Recommended Terminology for 

Microwave Radiometry 

James Randa 

Janne Lahtinen 

Adriano Camps 

A. J. Gasiewski 

Martti Hallikainen 

David M. Le Vine 

Manuel Martin-Neira 

Jeff Piepmeier 

Philip W. Rosenkranz 

Christopher S. Ruf 

James Shiue 

Niels Skou 

NIST Technical Note 1551

 

NIST Technical Note 1551 

Recommended Terminology for 

Microwave Radiometry 

August 2008 

U.S. Department of Commerce 

Carlos M. Gutierrez, Secretary 

National Institute of Standards and Technology 

James M. Turner, Deputy Director 

James Randa 

Electromagnetics Division

Electronics and Electrical Engineering 

Laboratory

National Institute of Standards and 

Technology

Boulder, CO 80305-3328 

Janne Lahtinen 

Harp Technologies Ltd. 

02150 Espoo, Finland 

Adriano Camps

Department of Signal Theory and 

Communications 

Universitat Politecnica de Catalunya 

Campus Nord 

08034 Barcelona, Spain 

A. J. Gasiewski

NOAA-CU Center for Environmental 

Technology (CET) 

Univ. of Colorado at Boulder 

Boulder, CO 80309 

Martti Hallikainen

Laboratory of Space Technology

Helsinki Univ. of Technology 

02150 Espoo, Finland 

David M. Le Vine 

James Shiue 

Jeff Piepmeier

NASA, Goddard Space Flight Center

Greenbelt, MD 20771 

Manuel Martin-Neira 

European Space Agency 

2200 AG Noordwijk 

The Netherlands 

Philip W. Rosenkranz 

Massachusetts Institute of Technology 

Cambridge, MA 02139 

Christopher S. Ruf 

Space Physics Research Laboratory 

Univ. of Michigan 

Ann Arbor, MI 48109 

Niels Skou

Technical University of Denmark 

B348, DK2800 Lyngby, Denmark 
U.S. Government Printing Office 

Washington: 2008 

For sale by the Superintendent of Documents, U.S. Government Printing Office 

Internet bookstore: gpo.gov Phone: 202-512-1800 Fax: 202-512-2250 

Mail: Stop SSOP, Washington, DC 20402-0001 

Certain commercial entities, equipment, or materials may be identified in this 

document in order to describe an experimental procedure or concept adequately. Such 

identification is not intended to imply recommendation or endorsement by the 

National Institute of Standards and Technology, nor is it intended to imply that the 

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

National Institute of Standards and Technology Technical Note 1551 

Natl. Inst. Stand. Technol. Tech. Note 1551, 32 pages (August 2008) 

CODEN: NTNOEF 
CONTENTS

 Page

1. BACKGROUND ....................................................................................................... 2 

2. CHAPTER I. GENERAL TERMINOLOGY ............................................................ 2 

3. CHAPTER II. REAL-APERTURE RADIOMETERS .............................................. 6 

4. CHAPTER III. POLARIMETRIC RADIOMETRY ................................................. 13 

5. REFERENCES .......................................................................................................... 26 

6. ALPHABETICAL INDEX ........................................................................................ 27 

iii

RECOMMENDED TERMINOLOGY FOR MICROWAVE 

RADIOMETRY

James Randa1

, Janne Lahtinen2, Adriano Camps3, A. J. Gasiewski4, 

Martti Hallikainen5, David M. Le Vine6, Manuel Martin-Neira7, Jeff Piepmeier6, 

Philip W. Rosenkranz8

, Christopher S. Ruf9, James Shiue6, and Niels Skou10 

1

Electromagnetics Division, NIST, Boulder, CO 80305 

2

 Harp Technologies Ltd, 02150 Espoo, Finland 

3

 Department of Signal Theory and Communications, Universitat Politecnica de 

Catalunya, Campus Nord, 08034 Barcelona, Spain 

4

 NOAA-CU Center for Environmental Technology (CET), Univ. of Colorado at 

Boulder, Boulder, CO 80309 

5

 Laboratory of Space Technology, Helsinki Univ. of Technology, 02150 Espoo, 

Finland 

6

 NASA, Goddard Space Flight Center, Greenbelt, MD 20771 

7

 European Space Agency, 2200 AG Noordwijk, The Netherlands 

8

 Massachusetts Institute of Technology, Cambridge, MA 02139 

9

 Space Physics Research Laboratory, Univ. of Michigan, Ann Arbor, MI 48109 

10 Technical University of Denmark, B348, DK2800 Lyngby, Denmark 

Abstract: We present recommended definitions for common terms in microwave remotesensing radiometry. Terms are grouped into three chapters: General Terminology, RealAperture Radiometers, and Polarimetric Radiometry. An alphabetical index lists the 

terms that are defined and the chapters in which the definitions are located. 

Keywords: definitions, microwave radiometry, remote sensing, standard terminology

1
BACKGROUND

 Several years ago, a project was begun under the auspices of the Committee on 

Earth Observation Satellites (CEOS), Calibration and Validation Working Group, to 

compile standard definitions for terms used in microwave remote-sensing radiometry. A 

parallel effort was begun at about the same time in the NIST Electromagnetics Division, 

and the two efforts soon merged. A compilation of standard terminology and practices 

had been published for radiometry at optical frequencies [1], and had been found to be 

very useful. Since much of the terminology for microwave radiometry differs from that 

used at optical frequencies, a project was begun to develop a corresponding set of 

standard definitions for microwave radiometry. The effort was divided into four 

chapters, covering General Terminology (Chapter 1, or I), Real-Aperture Radiometers 

(II), Polarimetric Radiometry (III), and Interferometric Radiometry (IV). 

Recommendations and suggestions were solicited from the microwave radiometry 

community, and a website [2] and e-mail circulation list were established for I and II, to 

accept suggestions and to post the current definitions. After an initial burst of activity, 

the definitions for I and II stabilized, and they were presented to CEOS in 2003. 

 A “Radiometry Dictionary” page has now been added to the website of the IEEE 

Geoscience and Remote Sensing Society (GRSS) [3], under “Technical Resources.” The 

purpose of the page is to increase the availability of the completed chapters, facilitate 

completion of the unfinished chapter, and raise awareness of the standard definitions. 

We expect that the list of standard definitions will evolve over time, with the completion 

of IV, as well as additions and modifications to I through III. The present paper contains 

the initial versions of I – III on the GRSS website. It thus constitutes a snapshot of I – III, 

documenting their state as of early 2008. 

 Many of the definitions have been drawn from other sources, in which case the 

source is referenced. Two of the most common sources, particularly for the terms in I, 

are the International Vocabulary of Basic and General Terms in Metrology (IVM) [4] and 

the Guide to the Expression of Uncertainty in Measurement (GUM) [5]. For terms 

contained in the IVM or GUM, we have tried to use those definitions. An alphabetical 

index follows the three chapters of definitions; it lists the terms defined and indicates in 

which chapter they are defined. 

CHAPTER I. GENERAL TERMINOLOGY 

Accuracy: closeness of the agreement between the result of a measurement and a true 

value of the measurand. Because the true value cannot be determined exactly, the 

measured or calculated value of highest available accuracy is typically taken to be the 

true value. If a value of higher accuracy than the value in question is not available, then 

the accuracy cannot be assigned a meaningful quantitative value. 

Allan standard deviation (ASD): the square root of Allan variance. Both the ASD and 

the Allan variance can be used to describe different noise types and drift of the response 

2
of a radiometer receiver. Compared to Allan variance, however, the ASD has the 

advantage that the units are in kelvins. 

Allan variance: a two-sample variance, where the variance is taken on the variable y. 

Each value of y in a set has been averaged over an interval τ and the y’s are taken in an 

adjacent series, i.e., no delay between the measurements of each: 

( ) ( )

2

2

2 1 y A τσ Δ= , 

where the brackets < > denote the expectation value, and Δy is the difference between 

sequential values of y. The Allan variance can be used to describe different noise types 

and drift of a radiometer receiver. See also: Allan Standard Deviation (ASD).

