequencies peaked at a lower
55
value than calculated, and their peaks were spread over two
decades Figures (5.16), (5.18), and (5.20) demonstrated that
they can be discerned for the particular turbulence generator
that was used in these experiments. Since the technique
depended only on the ability to recognize the peaks and not on
theirs shapes, the discrepencies mentioned above should not
downgrade the usefulness of the technique.
To obtain data for the path weighting measurements,
the heat gun was moved along the propagation path Figures
(5.22), (5.24) shows that the "normalized" power spectral
density of the 4mm and 8mm filter system, with the heat gun
set at 1/3 path length. Figures (5.23) and (5.25) shows the
"normalized" power spectral density of the 4mm and 8mm
filter system with the heat gun set at 2/3 of the path length
from the transmitter.
From consideration of their magnitudes at the
corresponding peak frequency responses, and a comparison of
their magnitudes, taken with the heat gun set at 1/2 of the
path length, it is clear that for the same filter wave length at
both termini, the most sensitive position is at 1/2 of the path
length.
U(f)
system with heat source at 1/3 of propagation path length.
tn
05
U(f)
0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000
Frequency (Hz)
Figure (5.23): "Normalized" power spectral density of the 4mm filter
system with heat source at 2/3 of propagation path length.
Ol
4
U(f)
0.6
0.5
0.4
0.3
0.2
0.1
0
0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000
Frequency (Hz)
Figure (5.24): "Normalized" power spectral density of the 8mm filter
system with heat source at 1/3 of propagation path length.
Ol
00
U (f)
Figure (5.25): "Normalized" power spectral density of the 8mm filter
system with heat source at 2/3 of propagation path length.
oi
co
60
6.2 Conclusion:
An experiment was performed in the laboratory with a
simulated turbulent atmosphere to verify the feasibility of
spatial filtering techniques for remote measurement of wind
velocity in the turbulent medium.
In the analysis bias of the path position and the effect
of the unmodulated aperture were observed. With the
assumption of constant uniform wind, the theory predicts
that the power spectral density of the receiving signal would
have a narrow peak at zero frequency and another peak at an
angular frequency given by the product of the transverse wind
velocity and the sum of the transmitter and receiver spatial
wavenumbers. From chapter 5, the stream velocity profile
shows that the assumption above was not valid, therefore, the
peaks were spread according to velocity fluctuations. Small
laboratory space imposed the size constraint on the system
resulting in poor resolution.
Although the overall results were not as good as
expected, the experiment showed that spatial filtering
techniques for remote measurement of wind velocity in the
turbulent atmosphere is a useful technique, and should be
further perfected.
BIBLIOGRAPHY
(1) S.F. Clifford and R.J. Lataitis, "Spatial and Temporal
Filtering of Scintillation in Remote Sensing, IEEE Trans.
Antennas Propag. AP35,597 (1987).
(2) Tingi Wang, S.F.Clifford, and G.R. Ochs, "Wind and
RefractiveTurbulence Sensing Using Crossed Laser
Beams," Appl. Opt. 13, 2602 (1974).
(3) S.F. Clifford and J.H. Churnside, "Refractive Turbulence
Profiling Using Synthetic Aperture Spatial Filtering of
Scintillation," Appl. Opt. 26, 1295 (1987).
(4) R.W.Lee, "Remote Probing Using Spatially Filtered
Apertures," J. Opt. Soc. Am. 64, 1295 (1974).
(5) R.E. Good, B.J. Watkins, A.F. Quesada, J.H. Brown, and G.B.
Loriot, "Radar and Optical Measurements of Cn2," Appl. Opt.
21, 3373 (1982).
(6) G.R. Ochs, Tingi Wang, R.S. Lawrence, and S.F. Clifford,
"Refractive Turbulence Profiles Measured by
OneDimensional Spatial Filtering of Scintillations," Appl.
Opt. 15, 2504 (1976).
(7) J.H. Churnside, R.J. Lataitis, and R.S. Lawrence, "Localized
measurements of Refractive Turbulence Using Spatial
Filtering of Scintillations," Appl. Opt. 27, 2199 (1988).
62
(8) R.W. Lee and J.C. Harp, "Weak Scattering in Random Media
with Applications to Remote Probing," Proc. IEEE 57, 375
(1969).

Full Text 
PAGE 1
SPATIAL FILTERING TECHNIQUE FOR REMOTE MEASUREMENT OF WIND VELOCITY IN TURBULENT ATMOSPHERE by John Giang Truong Vu B.S., University of Saigon, 1972 M.S., Roosevelt University, 1980 A thesis submitted to the Faculty of the Graduate School of the University of Colorado in partial fulfillment of the requirements for the degree of Master of Science Department of Electrical Engineering and Computer Science 1989
PAGE 2
This thesis for the Master of Science degree by John Giang Truong Vu has been approved for the Department of Electrical Engineering and Computer Science by Arun K. Majumdar Edward T. Wall Date 42...1 ?'C(
PAGE 3
ACKNOWLEOOEMENTS I would like to thank Dr. Arun K. Majumdar for his technical assistance and especially, for his support in obtaining essential equipment for this work.
