A Geometric Error Model for Misaligned Calibration Target in Passive
IEEE GEOSCIENCE AND REMOTE SENSING LETTERS, VOL. 10, NO. 6, NOVEMBER 2013
1597
A Geometric Error Model for Misaligned Calibration Target in Passive Microwave Remote-Sensing Systems Dazhen Gu, Senior Member, IEEE, James Randa, Senior Member, IEEE, and David K. Walker, Senior Member, IEEE
Abstract—We present a geometric error model associated with calibration-target misalignment in passive microwave remote-sensing systems. The developed analytic formulation is universally applicable to both lateral and rotational misalignment conditions. Numerical simulations are performed on two practical blackbody targets of different sizes used as radiation references for passive microwave remote sensing. The significance of this work is to furnish a framework of uncertainty analysis due to target misalignment and to provide a reference for alignment requirements based on passive radiometer measurement sensitivity. Index Terms—Blackbody target, microwave radiometer, misalignment, thermal radiation, uncertainty analysis.
uation of the remote-sensing observation. The measurement uncertainty of the microwave radiation from a blackbody target involves multiple sources, such as system sensitivity, environmental stability, hardware alignment, etc. Unlike other error sources known under the specific conditions, uncertainty arising from misalignment is often unclear and difficult to quantify. Pertinent theory is missing for users to assess such errors and set up appropriate alignment requirements. In this letter, we establish a theoretical formulation of measurement uncertainty caused by the misalignment of blackbody targets and simulate its effect in practical experiments.
I. I NTRODUCTION
II. T HEORY
P
ASSIVE microwave remote-sensing systems have become a critical component of climate and environmental observations globally. Applications of such instruments can be found in a wide range of disciplines, providing information on weather forecast [1], [2], climate change [3], precipitation rates [4], ocean salinity and roughness [5], and agricultural management [6], among others. Most spaceborne passive remotesensing systems are equipped with a blackbody target of a finite size that provides a hot temperature reference in addition to the cold reference provided by the cosmic background, while airborne systems may use multiple targets as references. At the National Institute of Standards and Technology, we are engaged in the research on standard blackbody targets at microwave frequencies and present here a related study in this field. The calibration of a blackbody target requires a thorough uncertainty analysis that ultimately affects the accuracy eval-
Manuscript received March 5, 2013; revised April 12, 2013; accepted May 7, 2013. Date of publication May 29, 2013; date of current version October 10, 2013. This work was supported by National Institute of Standards and Technology, an agency of the U.S. Government, and is not subject to U.S. copyright. D. Gu is with the Electromagnetics Division, National Institute of Standards and Technology, Boulder, CO 80305 USA, and also with the Department of Electrical, Computer, and Energy Engineering, University of Colorado, Boulder, CO 80309 USA (e-mail: [email protected]). J. Randa is with the Electromagnetics Division, National Institute of Standards and Technology, Boulder, CO 80305 USA, and also with the Department of Physics, University of Colorado, Boulder, CO 80309 USA (e-mail: [email protected]). D. K. Walker is with the Electromagnetics Division, National Institute of Standards and Technology, Boulder, CO 80305 USA (e-mail: david.walker@ nist.gov). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/LGRS.2013.2262471
We derive the Poynting vector from the coherence tensor between electric and magnetic fields of the blackbody radiation. The theoretical model is applied to the ideal alignment condition and then to different misalignment conditions to calculate the power received by a microwave radiometer. A. Radiometric Model of Blackbody The electromagnetic (EM)-mixed cross-spectral tensor at observation points r1 and r2 is defined as [7], [8] EM Wjk (r1 , r2 , ω)
1 = 2π
∞ Ej (r1 , t + τ ) −∞
×Hk∗ (r2 , t) exp(iωτ ) dτ
(1)
where Ej (r, t) and Hj (r, t)(j = x, y, z) symbolize the jcomponent of the electric- and magnetic-field vectors in the Cartesian coordinates at position r and at time t and the brackets · denote the ensemble average. For a blackbody, the EM-coherence tensor takes a simplified expression EM ˆ (r1 , r2 , ω) = −B(ω, T ) ˆj × k Wjk ˆs exp(iKˆs · r ) dΩ (2) · Ω
where r = r2 − r1 and ˆs is the unit vector viewing from ¯r (defined by (r1 + r2 )/2) to the surface of the blackbody target. Ω is the total solid angle subtended by the blackbody target viewing from ¯r, and K is the wavenumber in the experimental
U.S. Government work not protected by U.S. copyright.
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IEEE GEOSCIENCE AND REMOTE SENSING LETTERS, VOL. 10, NO. 6, NOVEMBER 2013
Fig. 1. Illustration of a receiving antenna located at the origin that collects radiation from a blackbody target centered at (0, 0, z0 ).