Bias (of a measuring instrument): systematic error of the indication of a measuring 

instrument . [4] 

Note: The bias of a measuring instrument is normally estimated by averaging the error of 

indication over an appropriate number of repeated measurements. 

Brightness: the radiance, the radiated power per unit solid angle per unit area normal to 

the direction defined by the solid angle Ω. 

Note: Brightness (of a surface) is often used for Spectral brightness density [6, 7]. The meaning 

is usually clear from the context. 

Calibration: set of operations that establish, under specified conditions, the relationship 

between sets of values of quantities indicated by a measuring instrument or measuring 

system and the corresponding values realized by standards. The result of a calibration 

permits either the assignment of values of measurands to the indications or the 

determination of corrections with respect to the indications. [4] 

Calibration equation (of a microwave radiometer): equation relating the primary 

measurand of the radiometer, e.g., the brightness temperature, to subsidiary measurands, 

such as powers, and to calibration quantities, such as standard values. (similar to [1]) 

Check standard: a standard that is used routinely to check measurements. 

Drift: slow change of a metrological characteristic of a measuring instrument. [4] 

Emissivity (of a surface): the spectral radiance of the surface relative to the spectral 

radiance of a perfect blackbody radiator at the same temperature, )(

),,( ),(

, TL

TL

e

BBf

f θ φ φθ = . 

Effective blackbody brightness temperature: the physical temperature of a perfect 

blackbody that would produce the same spectral brightness density or spectral radiance 

density as that under consideration. 

3
Error: result of a measurement minus a true value of the measurand. [4,5] 

Notes: Since a true value cannot be determined, in practice a conventional true value is used. 

When it is necessary to distinguish “error” from “relative error,” the former is sometimes called 

the “absolute error.” 

Experimental standard deviation: for a series of n measurements of the same 

measurand, the quantity s characterizing the dispersion of the results and given by the 

formula 

1

)(

1

2

−

−

=

∑=

n

xx

s

n

i

i

, 

xi being the result of the ith measurement and x being the arithmetic mean of the n results 

considered. [4,5] 

Notes: Considering the series of n values as a sample of a distribution, x is an unbiased estimate 

of the mean μ, and s

2

 is an unbiased estimate of the variance σ

2

 of that distribution. The 

expression / ns is an estimate of the standard deviation of the distribution of x and is called 

the experimental standard deviation of the mean. 

Noise temperature (at a reference plane in a microwave circuit/network, Tn): the 

available noise power spectral density at the reference plane divided by Boltzmann’s 

constant (kB); thus nB

av

n Tk

df

dP = by definition, where the subscript n denotes “noise.” For 

a resistor at physical temperature Tphys, the noise temperature is given by 

−1

=

Tk physB

nB hf

e

hf Tk , so that for <<1

Tk physB

hf , Tn ≈ Tphys . 

Quantization level: in the quantization process, the discrete value assigned to a 

particular subrange of the analog signal being quantized. (adapted from [8]) 

Radiance (L): the radiated power per unit solid angle per unit area normal to the 

direction defined by the solid angle Ω, 

⊥

= dAd

dP L

Ω , 

where dA⊥ = cosθ ′dA, and θ′ is the angle between the direction defined by the solid 

angle and the normal to the area element dA. 

Random error: result of a measurement minus the mean that would result from an 

infinite number of measurements of the same measurand carried out under repeatability 

conditions. [4,5] 

Notes: Random error is equal to error minus systematic error. Because only a finite number of 

measurements can be made, it is possible to determine only an estimate of random error. 

Reference temperature (T0): temperature used in ratios to set a scale, conventionally 

taken as T0 = 290 K. 

4
Relative error: error of measurement divided by a (conventional) true value of the 

measurand. [4,5] 

Relative uncertainty: uncertainty of the measurand divided by the nominal value of that 

measurand. 

Repeatability (of results of measurements): closeness of the agreement between the 

results of successive measurements of the same measurand carried out under the same 

conditions of measurement. [4] 

Notes: Repeatability conditions include: the same measurement procedure, the same observer, the 

same measuring instrument used under the same conditions, the same location, and repetition 

over a short period of time. Repeatability may be expressed quantitatively in terms of the 

dispersion characteristics of the results. 

Reproducibility (of results of measurements): closeness of the agreement between the 

results of measurements of the same measurand carried out under changed conditions of 

measurement. [4] 

Notes: A valid statement of reproducibility requires specification of the conditions changed. 

Reproducibility may be expressed quantitatively in terms of the dispersion characteristics of the 

results. 

Resolution: smallest difference between values of a measurand that can be meaningfully 

distinguished. (modified [4]) 

Response time: time interval between the instant when a stimulus is subjected to a 

specified abrupt change and the instant when the response reaches and remains within 

specified limits around its final steady value. [4]

Sensitivity: change in the response of a measuring instrument divided by the 

corresponding change in the stimulus. [4] 

Spectral brightness density: the brightness per unit frequency, the spectral radiance

density. 

Spectral radiance density (Lf): the radiance per unit frequency interval, df

dL Lf ≡

Stability: ability of a measuring instrument to maintain constant its metrological 

characteristics with time. [4] 

Notes: Stability may be quantified in several ways, for example: in terms of the time over which a 

metrological characteristic changes by a stated amount, or in terms of the change in a 

characteristic over a stated time. 

Systematic error: mean that would result from an infinite number of measurements of 

the same measurand carried out under repeatability conditions minus a true value of the 

measurand. [4,5] 

5
Notes: Systematic error is equal to error minus random error. Like true value, systematic error 

and its causes cannot be completely known. For a measuring instrument, see “bias.”

Traceability: property of the result of a measurement or the value of a standard whereby 

it can be related to stated references, usually national or international standards, through 

an unbroken chain of comparisons, all having stated uncertainties. [4] 

True value: value consistent with the definition of a given particular quantity. [4,5] 

Notes: This is a value that would be obtained by a perfect measurement. True values are by 

nature indeterminate. 

Type-A uncertainty: an uncertainty or component of uncertainty that is evaluated by the 

statistical analysis of series of observations. (adapted from [5]) 

Type-B uncertainty: an uncertainty or component of uncertainty that is evaluated by 

means other than the statistical analysis of series of observations. (adapted from [5]) 

Uncertainty budget: a detailed tabulation of the sources of uncertainty and their 

respective contributions to the standard uncertainty in the measurand of interest; 

sometimes referred to as “error budget.” 

Uncertainty of measurement: parameter, associated with the result of a measurement, 

that characterizes the dispersion of the values that could reasonably be attributed to the 

measurand; also a measure of the possible error in the estimated value of the measurand 

as provided by the result of a measurement. See [5] or [9] for details. 

Validation: the process of assessing, by independent means, the quality of the data 

products derived from the system inputs. 

Note: In a remote-sensing application, validation is commonly performed by comparing the 

value of a higher-order data product (e.g. sea surface temperature) inferred from the measured 

value of the brightness temperature, with a direct measurement of that quantity. 

Verification: the testing and analysis necessary to provide confirmation that all (e.g., 

instrument) requirements have been met. 

CHAPTER II. REAL-APERTURE RADIOMETERS

Absolute accuracy: commonly used to denote the total uncertainty, excluding the 

radiometric resolution. When the radiometric resolution is evaluated by statistical means 

(type-A uncertainty) and all other uncertainties are estimated by other means, the absolute 

accuracy corresponds to the total type-B uncertainty. 

Angular resolution: angular width of antenna beam, usually defined as the half-power 

beam width. See also antenna beam width. 

6
Antenna aperture temperature (TA,ap): the temperature of an ideal black body that 

would result in the same received power at the antenna aperture as in the actual case. In 

the Rayleigh-Jeans approximation, it is equal to the average incident brightness 

temperature weighted by the antenna directivity pattern, 

= ∫∫ −

π

ΩΩΩΩ

π

Ω

4

0, 0 )()( 4

1 T apA )( D b dT . 

The antenna aperture temperature corresponds to the antenna temperature at the 

aperture of the antenna. 

Antenna beam width: the (full) angle at which the antenna's radiation pattern (in power 

units) is at half its maximum value. It is also known as the “Half-power full width” 

(HPFW), or simply the "Half-power beam width," (HPBW). In engineering convention, 

it is also known as the "3 dB beam width." 

