A meteorological radar calibration technique is developed. It is found that the integrated, range corrected, received power saturates under intense rain conditions in a manner analogous to that encountered for the radiometric path temperature. Furthermore, it is found that this saturation condition establishes a bound which may be used to determine an absolution radar calibration for the case of radars operating at attenuating wavelengths. In the case of less intense rainfall or for radars at nonattenuating wavelengths, the relationship for direct calibration in terms of an independent measurement of radiometric path temperature is developed. This approach offers the advantage that the calibration is in terms of an independent measurement of the rainfall through the same elevated region as that viewed by the radar

Hodge, D. B.

English

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https://ntrs.nasa.gov/search.jsp?R=19780020357 2020-03-22T02:47:52+00:00Z
NASA M
METEOROLOGICAL RADAR CALIBRATIO3
D. B. Hodge
(WkSk-cR-156797) KZTNOROLOGICkL RkD1R	 978-28300
CALIBRATION (ohio State Univ., Columbus.)
Is P vC A02/FT 101	 CSCL 171
anclas
G3/32 27066
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Technical Report 784650-1
February 1978
4W
44a7a
National Aeronautics and Space Administration
GODDARD SPACE FLIGHT CEWER
Greenbelt, Maryland 20111
zL
l
NOTICE
r'
;:r^n Gcwern ant dr_..in;s, s:- c_cific:tiOns, or , other data are
fkk
usc' for any purpose other than in connection with a definitely
r':'	 r:. rt pro,:'.;rte•,?nt operation, tare United States
G;,.4rr; -ant thee:	incurs no responsi bi 1 i t : nor any cUl i ration
or, -.- •.d the fact that the Goverment may have formulated,
,:rr.isj mod, or in an:, way supplied Oie said dr,:::incs, s, :cifications,
or other data, is not to be reiarded by ir • plicaticn cr • oti,er,.:ise as
in	 r-anncr 1zC. ;S't' the hoi''_er or un y Oth r person or' corporation,
or con wing an:, riatlLs or permission to manufacture. use, or sell
patented in.crticn t -! t r y in ;ny ::ay be related thereto.
nRt^^NAL 4 4
AL ^ LITT
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CONTENTS
Page
I. INTRODUCTION	 1
II. THE RADAR RANGE EQUATION AND ITS SOLUTION	 2
III. THE RADAR SATURATION FACTOR	 9
IV. RADIOMETRIC TEMPERATURE	 11
V. RADAR CALIBRATION	 12
VI. SUMMARY	 13
REFERENCES	 14
1
l
I
1
7
7
I.	 INTRODUCTION
The use of radar for the remote quantitative measurement of rain
rate provides a powerful tool for those concerned with the distri-
bution and occurrence of rainfall in the atmosphere. A knowledge
of the spatial and temporal distributions of rainfall is essential
	
1	 in such diverse disciplines as meteorology, hydrology, and milli-
meter wavelength communications. Radar has been extensively used
for applications such as these since the development of meteorological
radar techniques in the 1940's. Nevertheless, the use of this approach
has been handicapped by a•lack of adequate precision in the resulting
quantitative rain rate data: for example, Harrold and Nicholass [1]
report average uncertainties in such radar measurements ranging from
30 to 1001. These uncertainties are largely attributable to the
difficulties encountered in performing an adequate calibration of
the radar measurement system.
Traditionally two distinct techniques have been used for radar
calibration. The first method utilizes a metallic sphere or other
target having a known radar cross section to establish an absolute
calibration. Unfortunately, this method is based upon the use of a
point target and does not compensate for the effects of beam-filled
targets such as those usually encountered in rain rate measurements.
Furthermore, it is often quite difficult to place such a standard
calibration target in the far y ield of the radar antenna and at a
height sufficient to eliminate ground reflections. The second common
method is to calibrate the radar observed rain rate against the rain
rate measured on the ground below a given radar resolution cell.
This approach obviously ignores differences resulting from both
vertical and horizontal wind components since the two measurements
to be compared are made at two different spatial locations. This dif-
ficulty is compounded by the fact that the radar measurement must
be elevated in order to avoid ground reflections.
This report addresses the problem of radar calibration from
several different directions. It is shown in the following that
	