medium, usually air or vacuum. B(ω, T ) is the spectral radiance of a blackbody at temperature T , given by B(ω, T ) =
ω 3 4π 3 c2 exp kω − 1 T B
(3)
where and kB are the Planck constant divided by 2π and the Boltzmann constant, respectively; c is the speed of light in vacuum. In (2), we assume an isothermal blackbody object with uniform temperature distribution across its surface. The Poynting vector S at the receiving antenna can be derived easily from (1) by forcing r1 = r2 = r or r = 0. The l-component of S is associated with the coherence tensor by Sl (r, ω) =
1 EM EM Wjk (r, r, ω) − Wkj (r, r, ω) 2
(4) C. Lateral Misalignment
where {·} denotes the real part of a complex variable. The order of jkl strictly follows the cyclic order of xyz, e.g., jkl can be either xyz, yzx, or zxy. We obtain a much simplified radiometric model of the blackbody radiation by substituting (2) into (4) (−ˆs) dΩ. (5) S(r, ω) = B(ω, T ) Ω
B. Perfectly Aligned Blackbody Target We assume that the blackbody radiation receiver, often consisting of an antenna as its front end, is fixed at the origin of the Cartesian coordinates as shown in Fig. 1. To a first-order approximation, we consider the receiving antenna as a point receiver: The variation across the antenna aperture is neglected. In addition, the antenna is sensitive only to the radiation flux along the −ˆ z-direction. The blackbody target is a flat circular disk with a radius of rtg located at (0, 0, z0 ). For perfect alignment, the normal of ˆ coincides with −ˆ the target surface n z. Due to the symmetry, only the z-component of S is nonzero, and it follows directly from the straightforward integration of (5) that S0z (ω) = −πB(ω, T )
2 rtg 2 + z2 . rtg 0
Fig. 2. (a) Lateral misalignment is characterized by two scalars: δx and δy . ˆ with two (b) Rotational misalignment shows a diverged surface normal vector n angles indicated by β and α.
(6)
For the remainder of this letter, the ensemble average over the Poynting vector and its components is implied, and the brackets enclosing S are dropped in the notation.
Now, we imagine that the misaligned target possesses only a small offset along the vertical direction. Its new position can be characterized by (0, δy , z0 ). The Poynting vector at the origin of the coordinates consists of not only a z-component but also a y-component, given by 2 rtg 2δy2 z02 Sz = − πB(ω, T ) 2 1−
(7a) 2 + z2 2 rtg + z02 rtg 0
2 2 δy rtg δy2 2z02 − rtg z0 Sy = − πB(ω, T )
1− 2 2 . (7b) 2 + z2 2 rtg rtg + z02 0 Although it is a very specific case of misalignment, such a result is generally applicable to more complicated misalignment cases. We represent (7a) and (7b) in a general formulation as 2 rtg 2δ 2 z 2 Sz = − πB(ω, T ) 2 1−
(8a) 2 + z2 2 rtg + z 2 rtg
2 2 δ 2 2z 2 − rtg z δrtg St = − πB(ω, T )
1− 2 2 . (8b) 2 + z2 2 rtg rtg + z 2 Equations (8a) and (8b) can be easily extended to deal with more general lateral misalignment, in which the target position is offset transversely and vertically to (δx , δy , z0 ), as shown in Fig. 2(a). Sz and St are calculated by assigning z = z0 and δ = δx2 + δy2 in (8a) and (8b). The unit vector ˆt is given by ˆ + δy y ˆ ). (1/δ)(δx x
GU et al.: GEOMETRIC ERROR MODEL FOR MISALIGNED CALIBRATION TARGET IN REMOTE-SENSING SYSTEMS
D. Rotational Misalignment
can be calculated as
A rotational misalignment can be characterized by the elevation angle β and the azimuth angle α, illustrated in Fig. 2(b). ˆ and z-axis z ˆ, β is the angle between the surface normal n where β = 180◦ corresponds to perfect alignment. α is the ˆ on the xy plane and the angle between the projected vector of n x-axis. To ease the problem, we rotate the xyz coordinates to x y z accordingly. The relationship between the two coordinate systems can be represented by ˆ = x ˆ sin α − y ˆ cos α x ˆ = −x ˆ cos β cos α − y ˆ cos β sin α + z ˆ sin β y ˆ = −x ˆ sin β cos α − y ˆ sin β sin α − z ˆ cos β. z
(9a) (9b) (9c)
Viewing from the x y z coordinates, we transform the rotational misalignment problem to the spatial misalignment problem. The center of the target is located at (0, δy , z0 ) in the new coordinates. The offset and the distance are given by δy = z0 sin β χ[0,π) (α) − χ[π,2π) (α) (10a) z0 = − z0 cos β (10b) where the indicator function χC (α) is defined as 1 if α ∈ C χC (α) ≡ 0 if α ∈ / C.
(11)
We now substitute z0 and δy for the variables z and δ, respectively, in (8a) and (8b) to calculate Sz and Sy . Note ˆ . We bear in mind that the receiver is still pointed that ˆt = y ˆ-direction rather than the z ˆ -direction. The quantity along the z of interest Sz in fact consists of both Sz - and Sy -components and is represented as Sz = −Sz cos β + Sy sin β.
(12)
E. General Misalignment A general misalignment case is composed of both spatial and rotational offsets. Physically, it can be characterized by two ˆ . We again first scalar offsets δx and δy and one vector offset n rotate the coordinate system to x y z in the same manner as we did in Section II-D and next translate the scalar offsets in the new coordinates. ˆ cos β cos α − z ˆ sin β cos α) ˆ = δx (ˆ x sin α − y (13a) δx x ˆ cos β sin α − z ˆ sin β sin α). (13b) ˆ = δy (−ˆ δy y x cos α − y The position of the target center in the x y z coordinates is (δx , δy , z0 ), with their specific values given by δx = δx sin α − δy cos α
1599
(14a)
δy = − δx cos β cos α − δy cos β sin α + z0 sin β χ[0,π) (α) − χ[π,2π) (α) (14b) z0 = − δx sin β cos α − δy sin β sin α − z0 cos β. (14c)
Similarly, we assign z = z0 and δ = δx2 + δy2 in (8a) and (8b) to compute Sz and St . Once again, Sz is of interest and
Sz = −Sz cos β +
δy δx2 + δy2
St sin β.