The antenna radiation pattern depends on the specific plane of cut, and two principal 

planes are generally given. For a linearly polarized antenna they are the E-plane (the 

plane containing the radiation maximum and the E-field) and the H-plane (the plane 

containing the radiation maximum and the H-field, which plane is perpendicular to the Eplane). 

Antenna gain (G(θ,φ)): the ratio of the radiation intensity in a given direction, to the 

radiation intensity that would be obtained if the antenna pattern were isotropic. (adapted 

from [10]) 

Note: This is equivalent to the commonly used definition that the antenna gain is the product of 

the directivity and the antenna ohmic efficiency, or radiation efficiency. If the direction is not 

specified, the direction of maximum radiation is implied.

Antenna ohmic efficiency (ηΩ): the ratio of the total radiated power divided by the total 

power accepted by the antenna. It is also equivalent to the ratio of the antenna radiation 

resistance (Rrad) divided by the sum of the antenna radiation resistance and the antenna 

ohmic resistance (RΩ): 

rad rad

in rad

P R

PRR η Ω

Ω

= = + . 

It is also commonly called the radiation efficiency [11]. 

Antenna output temperature (TA,out): the physical temperature of a matched impedance 

that delivers to the receiver the same noise power as the antenna. It includes two terms: 

the noise coming from the environment attenuated by the antenna ohmic efficiency and 

the thermal noise added by the antenna ohmic losses. In the Rayleigh-Jeans 

approximation, 

T outA T apA Tp )( )1()( Ω 0, =ηΩ Ω 0, + −ηΩ , 

7
where Tp is the physical temperature of the antenna and ηΩ is the ohmic efficiency of the 

antenna. The antenna output temperature is related to the input noise temperature of the 

receiver (Trec,in) by 

1 ,

,

−

=

outA kT

hf inrec

e

hf T . 

Antenna temperature (Ta): the temperature (K) equivalent of the power received with 

an antenna. The distinction between the antenna temperature, Ta, and the brightness 

temperature, Tb , is as follows. The antenna temperature is the equivalent of Tb after 

being spatially averaged by the direction-dependent antenna directivity D(Ω). To 

eliminate ambiguity, one should specify the point to which the calibrated antenna 

temperature is referenced. If no point is specified, it is assumed to be at the antenna 

aperture, in which case it corresponds to the antenna aperture temperature. 

For example, in traditional radio astronomy this point is usually the output port of the 

antenna, because that is the point in the system where the calibration loads are situated. 

In this case, the antenna temperature corresponds to the antenna output temperature. In 

other systems, particularly those with small antennas, the calibration targets are outside 

the antenna. In such a case, the calibrated antenna temperature refers to a point outside 

the antenna aperture, and the ohmic losses of the antenna are irrelevant. In still other 

cases, the calibration targets may lie between a large main reflector and a smaller feed 

antenna. Then the calibration of antenna temperature would remove feed antenna losses 

but not those of the main reflector. 

Blackbody: see perfect blackbody. 

Blackbody load: a microwave load with characteristics very close to those of a perfect 

blackbody within a certain frequency range. 

Boresight: the beam-maximum direction of a highly directive antenna. [6] 

Brightness Temperature (Tb): the temperature (K) equivalent, by the inverse Planck 

function, to a spectral radiance (W⋅Hz-1⋅m-2⋅sr-1), emitted by a blackbody at temperature 

Tb. 

In general, Tb is a function of frequency and direction Ω, where Ω stands for angles θ and 

φ. Thus, for unpolarized radiation, Tb is the temperature of a blackbody that would emit 

the same radiance at the specified frequency and direction. By extension, for polarized 

radiation, the brightness temperature of each polarization is the temperature of a 

blackbody that would emit the same radiance in the specified polarization at the specified 

frequency and direction. In the Rayleigh-Jeans limit, the microwave power per unit 

bandwidth received by a radiometer is P=k Tb, where k is Boltzmann’s constant. 

Close-coupling: the technique by which the receiving aperture of a radiometer is coupled 

to a blackbody load so that the system is closed to stray radiation. 

8
Directivity (of antenna in a given direction (D(θ,φ)): the ratio of the radiation intensity 

in a given direction from the antenna to the radiation intensity averaged over all 

directions. [6] 

Notes: The average radiation intensity is equal to the total power radiated by the antenna divided 

by 4π. If the direction is not specified, the direction of maximum radiation is implied. 

Dwell time: time interval required for the antenna beam to scan across one beam width, 

at constant speed, along the scan direction. 

Note: This and the definition of radiometric resolution per beamwidth assume one sample per 

footprint; if the radiometer samples at twice this rate, the integration time is halved and the 

radiometric resolution per beam width is increased by a factor of 2 . 

Effective Field of View (EFOV): When the radiometer antenna is scanning across the 

earth’s surface while the radiometer integrates, then the static instantaneous field of view

(IFOV, in length unit and along the scanning direction), which is based purely on 

geometrical projections of the antenna beam and the earth’s surface, is no longer valid. 

Instead, the Scanned EFOV or ESFOV along the scan direction is used in its place. The 

ESFOV takes into account the scan-induced “smear” effect, and in general is slightly 

larger than the static IFOV. The extent of the smear depends on the length of the 

integration time, the shape of the original antenna beam, and the scan speed with time. 

In order to be specific, the cross-track EFOV (EFOV-CT) is defined only for the case in 

which: (i) the scan speed is constant, and (ii) the integration time is equal to the time 

required to move the antenna beam one IFOV. Again CT implies the scanning direction, 

which may or may not be the cross-track direction. 

As an example, the EFOV-CT of a Gaussian shaped beam is 25 % larger than its IFOV, 

according to the above definition (i) and (ii). 

The EFOV-CT is determined as follows: 

First obtain the Exposure function E(x), which is the integral of the product of the 

antenna pattern G(x-m) and the integration time Δt, 

( , = ∫ − )() ΔtmxGxE

where m=vt is the coordinate of the antenna beam center, which is moving at a scan 

velocity of v. The integration is from t=0 to t=τ.

Following the same half-power point principle, the ESFOV-CT is determined by 

obtaining the points x1 and x2 (x2>x1 assumed) where the values of E(x) are half of its 

maximum value, and the ESFOV-CT value is equal to x2-x1. 

Effective input noise temperature (of receiver, Trec): the input termination noise 

temperature which, when the input termination is connected to a noise-free equivalent of 

the receiver, would result in the same output noise power as that of the actual receiver 

connected to a noise-free input termination. [12] It is related to the receiver noise figure

by 

9
⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛ + =

0

0

T

TT F rec rec , 

where T0 = 290 K. It is a function of frequency, and it depends on the impedance or 

reflection coefficient of the input termination. It also depends on the location of the input 

reference plane; unless otherwise stated, the receiver noise temperature is assumed to be 

defined at the receiver input. 

End-to-end calibration: calibration of the entire radiometer system as a unit, achieved 

by observing the values of output quantities (voltage, power, etc.) for known values of 

incident radiance at the antenna aperture. 

External calibration: a calibration method that applies reference signals from targets 

that lie outside the radiometer. If these targets illuminate the antenna of the radiometer, 

an end-to-end calibration is obtained. 

Half-power bandwidth (B3dB): the width at which the power response is half the 

maximum value; also called the 3-dB bandwidth. 

Instantaneous field of view (IFOV): the antenna beam width projected onto the earth’s 

surface. See also static IFOV. 

Since the antenna radiation pattern depends on the specific plane (containing the beam 

axis) of cut, so does the IFOV value. Conventionally, IFOVs at two principal planes are 

given: the E-plane (containing the E-vector of the equivalent dipole) and the H-plane 

(perpendicular to the E-plane). When a single IFOV value is used, it is the (arithmetic) 

mean of the IFOVs of the two principal planes. 

Internal calibration: calibration of a radiometer by connecting embedded reference 

noise sources to the receiver chain; a common example is injection calibration, in which a 

noise diode is used as the embedded reference noise source. 

Limiting radiometric resolution: the radiometric resolution evaluated for the case of 

zero input radiance or brightness temperature. Referred to as the “temperature sensitivity” 

in Radio Astronomy. 

