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NIST Technical Note 1539
I
NIS T Technical Note 1 539

Angular Errors in Polarimetric

Radar Cross Section Calibration

Using a Rotating Dihedral

Lorant A. Muth

Electromagnetics Division

Electronics and Electrical Engineering Laboratory

National Institute of Standards and Technology

325 Broadway

Boulder, Colorado 80305-3328

Catherine Johnson

Donald Morales

Timothy Conn

EG&G Technical Services, Inc. National Radar Cross Section Test Facility

46TG/TGR

Holloman Air Force Base, NM 88330

April 2005

^^ITES O*

'^

U.S. DEPARTMENT OF COMMERCE, Carlos M. Gutierrez, Secretary

TECHNOLOGY ADMINISTRATION, Phillip J. Bond, Under Secretary of Commerce for Technology

NATIONAL INSTITUTE OF STANDARDS AND TECHNOLOGY, Hratch G. Semerjian, Acting Director
Certain commercial entities, equipment, or materials may be identified in this dociunent in order to describe an experimental procedure or concept adequately. Such

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entities, materials, or equipment are necessarily the best available for the purpose.

National Institute of Standards and Technology Technical Note 1539

Natl. Inst. Stand. Technol. Tech. Note 1539, 20 pages (April 2005)

CODEN: NTNOEF

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ANGULAR ERRORS IN POLARIMETRIC RADAR CROSS SECTION

CALIBRATION USING A ROTATING DIHEDRAL

Lorant A. Muth, Catherine Johnson,*

Donald Morales,* and Timothy Conn*

Electronics and Electrical Engineering Laboratory

National Institute of Standards and Technology

325 Broadway

Boulder, Colorado 80305-3328

We examine how accurately a radar cross section measurement system's

transmit and receive parameters can be determined by use of a rotating

dihedral as the polarimetric calibration device. We derive expressions for

the errors due to misalignment in the angle of rotation of the dihedral. In

addition, we obtain expressions for the angles ao^hv and ao^yh for which

the measured cross-polarization ratios of a target vanish. Since the

theoretical cross-polarization ratio of a cylinder is 0, we can thus find the

calibration bias-correction angles. We use both simulated and real data

to demonstrate the robustness of this bias-angle correction technique.

We also derive expressions for the uncertainty in the polarimetric system

parameters.

Keywords: angular errors, calibration, polarimetric radar cross section,

rotating dihedrals

1 THEORETICAL FORMULATION

In general, the receive matrix in a polarimetric RCS measurement system is given by

^ 'vh 'vv '

where r^^ is the p-polarized electric field received when a q'-polarized electric field is illu- minating the target. Here p and q indicate the horizontal h or the vertical v polarizations.

Equation (1) can be normalized so that the receive matrix becomes

= {l ?)-• <^^'

EG&G Technical Services, Inc., at National Radar Cross Section Test Facility, 46TG/TGR,

Holloman Air Force Base, NM 88330
I

where t is the transmit matrix for reciprocal systems, and the cross-polarization ratios are defined as

Thv

e/i = 5 rhh

and

rvh

fvv

We represent the error-free receive matrix as

which has the inverse

'«^ = Ue.,„ rj/M='o'. (2)

where |ro| is the determinant,

We calibrate the polarimetric measurement system with a rotating dihedral [1], which has

the high-frequency scattering matrix

D{e) = ko(-':°f/ '"fj, (3) ^

^ V sm26' cos 20 J

^

'

where Q (with respect to the vertical) is the angle of rotation of the dihedral about the

line-of-sight from the radar to the dihedral; kn depends on the size of the dihedral and

can be obtained computationally. The measured signal is

M^'^^ = kkDrD{e)t, (4)

where /c, a phase-dependent constant, can be determined by a measurement on a cylinder

whose radar cross cross section has been computed accurately. The superscript ^^"^ in eq

(4) and elsewhere means that the quantity is derived from measured data. We write the

matrix components of eq (4) explicitly as

/i/r(d)^.. ( (-l + e2)cos26' + 2e/,sin26' (e/, - e„) cos2^ + (1 + e^e„) sin26'\

'^'^Dy^^^_^^^^^^2e + {l + ehey)Qm2e (l-e2)cos26' + 2e^sin2^ /'

(5)

Let

9 = do + a,

where a is the error in the angle of rotation. For a = 0, the ratios rj^ q and r^ q of the