(15)
III. E XAMPLES We apply the error model to practical blackbody targets that are used frequently in our laboratory for passive microwave remote-sensing applications. Numerical simulations are run on two calibration targets, one with a radius of 16.5 cm and the other with a radius of 9 cm. The radius values are obtained from two blackbody targets for practical passive remote-sensing applications [9]. We study the cases involving only the δy offset for the spatial misalignment, while the divergence of the surface normal is limited to the variation of β (α is kept at π/2, or equivalently, x = x) only for the rotational misalignment. It is relatively straightforward to extend the simulation to include the offsets of other variables by using the formulations in Section II for specific studies. We compute the error percentage, defined by Sz | − |S0z ||/|S0z |, to compare the variation of the z-component of the Poynting vector for different types of misalignment. The offset of δy is represented as a fraction of the target radius in Fig. 3. The normalized offset is limited to no more than 0.1rtg , and the divergence angle β is kept between 180◦ and 175◦ . We also include different target–antenna distances to show the error percentage variation. In the simulated range of offsets, the error percentage is smaller than 0.5% for both the small and large targets. The error percentage due to the lateral misalignment follows the second term in the square brackets of (7a). Hence, for a fixed δy , the value shows a decreasing trend as the distance increases. The error percentage caused by the rotational misalignment is more complicated, which has been discussed in Section II-D. In general, its trend for a fixed β as a function of the distance is not straightforward. In order to predict its behavior, the evaluation of the error percentage with the specific rtg value is required. In some additional simulations with different target sizes, we find that the error percentage can show a minimum at the location other than the boundaries in the simulated distance range. For microwave remote sensing, the knowledge of alignment requirements is important to a system designer. In the Rayleigh–Jeans approximation, the radiance B(ω, T ) is proportional to T , and so is the magnitude of the Poynting vector. Since the physical variable of interest in microwave remote sensing mostly comes down to the temperature, we can assess the requirement of the alignment based on the desired accuracy of the observed temperature. A recent experimental study of brightness-temperature standards shows a measurement uncertainty of about 1 K for a warm blackbody target at 350 K [10], which also meets some of the accuracy requirements of climate variables [11]. The measurement uncertainty of the brightness temperature corresponds to 0.3% in [10]. In order for the error arising from misalignment to be insignificant, we set the error percentage to 0.1% and evaluate the relevant alignment requirements from the simulation result. In other words,
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IEEE GEOSCIENCE AND REMOTE SENSING LETTERS, VOL. 10, NO. 6, NOVEMBER 2013
Fig. 3. Error percentage caused by misalignment at various target–antenna distances. (a) A 16.5-cm-radius target vertically offsets in units of the target radius. (b) The surface normal of the 16.5-cm-radius target diverges from 180◦ to 175◦ . (c) A 9-cm-radius target vertically offsets in units of the target radius. (d) The surface normal of the 9-cm-radius target diverges from 180◦ to 175◦ .
this additional misalignment error would make the brightnesstemperature uncertainty increase from 1 K to 1.05 K. For the small target, the lateral misalignment should be kept below 0.06rtg , and the rotational misalignment should be kept below 2.6◦ for less than 0.1% error in the simulated range. For the large target, the corresponding results are 0.046rtg and 2.7◦ , respectively. IV. D ISCUSSION Longitudinal misalignment, i.e., the variation of the target– antenna distance, is not explicitly studied in this letter. In practice, such misalignment is relatively easier to be minimized to a certain extent in comparison to other types of misalignment mentioned here. However, the inclusion of such a misalignment is also straightforward by replacing z0 with z0 + δz in (6). The significance of this study is to provide a guideline for the estimation of the measurement uncertainty associated with misalignment, which is not presently established to our knowledge. The uncertainty due to the misalignment can be included in the total uncertainty estimation of the target brightness temperature. In addition, the theoretical framework is valuable to users to determine the alignment requirement based on the radiometer sensitivity. Targets of an arbitrary shape may be encountered in practical remote-sensing systems. A direct analytic expression of the Poynting vector component for such misaligned targets may not be available. However, the integration can nube evaluated ˆ /r2 ) dA) merically by a similar treatment ( Ω dΩ → A (ˆs · n outlined in the Appendix.
V. C ONCLUSION By the use of the fundamental EM-coherence theory of blackbody radiation, a general error model is developed to compute the uncertainty arising from the geometric misalignment of the calibration target in passive microwave remotesensing systems. Such a model can be universally applied to particular lateral and rotational misalignments or a combination of the two. The error percentage is simulated on two practical blackbody targets for a range of distances between the target and the radiometric receiver. In order to maintain an insignificant impact on the overall uncertainty, the lateral misalignment should be held smaller than 0.06rtg and 0.046rtg for the small and large targets, respectively, while the offset angles should be kept below 2.6◦ and 2.7◦ for the small and large targets, respectively. This study provides a perspective on how to quantify the measurement uncertainty related to the target misalignment and how to determine the alignment requirement based on other uncertainty sources and required observation accuracy. A PPENDIX I NTEGRATION OVER AN O FF -C ENTERED C IRCULAR A PERTURE The integration over the solid angle is nontrivial when the target center is vertically offset. The regular transformation from Ω dΩ to dφ sin θdθ brings complications since the integral bounds of θ cannot be easily determined analytically. Rather, we need to represent the solid angle integration as ˆ (ˆ s · n /r2 )dA, where dA is the infinitesimal area of the A target, A is the total area of the target, and r is the distance
GU et al.: GEOMETRIC ERROR MODEL FOR MISALIGNED CALIBRATION TARGET IN REMOTE-SENSING SYSTEMS
from the origin to dA. Therefore, the integration to be solved in (5) is ˆs · n ˆ z0 B(ω, T ) −ˆs 2 dA = B(ω, T ) −ˆs 3 dx dy (16) r r A
A
ˆ). Due to the yz where ˆs is equivalent to (1/r)(xˆ x + yˆ y + z0 z plane symmetry, the x-component vanishes after the integration. We then need only to solve Sz and Sy z02 Sz = −B(ω, T ) 2 dx dy (x2 + y 2 + z02 )
1601
On the other hand, the target can now be regarded as a point source in the FF zone. The Poynting vector at the origin can be simply obtained from (6) ˆ S0z cos3 θ sin θ + z ˆS0z cos4 θ SF F = y
(19)
where θ is the angle between the zenith and the vector ˆs: θ = tan−1 (δy /z0 ). The order of 3 on cos θ comes from Lambert’s cosine law in addition to the equivalent distance being increased by a factor of 1/ cos θ. By the use of the Taylor series of trigonometric functions, it can be easily shown that (19) is in agreement with (18a) and (18b).