Noise-equivalent bandwidth (Bn): the width of an ideal rectangular filter (or other 

circuit or component) having the same maximum (available) gain as the real filter, and 

which produces the same output available power (integrated over frequency) as the real 

one when the input is white noise. In terms of the filter (available) gain G, 

max

0

)(

G

dffG

Bn

∫

=

∞

 . 

Noise figure (of receiver): at a specified input frequency, the ratio of 1) the total noise 

spectral power density at a corresponding output frequency available at the output port 

when the noise temperature of the input termination is T0 = 290 K, to 2) that portion of 

10
the total output noise spectral power density that is engendered by the input termination. 

[12] The noise figure depends on the impedance or reflection coefficient of the input 

termination and on the location of the input reference plane. Unless otherwise stated, the 

receiver noise figure is assumed to be defined at the receiver input. 

(Normalized) Antenna pattern (t(θ,φ)): the antenna gain (or directivity) in a specified 

direction divided by the maximum value of the antenna gain (or directivity). See 

antenna gain for the relationship between antenna gain and directivity. 

Operating noise figure (of receiver): at a specified input frequency, the ratio of 1) the 

total noise spectral power density at a corresponding output frequency available at the 

output port when the noise temperature of the input termination is the actual input noise 

temperature (or brightness temperature, for a system), to 2) that portion of the total output 

noise spectral power density that is engendered by the input termination. (adapted from 

[13]) In terms of the effective input noise temperature, the operating noise figure is given 

by 

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛ + =

in

inrec

Op T

TT F , 

where Tin is the nominal value of the input noise temperature. 

Perfect blackbody: a perfect absorber (and therefore the best possible emitter) of 

thermal electromagnetic radiation, whose spectral radiance density (or spectral 

brightness density) is given by the Planck formula, 

( ),

1

2

2 /

3

− = Tkhf f B ec

hf L

where f is the frequency, h is Planck’s constant, kB is Boltzmann’s constant, and T is the 

physical temperature of the blackbody. (adapted from [7]) 

Pixel-to-pixel accuracy: the brightness temperature accuracy in measuring only one 

resolution element; the values of other resolution elements are not used for correction. 

(Equivalent) Predetection bandwidth (Bpre): the effective bandwidth of the 

predetection section of a radiometer, used in computing the ideal radiometer resolution. 

In terms of the gain G of the predetection section, 

∫

∫

∞

∞

⎭

⎬

⎫

⎩

⎨

⎧

=

0

2

2

0

)(

)(

dffG

dffG

Bpre .

Radiation efficiency: see antenna ohmic efficiency. [11]

Radiation intensity: the magnitude of the Poynting vector, i.e., the flux density of the 

radiation. Its dimensions are power per unit area. 

11
Radiometer: a very sensitive receiver, typically with an antenna input, that is used to 

measure radiated electromagnetic power. 

Radiometric (or radiometer) resolution: the smallest change in input brightness 

temperature or radiance that can be detected in the system output. The radiometric 

resolution can be measured by computing the Type-A uncertainty, e.g., the standard 

deviation of the mean of a number of measurements of the same quantity made over a 

short enough time period that the system can be considered to be stable. It is also often 

estimated by using the ideal equation for a total-power radiometer, 

τ

Δ

pre

Sys

B

T

Tmin = , or 

the variant of this equation that is appropriate for the particular radiometer in question. 

Radiometric resolution per beam width: (for a continuously scanning radiometer) the 

radiometric sensitivity evaluated for an integration time equal to the dwell time. 

Radiometric sensitivity: often used to mean radiometric resolution, but this use is 

discouraged in view of the definition of sensitivity. 

Spatial resolution: the length of the major and/or minor axes diameters of the 3 dB 

contour of the antenna pattern projected onto the earth’s surface. These two diameters 

may differ. See also IFOV. 

Static IFOV: When the IFOV is expressed in length units, it is used to express the 

diameter (arc length) of a circular area that is the intersection of the antenna beam 

(assumed to be a right circular cone) and the earth’s surface (assumed to be a spherical 

shell). This is valid in the case of normal incidence.

When the radiometer antenna is viewing the earth at an oblique angle, then the 

intersection area (between the right circular cone of the antenna beam and the spherical 

earth surface) is a pear-shaped figure. In this case two arc lengths are used to represent 

the IFOV, a Major diameter or IFOV-Maj, and a Minor diameter or IFOV-Min. 

Static IFOV-AT, static IFOV-CT: For a cross-track scanning beam, the words Minor 

and Major diameters can be replaced by Along-Track and Cross-Track, respectively. 

Here A-T and C-T also signify the directions parallel to, and perpendicular to the subsatellite track direction, respectively. 

For a conical scanning beam, the terms A-T and C-T still apply. 

 

System gain: There are three different definitions of power gain: 

• Power gain: L

p

i

P G

P = , 

• Transfer power gain: L

T

av G

P G

P = , 

12
• Available power gain: av L

a

av G

P

G

P = , 

where PL is the power that is delivered to the load, Pi is the power that is delivered by the 

source to the input port, and Pav G and Pav L are the available powers from the generator 

and to the load, respectively. 

These three definitions of power gain are related by the following relationships: 

( )( )

( )( ),

1

11

,

1

11

,

2

2 2

2

2 2

Lo

o L

L

Gi

i G

g

LagpT

M

M

MGMGG

ΓΓ

ΓΓ

ΓΓ

ΓΓ

−

−− =

−

−− =

= =

and 

( ) ( ) 22

22

2 2

21

1 1

1

iL

L

p

S

S

G

ΓΓ

Γ

− −

− = , 

where Γi, Γo, ΓG, and ΓL are respectively the receiver (amplifier) input (i) and output (o) 

reflection coefficients, and the source (G) and load (L) reflection coefficients. Mg and ML

are the mismatch factors [14] at the input (g) and output (L) reference planes. 

We usually refer to Ga, since in practice the amplifiers, filters and other subsystems are 

never perfectly matched. 

Tip-curve calibration (or tipping-curve calibration): N-point calibration (N≥2) using 

the atmosphere at different zenith angles as calibration target. 

Two-point calibration: fixing the relationship between the input signal and the output 

response of a radiometer using two distinct input stimuli. Assuming a linear receiver, all 

possible input signal levels can now be retrieved from the radiometer output responses. In 

the case of an external end-to-end calibration, the input signal equals the antenna 

aperture temperature of the radiometer. 

CHAPTER III. POLARIMETRIC RADIOMETRY

Note: In this chapter, the signals are assumed to be constant over frequency unless 

otherwise stated. 

Adding analog correlator: a cross-correlator that measures additional Stokes 

parameters / modified Stokes parameters by post-detection summing of distinct principal 

13
polarizations as well as the squared sums of them. In a polarization frame defined by 

vertical polarization and horizontal polarization, for example, the third and fourth Stokes 

parameters / modified Stokes parameters are measured in the following way: The sum of 

the orthogonal waves is square detected, respectively, in phase and in quadrature, 

followed by the subtraction of orthogonal autocorrelations. Using Stokes parameters in 

brightness temperature / modified Stokes parameters in brightness temperature: 

,Im 2

2

Re 2

*

2 2 2

2 2

4

*

2

2 2 2 2

3

hv

B

hv v h

B

V

hv

B

hv v h

B

U

EE

Bk

EE EE

Bk jTT

EE

Bk

EEEE

Bk

TT

⋅ ⋅⋅

⋅ =

⎥

⎥

⎦

⎤

⎢

⎢

⎣

⎡

⎟ −− ⎠

⎞ ⎜

⎝

⎛

−+⋅ ⋅⋅ ⋅==

⋅ ⋅⋅

⋅ =⎥⎦

⎤ ⎢⎣⎡ −−+⋅ ⋅⋅ ==

η

π λ

η

λ

η

λ

η

λ

where η is the wave impedance of the medium, Ev is the vertical component of the 

electric field, and Eh and Eh ( 2

π

− ) are, respectively, the horizontal component of the 

electric field in phase and in quadrature with respect to the vertical one. Wavelength, the 

wave impedance of the medium, bandwidth, and Boltzmann’s constant are denoted by λ, 

η, B, and kB, respectively. See also: coherent detection, direct correlating correlator. 