Fourier coefficients of the matrix components Mf^jj and MyJ are

,2 {d) _ 1 ^h,0 ~

2 e/i,o ^/i,o — - (6)
and

For Q; ^ 0, these ratios are

<,o=9— • (7)

^{d) ^

(glo-l)cos2a + 2e/,,osin2Q

'^'^'^ ~ 2e,,,o cos 2a + (1 - e^ q) sin 2a ^ ^

and

(d) _ (1 - g^,o) cos 2a + 2e^,o sin 2a

For known r\^ q and r^ q (obtained from the Fourier coefficients of the data), we can readily

show by substitution that

e/j,o, and (10)

and

e^;,o, and (11)

are solutions to eqs (6-7). We note that these solution pairs are negative reciprocals of

each other. In practice, we want a radar system with cross-polarization ratios less than

unity. Without loss of generality, we assume that eh^Q << 1 and ty^Q << 1. The solutions

to eqs (8-9) in terms of the solutions of eqs (6-7) are

(l + e|o)sin2a + 2e/i,o

(l + <o)cos2a + l-e2o'

(12)

(1 + e^o)sin2a - 2ehfi

^h,a =

and

(l + eM)cos2a-l + e2o'

-(l + e2o)sin2a + 2e^;,o

€^^1 n/ —

(l + <o)cos2a + l-e;o'

(13)

-(1 -f eJo)sin2a - 2ey,o

{l + elQ)cos2a-l + elo'

We can readily verify that the solution pairs in eqs (12-13) are negative reciprocals of each

other, as in the error-free case. For a = 0, the first solutions in eqs (12-13) reduce to €fi,o and e^; 0, respectively; and for a ~ 0, these same solutions are still less than unity.

If a target's scattering matrix is given by
then the error-free received signal from the target is

Mi'^^ = fcVoTfo, (15)

where k' is a complex constant that depends on the distance r between the target and

the radar. If the target is at the same location as the dihedral, then k' — /c, as in eq

(4). However, we do not know rg, the error-free receive matrix, since there is always an

angular bias error in a calibration with a rotating dihedral. Equation (15) can be inverted

to obtain an error-contaminated target scattering matrix Tq,, when we use the known

error-contaminated system receive and transmit matrices

Here the superscript ^^^ indicates that we are using the first solutions in eqs (12-13). These

solutions let us recover the error-free target matrix in eq (15), if we set a — 0. The target's error-contaminated scattering matrix obtained from the measurements M^

is then

k'T^=r-'Mi'^r-\ (17)

We can expand the components of T^ in a second-order Taylor series to exhibit explicitly

the dependence on the small error-parameter a. The details of these expansions are given

in the appendix.

Here, we concentrate on the cross-polarization ratios,

hv _ '^hv,a ,-.Q\

and

.vh ^ tvh,a /-.qx vv,a — - ' V-^'^;

which can be used to determine the small angular-bias error a present in the calibration

procedure that has been propagated into the system cross-polarization ratios eh,a and Cy^aAgain, to simplify these ratios for small errors a, we expand them in a second-order Taylor

series of the form

7 = 7 \-(^Qhv,0 + O{a ), (20)

thh,a T'hhfi and similarly for T^y ^. The explicit forms of Qhv.o and Qvh.o in terms of all the error-free

system and target parameters are to be found in the appendix.

An important special case of eq (20) is when the cross-polarization ratio on the left vanishes.

We then solve for ao,/iu:

U + ^h,o)^vvfi ~ i/i/i,o(l + e^,o)
and, similarly,

i-'^ + ^/i 0>'''^'w,0 - c/i/i,oi-l- + e„ oj

Theoretically, both eqs (21) and (22) should give the same angle a for reciprocal targets

with thv,o = ivh,o- However, in practice we might obtain slightly different results due

to system noise and target misalignments. We can neglect all second-order terms in eqs

(21-22) to drive a simple zero-order expression for a:

(^o,hv ~ 7 , (23)

tuv,0 ~ i^hh,0

which is independent of the system parameters. Similarly, we can easily obtain an expression for ao^yh- We have assumed in eq (23) that tvv,o — thh,o ¥" 0- Note that for a sphere

this condition is not satisfied; however, a sphere is not depolarizing, therefore, thv,o ~ 0,

and, hence, a is undefined.

2 SIMULATIONS

We used computer simulations to verify the analysis presented in the previous section and

to demonstrate that the bias angle a, which represents the angular misalignment between

the calibration dihedral and a target, can be accurately determined from measurements.