A
= −B(ω, T )
z02 +
+ δy ) +
(x2 +
z02 y 2
+ z02 )
≈ −B(ω, T ) A
2
(y
A
x2
z02
2 dx dy
ACKNOWLEDGMENT
2
2δy2 x2 − 5y 2 + z02 4y δy dx dy · 1 − 2 − 2 x + y 2 + z02 (x2 + y 2 + z02 )
2π = −B(ω, T )z02 ×
rtg dφ
0
1 (ρ2 + z02 )
2
ρ dρ 0
−
2δy2
ρ2 cos2 φ − 5ρ2 sin2 φ + z02
4
(ρ2 + z02 ) 2 rtg 2δy2 z02 = −πB(ω, T ) 2 1−
. 2 + z2 2 rtg + z02 rtg 0
(17)
The coordinates x y facilitate the transition from xy to ρφ. In addition, the integration of any odd function of y results in zero as implicitly shown from the third step to the fourth step in (17). Following a similar principle, we can reproduce (7b) for Sy . To ascertain that the criterion δ z0 holds, we evaluate δ/z0 in the simulation of this work and confirm that its value is below 0.083 for all cases. Furthermore, all the plots reported in this letter have been verified by the numerical computation of the integral without any approximation. To further validate the derivation, we consider the far field (FF) condition where rtg z0 . On the one hand, the FF values of Sz and Sy can be computed from (7a) and (7b) by taking the FF approximation 2 2 r 2δ tg y (18a) SzF F = − πB(ω, T ) 2 1 − 2 z0 z0 2 rtg 2δy3 δy FF Sy = − πB(ω, T ) 2 (18b) − 3 . z0 z0 z0
The authors would like to thank P. Racette at the National Aeronautics and Space Administration Goddard Space Flight Center for loaning the calibration targets. D. Gu would like to thank Y. Liu of National Instruments for his technical inputs. R EFERENCES [1] S.-A. Boukabara, F. Weng, and Q. Liu, “Passive microwave remote sensing of extreme weather events using NOAA-18 AMSUA and MHS,” IEEE Trans. Geosci. Remote Sens., vol. 45, no. 7, pp. 2228–2246, Jul. 2007. [2] Z. Yang, N. Lu, J. Shi, P. Zhang, C. Dong, and J. Yang, “Overview of FY-3 payload and ground application system,” IEEE Trans. Geosci. Remote Sens., vol. 50, no. 12, pp. 4846–4853, Dec. 2012. [3] J. S. Kimball, L. A. Jones, K. Zhang, F. A. Heinsch, K. C. McDonald, and W. C. Oechel, “A satellite approach to estimate land atmosphere CO2 exchange for boreal and Arctic biomes using MODIS and AMSR-E,” IEEE Trans. Geosci. Remote Sens., vol. 47, no. 2, pp. 569– 587, Feb. 2009. [4] C. Surussavadee and D. H. Staelin, “NPOESS precipitation retrievals using the ATMS passive microwave spectrometer,” IEEE Geosci. Remote Sens. Lett., vol. 7, no. 3, pp. 440–444, Jul. 2010. [5] J. L. Garrison, J. K. Voo, S. H. Yueh, M. S. Grant, A. G. Fore, and J. S. Haase, “Estimation of sea surface roughness effects in microwave radiometric measurements of salinity using reflected global navigation satellite system signals,” IEEE Geosci. Remote Sens. Lett., vol. 8, no. 6, pp. 1170–1174, Nov. 2011. [6] R. Bindlish, T. Jackson, R. Sun, M. Cosh, S. Yueh, and S. Dinardo, “Combined passive and active microwave observations of soil moisture during CLASIC,” IEEE Geosci. Remote Sens. Lett., vol. 6, no. 4, pp. 644– 648, Oct. 2009. [7] W. Eckardt, “Macroscopic theory of electromagnetic fluctuations and stationary radiative heat transfer,” Phys. Rev. A, vol. 29, no. 4, pp. 1991– 2003, Apr. 1984. [8] D. C. Bertilone, “On the cross-spectral tensors for black-body emission into space,” J. Mod. Opt., vol. 43, no. 1, pp. 207–218, Jan. 1996. [9] D. Gu, D. Houtz, J. Randa, and D. K. Walker, “Reflectivity study of microwave blackbody target,” IEEE Trans. Geosci. Remote Sens., vol. 49, no. 9, pp. 3443–3451, Sep. 2011. [10] D. Gu, D. Houtz, J. Randa, and D. K. Walker, “Extraction of illumination efficiency by solely radiometric measurements for improved brightness-temperature characterization of microwave blackbody target,” IEEE Trans. Geosci. Remote Sens., vol. 50, no. 11, pp. 4575–4583, Nov. 2012. [11] R. Datla, B. Emery, G. Ohring, R. Spenser, and B. Wielicki, “Stability and accuracy requirements for satellite remote sensing instrumentation for global climate change monitoring,” in Proc. ISPRS, Jul. 2004, pp. 17–27.