PAGE 4
iv Vu, John Giang Truong (M.S., Electrical Engineering) Spatial Filtering Technique for Remote Measurement of Wind Velocity in Turbulent Atmosphere Thesis directed by Professor Arun K. Majumdar Spatial filters applied on transmitting and receiving aperture. s are used in lineofsight measurement of transverse flow velocity of the wind in turbulent transmission medium, exhibiting small random fluctuations in its refractive index which is produced by irregularities across the probe wave propagation path, provided that the weak scattering approximation is appropriate. The technique uses spatial filtering of scintillation from a spatially filtered incoherent optical scource. A simulated disturbance of the transmission medium was generated in the laboratory, and data were recorded and processed through a personal computer. The form and content of this abstract are approved. recommend its publication. .At_ ](_ Signed ______ ____ Faculty member in charge of thesis
PAGE 5
v CONTENTS CHAPTER . .. 1. INTRODUCTION ................... ............ ................... ; .......................... 1 2. THEORY OF SPATIAL FILTER OPERATION ......................... 3 3. EFFECT OF SPATIALLY FILTERED APERTURE. .................. .? 3.1 Spatial filter at the receiver terminal.. .............. 7 3.2 Spatial filter at both termini. ..... ........................ 1 2 4. WEIGHTING FUNCTION .............................. ........... ..... .............. 17 4.1 Path weighting function .......................................... 1 7 4.2 Wave number weighting function ........................ 2 3 5. EXPERIMENT ................................................................................ 2 8 5.1 Spatial filters ............................................ .... ............ 28 5.2 Transmitter source ................... ..... ........................ . 28 5.3 Turbulence source .......................................... ........... 3 0 5.4 Receiver system .......................................................... 31 5.5 Recording and Processing lnstruments ............ 31 5.6 Procedure ....................................................................... 3 3 6. ANALYSIS AND CONCLUSIONS .......... .................................. 4 5 6.1 Analysis ........................................................................... 4 5 6.2 Conclusions ................................................................... 6 0 BIBLIOGRAPHY .................................................... ................................... 61
PAGE 6
vi AGURES FIGURE 3.1 Sinusoidal turbulence at the receiving plane ............ 8 3.2 Two transmitters configuration ..... or 1 0 3.3 Zerosum receiving aperture ..... ........ ........ ...... ........ ... 11 3.4 Zerosum spatial filter function and the corresponding approximation as a function of the coordinate x for a circular aperture of radius r ............................................................................. . 1 3 3.5 Geometry of two filters at both termini. ............. ... 1 4 3.6 Nonzerosum spatial filter function and the corresponding approximation as a function of the coordinate x for a circular aperture of radius r ............................................................................... 1 6 4.1 Normalized path weighting as a function of the normalized path position for 6 cycles transmitter and 2 cycles receiver filters ............... 21 4.2 Normalized path weighting as a function of the normalized path position for transmitter and receiver with 6 cycles filters .............................. 2 2 4.3 Normalized path weighting as a function of the normalized path position for 6 cycles transmitter and 18 cycles receiver ............................ 22 4.4 Power spectral density of the receiver filter
PAGE 7
vii as function of wave number '(2pi/2.54 em) .............. 26 4.5 Power spectral density of the receiver filter as function of wave number (2pi/1.52 em) .............. 27 4.6 Power spectral density of the receiver filter as function of wave number (2pi/10.2 cm) .............. 27 5.1 Diagram of the experiment con figuratiori ....... ........ 2 9 5.2 4 mm spatial filter ............................................................. 30 5.3 Temperature and velocity profile of the heat source ............................................................................. 3 2 5.4 Output signal of the 2mm filter system ................... 3 5 5.5 Output signal of the 4mm filter system : ................. 3 6 5.6 Output signal of the 6mm filter system ................... 3 7 5. 7 Output signal of the Smm filter system ................... 3 8 5.8 Output signal of the unmodulated aperture ............. 3 9 5.9 Power spectral density of the unmodulated aperture ................................................................................... 4 0 5.1 0 Noise of the system ........................................................... .41 5.11 Power spectral density of the system noise .......... 4 2 5. 12 "Normalized .. power spectral density of the unmodulated aperture ........................................................ 4 3 5.13 "Normalized" power spectral density of the system noise ........................... ............................................... 44 5.14 Signal power spectral density of the 2mm filter system ... ............. ........... : ........................................... 4 6 5.15 .. Normalized" signal power spectral density
PAGE 8
viii of the 2mm filter system.:; ......................... ......... .4 7 5.16 Signal power spectral density of the 4mm filter system ................................................ ........................ 4 8 5.17 "Normalized" signal power spectral density of the 4mm filter system ............................................... .4 9 5.18 Signal power spectral density of the 6mm filter system .................................................................... ..... 5 0 5.19 "Normalized" signal power spectral density of the 6mm filter systeri1 ............ : ................................... 51 5.20 Signal power spectral density of the 8mm filter system ......................................................................... 5 2 5.21 "Normalized" signal power spectral density of the 8mm filter system .................... ....................... ..... 5 3 5.22 Power spectral density of the 4mm filter system with heat source at 1/3 of propagation path length .................... ............................... 56 5.23 Power spectral density of the 4mm filter system with heat source at 2/3 of propagation path length .................................................... 5 7 5.24 Power spectral density of the 8mm filter system with heat source at 1/3 of propagation path length .................................................... 5 8 5.25 Power spectral density of the 8mm filter system with heat source at 2/3 of propagation path length .................................................... 59
PAGE 9
CHAPTER 1 INTRODUCTION The performance of optical systems that operate in the atmosphere is affected by the presence of turbulent refractive index fluctuations. Accurate prediction of system performance depends on knowledge of the strength of those fluctuations along the propagation path. Since the index of refraction fluctuations in the optical portion of the spectrum are dominated by temperature fluctuations, one can measures the temperature fluctuations with an array of small, fast thermometers and infer the refractive index fluctuations and other parameter, such as the wind velocity transverse to the propagation path of the optical system. However, this method is inconvenient where measurements are carried out at different locations or when the location at which measurements are desired is inaccessible. For this reason, acoustic, radar, and optical remote sensors have been used to measure profiles of refractive turbulence. The method of acoustic and radar for refractive index measurements continue to be a useful technique. However, it has a number of serious limitations. For acoustic wave, it is generally useful at ranges of a few hundred meters or less because of the strong absorption of acoustic energy by
PAGE 10
2 the atmosphere. It is also limited to obtaining vertical profiles of turbulence because of ground reflection problem. For radar, the range can be much greater than that of acoustic sensor. But it shares the ground clutter problem of the acoustic sensor. Therefore, radar sensors have been used exclusively for obtaining vertical profiles of turbulence. Another drawback of radar sensors are poorer spatial resolution and influence of humidity effects. The magnitude of the humidity contribution must be known to make predictions of optical effects based on radar Cn 2 strength of the turbulence parameter measurements. The final class of remote sensing techniques that has been used is based on lineofsight optical propagation through turbulence. It used the relationship between the position of a turbulent scattering layer and the spatial scale of the scintillation produced by that scattering layer at a receiver some distance away. By installing the proper spatial filters at both termini, one can extract information to measure the desired quantities of the transmission medium such as its refractive index and related parameters. For this thesis, the desired quantity to be measured is the wind velocity.