In particular, we designed the simulations to understand eq (23) in two ways:

(1) explore the range of a for which eq (23) recovers the correct value, and

(2) explore the sensitivity of eq (23) to the condition that its denominator must not vanish.

We assume various values for the receive parameters in eq (1), and use eqs (3) and (4)

to simulate measured dihedral data. Without any loss of generality, we assume that the

product kkd in eq (4) is 1. We then introduce a range of bias angles a into eq (5) to

obtain biased dihedral measurements, and use Fourier analysis to solve for biased system

parameters. Next we simulate various target measurements using the scattering matrix

defined in eq (14). The biased scattering-matrix elements are recovered using eq (17), and

then use eq (23) to recover the known bias angle a. We performed simulations with various cross-polarization ratios e^ and ey in eq (Ic), and

various test target matrices T in eq (14). We used the following combinations of parameters:

(1) for the receive and transmit matrices.

1 0.1

(-0 15 7)=*- (^^)

and

-(-0.015 T)-^^' '-)

which represent average and good channel isolations, respectively, and
(2) for the test targets matrices,

0.8

ir (^*)

and

0.98 0\

J- (^^^

Since all the ofF-diagonal scattering matrix elements are 0, these matrices represent nondepolarizing targets with various copolar ratios that satisfy the condition tyy^o — thh,o 7^ to

different degrees. When the copolar components are nearly equal, as in case TS, we have

a sphere-like target and expect the observed (biased) cross-polarization to be very low.

Figures 1 and 2 show the results of simulations. In Figure 1 we see that errors in the

computed bias angle are very small in all cases except T5 (the sphere-like target). For a

well isolated system R2, these errors are significantly smaller than in the poorly isolated

system Rl. Overall, however, the errors in the predicted bias angles are insignificant,

especially at very small bias angles. For TS, however, the errors quickly become very

large. For Rl the error curves are linear in a; for the case R2, the presence of second and

higher-order terms is apparent.

In Figure 2 we show the measured cross-polarization ratio in decibels as a function of

known bias angles a for the cases Tl • • T5. As a —) 0, this measured ratio approaches

—(X), which is the correct theoretical value (assumed). The dependence on a is very strong

in all cases. The dependence on the ratio of copolar matrix components is also very strong;

we get the lowest measured cross-polarization ratios for T5, as expected. The dependence on system isolation {Rl vs. R2) is very weak.

These figures show that our technique to remove angular error biases in polarimetric calibration using a rotating dihedral is robust. The simulations demonstrate that

(1) a wide range of bias angles are recovered accurately, and

(2) even if the the denominator in eq (23) nearly vanishes, we still get satisfactory results.

This implies that the correction technique can be applied to measurements on many

types of targets.

3 EXPERIMENTAL DETERMINATION OF ALIGNMENT BIAS

We have also demonstrated the robustness of our technique using real data. Figure 3 shows

the calibration fixture used to rotate the calibration dihedral. The dihedral was aligned

with the vertical by means of a manual inclinometer. The rotation was continuous, and the
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Figure 3. The fixture used to rotate a calibration dihedral for polarimetric measurements to obtain the full transmit-receive matrix of the radar system. The cross-polarization response

of the fixture without the dihedral was also measured.

50 100 150 200 250 300 350

Figure 4. The uncalibrated polarimetric signals at 9.6 GHz received from the

rotating dihedral shown in Figure 3.
angle of rotation was recorded in real time as the radar was transmitting and receiving the

calibration pulses. The uncalibrated data, shown in Figure 4, were then used to determine

the system transmit and receive parameters using the method developed in Reference [2], which, in principle, is equivalent to using eqs (6), (7) and (17).

Next, measurements were made on a cylinder, because, theoretically, it has zero cross- polarization response. The cylinder was also aligned with the vertical by means of the

manual inclinometer. In case of perfect alignment with the dihedral, the measured cross- polarized response of the cylinder would be zero theoretically, or at the level of noise

experimentally. We make the realistic assumption that the cylinder is much better aligned

than the dihedral. Then, any nonvanishing cross-polar measurement on the cylinder can be attributed to the misalignment of the dihedral. This bias in the rotational angle of the

dihedral, relative to the coordinate system in which the cylinder is aligned, is a. Any small

cross-polarized measurement (beyond noise) on the cylinder indicates the presence of bias,

and a succesful correction technique should eliminate this.