1597
A Geometric Error Model for Misaligned Calibration Target in Passive Microwave Remote-Sensing Systems Dazhen Gu, Senior Member, IEEE, James Randa, Senior Member, IEEE, and David K. Walker, Senior Member, IEEE
Abstract—We present a geometric error model associated with calibration-target misalignment in passive microwave remote-sensing systems. The developed analytic formulation is universally applicable to both lateral and rotational misalignment conditions. Numerical simulations are performed on two practical blackbody targets of different sizes used as radiation references for passive microwave remote sensing. The significance of this work is to furnish a framework of uncertainty analysis due to target misalignment and to provide a reference for alignment requirements based on passive radiometer measurement sensitivity. Index Terms—Blackbody target, microwave radiometer, misalignment, thermal radiation, uncertainty analysis.
uation of the remote-sensing observation. The measurement uncertainty of the microwave radiation from a blackbody target involves multiple sources, such as system sensitivity, environmental stability, hardware alignment, etc. Unlike other error sources known under the specific conditions, uncertainty arising from misalignment is often unclear and difficult to quantify. Pertinent theory is missing for users to assess such errors and set up appropriate alignment requirements. In this letter, we establish a theoretical formulation of measurement uncertainty caused by the misalignment of blackbody targets and simulate its effect in practical experiments.
I. I NTRODUCTION
II. T HEORY
P
ASSIVE microwave remote-sensing systems have become a critical component of climate and environmental observations globally. Applications of such instruments can be found in a wide range of disciplines, providing information on weather forecast [1], [2], climate change [3], precipitation rates [4], ocean salinity and roughness [5], and agricultural management [6], among others. Most spaceborne passive remotesensing systems are equipped with a blackbody target of a finite size that provides a hot temperature reference in addition to the cold reference provided by the cosmic background, while airborne systems may use multiple targets as references. At the National Institute of Standards and Technology, we are engaged in the research on standard blackbody targets at microwave frequencies and present here a related study in this field. The calibration of a blackbody target requires a thorough uncertainty analysis that ultimately affects the accuracy eval-
Manuscript received March 5, 2013; revised April 12, 2013; accepted May 7, 2013. Date of publication May 29, 2013; date of current version October 10, 2013. This work was supported by National Institute of Standards and Technology, an agency of the U.S. Government, and is not subject to U.S. copyright. D. Gu is with the Electromagnetics Division, National Institute of Standards and Technology, Boulder, CO 80305 USA, and also with the Department of Electrical, Computer, and Energy Engineering, University of Colorado, Boulder, CO 80309 USA (e-mail: [email protected]). J. Randa is with the Electromagnetics Division, National Institute of Standards and Technology, Boulder, CO 80305 USA, and also with the Department of Physics, University of Colorado, Boulder, CO 80309 USA (e-mail: [email protected]). D. K. Walker is with the Electromagnetics Division, National Institute of Standards and Technology, Boulder, CO 80305 USA (e-mail: david.walker@ nist.gov). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/LGRS.2013.2262471
We derive the Poynting vector from the coherence tensor between electric and magnetic fields of the blackbody radiation. The theoretical model is applied to the ideal alignment condition and then to different misalignment conditions to calculate the power received by a microwave radiometer. A. Radiometric Model of Blackbody The electromagnetic (EM)-mixed cross-spectral tensor at observation points r1 and r2 is defined as [7], [8] EM Wjk (r1 , r2 , ω)
1 = 2π
∞ Ej (r1 , t + τ ) −∞
×Hk∗ (r2 , t) exp(iωτ ) dτ
(1)
where Ej (r, t) and Hj (r, t)(j = x, y, z) symbolize the jcomponent of the electric- and magnetic-field vectors in the Cartesian coordinates at position r and at time t and the brackets · denote the ensemble average. For a blackbody, the EM-coherence tensor takes a simplified expression EM ˆ (r1 , r2 , ω) = −B(ω, T ) ˆj × k Wjk ˆs exp(iKˆs · r ) dΩ (2) · Ω
where r = r2 − r1 and ˆs is the unit vector viewing from ¯r (defined by (r1 + r2 )/2) to the surface of the blackbody target. Ω is the total solid angle subtended by the blackbody target viewing from ¯r, and K is the wavenumber in the experimental
U.S. Government work not protected by U.S. copyright.
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IEEE GEOSCIENCE AND REMOTE SENSING LETTERS, VOL. 10, NO. 6, NOVEMBER 2013
Fig. 1. Illustration of a receiving antenna located at the origin that collects radiation from a blackbody target centered at (0, 0, z0 ).
medium, usually air or vacuum. B(ω, T ) is the spectral radiance of a blackbody at temperature T , given by B(ω, T ) =
ω 3 4π 3 c2 exp kω − 1 T B
(3)
where and kB are the Planck constant divided by 2π and the Boltzmann constant, respectively; c is the speed of light in vacuum. In (2), we assume an isothermal blackbody object with uniform temperature distribution across its surface. The Poynting vector S at the receiving antenna can be derived easily from (1) by forcing r1 = r2 = r or r = 0. The l-component of S is associated with the coherence tensor by Sl (r, ω) =
1 EM EM Wjk (r, r, ω) − Wkj (r, r, ω) 2
(4) C. Lateral Misalignment
where {·} denotes the real part of a complex variable. The order of jkl strictly follows the cyclic order of xyz, e.g., jkl can be either xyz, yzx, or zxy. We obtain a much simplified radiometric model of the blackbody radiation by substituting (2) into (4) (−ˆs) dΩ. (5) S(r, ω) = B(ω, T ) Ω
B. Perfectly Aligned Blackbody Target We assume that the blackbody radiation receiver, often consisting of an antenna as its front end, is fixed at the origin of the Cartesian coordinates as shown in Fig. 1. To a first-order approximation, we consider the receiving antenna as a point receiver: The variation across the antenna aperture is neglected. In addition, the antenna is sensitive only to the radiation flux along the −ˆ z-direction. The blackbody target is a flat circular disk with a radius of rtg located at (0, 0, z0 ). For perfect alignment, the normal of ˆ coincides with −ˆ the target surface n z. Due to the symmetry, only the z-component of S is nonzero, and it follows directly from the straightforward integration of (5) that S0z (ω) = −πB(ω, T )
2 rtg 2 + z2 . rtg 0
Fig. 2. (a) Lateral misalignment is characterized by two scalars: δx and δy . ˆ with two (b) Rotational misalignment shows a diverged surface normal vector n angles indicated by β and α.