^j	 the integrated, range-corrected radar backscatter saturates in a
	
J	 manner analogous to that found in radiometric temperature measure-
ments. Furthermore, it is shown that this saturation leads to an
absolute calibration method for radars operating at attenuating wave-
lengths. The method for calibration of meteorological radars in terms
of independent radiometric temperature measurements is also developed.
This technique offers the advantage that the calibration is based
	
j	 upon an independent measurement made throughout the same spatial
	J	 region as that viewed by the radar.
..
Q = n 5 1 kl 2 f2 G T Pt
210. Rn2 • c
(2) [watts/meter]
II. THE RADAR RANGE EQUATION AND ITS SOLUTION
The development below follows closely that originally presented
by Hitschfeld and Burden [2]. Here it i s assumed that a narrow
beam, short pulse radar is viewing a region of rainfall which fills
each radar resolution cell homogeneously. Then, if the power pattern
of the radar antenna is assumed to be Gaussian, the following radar
range equation results [3]:
(1)	 Pr = 2- Z e-2ARD [watts]
r
where Pr is the time average received power backscattered from a
resolution cell located at a range of r meters from the radar. The
radar calibration constant, Q, is
where
102  = (n2-1 12
n2+2
n = the complex refractive index of liquid water
(3) f = radar frequency [HZ]
G = maximum antenna gain
T = radar pulse length [sec.]
Pt = peak transmitted power [watts]
c = velocity of propagation [m/sec].
The equivalent reflectivity, Z, has been defined in terms of the
volume6.• -ic radar cross section, [m2/m3]:
(4) Z =	 c4n	 [m6/m3].
n5f4 l kj 2
ARD is the one-way attenuation between the radar and the resolution cell
being observed expressed in terms of specific attenuation, apD[nepers/m],
as
2
lJ
7
7
a
(5) ARD = jr aRDdr [nepers].
0
jSeveral additional assumptions will be incorporated in the following
development. These include the assumption that both the specific at-
tenuation, a, and the reflectivity, Z, may be represented at any fre-
quency by a power law dependence on the rain rate, R [nm/hr], i.e.,
b RFi(6) -
aRD - a RD R
RM	 RM
and
(7) Z=azR 
b
z
where the subscripts RD and RM indicate that the subscripted quantity
is to be evaluated at either the radar or radiometer frequency, re-
spectively. Calculations based on Mie theory for various drop size
distributions [4,5] indicate that this is a reasonable assumption
even for frequencies where the drop sizes are resonant; this is dem-
onstrated by Figures 1 and 2 which show regression power low fits to
Mie calculations at frequencies from 1 to 500 GHz for Marshall-Palmer
drop size distributions at 0°C. The a and b constants in Equations (6)
and (7) are, of course, frequency dependent and have been tabulated for
extensive frequency ranges in Reference 4 for the case of specific
attenuation and in Reference 5 for both specific attenuation and
reflectivity.
It will also be assumed that attenuation and radiometric temperature,
Ts [°K], are related by
TM(8) A,. = Zn ^Tm - Ts
where Tm [°K] is the mean absorption temperature of the medium and the
subscript - indicates that the attenuation is associated with a path
extending to a resolution cell located at infinity [6]. This theoretical
relationship is strictly valid for media where scattering is negligible;
however, it has been demonstrated empirically that it is quite adequate
for the rainfall case at frequencies at least up to 15 GHz [7,8]. An
example of this agreement is shown in Figure 3 for the case of simul-
taneous 15 GHz radiometric temperature measurements and attenuation
measurements using the ATS-5 synchronous satellite beacon.
Now, proceeding with the solution of the radar range equation, the
rain rate, R, is elim i nated from Equations (6) and (7) yielding
J
3
•
lb 	ATTN. VS- RAIN-RATE
b
^/ / • e
max, ^^ '	 " / •	 A10.
b	 '
./	 x/	 ,J+
Im •/	 •-o	 •
c06
t
-
X
x	 ,•	 /fit
x• x/
g•
L
TO
RAIN RATE (MM/HR)
Figure 1. Specific attenuation, a , vs. rain rate
at several frequencies.
4
i1
1
7
D
1
7
b	 Z-EO VS. RAIN-RATE
	