The bias-correction angle was determined numerically from the biased calibration data. We

start by assuming a value for a and determine the system parameters. Then we compute

the cylinder scattering matrix elements. Figure 5 shows the cross-polar (hv) element of the

cylinder's scattering matrix as a function of the bias angle. The bias correction angle was

determined to be 0.18°, compared to 0.19° calculated from eq (23). The 0.01° difference

is consistent with the error curves presented in Figure 1. When we compare the cross- polar ratio levels at a = 0.2 in Figure 2 and at the bias-corrected angle a' = in Figure

5, we can estimate that we are dealing with a measurement system configured similarly

to case R2; T2. However, case Rl cannot be eliminated conclusively with such a simple

comparison.

In Figure 6 we show the co-polar and cross-polar vh scattering matrix elements of the

supporting test target (shown in Figure 3) as a function of azimuthal rotation. The two

lower curves show the computed cross-polar responses before and after the calibration bias

angle has been removed from the data to obtain bias-free system parameters. We observe a significantly lower cross-polar ratio, in agreement with computational predictions.

4 CALIBRATION UNCERTAINTY

After the calibration data have been corrected for the bias, expressions (A8) through (A12) can be used to estimate the uncertainties in the scattering matrix elements of the target,

and expressions (A13) and (A14) can be used to estimate the uncertainties in the system

parameters. We let the a = terms be the bias-corrected values, and redefine a as the

estimated upper bound of uncertainy in the computed bias angle obtained using eq (23).

The main sources of uncertainty in a are:

(1) computational errors, especially due to the approximations used,

(2) the presence of noise in the measurements, and

(3) drift.

Special techniques can be used to correct the data for drift by exploiting the rotational

symmetries of the dihedral [3, Section 4]. We will not discuss the details here, but assume

10

1
bias-correction angle (deg)

Figure 5. The measured cross-polarization ratio (decibels)

of a cylinder as a function of the bias-correction angle.

The deep null indicates that the calibration bias angle has

been removed to obtain bias-free system parameters.

d

o

o

!/5 o

o

-a

-150 -100 -50 50 azimuth (deg)

100 150

Figure 6. The polarimetric response of the test body shown

in Figure 3 as a function of azimuthal rotation. The top curve is the HH co-polar response; the middle and bottom

curves are the cross-polar responses without and with

calibration bias-angle correction, respectively.

11
that the effect of drift has been adequately removed. According to Figure 1, we can

estimate the upper bound of computational errors Sa ~ a/ 10, which is then used as the

estimated uncertainty in a. To account for the effect of noise, we include a coverage factor

of 2 and round upwards. Hence, in our simulations,

6a = ±a/5. (24)

For example, in Figure 5 the bias-correction angle was found to be a ~ 0.18°; hence,

da = 0.04°, and a = 0.18° ± 04°. This estimate is valid for systems with parameters

in the range of Rl to R2 presented in Figure 1. To determine the uncertainties in the

measurements, we use the first-order expressions (A8) through (A13) with a = zt^a, and

substitute the bias-corrected measurements for all quantities with subscripts. For the

case R2 : T2, we get

i/^/^ = 1 ± 0.000007; thv = ± 0.0007; t^h = ± 0.0007; t^„ = -1 ± 0.00001, (25)

and

eh = 0.01 ± 0.0004; e„ = -0.015 ± 0.0004. (26)

We emphasize that the simple method of estimating the uncertainty 5a in a presented

here need not be adopted. Any fully described method, with well-defined operational

steps, supported with clearly stated and easily defensible assumptions, would be adequate

to the task.

5 SUMMARY AND CONCLUSION

We have presented a theoretical procedure to remove angular misalignment errors in the

calibration of polarimetric RCS measurement systems. The calibration is accomplished

with a rotating dihedral which might not be perfectly aligned in the vertical direction. We

define the vertical as the vertical alignment of a calibration cylinder. In practice, the cylinder can be aligned with a manual inclinometer much more accurately than can a dihedral.

We exploit the vanishing of the cross-polar response of a cylinder to derive a simple expression for the misalignment bias angle. The results of theoretical simulations agree with real

measurements. We show excellent agreement between the bias angles obtained numerically

and theoretically using the expression derived in this study. The expressions derived for

the target polarimetric scattering matrix components and for the system parameters as functions of the bias angle can also be used to obtain the corresponding uncertainties by

use of an estimated upper-bound uncertainty in the bias angle. The results indicate that

the theoretical bias-correction technique presented in this study is very robust.

This work was supported by the DoD Triservice Calibration Coordination Group (CCG).