PAGE 11
CHAPTER2 THEORY OF SPATIAL FILTER OPERATION Local measurements of turbulence along a lineof sight path could be made by using an incoherent receiver with spatially modulated aperture response (a spatial filter receiver) to detect the scintillation pattern produced by an incoherent transmitter across which the intensity distribution had been modulated (a spatial filter transmitter). Clifford and Lataitis (1) predict that for a weak homogeneous turbulence,. the temporal auto covariance of the detected signal normalized to its mean unfiltered value is given by Where C('t) = 4 Cx('t) Cx('t) = 27tk2 L J dsJ d 2 K q, (K,O,s) sin2[K2 Ls(1s)/2k] exp(iKv(s)'t) IFt[K(1s)]l 2 IF r(Ks)l 2 (1) (2) is the spatially filtered, temporal, spherical wave, logamplitude covariance. 't : time delay k : optical wave number L : path length
PAGE 12
s : path position of the turbulence v : transverse turbulent velocity K : wave number of the turbulent refractive index 4 q,(K,O,s) = path dependence refractiveindex spectrum fr(p),ft(P) :transmitter and receiver intensity modulation 1Fr12 1Ftl2 :power spectra of the transmitter and receiver intensity modulation and are defined by (3) Equation (2) is valid only if the refractive index irregularities can be treated as conservative additives that are advected by the flow without changes in their spatial characteristics. Considering the ideal case of a spatial filter transmitter and spatial filter receiver of infinite transverse extent with sinusoidal intensity modulation function ft(P) a cos(KtP) and fr(P) a cos(Kr.p), respectively. The filter are by their spatial wave vectors Kt = at 27t/A.t and K r = ar21t/A.r, where at and ar are unit vector defining the orientation of the spatial filters, "t and A. r are the transmitter and receiver spatial wavelengths. The modulated
PAGE 13
power spectra are given by 1Ft,r(r)l2 a S(rKt,r> + S(r +Kt,r> for S is the delta function, Equation (1) reduces to 5 (4) e(1:) = cos[K.v(s)1:](S) where K : wave vector of the turbulence and defined as K = Kt+ Kr (6) (7) s : position of the turbulence along the propagation path and defined as s = Kr/(Kt + Kr) (8) With an assumption that the refractive turbulence is isotropic and that Kt and Kr lie along the same direction (i.e., the rotational orientation of the filters is the same). Equation (5) indicates that the intergral over the frequency spectrum yields the signal variance, which is proportional to the refractive index spectrum evaluated at path position s and single turbulence wave vector K. If only the level of the spectrum as measured by en 2 changes along the path, the spectrum can be written as the product of a pathdependent term en 2(s), and a term cp'(K) that describes the shape of the spectrum. Thus, the sigal variance
PAGE 14
6 can then be used to extract an estimate of Cn 2 at path position s and wave vector K. Further, the temporal frequency w produced by each such K(s) is simply w(s) = K(s). v(s) (9) where v(s) linear velocity of the transverse turbulence at path position s By varying Kt and Kr so that s changes, one can profile C n 2 along the path by observing the variations in cr2 and measure the turbulence velocity from the temporal frequency w.
PAGE 15
CHAPTER3 EFFECT OF A SPATIALLY FILTERED APERTURE 3.1 Spatjal Filter at the Recejyer Terminal Aperture filtering has generally been considered in a limited sense, as the low pass filter action of a large aperture, in removing the effects of small scatterers from the receiver output. In fact, large apertures can be used to emphasize such small scatterers when the aperture illumination function is properly designed to provide a band pass spatial filter. To illustrate the effects of a spatially filtered aperture, assume the turbulence wavenumber K(s) at position s between the transmitter and the receiver, contains a single component K(s) as shown in Figure (3.1) and acts as a weak sinusoidal lens, imposing a sinusoidal phase modulation upon the wavefront of the probing beam. It is obvious that the effect of this component at the receiving plane (L), is a similar sinusoidal modulation of the received field, which spatial period is the projection of turbulence from the source to the receiving plane. If the turbulance has a velocity v(s), the disturbance at the receiving plane has a velocity greater by the factor of Us.