Analog correlator: see analog cross-correlator. 

Analog cross-correlator: a cross-correlator based on analog technology, i.e., the signals 

are not quantized prior to correlation. See also: adding analog correlator, coherent 

detection, digital cross-correlator, direct correlating correlator. 

Calibration coefficients: elements of the gain matrix and the offset vector. See also: 

response of a fully polarimetric radiometer. 

Channel (of a radiometer): a receiver chain that (in the ideal case) is sensitive to a 

certain Stokes parameter / modified Stokes parameter. See also: Stokes parameter mixing. 

Circular polarization: see polarization. 

Coherence matrix (sometimes called the coherency matrix): matrix representing the 

auto- and cross-correlation terms of two distinct polarizations. For example, based on 

vertical polarization and horizontal polarization, the coherence matrix is 

2 *

2 *

v v

vh h

E EE

J

EE E

h

⎡ ⎤ ⎢ ⎥ = ⎢ ⎥ ⎢ ⎥ ⎣ ⎦

, 

where Ev and Eh are vertical and horizontal electric field components, respectively, and * 

denotes complex conjugate. 

14
15

More conveniently for radiometric applications, the coherence matrix can also be 

expressed in terms of brightness temperature, i.e. multiplying the above defined 

parameters by the factor 

2

kB

1



1

B1, where  is the wavelength of the wave, kB is 

Boltzmann’s constant,  is the wave impedance of the medium, and B is the bandwidth: 

 

 

3 4

3 4

1

2

1

2

v

h

T T jT

J

T jT T

            

. 

Coherent detection: measurement of Stokes parameters / modified Stokes parameters by 

the cross-correlation of signals collected at principal polarizations; the technique requires 

the preservation of phase coherence between the signals that are correlated. Either adding 

analog correlator or directcorrelating correlator (analog or digital) can be used. In a 

polarization frame defined by two orthogonal linear polarizations, for example, the third 

and fourth Stokes parameters / modified Stokes parameters are measured by complex 

cross-correlation of waves at these polarizations. The first and second modified Stokes 

parameters are obtained by the autocorrelation of orthogonal signals; the retrieval of the 

first and second Stokes parameters requires additional post-detection summing. See also: 

direct correlating polarimetric radiometer. 

Completely polarized wave: an electromagnetic wave with completely correlated 

orthogonal polarizations, i.e., the determinant of its coherence matrix is zero. Using 

Stokes parameters it can be written 

2222  VUQI , 

where I, Q, U, and V are the first, second, third, and fourth Stokes parameters, 

respectively. Using modified Stokes parameters, on the other hand, 

    2

4

2

3

2 2 SSSSSS hv hv  , 

where Sv, Sh, S3, and S4 are the first, second, third, and fourth modified Stokes parameters, 

respectively. 

See also: completely unpolarized wave, partially polarized wave. 

Completely unpolarized wave: regardless of the polarization frame, the first and second 

modified Stokes parameters are equal (and the difference is thus zero) and the third and 

fourth Stokes parameters / modified Stokes parameters are zero. Using the terminology of 

Stokes parameters: Q = U = V = 0. The determinant of its coherency matrix is equal to 

the wave’s intensity. See also: completely polarized wave, partially polarized wave. 
Complex correlator: complex cross-correlator. 

Complex cross-correlator: a device that estimates the time average of the product of the 

in-phase and quadrature components of two time-varying signals, x(t) and y(t). The inphase component of a signal, I, is its even, or Fourier cosine, component; and the 

quadrature component, Q, is its odd, or Fourier sine, component. If we use x(t) to set the 

phase, the real part of the complex cross-correlation estimates <IxIy>, and the imaginary 

part estimates <IxQy>. There are both digital and analog methods of implementing the 

complex correlation of band-limited signals, either while they are modulated by a carrier 

or at baseband. 

Complex degree of coherence: the off-diagonal terms of the coherency matrix

normalized by the geometric mean of the diagonal terms. Using vertical polarization and 

horizontal polarization, for example, 

*

2 2

v h

vh

v h

E E

E E

μ = , 

where Ev and Eh are vertical and horizontal electric field components, respectively. 

Correlator: see: cross-correlator. 

Cross-correlator: a device that calculates the time average of the product of two distinct 

input signals. 

Degradation factor: a quantity representing the degradation of the signal-to-noise ratio 

of polarimetric channels due to channel non-idealities: 

() ()

( ) ( ) dffHfH

dffHfH

D

∫

∫

∞

∞

⋅

=

0

2 *

2

2

1

0

*

21

β

, 

where H1(f) and H2(f) are the complex voltage transfer functions of channels 1 and 2, 

respectively, prior to correlation, and β is the width of the pass-band. Note that the 

degradation factor includes the fringe-washing effects generated by transfer function 

differences, but not those generated by propagation time differences. 

16
Degree of polarization: the ratio of completely polarized power to total power: 

( )

I

VUQ

SS

SSSS

d

hv

hv

2 222

4

2

3

2 ++ = +

++− = , 

where Sv, Sh, S3, S4 are the first, second, third, and fourth modified Stokes parameters, 

respectively, and I, Q, U, and V are the first, second, third, and fourth Stokes parameters, 

respectively. See also: partially polarized wave. 

Delay line: a microwave component used to generate time delay. In a correlating 

radiometer it is used to equalize the propagation times of distinct signals prior to crosscorrelation and reduce fringe-washing effects, or to introduce a specific delay in order to 

compute the cross-correlation at different time lags. 

Digital correlator: see: digital cross-correlator. 

Digital cross-correlator: a cross-correlator based on digital technology, i.e., the 

correlation of quantized signals is performed. Depending on the correlator configuration, 

two or more quantization levels are applied. See also: analog cross-correlator, coherent 

detection, direct correlating correlator. 

Direct correlating correlator: an analog or digital device that directly estimates 

additional Stokes parameters / modified Stokes parameters by cross-correlation of the 

principal polarizations without any post-detection addition. Conventionally, the third and 

fourth Stokes parameters / modified Stokes parameters are estimated by correlating 

vertical polarization and horizontal polarization with each other. See also: adding analog 

correlator, digital cross-correlator, analog cross-correlator. 

Direct correlating polarimetric radiometer: a polarimetric radiometer based on 

coherent detection and a direct correlating correlator. See also: polarization combining

radiometer. 

Fringe-washing effect: degradation in the cross-correlation of two non-zero bandwidth 

signals that can be caused by: 1) different transit times from the signal source to the 

antennas; and/or 2) different group delay responses (time delay at each frequency) of the 

receivers/correlators. The group delay response may include instrumental delays added in 

the receiving chain, and it is uniquely related to the frequency response through the 

following equation: 

( ) ( )

g

d

d

ω

τ ω

ω

Φ = − , 

where ω is radian frequency and Φ(ω) is the phase of the receiver's frequency response. 

17
Fully polarimetric calibration: the determination of the offset and gain matrix elements 

of a fully polarimetric radiometer. See also: response of a fully polarimetric radiometer, 

tri-polarimetric calibration. 

Fully polarimetric calibration standard: a device for performing external calibration of 

polarimetric radiometers. It can generate at least four linearly independent polarized 

reference signals. Along with an unpolarized calibration reference, a fully polarimetric 

calibration standard enables the fully polarimetric calibration of fully polarimetric 

radiometers. See also: tri-polarimetric calibration standard. 

Fully polarimetric radiometer: a radiometer capable of directly or indirectly measuring 

all four Stokes parameters / modified Stokes parameters. 

Gain matrix: matrix that contains the amplification terms relating the observed Stokes 

parameters in brightness temperature / modified Stokes parameters in brightness 

temperature and the outputs of the radiometer channels. Diagonal elements of the matrix 

represent autocorrelation channel gains; off-diagonal elements represent crosspolarization terms generated by interchannel cross-talk and antenna cross-polarization. 

See also: offset vector, response of a fully polarimetric radiometer. 

Horizontal brightness temperature: brightness temperature derived from field 

components in the direction of horizontal polarization. See also: vertical brightness 

temperature. 