Special thanks are extended to H. Click, M. Cunevalis, and R. Cleland of the Microwave

Subgroup for their long-term support of this project.

12
REFERENCES

[1] L. A. Muth, R. C. Wittman and R. L. Lewis, " Polarimetric calibration of reciprocal

antenna radars," Antenna Measurement Techniques Assoc. (AMTA) 1995 Proceedings^

1995.

[2] M. W. Whitt and F. T. Ulaby, "A polarimetric radar calibration technique with insensitivity to target orientation," Radio Science., vol. 25, pp. 1137 - 1143, 1990.

[3] L. A. Muth, R D. Domich, B. M. Welsh, A. L. Buterbaugh and B. M. Kent, "Polarimetric calibration standards for monostatic radar systems," Proceedings of the 1998 Workshop

and Symposium., NCSL, 1998.

13
APPENDIX

Here we explicitly exhibit the matrix elements of Tf^ in terms of the error-free system

parameters to a first order in a:

thh,a ^ thh,o - a \-0{a), [Al)

((1 + 4,o)^^',o - eh,o(l + e5,o))^/iu,o + (1 + e?,o)*^^,o -^wv,o(l + 4,o) ^, 2\ / ^^n

a +0{a^), {A2)

{i^ + 4^o)'^v,o-(^h,o{'^ + (^l,o))tvh,o + {^ + elQ)thh,o-tyy,o{l + elQ)

a \-0{a ), {A3)

and

tvv,a = tvv,o + tt —

\- 0{a ). {A4)

The hv cross-polarization ratio is, to a first order in a (see eq (20)),

7 = 7 + <^Qhv,Q + C>(a ),

and a similar expression can be written for the vh polarization.

Explicitly, Qhvfl is given by

2 /I I 2 \ 'vw,0

(1 — ^h,o^v,o)Qhv,o = 1 + e^ ~ (1 + £/i,o)

^/i/i,0 ' "''" ^thh,0 thh,0 ^hh,0

+ e,,,o(l + e^,o) + et;,o(l + e^,o) l + (1 + ^h,o) ( 7 7 + 72—J

' i^^)

thh n \thh n C^iA, n t. . r, /

and Q^,/^,o is given by

(1 - eh,oey^o)Qyh,o = -(1 + e^,o) + (1 + 4,o)

2 \ I /I , 2 N^/i/i,0

- [e/^,o(l + e^,,o) + e«,o(l + e/,,o)j 7 (1 + e^,o) ( T

7 + 72—

We observe that the leading terms in eqs (A5) and (A6) are

{A6)

1_!e^, and -1 + ^^^, {A7)

thh,0 tvv,0

14
which is consistent with eq (23). For a dihedral, the ratios in eq (A7) evaluate to —1.

For a system with good polarimetric isolation, second-order terms can be safely neglected.

The target scattering matrix elements then simplify as

thh,a ~ thh,o{^ - 2Q;e^,o) - Oi{thv,o + tyh,o) + Oia"^] e^), (A8)

thv,a ~ thv,o{l + a{(^v,o - e/i,o)) + a{thh,o - tvv,o) + 0{a^\ e^), (A9)

tvh,a ~ tyh,o(^ + (^{^v,o - e/i,o)) + (^{thh,o - tvv,o) + 0{a^; e^), (^10)

and

tvv,a ~ ivt;,o(l + 2aey,o) + a{Uhfi + ^hw,o) + 0(a^; e^). (^H)

The cross-polarization ratios simplify similarly; for example, —^- « —-^ +a(l - h{eh,o + ey^o)- ) + 0{a ; e ;t ). {A12)

thh,a thh,0 '^hh,0 '^hh,0

If in eq (A12) we further neglect terms with e-s and set the left side to 0, we obtain eq

(23).

Finally, we obtain the sytem parameters to first order in a from eqs (12-13):

and

eh,a = €h,o + ail + e^,o) + 0{a^), {A13)

ev,a = e«,o - a(l + ej n) + 0{a^). (AU)

15

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Federal Information Processing Standards Publications (FIPS PUB)—Publications in this series collectively

constitute the Federal Information Processing Standards Register. The Register serves as the official source of

information in the Federal Government regarding standards issued by NIST pursuant to the Federal Property and

Administrative Services Act of 1949 as amended, Pubhc Law 89-306 (79 Stat. 1 127), and as implemented by

Executive Order 1 1717 (38 FR 12315, dated May 1 1,1973) and Part 6 of Title 15 CFR (Code of Federal

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