PAGE 16
8 T 2pi/Kr 1 Fig.(3.1) Sinusoidal turbulence at the receiving plane, produced by a sinusoidal refractive turbulence at s, of wave number K(s). The relative velocity and scale of the turbulence at the receiving plane are the projections of the turbulence at s from the transmitter A particular path has been set up with a point source at L=O, and a receiving aperture at L of such a nature that only a disturbance with a certain spatial wavenumber K causes variance at the receiver output. If the refractivity disturbance of all wavenumbers exist at all points along the path, only certain of these will produce an output at the receiver. Specifically, only those wave numbers K(s) that satisfy Kr =K(s) s/L, (1 0) because only these wavenumbers have K as their projection. The temporal frequency w produced by each K(s) is simply
PAGE 17
9 w(s) = K(s) v(s) (11) From the reasoning above, it is clear that the spatial filter allows only one wave number K at each position to contribute to the receiver output; this wave number K is different for each position. If the velocity is uniform, each of these wavenumbers may be associated with a unique temporal frequency in the receiver output. Thus, the position variable along the path is translated into a frequency variable. In order to calculate the disturbance position and its velocity, further information is needed and can be obtained by placing a second source in the same plane but at a distance d away from the first" source as shown in Figure (3.2). The fluctuations from these two sources are recorded separately, and the relative phase ( q>) of these fluctuation is determined for each frequency. From Figure (3.2), the separation of the two paths at position s is d' = d (Ls) I L Therefore, the magnitude of the phase difference is lq>(w)l = Kd (Ls) I L (12) (13) Using equation (1 0), (11 ), and (12) to solve for the path position s, the turbulence wavenumber K(s), and the turbulence velovity v(s) S= U(1 +lq>(w)IIKrd) K(s)= +lq>(w)I/Krd (14) (15)
PAGE 18
1 0 i d 2pi/K(s) R i Fig.(3.2) Two Transmitters configuration v(s)= w/(K+Iq>(w)l/d) (16) From chapter 2, it showed how the spatial filter interrogates the turbulent medium. In an actual implementation of a single spatially filtered aperture (at the receiver terminal), the ideal filter that was discussed can only be approximated. Consequently, the receiver will respond not only to a single wavenumber at a single path position, but to some finite resolution cell in both wave number and in path position. Also, a certain amount of measurement noise is unavoidable. This will limit the precision to which the measurement can be made. The receiver that most closely approximates the ideal receiver is probably the zero sum receiver shown in
PAGE 19
Log amplifier log amplifier 1 1 To Spectrum Analyzer Differential amplifier Fig.(3.3) Zerosum receiver configuration Figure (3.3). The incident light is collected by an aperture of radius r Behind the aperture is a spatial filter consisting of reflective stripes alternating with transmissive stripes. These stripes are of equal width, given by x/Kr so that the filter has a fundamental frequency of Kr The light transmitted by the filter is collected by a photodetector followed by a logarithmic amplifier. The reflected light is similarly detected by a second detector. The difference between these two detector outputs is then fed into a spectrum analyzer to provide turbulence
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12 strength as a function of height. To sample different regions of the turbulence spectrum, the effective frequency of the spatial filter would be changed. The filter just described in zerosum receiver is extremely difficult to treat analytically and is quite hard to implement. Thus, it is approximated by the normalized function fr(x,y) = (xr 2r1 exp[(x 2+y2)/2r2]cos(Krx) (17) An example of this function is presented in Figure (3.4). 3.2 Spatial Filter at Both Termini The scheme described in 3.1 has one drawback in the use of a second source at a known distance away from the first source, and a special configuration for the receiver. This arrangement may prove to be difficult and impossible for some applications. The most satisfactory solution, however, is by using spatial filters at both the receiver and transmitter terminals. This configuration is called a nonzerosum receiver. From the general geometry of Figure (3.5), the two filters act independently to weigh the effects of turbulence on the receiveing aperture, providing the apertures to be considered incoherent, so that powers may be summed without regard to carrier phase.
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v r1 I l, I I I I I I I I I I I I I I IV I I I I I I I I I .. r I I I I I I I I I I I I I : I I I LJ fA (x,o) rr ., 1 I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I 1 I I I I : I 111 I I I I I I I /' I I I I I I I I 1 1 I v I I I 1 I 1 1 I I : I I I I I 1 I I I I I I I I I vI I I I I I I 1 I 1 1 I I I 1 I : I I I I I 1 I 1 I I I I _.. LJ L_l 1__1 r 13 Figure (3.4) Zerosum spatial filter function(dashed line) and the corresponding approximation (solid line) as a function of the horizontal coordinate x for a circular aperture of radius r The receiving spatial filter acts on the projection Ks/L of the turbulence wave number and the transmitting acts on the projection in the reverse direction, K(Ls)/L, reflecting the reciprocity of the situation; the disturbance of energy from each point on each aperture must be weighted by the entire opposite filter.
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14 Kt=K(s)(Ls)/L Kr=K(s)s/L 4 s Fig. (3.5) Geometry of two filter at both termini If the two filters are resonated at spatial wave number Kt and Kr then and Kt = K(s)(Ls)/L Kr = K(s) s/L From equations (17) and (18) s = U [1 +(KrKr)] K(s) = Kt+Kr (17) (18) (19) (20)
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15 From equations (19) and (20), the resonant wave number and the location of the turbulence are determined by simple relationships that involve only the resonant wave numbers of the filters; the location of the turbulence can be found along the propagation path by simply varying the filter wavenumber Kt or Kr, or both Kt and Kr may be varied proportionally to vary K(s) while s remains constant. For ideal filters, all variance observed at the receiver output must originate at the known location s and resonant wave number K(s) of the turbulence. Also, since the temporal frequency appropriate to the resonant wavenumber K(s) of the turbulence is simply K(s)v(s), measurement of the fluctuation frequency leads immediately to the determination of the turbulence velocity. The configuration of spatial filters at both termini (nonzerosum receiver) has several advantages over the zerosum receiver (singl e spatially filtered aperture) configuration. In this configuration, the receiver consists of a single detector and a spatial filter that. transmits alternate strips. The approximate normalized function for this receiver is given by fr(x,y) = (2xr2r1 exp[(x 2 +y 2 )/2r 2 ][1 +cosKrxl (21) Which is plotted along with the actual function in Figure (3.6)
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16 fR(x,o) r, r, r r, ,., I I I I I I I I I I I" I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I X r Fig.(3.6) nonzerosum receiver filter function (dashed line) and its approximation (solid line). Since there is only one detector to contribute to the noise of the nonzerosum receiver, the signaltonoise ratio could be improved and the actual implementation of the system is much simpler than the zerosum receiver configuration.