Horizontal polarization: see polarization. 

Incoherent detection: measurement of Stokes parameters / modified Stokes parameters

by post-detection summing of flux densities at distinct principal polarizations. Phase 

coherence among channels is not required. In a polarization frame defined by vertical 

polarization and horizontal polarization, for example, the third Stokes parameter / 

modified Stokes parameter can be obtained as a difference between flux densities at +45°

and −45° skewed linear polarizations and the fourth Stokes parameter / modified Stokes 

parameter as a difference between flux densities at left-handed and right-handed circular 

polarizations. The first and second Stokes parameters are, respectively the sum and 

difference of the detected vertical and horizontal flux densities; the first and second 

modified Stokes parameters are, respectively, the detected vertical and horizontal flux 

densities. See also: coherent detection, polarization combining radiometer. 

Inter-channel cross-talk: non-ideality of a radiometer receiver leading to signal leakage 

between channels. See also: Stokes parameter mixing. 

Linear polarization: see: polarization. 

Linearly polarized calibration standard: tri-polarimetric calibration standard, which 

is capable of generating reference signals at orthogonal linear polarizations and a linearly 

polarized reference signal. See also: fully polarimetric calibration standard. 

18
Modified Stokes parameters: the polarization of an electromagnetic wave is completely 

described by the four Stokes parameters. However, in remote sensing the so-called 

modified Stokes parameters are frequently used, where the first and second parameters 

represent, respectively, the flux density at vertical polarization and horizontal 

polarization. Using Stokes parameters, modified Stokes parameters are defined as: 

2

QI Sv

+ =

2

QI Sh

− =

3 = SS U

4 = SS V , 

where I, Q, U, and V are, respectively, the first, second, third, and fourth Stokes 

parameters. Note that in polarimetric remote-sensing radiometry, the modified Stokes 

parameters are conventionally expressed in terms of brightness temperature, i.e., modified 

Stokes parameters in brightness temperature. 

Modified Stokes parameters in brightness temperature: modified Stokes parameters 

expressed in terms of brightness temperature (in kelvins): 

v

B

v S

Bk

T ⋅ ⋅ =

2 λ

h

B

h S

Bk

T ⋅ ⋅ =

2 λ

3

2

3 S

Bk

T

B

⋅ ⋅ = λ

4

2

4 S

Bk

T

B

⋅ ⋅ = λ , 

where λ is the wavelength of the wave, kB is Boltzmann’s constant, and B is the noiseequivalent bandwidth. The first, second, third and fourth modified Stokes parameters are 

denoted by Sv, Sh, S3, and S4, respectively. See also: modified Stokes parameters, modified 

Stokes vector in brightness temperature, Stokes parameters in brightness temperature. 

Modified Stokes vector in brightness temperature: vector containing all four modified 

Stokes parameters in brightness temperature of a certain brightness temperature scene: 

19
⎥

⎥

⎥

⎥

⎦

⎤

⎢

⎢

⎢

⎢

⎣

⎡

=

4

3

mod,

T

T

T

T

T h

v

B , 

where Tv, Th, T3, and T4 denote the first, second, third and fourth modified Stokes 

parameters, respectively. See also: Stokes vector in brightness temperature.

Offset vector: vector that contains the responses of the radiometer channels associated 

with the zero input signal of each Stokes parameter. See also: gain matrix, response of a 

fully polarimetric radiometer. 

Partially polarized wave: an electromagnetic wave consisting of completely polarized 

wave(s) and completely unpolarized wave(s), i.e., the degree of polarization is between 0 

and 1. 

Phase modulation: an incoherent detection technique that enables the measurement of 

all Stokes parameters / modified Stokes parameters by a single-channel radiometer 

receiver. The phase modulating component can be installed prior to or after an antenna. 

In the former case a quasi-optical device based, e.g., on a phase retardation plate or a 

polarizing grid can be applied as a phase shifter. 

Phase retardation plate: a quasi-optical microwave device that generates a phase shift 

between orthogonal linear polarizations passing through the plate. Special cases are a 

quarter-wave plate and a half-wave plate, which generate π/2 and π phase shifts, 

respectively. Retardation plates can be applied in fully polarimetric calibration standards 

to generate a circularly polarized signal from a linearly polarized signal, or in 

polarimetric microwave receivers to generate phase modulation. 

Platform motion compensation: a procedure for compensating the changes of the 

observed Stokes parameters / modified Stokes parameters that are generated by the 

motion of the radiometer platform. These changes can be caused, e.g., by the rotation of 

the polarization frame of the antenna(s), which may lead to Stokes parameter mixing

between the first three Stokes parameters / modified Stokes parameters, and variations in 

incidence angle. 

Poincaré sphere: the polarization of any completely polarized electromagnetic wave can 

be expressed by two angles, which define the position on the surface of the so-called 

Poincaré sphere (see figure below). In this representation, left and right hand circular 

polarizations are on the North and South poles, respectively; and the four linear principal 

polarizations are located symmetrically on the equator. 

20
Polarimetric brightness temperatures: common name for the third and fourth Stokes 

parameters in brightness temperature / modified Stokes parameters in brightness 

temperature. 

Polarimetric calibration: common name for tri-polarimetric calibration and fully 

polarimetric calibration. 

Polarimetric calibration standard: common name for fully polarimetric calibration 

standards and tri-polarimetric calibration standards. 

Polarimetric channels: common name for the radiometer channels that estimate the 

third and/or fourth Stokes parameters / modified Stokes parameters. 

Polarimetric radiometer: a radiometer that measures additional Stokes parameters

beyond the first and second Stokes parameters / modified Stokes parameters. Common 

name for tri-polarimetric radiometers and fully polarimetric radiometers. 

Polarization: the direction of the electric (E) or magnetic (H) field in a propagating 

electromagnetic wave. A general, elliptically polarized electromagnetic plane wave 

propagating in the r

v direction can have its electric field expressed in phasor form as 

jkr

qp eEqEpE − ( += ˆˆ ) v , 

where pˆ and qˆ are unit vectors oriented perpendicular to r

v and satisfying 

qp rˆ = rr

vv ˆˆ =× / , Ep and Eq are the complex amplitudes of the electric field in the and 

 directions, respectively, k is the wavenumber of the propagating wave, and 

pˆ

qˆ r r

v = . 

The physical time-varying electric field is given by { } jωt eEtrEv v = Re),( , where ω is the 

21
frequency in rad/s. There are a number of special cases of polarization that are 

commonly measured in radiometry: 

Horizontal linear polarization satisfies: 1) is oriented tangential to the earth’s 

surface; 2) has a positive projection onto the normal of the surface; 3) ; and 

4) . 

pˆ

qˆ Ep ≠ 0

Eq = 0

Vertical linear polarization satisfies: 1) is oriented tangential to the earth’s surface; 

2) has a positive projection onto the normal of the surface; 3) ; and 4) 

. 

pˆ

qˆ

≠

Ep = 0

Eq 0

+45o

 slant linear polarization satisfies: 1) is oriented tangential to the earth’s 

surface; 2) has a positive projection onto the normal of the surface; and 3) 

. 

pˆ

qˆ

EE qp ≠= 0

–45o

 slant linear polarization satisfies: 1) is oriented tangential to the earth’s 

surface; 2) has a positive projection onto the normal of the surface; and 3) 

. 

pˆ

qˆ

≠−= 0 EE qp

Right hand circular polarization satisfies: 1) Ep ≠ 0 ; and 2) . jπ 2/

pq eEE − =

Left hand circular polarization satisfies: 1) Ep ≠ 0 ; and 2) . jπ 2/

pq = eEE

Polarization combining network: a microwave circuit that computes the sum or 

difference of two principal polarizations in-phase and/or quadrature to produce other 

principal polarizations. See also: incoherent detection, polarization combining 

radiometer. 

Polarization combining radiometer: a polarimetric radiometer based on incoherent 

detection; post-detection summing of principal polarizations is applied to measure 

additional Stokes parameters / modified Stokes parameters. See also: direct correlating 

polarimetric radiometer.

Polarization frame: any orthogonal coordinate basis relating polarizations to each other. 