(6)
For the remainder of this letter, the ensemble average over the Poynting vector and its components is implied, and the brackets enclosing S are dropped in the notation.
Now, we imagine that the misaligned target possesses only a small offset along the vertical direction. Its new position can be characterized by (0, δy , z0 ). The Poynting vector at the origin of the coordinates consists of not only a z-component but also a y-component, given by 2 rtg 2δy2 z02 Sz = − πB(ω, T ) 2 1−
(7a) 2 + z2 2 rtg + z02 rtg 0
2 2 δy rtg δy2 2z02 − rtg z0 Sy = − πB(ω, T )
1− 2 2 . (7b) 2 + z2 2 rtg rtg + z02 0 Although it is a very specific case of misalignment, such a result is generally applicable to more complicated misalignment cases. We represent (7a) and (7b) in a general formulation as 2 rtg 2δ 2 z 2 Sz = − πB(ω, T ) 2 1−
(8a) 2 + z2 2 rtg + z 2 rtg
2 2 δ 2 2z 2 − rtg z δrtg St = − πB(ω, T )
1− 2 2 . (8b) 2 + z2 2 rtg rtg + z 2 Equations (8a) and (8b) can be easily extended to deal with more general lateral misalignment, in which the target position is offset transversely and vertically to (δx , δy , z0 ), as shown in Fig. 2(a). Sz and St are calculated by assigning z = z0 and δ = δx2 + δy2 in (8a) and (8b). The unit vector ˆt is given by ˆ + δy y ˆ ). (1/δ)(δx x
GU et al.: GEOMETRIC ERROR MODEL FOR MISALIGNED CALIBRATION TARGET IN REMOTE-SENSING SYSTEMS
D. Rotational Misalignment
can be calculated as
A rotational misalignment can be characterized by the elevation angle β and the azimuth angle α, illustrated in Fig. 2(b). ˆ and z-axis z ˆ, β is the angle between the surface normal n where β = 180◦ corresponds to perfect alignment. α is the ˆ on the xy plane and the angle between the projected vector of n x-axis. To ease the problem, we rotate the xyz coordinates to x y z accordingly. The relationship between the two coordinate systems can be represented by ˆ = x ˆ sin α − y ˆ cos α x ˆ = −x ˆ cos β cos α − y ˆ cos β sin α + z ˆ sin β y ˆ = −x ˆ sin β cos α − y ˆ sin β sin α − z ˆ cos β. z
(9a) (9b) (9c)
Viewing from the x y z coordinates, we transform the rotational misalignment problem to the spatial misalignment problem. The center of the target is located at (0, δy , z0 ) in the new coordinates. The offset and the distance are given by δy = z0 sin β χ[0,π) (α) − χ[π,2π) (α) (10a) z0 = − z0 cos β (10b) where the indicator function χC (α) is defined as 1 if α ∈ C χC (α) ≡ 0 if α ∈ / C.
(11)
We now substitute z0 and δy for the variables z and δ, respectively, in (8a) and (8b) to calculate Sz and Sy . Note ˆ . We bear in mind that the receiver is still pointed that ˆt = y ˆ-direction rather than the z ˆ -direction. The quantity along the z of interest Sz in fact consists of both Sz - and Sy -components and is represented as Sz = −Sz cos β + Sy sin β.
(12)
E. General Misalignment A general misalignment case is composed of both spatial and rotational offsets. Physically, it can be characterized by two ˆ . We again first scalar offsets δx and δy and one vector offset n rotate the coordinate system to x y z in the same manner as we did in Section II-D and next translate the scalar offsets in the new coordinates. ˆ cos β cos α − z ˆ sin β cos α) ˆ = δx (ˆ x sin α − y (13a) δx x ˆ cos β sin α − z ˆ sin β sin α). (13b) ˆ = δy (−ˆ δy y x cos α − y The position of the target center in the x y z coordinates is (δx , δy , z0 ), with their specific values given by δx = δx sin α − δy cos α
1599
(14a)
δy = − δx cos β cos α − δy cos β sin α + z0 sin β χ[0,π) (α) − χ[π,2π) (α) (14b) z0 = − δx sin β cos α − δy sin β sin α − z0 cos β. (14c)
Similarly, we assign z = z0 and δ = δx2 + δy2 in (8a) and (8b) to compute Sz and St . Once again, Sz is of interest and
Sz = −Sz cos β +
δy δx2 + δy2
St sin β.