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CHAPTER4 WEIGHTING FUNCTION To investigate the expected performance of a given spatial filter system theoretically, it is necessary to calculate the corresponding path and wave number weighting functions that describe the sensitivity of the system to difference path positions and spatial wave numbers of the refractive turbulence spectrum. 4.1 Path Weighting Eunctjon With an assumption that the path dependence of the refractive index spectrum is due solely to Cn 2 variations along the path, then (K,O,s) = Cn 2(s) '(K,O) (22) By rewriting Eq.(1 ), a path weighting function P(s) for en 2 profiling is defined as rl= 4Cx(O) = J ds Cn 2 (s).P(s) (23) Where P(s) = 81tk2 L J d2K'(K,O) sin2[K2Ls(1s)/2k] lft[K(1s)]I 21Er[Ks]l2 (24)
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18 Calculation for the modulation power spectra of the transmitter 1Ft12 and rece iver 1Fr12 is needed to further evaluate of Eq.(24). A simple way to generate spatial filters is to mask the transmitter and receiver apertures with equally spaced parallel strips of tape. The resulting modulation function f(p) is zero outside the aperture and wherever there is a strip of masking tape, and is one elsewhere. The spatial wavelength A. of the filter is the distance between the centers of the stripes of tape. For striped cicular apertures, which are the most likely to be encountered in practice, the spatial power spectrum is found to be Where IF(r)l 2 = IF(r xr y)l2 =I :Esinc(n7r12) exp(inKtA) r : radius of the circular aperture : spatial shift of the stripes Kf : spatial wave number of the filter n : number of stripes across the aperture J1 c(X) = J1 (X) I X (25) Equation (25) can be used in equations (24) and (33) to generate the path weighting and wave number weighting
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19 function for both the zerosum and nonzerosum filter system. The resulting expressions are complex and require numerical analysis technique for evaluation. Analytical tractable expressions that yield some physical insight can be derived by using an approximate form for the aperture modulation functions as mentioned in Chapter 3. Consider the case of a nonzerosum circular Gaussian aperture with a perfectly sinusoidal modulation and arbitrary rotational orientation with modulation filter function as shown by Fig.(3.6). IF(r)l 2 = 1/4(exp(r2r2t2) + 1/4(exp[(r+K) 2 r2t2] + exp[{rK)2r2/2])) {26) Equation {26) is an approximation to the modulation power spectrum of the nonzerosum vertically strip ed spatial filter described by Eq.(25). It is reasonably accurate, provided there are many spatial cycles acrsoss the aperture. With the assumption that the Cn 2 distribution does not favor the ends of the path and there is no rotational mismatch of the spatial filters, the path weighting function can be approximated by P{s):: 8.48*1 o2L 2{s){s2(1s)2/{s2+{1s)2) 7/6 + C'{s) exp [{ss')2t..f(s)]} (27) Where
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20 y(s) = {2[s2+(1s) 2 ]} 1 2/[(1 Kt + Kr I) r] (28) C'(s) = 0.48[r/(Uk) 1/211/3 [(Kt+Kr)(Uk) 1/2r11/3 [(1s)2+s2]8/3 {1+RI[1+s(R1)]}11/3 sin2 {1/2[(Kt+Kr)(Uk) 1/2]2 *[1+s(R1 )/(1 +R)]2s(1s)/[s2+(1s)2]2} (29) R = (30) (31) The relative contribution of each term in Equation (27) to the signal variance is depended on the size of C'(s) and the width y(s). C'(s) is a strong function of the quantity HKt+Kr)(L/k) 112r11/3 which reflects the refractive index spectral weight associated with the sampled wave number K=Kt+ K r For smaller transmitter and receiver spatial wave lengths, the magnitude of the second term relative to the first term decreases because the spatial filter component of the signal comes from smaller scales in the turbulent flow. The width 2y(s) = 2[(Kt+Kr)rr1 of the also decreases, reflecting the reduction in size of the scattering volume that contributes to the spatially filtered component of the signal. Figures (4.1 ), (4.2), and (4.3) show the approximated
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21 normalized path weighting function P(s) with 6 spatial cycles across the transmitter and 2, 6, and 18 spatial cycles across the receiver. ... ... ... The approximated path weighting functions exhibit an enhancement at the expected path position of 1/4, 1/2, and 3/4 path length. .c 0 a; 3: cG "CD ... 0 .25 .5 .75 normalized path position Fig.(4.1) Normalized path weighting as a function of the normalized path position for 6 cycles transmitter and 2 cycles receiver spatial filters
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22 1 0 .25 .5 .75 normalized path position Fig.(4.2) Normalized path weighting as a function of the normalized path position for transmitter and receiver with 6 cycles spatial filters ..r:. 0) a; == Q).5 > ::::: CG normalized path position Fig.(4.3) Normalized path weighting as a function of the normalized path position for 6 cycles transmitter and eighteen cycles receiver
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23 4.2 Wave number wejghtjng fynctjon . The wave number weighting function can be defined as (32) Where W(K) = 8xk2Ltl>'(K,O) J ds C0 2(s) sin2[K2Ls(1 s)/2k] *1Ft[ K(1s)]I 21Fr[Ks]l2 (33) A probJem with this definition for W(K) is that it is a function of the distribution at Co2 along the path. If the distribution is known, then the resulting expJession for W(K) will accurately describe the contribution of different wave numbers to the spatially filtered signal. However, the of Co2 is not known. avoid the problem, assume that Co2 is constant across the resolution cell of the system, then replace it with its value at the expected sampling position s, and remove it from the intergrand. Subject to the same approximations used in 4.1 to obtain Equation (27), the wave number weighting function is Where W(K) = 5.18*102k2Lc02K11/3{D(K)exp(K2/A2) +H(K)(exp[[K(Kt+Kr)]2/A2) (34)
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D (K)=1t 1 /2/Kr[erf(Kr/2)]sin2 (K.2L/8k) H (K) = 1 /32{(7t) 1 /2/Kr[erf[Kr(1s')]+erf(Krs']} sin 2[K2Ls'(1s)/2k] 24 *exp[ r2(KtKr)2 si n 2 e/4], 0, otherwise 0 s s' s 1 d = 2/r s' = K[K(KtKr)]/2K2 e =angle between K and (KrKr) K : magnetude of K erf : error function (35) (36) The first term in Equation (30) describes the unmodulated whole aperture response and indicates that only scales larger than the aperture (i.e; K s 2/r) contribute to this component of the signal. These larger spatial scales correspond to the background level in Figures (4.1 ), (4.2), and (4.3). The second term describes the effect of the aperture modulation, which samples the flow at spatial scales corresponding to K=_Kt+Kr These scales are much smaller than those that contribute to the first term, and correspond to the spatial filter enhancement of the path weighting functions in Figures (4.1 ), (4.2), and (4.3). The spectral resolution associated with the spatial filter component of the