i	 . x'
/
	
X. •	 x80•
	
,P •	 r Itoo.
.•	 Xr
m	 le op	 , /	 39 00	 X100.
L	 .x'^	 •
	
^•	 r.r ^r
	
r	 ' r	 X150•
	
ow., 
r	
Ix r	
Agoo.
Cy
LLJ-b
	
^^^+'•r,XI
X^ x r •
. 
• /	
'loo.
b^•r.x^
.^.
• •
b00-1 • •
Tb
y d°	 3 t 6fbby0' ^§ u 1 obij(f i
RAIN RATE (MM/MR)
Figure 2. Equivalent reflectivity, Zeq, vs. rain rate
at several frequencies.
5
i
^i
i
R
-122
—124
—126
— 128
—14,
SIGNAL LEVEL	 • o •
	
vs	 %
RADIOMETRIC TEMPERATURE
	 •
f=15.3 GHz	 .•,
'	 SITE NO. 1
JUNE 17, 1970
2315z — 2330z
10 SAMPLE AVERAGE
A=In( TM3	 Tm Ts
A • ATTENUATION
Ti =RADIOMETRIC TEMPERATURE
Tm = MEAN ABSORPTION TEMPERATURE
(TM = 278 • K)
E —13(
m
v
w —135
WJ
a —131
z0
N
—13
—14
—14
0	 100	 200	 300
T (•K)
Figure 3. A comparison between measured attenuation and measured
radiometric temperature at 15 GHz [8].
6
j	 bRD
Z	 z
(9) aRD = aRD az
Combining Equations (1), (5), and (9), the radar range equation is
expressed in terms of equivalent reflectivity:
-2r a
RD	 fr ZbRD/bzdr*
(10) r Pr
2 
= QZe laz bRD/bz 	o	 J1
Making the substitutions
aRD_(11) 9 2 az bRD/bz
and
(12) Y = Z-bRD/bz
for simplicity, yields
r 1
(13) r2Pr = Q y
-bz/bRD -ga o y dr.
Now, the logarithm is taken of Equation ( 13); the result is then dif-
ferentiated with respect to r and rearranged:
I	 (14)	 1 + b D^ , d[knd(r2Pr)] 	 bRDz	 y = -	 g.
The solution of Equation (14) is
bR0
b	 (r
1	 (15)	
c - ^' g o (r2 Pr)Z dr
Y a
(r2P 
r)
bRD/ z
.1	 where c is an unknown integration constant. Substituting for g and y,
the solution,
Ir2Pr(16)	 Z =
[c-2 bRD aRD
fo
r 2 bRD/bz bz/bRD
'^ azRD zb /b 	
(r Pr)	 dr]
is obtained.
1
Z4
i
The integration constant, C, may be evaluated by considering the
simple hypothetical reflectivity distribution shown in Figure 4.
ro
	