(15)
III. E XAMPLES We apply the error model to practical blackbody targets that are used frequently in our laboratory for passive microwave remote-sensing applications. Numerical simulations are run on two calibration targets, one with a radius of 16.5 cm and the other with a radius of 9 cm. The radius values are obtained from two blackbody targets for practical passive remote-sensing applications [9]. We study the cases involving only the δy offset for the spatial misalignment, while the divergence of the surface normal is limited to the variation of β (α is kept at π/2, or equivalently, x = x) only for the rotational misalignment. It is relatively straightforward to extend the simulation to include the offsets of other variables by using the formulations in Section II for specific studies. We compute the error percentage, defined by Sz | − |S0z ||/|S0z |, to compare the variation of the z-component of the Poynting vector for different types of misalignment. The offset of δy is represented as a fraction of the target radius in Fig. 3. The normalized offset is limited to no more than 0.1rtg , and the divergence angle β is kept between 180◦ and 175◦ . We also include different target–antenna distances to show the error percentage variation. In the simulated range of offsets, the error percentage is smaller than 0.5% for both the small and large targets. The error percentage due to the lateral misalignment follows the second term in the square brackets of (7a). Hence, for a fixed δy , the value shows a decreasing trend as the distance increases. The error percentage caused by the rotational misalignment is more complicated, which has been discussed in Section II-D. In general, its trend for a fixed β as a function of the distance is not straightforward. In order to predict its behavior, the evaluation of the error percentage with the specific rtg value is required. In some additional simulations with different target sizes, we find that the error percentage can show a minimum at the location other than the boundaries in the simulated distance range. For microwave remote sensing, the knowledge of alignment requirements is important to a system designer. In the Rayleigh–Jeans approximation, the radiance B(ω, T ) is proportional to T , and so is the magnitude of the Poynting vector. Since the physical variable of interest in microwave remote sensing mostly comes down to the temperature, we can assess the requirement of the alignment based on the desired accuracy of the observed temperature. A recent experimental study of brightness-temperature standards shows a measurement uncertainty of about 1 K for a warm blackbody target at 350 K [10], which also meets some of the accuracy requirements of climate variables [11]. The measurement uncertainty of the brightness temperature corresponds to 0.3% in [10]. In order for the error arising from misalignment to be insignificant, we set the error percentage to 0.1% and evaluate the relevant alignment requirements from the simulation result. In other words,
1600
IEEE GEOSCIENCE AND REMOTE SENSING LETTERS, VOL. 10, NO. 6, NOVEMBER 2013
Fig. 3. Error percentage caused by misalignment at various target–antenna distances. (a) A 16.5-cm-radius target vertically offsets in units of the target radius. (b) The surface normal of the 16.5-cm-radius target diverges from 180◦ to 175◦ . (c) A 9-cm-radius target vertically offsets in units of the target radius. (d) The surface normal of the 9-cm-radius target diverges from 180◦ to 175◦ .
this additional misalignment error would make the brightnesstemperature uncertainty increase from 1 K to 1.05 K. For the small target, the lateral misalignment should be kept below 0.06rtg , and the rotational misalignment should be kept below 2.6◦ for less than 0.1% error in the simulated range. For the large target, the corresponding results are 0.046rtg and 2.7◦ , respectively. IV. D ISCUSSION Longitudinal misalignment, i.e., the variation of the target– antenna distance, is not explicitly studied in this letter. In practice, such misalignment is relatively easier to be minimized to a certain extent in comparison to other types of misalignment mentioned here. However, the inclusion of such a misalignment is also straightforward by replacing z0 with z0 + δz in (6). The significance of this study is to provide a guideline for the estimation of the measurement uncertainty associated with misalignment, which is not presently established to our knowledge. The uncertainty due to the misalignment can be included in the total uncertainty estimation of the target brightness temperature. In addition, the theoretical framework is valuable to users to determine the alignment requirement based on the radiometer sensitivity. Targets of an arbitrary shape may be encountered in practical remote-sensing systems. A direct analytic expression of the Poynting vector component for such misaligned targets may not be available. However, the integration can nube evaluated ˆ /r2 ) dA) merically by a similar treatment ( Ω dΩ → A (ˆs · n outlined in the Appendix.
V. C ONCLUSION By the use of the fundamental EM-coherence theory of blackbody radiation, a general error model is developed to compute the uncertainty arising from the geometric misalignment of the calibration target in passive microwave remotesensing systems. Such a model can be universally applied to particular lateral and rotational misalignments or a combination of the two. The error percentage is simulated on two practical blackbody targets for a range of distances between the target and the radiometric receiver. In order to maintain an insignificant impact on the overall uncertainty, the lateral misalignment should be held smaller than 0.06rtg and 0.046rtg for the small and large targets, respectively, while the offset angles should be kept below 2.6◦ and 2.7◦ for the small and large targets, respectively. This study provides a perspective on how to quantify the measurement uncertainty related to the target misalignment and how to determine the alignment requirement based on other uncertainty sources and required observation accuracy. A PPENDIX I NTEGRATION OVER AN O FF -C ENTERED C IRCULAR A PERTURE The integration over the solid angle is nontrivial when the target center is vertically offset. The regular transformation from Ω dΩ to dφ sin θdθ brings complications since the integral bounds of θ cannot be easily determined analytically. Rather, we need to represent the solid angle integration as ˆ (ˆ s · n /r2 )dA, where dA is the infinitesimal area of the A target, A is the total area of the target, and r is the distance
GU et al.: GEOMETRIC ERROR MODEL FOR MISALIGNED CALIBRATION TARGET IN REMOTE-SENSING SYSTEMS
from the origin to dA. Therefore, the integration to be solved in (5) is ˆs · n ˆ z0 B(ω, T ) −ˆs 2 dA = B(ω, T ) −ˆs 3 dx dy (16) r r A
A
ˆ). Due to the yz where ˆs is equivalent to (1/r)(xˆ x + yˆ y + z0 z plane symmetry, the x-component vanishes after the integration. We then need only to solve Sz and Sy z02 Sz = −B(ω, T ) 2 dx dy (x2 + y 2 + z02 )
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On the other hand, the target can now be regarded as a point source in the FF zone. The Poynting vector at the origin can be simply obtained from (6) ˆ S0z cos3 θ sin θ + z ˆS0z cos4 θ SF F = y
(19)
where θ is the angle between the zenith and the vector ˆs: θ = tan−1 (δy /z0 ). The order of 3 on cos θ comes from Lambert’s cosine law in addition to the equivalent distance being increased by a factor of 1/ cos θ. By the use of the Taylor series of trigonometric functions, it can be easily shown that (19) is in agreement with (18a) and (18b).