Polarized brightness temperature scene: a scene that emits electromagnetic waves that 

are not completely randomly polarized. See also: degree of polarization, unpolarized 

brightness temperature scene. 

Polarizing grid: a quasi-optical device that consists of parallel conductors. In the ideal 

case, in which the conductors have no ohmic losses and the grid is dense compared to the 

wavelength, a wave with electric field perpendicular to the conductors passes through the 

22
grid without attenuation, whereas a wave with electric field parallel to the conductors is 

completely reflected. In other orientations part of the wave is passed, part is reflected, 

and a cross-correlated (linearly polarized) wave component is generated. These 

characteristics can be applied to fabricate a linearly polarized calibration standard or a 

polarimetric radiometer based on phase modulation. 

Principal polarizations: six polarizations by which an electromagnetic wave can be 

completely characterized at a certain frequency: four linear polarizations skewed in 45°

intervals (0°, 45°, 90° 135° rotated with respect to the reference polarization frame) and 

left- and right-handed circular polarizations. In remote sensing applications it is often 

convenient to select an orthogonal linear polarization frame with 0° and 90° aligned with

horizontal polarization and vertical polarization, respectively. Then, 45° and 135°

skewed linear polarizations can be referred to as +45° and −45° skewed linear 

polarizations, respectively. 

Response of a fully polarimetric radiometer: the response of a radiometer is dependent 

on the brightness temperature scene and the characteristics of the radiometer. In terms of 

the modified Stokes vector in brightness temperature, this relationship for a fully 

polarimetric radiometer is 

mod mod mod, mod mod

4

3

4

3

44 4443

33 3433

43

43

4

3

mod noTgn

o

o

o

o

T

T

T

T

gggg

gggg

gggg

gggg

r

r

r

r

r B

h

v

h

v

hv

hv

hhhhhv

vvvhvv

h

v

=+ ++

⎥

⎥

⎥

⎥

⎦

⎤

⎢

⎢

⎢

⎢

⎣

⎡

+

⎥

⎥

⎥

⎥

⎦

⎤

⎢

⎢

⎢

⎢

⎣

⎡

⎥

⎥

⎥

⎥

⎦

⎤

⎢

⎢

⎢

⎢

⎣

⎡

=

⎥

⎥

⎥

⎥

⎦

⎤

⎢

⎢

⎢

⎢

⎣

⎡

= , 

where rmod is the low frequency output response vector, mod g is the gain matrix, 

T B mod, is the modified Stokes vector in brightness temperature, omod is the offset 

vector, and nmod is the zero-mean Gaussian instrument noise at the channel outputs. The 

off-diagonal terms of the gain-matrix represent interchannel cross-talk. In terms of the 

Stokes vector in brightness temperature, this relationship can be written as 

noTgn

o

o

o

o

T

T

T

T

gggg

gggg

gggg

gggg

r

r

r

r

r B

V

U

Q

I

V

U

Q

I

VI VQ VU VV

UI UQ UU UV

QI QQ QU QV

II IQ IU IV

V

U

Q

I

++=+

⎥

⎥

⎥

⎥

⎦

⎤

⎢

⎢

⎢

⎢

⎣

⎡

+

⎥

⎥

⎥

⎥

⎦

⎤

⎢

⎢

⎢

⎢

⎣

⎡

⎥

⎥

⎥

⎥

⎥

⎦

⎤

⎢

⎢

⎢

⎢

⎢

⎣

⎡

=

⎥

⎥

⎥

⎥

⎦

⎤

⎢

⎢

⎢

⎢

⎣

⎡

= . 

Sense of polarization: for an elliptical or circularly polarized field vector, the sense of 

rotation of the extremity of the field vector when its origin is fixed [6]. 

Stokes parameter mixing: contamination of a measured Stokes parameter / modified 

Stokes parameter value by other parameters, with the result that a certain radiometer 

channel is sensitive to more than one Stokes parameter / modified Stokes parameter. 

23
Stokes parameter mixing can be generated by, e.g., inter-channel cross-talk, antenna 

cross-polarization, and/or platform motion. See also: platform motion compensation, 

response of a fully polarimetric radiometer. 

Stokes parameters: four parameters that completely characterize an electromagnetic 

wave at a certain frequency. The first Stokes parameter (I) denotes the total Poynting 

vector or flux density of the wave. Expressing the other parameters in terms of linearly 

polarized components, the second Stokes parameter (Q) is the difference between the flux 

densities polarized in two orthogonal directions in a given polarization frame, e.g., 

vertical polarization and horizontal polarization. The third (U) and fourth (V) Stokes 

parameters respectively represent twice the real and imaginary parts of the cross 

correlation between these orthogonal polarizations. The third Stokes parameter can also 

be defined as a difference between +45° and −45° skewed linearly polarized flux 

densities, and the fourth parameter as a difference between left-handed and right-handed 

circularly polarized flux densities: 

hv

v h

SS

EE

I +=

+

= η

2 2

, 

hv

v h

SS

EE

Q −=

−

= η

2 2

, 

45 45

* Re2 −= −+

⋅ = SS

EE

U hv

η , 

rclc

hv

SS

EE

V −= ⋅ = η

* Im2 , 

where S is the flux density of the wave (in W⋅m−2

). The subscripts v, h, +45, −45, lc, and 

rc represent respectively the vertically, horizontally, +45° skewed linearly, −45° skewed 

linearly, left-handed circularly, and right-handed circularly polarized waves, η is the 

wave impedance of the medium, and Ev and Eh are the electric field components with 

vertical and horizontal polarizations, respectively. Note that in polarimetric remote 

sensing radiometry the Stokes parameters are conventionally expressed in terms of 

brightness temperature, i.e., Stokes parameters in brightness temperature. See also:

modified Stokes parameters, Stokes vector in brightness temperature. 

Stokes parameters in brightness temperature: Stokes parameters expressed in terms of 

brightness temperature (in kelvins): 

I

Bk

T

B

I ⋅ ⋅ =

2 λ , 

Q BkT

B

Q ⋅ ⋅ =

2 λ , 

24
U

Bk

T

B

U ⋅ ⋅ =

2 λ , 

V

Bk

T

B

V ⋅ ⋅ =

2 λ , 

where wavelength, Boltzmann’s constant, and noise-equivalent bandwidth are denoted by 

λ, kB, and B, respectively. The first, second, third and fourth Stokes parameters are 

denoted by I, Q, U, and V. Note that in the published literature on polarimetric remotesensing radiometry, I, Q, U, and V have also been used for the Stokes parameters in 

brightness temperature (units kelvins). This practice may generate confusion and is 

therefore discouraged. See also: modified Stokes parameters in brightness temperature, 

Stokes vector in brightness temperature. 

Stokes vector in brightness temperature: vector containing all four Stokes parameters 

in brightness temperature of a certain brightness temperature scene: 

⎥

⎥

⎥

⎥

⎦

⎤

⎢

⎢

⎢

⎢

⎣

⎡

=

V

U

Q

I

B

T

T

T

T

T , 

where TI, TQ, TU, and TV denote the first, second, third and fourth Stokes parameters in 

brightness temperature, respectively. See also: modified Stokes vector in brightness 

temperature. 

Tri-polarimetric calibration: the determination of the offset and gain terms (see:

response of a fully polarimetric radiometer) of a tri-polarimetric radiometer. See also:

fully polarimetric calibration. 

Tri-polarimetric calibration standard: a device for external calibration of radiometers 

that can generate three linearly independent polarized brightness temperature scenes. 

Along with an unpolarized calibration standard, a tri-polarimetric calibration standard 

enables tri-polarimetric calibration. See also: fully polarimetric calibration standard. 

Tri-polarimetric radiometer: a radiometer that measures three of the four Stokes 

parameters / modified Stokes parameters. 

Unpolarized brightness temperature scene: a brightness temperature scene that emits 

completely unpolarized waves, i.e., the waves are completely randomly polarized. 

Unpolarized calibration: a calibration procedure applying only unpolarized reference 

signals. Enables the determination of offset and diagonal gain terms (see: response of a 

fully polarimetric radiometer) of those radiometer channels, which are sensitive to 

unpolarized signals. See also: unpolarized reference standard. 