r
Figure 4. Reflectivity distribution used to evaluate the
unknown constant of integration, c.
Since the attenuation over the interval from r=0 to ro is zero,
Equation (1) becomes
(17) Pr = Q2 Z 
0
when evaluated for r=ro. Similarly, Pr=O for 0<r<ro and, therefore,
Equation (16) becomes
2
(18) Zo = r o
P
rc z_6/bRD
when evaluated for r=ro. Consequently,
(19) c a QbRD/bz .
Substituting Equation (19) into Equation (16), the following
compact form of the solution of the range equation is obtained
(20) Z = r2P r
Q(1 -Id bz/bRD
8
1 .
a
where
(21) Ir = 2aRDbRD	 (r (r2pr)bRD/bz dr .
p
z(
a
zQ ) 
b
RD 
/b2 
10
III. THE RADAR SATUR- ­1ON FACTOR
The quantity Ir has a number of interesting properties. For example,
comparing Equations (1) and (201 it is noted that
(22) ARD=An_1
(1-I0bz/2bRD
Thus, the quantity Ir is intimately related with the path attenuation.
Furthermore, equating Equations (8) and (2[) for a resolution cell
located at infinity, it is found that
(23) 1 - T$	 (1-I^)bz/2bRDTM
where, again, the subscript W indicates that the integration is to be
performed over the entire range from r=0 to -. Here it is seen that
Ie
 is closely related to the ratio of the radiometric sky temperature,
Ts ,to the mean absorption temperature, Tm. Since the ratio T S/Tm is
a measure of the degree of saturation associated with the radiometric
temperature measurement, it is reasonable to associate that inter-
pretation with the radar case as well. Thus, the quantity Ir will be
referred to as the radar saturation factor.
The radar saturation factor, Ir, is unitless, ranges from zero to
one, and increases monotonically with range. Examining Equation ;22)
it will be noted that, as Ir approaches unity, the path attenuation
becomes extremely large.
Examining Equation (21), it is noted th4t the calculation of Ir
involves a very simple range correction and integration of the received
backscattered radar power. Thus, the real time calculation of Ir
provides a simple means of determining, in real time, whether a radar
has become "blind" due to range attenuation and, furthermore,  at
what range the "blindness" occurs.
9
rNow consider an extended region of uniform rain rate and, hence, (f
uniform reflectivity as depicted in Figure S. Examining Equation (20) 	 E
it is clear tnat the quantity
(24)	 r
---z^ b = QZ
(1-Ir) 
z^ RO	 j
is a constant independent of ran ge since both Q and Z are independent	 j
of range. Furthermore, since the attenuation will increase. without
bound as r increases, the range corrected received power, r4 r, must
approach zero (see Equation (1)). Therefore, the radar saturation
factor must approach unity as the range increases.
R
I
Ro
l
r
z
r
r 2 Pr
._
II ^--
r
Figure 5. The case of extended uniform rain rate.
E
10
r
4	
_
7
11
Similar reasoning leads to the same conclusion even for the case
of an inhomogeneous rain rate distribution if either the rain rate
or extent is sufficiently large, i.e.,
(25) Ir i 1
for any case where the path attenuation becomes sufficiently large.
Substituting Equation (21) into (25) and rearranging yields
(26) a	
a
	
fmo
(r2 Pr) bRD/bzdr bz/bRD .Q
z 
for sufficiently high or extended rain rates. The left side of
Equation ( 26) has an absolute bound which is simply the calibration
constant of the radar. Thus, if the left side of Equation (26) is
evaluated and the maximum value found for experimental data obtained
using an attenuating wavelength radar viewing a sufficiently intense
storm, this value establishes an absolute radar calibration constant
independent of any other measurements.
7	 IV. RADIOMETRIC TEMPERATURE
Since the reflectivity, Z, can be found as t -junction of range,
Equation (20), the rain rate, R, is also known, equation (7):
2 I^bz(27) R = r P r	1
	
a7Q	 (1-I01JbRD
Similarly, using Equation (6) the specific attenuation can be found
at any other frequency. In particular, let the specific attenuation
be evaluated for some radiometer frequency other than the radar fre-
quency; then,
( r2pr )bRM/bz
(28) aRM = aRM  1
azQ(
1-Ir ) 
RM RD
and the path attenuation at the radiometer frequency is
aRM (r2Pr)
 bRM/bz
f
(29) ARC°(a
 Q)bRM/bz 10 	RM RD dr.
z	 (1-Ir)
Combining Equations (8) and (29) yields
aRM	 W ( r2Pr)
	
((	
bRM/b
z
(30)	 Ts = Tmt-exp _	 b b 1	 b	 dr(azQ) RNI/ z o 0-I )b
	
RD
so that the radiometric sky temperature can be obtained at any frequency
directly from the radar data.
V. RADAR CALIBRATION
If Equation (30) is rearranged, the radar calibration constant,
Q, can be expressed in terms of the backscattered radar power and the
radiometric temperature as:
bz/bRM
aRM	
(r2 
Pr) 
bRM/bz dr
31	 =^ " 1^RD( )
	