A
= −B(ω, T )
z02 +
+ δy ) +
(x2 +
z02 y 2
+ z02 )
≈ −B(ω, T ) A
2
(y
A
x2
z02
2 dx dy
ACKNOWLEDGMENT
2
2δy2 x2 − 5y 2 + z02 4y δy dx dy · 1 − 2 − 2 x + y 2 + z02 (x2 + y 2 + z02 )
2π = −B(ω, T )z02 ×
rtg dφ
0
1 (ρ2 + z02 )
2
ρ dρ 0
−
2δy2
ρ2 cos2 φ − 5ρ2 sin2 φ + z02
4
(ρ2 + z02 ) 2 rtg 2δy2 z02 = −πB(ω, T ) 2 1−
. 2 + z2 2 rtg + z02 rtg 0
(17)
The coordinates x y facilitate the transition from xy to ρφ. In addition, the integration of any odd function of y results in zero as implicitly shown from the third step to the fourth step in (17). Following a similar principle, we can reproduce (7b) for Sy . To ascertain that the criterion δ z0 holds, we evaluate δ/z0 in the simulation of this work and confirm that its value is below 0.083 for all cases. Furthermore, all the plots reported in this letter have been verified by the numerical computation of the integral without any approximation. To further validate the derivation, we consider the far field (FF) condition where rtg z0 . On the one hand, the FF values of Sz and Sy can be computed from (7a) and (7b) by taking the FF approximation 2 2 r 2δ tg y (18a) SzF F = − πB(ω, T ) 2 1 − 2 z0 z0 2 rtg 2δy3 δy FF Sy = − πB(ω, T ) 2 (18b) − 3 . z0 z0 z0
The authors would like to thank P. Racette at the National Aeronautics and Space Administration Goddard Space Flight Center for loaning the calibration targets. D. Gu would like to thank Y. Liu of National Instruments for his technical inputs. R EFERENCES [1] S.-A. Boukabara, F. Weng, and Q. Liu, “Passive microwave remote sensing of extreme weather events using NOAA-18 AMSUA and MHS,” IEEE Trans. Geosci. Remote Sens., vol. 45, no. 7, pp. 2228–2246, Jul. 2007. [2] Z. Yang, N. Lu, J. Shi, P. Zhang, C. Dong, and J. Yang, “Overview of FY-3 payload and ground application system,” IEEE Trans. Geosci. Remote Sens., vol. 50, no. 12, pp. 4846–4853, Dec. 2012. [3] J. S. Kimball, L. A. Jones, K. Zhang, F. A. Heinsch, K. C. McDonald, and W. C. Oechel, “A satellite approach to estimate land atmosphere CO2 exchange for boreal and Arctic biomes using MODIS and AMSR-E,” IEEE Trans. Geosci. Remote Sens., vol. 47, no. 2, pp. 569– 587, Feb. 2009. [4] C. Surussavadee and D. H. Staelin, “NPOESS precipitation retrievals using the ATMS passive microwave spectrometer,” IEEE Geosci. Remote Sens. Lett., vol. 7, no. 3, pp. 440–444, Jul. 2010. [5] J. L. Garrison, J. K. Voo, S. H. Yueh, M. S. Grant, A. G. Fore, and J. S. Haase, “Estimation of sea surface roughness effects in microwave radiometric measurements of salinity using reflected global navigation satellite system signals,” IEEE Geosci. Remote Sens. Lett., vol. 8, no. 6, pp. 1170–1174, Nov. 2011. [6] R. Bindlish, T. Jackson, R. Sun, M. Cosh, S. Yueh, and S. Dinardo, “Combined passive and active microwave observations of soil moisture during CLASIC,” IEEE Geosci. Remote Sens. Lett., vol. 6, no. 4, pp. 644– 648, Oct. 2009. [7] W. Eckardt, “Macroscopic theory of electromagnetic fluctuations and stationary radiative heat transfer,” Phys. Rev. A, vol. 29, no. 4, pp. 1991– 2003, Apr. 1984. [8] D. C. Bertilone, “On the cross-spectral tensors for black-body emission into space,” J. Mod. Opt., vol. 43, no. 1, pp. 207–218, Jan. 1996. [9] D. Gu, D. Houtz, J. Randa, and D. K. Walker, “Reflectivity study of microwave blackbody target,” IEEE Trans. Geosci. Remote Sens., vol. 49, no. 9, pp. 3443–3451, Sep. 2011. [10] D. Gu, D. Houtz, J. Randa, and D. K. Walker, “Extraction of illumination efficiency by solely radiometric measurements for improved brightness-temperature characterization of microwave blackbody target,” IEEE Trans. Geosci. Remote Sens., vol. 50, no. 11, pp. 4575–4583, Nov. 2012. [11] R. Datla, B. Emery, G. Ohring, R. Spenser, and B. Wielicki, “Stability and accuracy requirements for satellite remote sensing instrumentation for global climate change monitoring,” in Proc. ISPRS, Jul. 2004, pp. 17–27.