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25 signal is .& = 2/r. The location of the peak response is at K=Kt+Kr so that if rlK(s)+Krl >> 1, the low wave number and spatial filter components will be well separated. For a vertically oriented and match spatial filters system, Equation (34) is reduced to Where With W(Kx,Ky) = 5.8*1 o2k2Lcn 2
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26 Kx,Ky describe the horizontal and vertical wave number. To obtain a clearer view of the effect of the filter wave number upon a spatial filter system, the output power spectrum as a function of the filter wave nu. mber should be observed. This is shown by Figures (4.4), (4.5), and (4.6), providing the transmission function of the aperture contains only the horizontal component and the transmitter fundamental wavelength of 2.54cm, which corresponds to the receiver fundamental wavelength of 2.54cm, 1.51 em, and 15.2cm respectly . 31E 2 .210 Q) a. en .1r1 0 A I 5 1 L A I 0 5 10 Wave number Fig.(4.4) Power spectral density of the receiver filter as function of wave number (2 pi/2.54 em)
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E 2 .210 Q) c. C/) .111 0 I 5 I I 0 5 1 0 Wave number Fig.(4.5) Power spectral density of the receiver filter as function of wave number (2 pi/1.52 em) 1 0 10 Fig.(4.6) Power spectral density of the receiver filter as function of wave number (2 pi/15.2 em) 27
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CHAPTERS EXPERIMENT An experiment was performed in the Laboratory to verify the theory described in previous chapters. A diagram of the experimental configuration is given in Figure (5.1 ). 5.1 Spatial Filters Several attempts to use masking tape on a piece of glass proved to be futile due to the size constraint. As an alternative, a Me Draw program was used to generate the needed stripes for different wavele ngths. The hard copies were then transferred to the transparencies. This method was quite good for spatial wavelengths greater than 4mm, but proved to be less than satisfactory for smaller wavelengths. Filters with wavelengths of 2mm, 4mm, 6mm, Smm, 12inm, and 16mm were generated using the described method, Figure (5.2) showed 4mm spatial filter. 5.2 Transmitter Source Any incoherent light (incandescent, L.e.d, etc .. ) could be used for a transmitter source, but for possibility future longer range experiment, a Laser source was chosen.
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colimated incoherent beam ( grounded glass I transmitter L 8 s e r density filter computer [j][j] Osciloscope spectrum analyzer X as C) c ::! spatial filter heat source ,J' receiver spatial filter ............. interference filter Fig.(5.1) Diagram of experiment configuration 29
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4mm Fig.(5.2) 4mm spatial filter 30 In this experiment, a HeNe Laser ( Spectra Physic Stabilite, model 124A ) was used. This Laser put out 15mW of optical power at a wavelength of 632.8nm. To convert a coherent Laser beam into a incoherent light source, a grounded glass portion of a microscope slide was put at the Laser output and placed at the focus of a 25cm focallength lens. The clear aperture of the lens was a round of 1 Ocm in diameter. An adjustable aperture was used to reduce the collimated incoherent beam from 1 Ocm to Scm cross section diameter. To avoid saturation and to improve signal to noise ratio, a neutral density filter (16% transmission) was placed in front of the grounded glass. 5.3 Turbulence Source
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31 The turbulence source selected was a heat gun that blows hot air out through a 5 em diameter barrel. This was placed at the same height as the optical path at a horizontal distance of 6 inches. The jet from the heater was characterized roughly with an thermometer and a 6 em diameter impeller anemometer as shown in Figure (5.2). 5.4 Receiver system The receiver system consisted of a 6 em diameter lens with a focal length of 25' em to focus the optical signal onto the photodetector. The detector was a photodiode with a JFET operational amplifier builtin (EG&G HUV11 008), the opamp was configured as a transimpedance amplifier with feed back resistor of 2 Mohm. The output of the amplified detector was coupled to a voltage amplifier to increase the output signal amplitude. To block background noise, the collected optical signal from the collecting lens was passed through an interference filter with a passband of 2Qnm centered at an optical wavelength of 632.8 nm. 5.5 Recording and Processing InstrUments Electrical output of the signal was then connected to an Osciloscope, a Spectrum analyzer, and a PC with builtin AID board (DT 2814).
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32 V(m/s) 45"C 145"C (5 m/s) .,.._scm_,. Heat gun Fig. (5.3): Temperature and velocity profile of the heat source
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33 The Osciloscope was used to align the system, primary evaluation of the system feasibility was done by the Spectrum analyzer. The PC was then used to record and process the signal. 5 6 Procedure Data was recorded for the Spatial Filter of 2mm, 4mm, 6mm, and 8mm wavelength which shown by Figures (5.4), (5.5}, (5.6}, and (5.7), respectively. Each data run consisted of 1 000 sample points at the sampling rate of 20KHz. Before the recorded data got processed further, a base line of the system response was established to observe the response of the unmodulated aperture on the turbulence (no filter, heat gun on) in the time domain, shown in Figure (5. 8), and its power spectral density, which is shown in Figure (5.9). To check the noise level of the system, a dry run (no filter, heat gun off) was performed and is shown in Figures (5.1 0) and (5.11 ). To enhance the peak response visually, a "normalized" power spectral density function U(f} was defined as: Where U(f) = W(f)*f/ f W(f) df f : frequency in Hertz W(f} : power sectral density of the signal Integration upper bound = 1OKhz
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34 Integration lower bound OHz Figures (5.12) and (5.13) showed the graphs of U(f) v.s frequency for the unmodulated aperture and the system noise. Figure (5.9) shows that the unmodulated aperture acts as a low pass filter with a cut off frequency at about 500Hz. Figure (5.11) shows that the system noise is a periodic function riched in harmonics with the fundamental frequency is at about 60Hz and its harmonics are distributed to the full spectrum of the system with bias in the low (<500Hz) and high end (>2500Hz). For this reason, the experiment was limited to spatial filter with a wavelength of 2mm, 4mm, 6mm, and Smm. A signal of the system that has wavelength longer than Smm or shorter than 2mm would be buried under the noise floor.