Q	 0a	 (	 r)z
^,n
 (
Tm
Tm-Ts
The right hand side of this expression is implicitly dependent upon Q
because of the presence of Ir. Nevertheless, if a reasonable estimate
of Q can be made, this expression can be used to solve for Q in a
stable, iterative manner. The astimate of Q should be used to cal-
culate the radar saturation factor, Ir, and then the improved estimate
of Q is calculated using Equation (31). Thus, a bore-sight radiometer
operating at a frequency either the same as or different than the radar
frequency can be used to calibrate the radar on an instantaneous basis.
Obviously, if the radar is operating at a non-attenuating wavelength
or if the rainfall observed is very light, the radar saturation may
be very small or even negligible; in this case, neglecting Ir in
Equation (31) eliminates the need for an iterative solution of this
equation.
Conversely, if very intense rain is observed at an attenuating
wavelength, the radar saturation factor, Ir, will approach unity for
large r; in this case the integrand in Equation (31) will become in-
determinate and, thus, may generate significant errors. However,
this is precisely the case where the bound given by Equation (26) is
applicable; thus, the radiometric temperature measurement and the
use of Equation (31) are not required.
12
1
13
VI. SUMMARY
Using the Hitschfeld-Borden approach, the radar reflectivity, Z, at
a given range has been obtained in terms of the received radar power,
the radar calibration constant, and the parameters of the power law
relations between specific attenuation, reflectivity, and rain rate.
This solution of the radar range equation contains a factor, Ir, re-
ferred to as the radar saturation factor, which is dependent upon the
integral of the range corrected received radar power. This radar
saturation factor, in turn, is shown to exhibit many interesting char-
acteristics; for example, this factor is very simply related to the path
attenuation and the radiometric temperature. Furthermore, this factor
is shown to saturate, i.e., approach a known bound, in a manner analo-
gous to that in which radiometric temperature saturates. Finally, these
properties are used to develop two alternative techniques useful for the
determination of the radar calibration constant. The first technique,
applicable to radars operating at attenuating wavelengths, permits a
direct, absolute calibration of the radar in terms of the measured radar
data itself. The second technique, employing independent radiometric
temperature measurements, permits continuous radar calibration during
data acquisition.
J
REFERENCES
[1] T. W. Harrold and C. A. Nicholass, "The Accuracy of Sr,me
Recent Radar Estimates of Surface Precipitation,"
Meteorological Magazine, v. 101, p. 193, 1972.
[2] W. Hitschfeld and J. Borden, "Errors Inherent in the Radar
Measurement of Rainfall at Attenuating Wavelengths," Journal
of Meteorology., v. 11, p. 58, 1954.
[3] J. R. Probert-Jones, "The Radar Equation in Meteorology,"
Quaterly Journal of the Royal Meteorological Society,
v. 88, p. 485, 1962.
[4] R. L. Olsen, D. V. Rogers, D. B. Hodge, "The aR b Relation
in the Calculation of Rain Attenuation," accepted for
publication in IEEE Transactions on Antennas and Propagation.
[5] D. M. J. Devasirvatham and D. B. Hodge, "Power Law Relationships
for Rain Attenuation and REflectivity," Report 784650-2,
January 1978, The Ohio State University ElectroScience
Laboratory, Department of Electrical Engineering; prepared
under Contract NAS5-23850 for NASA, GSFC, Greenbelt, Maryland.
[6] K. N. Wulfsberg, "Sky Noise Measurements at Millimeter Wave-
lengths," Proceedings of the IEEE, v. 52, p. 321, 1964.
[7] J. I. Strickland, "The Measurement of Slant Path Attenuation
Using Radar, Radiometers and a Satellite Beacon," Journal de
Recherches Atmospheriques, v. 8, p. 347, 1974.
[8] D. B. Hodge, "A 15.3 GHz Satellite-to-Ground Diversity
Experiment Utilizing the ATS-5 Satellite," Report 2374-11,
October 1972, The Ohio State University ElectroScience
Laboratory, Department of Electrical Engineering; prepared
under Grant NGR 36-008-080 for National Aeronautics and
Space Administration. (N73-12144)
14


Meteorological radar calibration

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