25
Unpolarized calibration standard: a calibration reference that generates unpolarized

signals. See also: polarimetric calibration standard, unpolarized calibration. 

Vertical brightness temperature: a brightness temperature derived from field 

components in the direction of vertical polarization. See also: horizontal brightness 

temperature. 

Vertical polarization: see polarization. 

________________________________ 

 We are grateful to Dr. Andreas Colliander of the European Space Agency and 

Professor Steven Reising of Colorado State University for very thorough and detailed 

reviews. This document is significantly better than it would otherwise have been due to 

their efforts. We also thank Dr. Paul Racette of NASA for early interest in and 

contributions to this effort. 

REFERENCES

[1] C. Wyatt, V. Provalsky, and R. Datla, “Recommended practice; symbols, terms, 

units and uncertainty analysis for radiometric sensor calibration,” Natl. Inst. 

Stand. Technol. Handbook 152 (1998 September). 

[2] http://boulder.nist.gov/div818/stdterms/index.htm

[3] http://www.grss-ieee.org/

[4] International Vocabulary of Basic and General Terms, Second edition, 

International Organization for Standardization (ISO), Switzerland (1993). 

[5] Guide to the Expression of Uncertainty in Measurement, International 

Organization for Standardization (ISO), Switzerland (1993). 

[6] IEEE Standard Dictionary of Electrical and Electronics Terms, 4th ed., 

ANSI/IEEE Std 100-1988. 

[7] J.D. Krauss, Radio Astronomy, chapter 3, McGraw-Hill, New York (1966). 

[8] Adapted from U.S. Federal Standard 1037C. 

[9] B.N. Taylor and C.E. Kuyatt, “Guidelines for evaluating and expressing the 

uncertainty of NIST measurement results,” Natl. Inst. Stand. Technol. Tech. Note 

1297 (1994 September). 

26
[10] IEEE Standard 145-1993 (Standard Definitions of Terms for Antennas). 

[11] D.M. Pozar, Microwave Engineering, chapter 12, John Wiley & Sons, New York 

(1998). 

[12] H.A. Haus et al., “IRE standards on methods of measuring noise in linear 

twoports, 1959,” Proc. IRE, vol. 48, pp. 60 – 68 (1960). 

[13] H.A. Haus et al., “Description of the noise performance of amplifiers and 

receiving systems,” Proc. IEEE, vol. 51, pp. 436 – 442 (1963). 

[14] G.F. Engen, Microwave Circuit Theory and Foundations of Microwave 

Metrology, Ch. 3, Peter Perigrinus Ltd., London (1992). 

ALPHABETICAL INDEX

Note: I, II, and III refer to chapters of the text. IVM and GUM indicate that the definition 

is given in the IVM [4] and/or GUM [5] and is not repeated in this document. 

Absolute accuracy, II 

Accuracy, I; see also Absolute accuracy, Pixel-to-pixel accuracy 

Accuracy of a measuring instrument, I 

Adding analog correlator, III 

Allan standard deviation, I 

Allan variance, I 

Analog correlator, III 

Analog cross correlator, III 

Angular resolution, II 

Antenna beam width, II 

Antenna gain, II 

Antenna ohmic efficiency, II 

Antenna pattern, see (Normalized) antenna pattern 

Antenna temperature, II 

Antenna aperture temperature, II 

Antenna output temperature, II 

Bandwidth 

 Half-power bandwidth, II 

 Noise-equivalent bandwidth, II 

 (Equivalent) predetection bandwidth, II 

Bias, I 

Blackbody, II 

Blackbody load, II 

Boresight, II 

Brightness, I 

Brightness temperature, I; see also Effective black-body brightness temperature. 

27
Calibration, I 

End-to-end calibration, II 

External calibration, II 

Internal calibration, II 

Tip-curve calibration, II 

Two-point calibration, II 

Calibration coefficients, III 

Calibration equation, I 

Channel (of a radiometer), III 

Circular polarization, III 

Close-couple, II 

Coherency matrix, III 

Coherent detection, III 

Completely polarized wave, III 

Completely unpolarized wave, III 

Complex correlator, III 

Complex cros-correlator, III 

Complex degree of coherence, III 

Corrected result, IVM, GUM 

Correlator, III 

Cross-correlator, III 

Degradation factor, III 

Degree of polarization, III 

Delay line, III 

Digital correlator, III 

Digital cross-correlator, III 

Direct correlating correlator, III 

Direct correlating polarimetric radiometer, III 

Directivity, II 

Drift, I 

Dwell time, II 

Effective black-body brightness temperature, I 

Effective noise bandwidth, II 

Emissivity of surface, I 

Error, I 

Experimental standard deviation, I 

Field of view 

 Effective field of view, II 

 Instantaneous field of view (IFOV), II 

 Static IFOV, II 

 Static IFOV-Cross Track, II 

 Static IFOV-Down Track, II 

Fringe-washing effect, III 

Fully polarimetric calibration, III 

Fully polarimetric calibration standard, III 

28
Fully polarimetric radiometer, III 

Gain, see Antenna gain, Receiver gain, System gain 

Gain matrix, III 

Horizontal brightness temperature, III 

Horizontal polarization, III 

Incoherent detection, III 

Inter-channel cross-talk, III 

Limiting radiometric resolution, II 

Linear polarization, III 

Linearly polarized calibration standard, III 

Measurand, IVM, GUM 

Modified Stokes parameters, III 

Modified Stokes parameters in brightness temperature, III 

Modified Stokes vector in brightness temperature, III 

Noise-equivalent Bandwidth, II 

Noise figure, II 

Noise temperature, I; see also Receiver noise temperature, Reference temperature, 

System noise temperature 

(Normalized) antenna pattern, II 

Offset vector, III 

Operating noise figure, II 

Partially polarized wave, III 

Perfect blackbody, II 

Phase modulation, III 

Phase retardation plane, III 

Pixel-to-pixel accuracy, II 

Platform motion compensation, III 

Poincaré sphere, III 

Polarimetric brightness temperautres, III 

Polarimetric calibration, III 

Polarimetric calibration standard, III 

Polarimetric channels, III 

Polarimetric radiometer, III 

Polarization, III 

Polarization combining network, III 

Polarization combining radiometer, III 

Polarization frame, III 

Polarized brightness temperature scene, III 

Polarizing grid, III 

Principal polarizations, III 

Quantization level, I 

Radiance, I 

Radiation intensity, II 

Radiometer, II 

Radiometric resolution, II 

Radiometric sensitivity, II 

29
Random error, I 

Receiver gain, II 

Receiver noise figure, II 

Receiver noise temperature, II 

Reference temperature, I 

Relative error, I 

Relative uncertainty, I 

Repeatability, I 

Reproducibility, I 

Resolution, I 

 Limiting radiometric resolution, II 

 Radiometric resolution, II 

 Radiometric resolution per beam width, II 

 Spatial resolution, II 

Response of a fully polarimetric radiometer, III 

Response time, I 

Result of a measurement, IVM, GUM 

Sense of polarization, III 

Sensitivity, I 

Spectral brightness density, I 

Spectral radiance density, I 

Stability, I 

Standard 

 Check standard, I 

 Measurement standard, IVM 

 Primary standard, IVM 

 Secondary standard, IVM 

 Reference standard, IVM 

 Working standard, IVM 

 Transfer standard, IVM 

Stokes parameter mixing, III 

Stokes parameters, III 

Stokes parameters in brightness temperature, III 

Stokes vector in brightness temperature, III 

Systematic error, I 

System gain, II 

System noise figure, II 

System noise temperature, II 

Traceability, I 

Tri-polarimetric calibration, III 

Tri-polarimetric calibration standard, III 

Tri-polarimetric radiometer, III 

True value, I 

Type-A uncertainty, I 

Type-B uncertainty, I 

Uncertainty budget, I 

30
31

Uncertainty of measurement, I 

Unpolarized brightness temperature scene, III 

Unpolarized calibration, III 

Unpolarized calibration standard, III 

Validation, I 

Verification, I 

Vertical brightness temperature, III 

Vertical polarization, III 

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