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0. 15 tIa ll : Ill .Ill O. li'i .M I Q) 'C :::J i 0. 05 M 1'11JI 'IIIII .,,.... 111111111 Ill llll llllllllll .. If J jl I ,.1 I II II II 0 100 200 300 400 500 600 700 800 900 1000 Number of data Figure (5.4): Output signal of the 2mm filter system. VJ (]1
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0.3 0.25 0.2 I .I I II II I I 0.15 .... I I l ..... 0 0.1 Q) L I "C N I ::I tl 0.05 I ..... C. e < I' I !liN II I II U' r 1m q 0 0.05 I I I I I I 0.1 0.15 0 100 200 300 400 500 600 700 BOO 900 1000 Number of data Figure (5.5): Output signal of the'4mm filter system. w 0)
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". (/) 'Q) .:: .= E E <0 Q) ..c ..... 0 ca 1:: 0) u; ..... a. ..... 0 (0 1.() Q) '0) u:::
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0.4 ...., 0.3 to 2 !t II I I Ill U Ill 'j 11 111 n 11 llllflll I 11111111111 I fi I' I 1111 II IJ I filii I rt 0.1 l Ni M II \ CIJ 'C :l .. ;:: 0 r tn i Q. e < 0.1 I I 0.2 t0 100 200 300 400 500 600 700 BOO 900 1000 Number of data Figure (5.7): Output signal of the 8mm filter system. w (X)
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2 1.5 1 0.5 ri 0 0 > ..... Q) 'C :J .... 0.5 ri c: e < 1 1.5 2 2.5 0 100 200 300 400 500 600 700 BOO Number of data Figure (5 8): Output signal of the unmodulated aperture. 900 1000 w c.o
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e :I c.. ., u Q) U) x104 2 1.8 1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 0 r w eh A h b I I I I I I I 0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 Frequency (Hz) Figure (5.9): Power spectral density of the unmodulated aperture. 0
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0 0 0 ..... 0 0 01 0 0 IX) 0 0 0 0 10 0 0 10 0 0 "'f 0 0 ('I') 0 0 (\J 0 0 ..... 0 ('I') 0 10 (\J 0 (\J 0 10 ..... 0 0 10 0 0 0 ('UOA) apnntdW\f 10 0 0 I 0 I 0 I (\J 0 I 41 E Q) (/) >. (/) 10 Q) ...., ..c 10 'C 0 .... Q) 0 'Ql 0 .Q z e :l z 0 T'"'" lt) .......... Q) .... ::l C> u:::
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100 90 80 70 60 e :J '50 u Q) a. (I] 40 30 20 10 0 0 500 1000 1500 .2000 2500 3000 3500 4000 4500 5000 Frequency (Hz) Figure (5.11 ): Power spectral density of the system noise. 1\)
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1 ..0.9 0.8 0.7 0.6 ... 0.5 ::> 0.4 j 0 .. ,. .A.Je\.,M,...Ln+A' 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 Frequency (Hz) Figure {5.12): "Normalized" power spectral density of the unmodulated apperture. w
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0.5 0.45 0,4 0.35 0.3 .... !: 0.25 ::l 0.2 0.15 0.1 0.05 0 0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 Frequency (Hz) Figure (5.13): "Normalized" power spectral density of the system noise.
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CHAPTERS ANALYSIS AND CONCLUSION 6.1 Analysis For the described spatial filter system (double spatially filtered apertures, non zerosum receiver), with the wind velocity of Sm/s. The theoritically calculated peak frequency responses are: Filter wave length(mm) 2 4 6 8 Frequency(KHz) 5 2.5 1.67 1.25 Figures (5.14) and (5.15), (5.16) and (5.17), (5.18) and (5.19), (5.20) and (5.21) showed the power spectral density and its "normalized" for the 2mm, 4mm, 6mm, and 8mm filter, respectively. For the 2mm wavelength spatial filter, a peak at 5KHz was expected. Inspecting Figures (5.14) and (5.15) shows no such peak. In Chapter 3, it was proved that a spatial filter allowed only one wave number of the turbulence at each position s to contribute to the receiver output, and the turbulence wave number must satisfy the relation
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100 90 80 70 60 e ;;, c.. +J 50 u Q) c. en 40 30 20 10 0 0 500 1000 1500 2000 2500 3000 3500 4000 4500 Frequency (Hz) Figure (5.14): Signal power spectral 9ensity of the 2mm filter system. 5000 0')
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0.12 0.1 0.08 !; 0.06 ;:) 0.04 0.02 0 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 Frequency (Hz) Figure (5.15): "Normalized" signal power spectral density of the 2mm filter system. .....1
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E :l c.. +' u cu c. en 102 101 100 0 500 1000 1500 2000 2500 3000 3500' 4000 4500 Frequency (Hz) Figure (5.16): Signal power spectral density of the 4mm filter system. l ..., l I .., 5000 00
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0.3 rl 0.25 0.2 ..._ .:= 0. 15 :::J 0.1 0.05 : 0 .. n r 1 ', ,, .. ,, ., y I''"Y ''""l..fttAJLI vnu vu vu ,LAY IV \IM\J!\AVJI 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 Frequency (Hz) Figure (5.17): "Normalized" signal power spectral density of the 4mm filter system. (.0'
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E ::J t. ..... u Q) a. en 103 r102 r101 100 0 ,., I I ' 1 500 1000 1500 2000 2500 3000 3500 4000 4500 FreQuency (Hz) Figure (5.18): Signal power spectral density of the 6mm filter system. ' !11 5000 01 0
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..... .... ..., ::J 0. 25 r.rr ,I 0.2 0.15 0.1 0.05 O Ul t I I' II j I Ill ,Ill I ... I J"t'jY''. 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 Frequency (Hz) Figure (5.19): "Normalized" signal power spectral density of the 6mm filter system. 01 _..
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e :J c.. ...., u QJ a. en 103 102 101 100 0 r" II It I 1. I II 500 1000 1500 2000 2500 3000 3500 4000 4500 Frequency (Hz) Figure (5.20): Signal power spectral density of the 8mm filter system : : . 5000 01 1\.)
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0.35 0.3 0.25 .... 0.2 ::J 0.15 0.1 0.05 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 Frequency (Hz) Figure (5.21): "Normalized" signal power spectral density of the 8mm filter system. 01 (...)
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54 K(s) = Kt + Kr In the case of a 2mm filter, K(s) must equal 47t/.002 m1 and from the above observation, it is known that the turbulence did not contain such wave number.Visual inspection of Figure (5.17) shows a hump starting at about 1.5 KHz, peaking at about 2.3 KHz and then gradually falling off at higher frequencies. Inspection of Figure (5.18) shows a fast rise at about 800Hz, peaking at about 1.3 KHz and falling off slowly at higher frequencies. Inspection of Figure (5.20) shows a fast rise at about 500 Hz, peaking at about 1KHz and tapering off at higher frequencies. To understand these discrepancies, recall that a uniform wind velocity was assumed and with this assumption the high frequency cut off is expected to be sharp The turbulence generator (heat gun) has some maximum velocity in the center of the stream and its velocity decreased at the edges as shown in Figure (5.3). Also, the velocity of the heat gun decreased when the heating element kickedin, and increased when the heating element kickingoff. Therefore, the measured values showed a slightly shift in the peak frequency responses and slowly rolled off at higher frequencies Although the output frequencies peaked at a lower
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55 value than calculated, and their peaks were spread over two decades Figures (5.16), (5.18), and (5.20) demonstrated that they can be discerned for the particular turbulence generator that was used in these experiments. Since the technique depended only on the ability to recognize the peaks and not on theirs shapes, the discrepencies mentioned above should not downgrade the usefullness of the technique. To obtain data for the path weighting measurements, the heat gun was moved along the propagation path Figures (5.22), (5.24) shows that the "normalized" power spectral density of the 4mm and 8mm filter system, with the heat gun set at 1/3 path length. Figures (5.23) and (5.25) shows the "normalized" power spectral density of the 4mm and 8mm filter system with the heat gun set at 2/3 of the path length from the transmitter. From consideration of their magnitudes at the corresponding peak frequency responses, and a comparison of their magnitudes, taken with the heat gun set at 1/2 of the path length, it is clear that for the same filter wave length at both termini, the most sensitive position is at 1/2 of the path length.
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..... ..... ..... ::l 0 25 ., 0.2 0.15 0 1 0.05 0 '" ,.. 1 w 1 ,,,, 1 I r 1 I 1 n r rt r\f u Y"' !\ 1v ... u JW lJNI 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 FreQuency (Hz) Figure (5 22): "Normalized" power spectral density of the 4mm filter system with heat source at 1/3 of propagation path length. 01 0)
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.... .... ...... 0.2 0.15 0.1 0.05 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 Frequency (Hz) Figure (5.23): "Normalized" power spectral density of the 4mm filter system with heat source at 213 of propagation path length. (J1 ....,J
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0.6 .,0.5 0.4 0.3 => 0.2 0.1 om . ,.,.Ill, ... I ................. .. ........ 0 nnn "'""" .,.,,... __ _. ___ 10vv 2000 3000 4000 5000 6000 7000 8000 9000 10000 Frequency (Hz) Figure (5.24): "Normalized" power spectral density of the 8mm filter system with heat source at 1/3 of propagation path length. U1 (X)
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0.3 0.25 0.2 .... !; 0.15 ::> 0.1 0.05 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 Frequency (Hz) Figure (5.25): "Normalized" power spectral density of the Bmm filter system with heat source at 2/3 of propagation path length. 01 c.o
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60 6.2 Conclusion: An experiment was performed in the laboratory with a simulated turbulent atmosphere to verify the feasibility of spatial filtering techniques for remote measurement of wind velocity in the turbulent medium. In the analysis bias of the path position and the effect of the unmodulated aperture were observed. With the assumption of constant uniform wind, the theory predicts that the power spectral density of the receiving signal would have a narrow peak at zero frequency and another peak at an angular frequency given by the product of the transverse wind velocity and the sum of the transmitter and receiver spatial wavenumbers. From chapter 5, the stream velocity profile shows that the assumption above was not valid, therefore, the peaks were spread according to velocity fluctuations. Small laboratory space imposed the size constraint on the system resulting in poor resolution. Although the overall results were not as good as expected, the experiment showed that spatial filtering techniques for remote measurement of wind velocity in the turbulent atmosphere is a useful technique, and should be further perfected.
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BIBLIOGRAPHY (1) S.F. Clifford and R.J. Lataitis, "Spatial and Temporal Filtering of Scintillation in Remote Sensing," IEEE Trans. Antennas Propag. AP35,597 (1987). (2) Tingi Wang, S.F.Ciifford, and G.R. Ochs, "Wind and RefractiveTurbulence Sensing Using Crossed Laser Beams," Appl. Opt. 13, 2602 (1974). (3) S.F. Clifford and J.H. Churnside, "Refractive Turbulence Profiling Using Synthetic Aperture Spatial Filtering of Scintillation," AppL Opt. 26, 1295 (1987). (4) R.W.Lee, "Remote Probing Using Spatially Filtered Apertures," J. Opt. Soc. Am. 64, 1295 (1974). (5) R.E. Good, B.J. Watkins, A.F. Quesada, J.H. Brown, and G.B. Loriot, "Radar and Optical Measurements of Cn 2," Appl. Opt. 21' 3373 (1982). (6) G.R. Ochs, Tingi Wang, R.S. Lawrence, and S.F. Clifford, "Refractive Turbulence Profiles Measured by OneDimensional Spatial Filtering of Scintillations," Appl. Opt. 15, 2504 (1976). (7) J.H. Churnside, R.J. Lataitis, and R.S. Lawrence, "Localized measurements of Refractive Turbulence Using Spatial Filtering of Scintillations," Appl. Opt. 27, 2199 (1988).
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62 (8) R.W. Lee and J.C. Harp, "Weak Scattering in Random Media with Applications to Remote Probing," Proc. IEEE 57, 375 (1969).

