680 IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 36, NO. 2, MARCH 1998
(a)
(b)
Fig. 3. (a) Obse a ion o
"
00
change o ice slab on du aluminum mold a
13.5 GHz. Poin s (1–3) co espond o he ice ime exis ence in one day; (4)
wo days; (5) 32 days; (9–10) 47 days; (11) 66 days; and (12) 85 days. (b)
The measu emen s analogous o (a) a 37.5 GHz.
he o he hand, quick eezing may p oduce a speci ic dis ibu ion o
sal concen a ion in ice olume.
Howe e , he main esul o he measu emen s, as i ollows
om Figs. 2 and 3, is he hys e esis o
"
00
in he case o slow-
cycle empe a u e change o many days. Ou o he expe imen s wi h
di e en empe a u e condi ions and wi h long- ime measu emen s
also indica ed he e ec o ambigui y o elec omagne ic losses.
The discussion on hys e esis o he eal pa o ice dielec ic
pe mi i i y a 1 kHz was published ea lie in [7]. I was obse ed
o he ice ha o med he 2% NaCl solu ion. This expe imen las ed
o abou an hou . Howe e , his obse a ion was in e p e ed as he
in luence o he hea o phase ansi ion. In ou expe imen s, he
analogous explana ion is no ue because he e was a small amoun
o sal and a long pe iod o keeping ice a a low-ambien empe a u e.
We conclude ha he hys e esis e ec in ou expe imen s may
be connec ed wi h he exis ence o supe cooling sal mic oscopic
inclusions a empe a u es lowe han eu ec ic poin , whe e hey
a e in a liquid s a e. The possible supe cooling empe a u e is
de e mined by he shape and he size o liquid inclusions. I seems
ha hei me amo phism de e mine he olume-liquid concen a ion
and, consequen ly, he imagina y pa o dielec ic pe mi i i y. The
de e mina ion o he exac o igin o elec omagne ic-loss ambigui y
equi es a mo e de ailed in es iga ion o eshwa e ice s uc u e and
physical and chemical peculia i ies o inclusions.
V. CONCLUSIONS
The e a e signi ican ime changes o he imagina y pa o dielec-
ic pe mi i i y e en o cons an - alue sal -impu i y concen a ion
in eshwa e ice. Di e en alues o elec omagne ic loss we e
expe imen ally obse ed a he iden ical empe a u e. Thus, o he
de e mina ion o ice dielec ic loss, we mus ake in o accoun he
ice- ime exis ence a e wa e is ozen, he ice- empe a u e his o y,
and he inclusions cha ac e is ics. Dis ega ding hese condi ions in
p e ious pape s led no so much o measu emen e o s, bu o he
desc ip ion o ice wi h di e en s uc u es. The e o e, he c yosphe e
emo e sensing equi es aking in o accoun he ime changeabili y o
ice elec omagne ic p ope ies.
REFERENCES
[1] A. S og yn, “A s udy o he mic owa e b igh ness empe a u e o snow
om he poin o s ong luc ua ion heo y,” IEEE T ans. Geosci. Remo e
Sensing, ol. GE-24, pp. 220–231, Ma . 1986.
[2] S. C. Wa en, “Op ical cons an s o ice om he ul a iole o he
mic owa e,” Appl. Op ., ol. 23, no. 8, pp. 1206–1225, 1984.
[3] C. Ma zle and U. Wegmulle , “Dielec ic p ope ies o eshwa e ice a
mic owa e equencies,” J. Phys. D, Appl. Phys., ol. 20, pp. 1623–1630,
1987; E a a in J. Phys. D, Appl. Phys., ol. 21, p. 1660, 1988.
[4] L. Le i and L. Luba , “On he elec ic p ope ies o ice doped wi h
NH
4
F,” Phys. Kondens. Ma e ie, ol. 7, pp. 368–371, 1968.
[5] I. G. Young and R. E. Salomon, “Dielec ic beha io o ice wi h HCl
impu i y,” J. Chem. Phys., ol. 48, no. 4, pp. 1635–1644, 1968.
[6] G. S. Bo donski and S. D. K ylo , “Radio b igh ness a ia ion o
eshwa e ice co e in win e pe iod,” Iz es ya Academii Nauk Rossii
Se ia Fiziki A mos e y i Okeana, ol. 29, no. 6, pp. 842–847, 1993.
[7] V. V. Bogo odsky and G. P. Hohlo , “In luence o some sal componen s
and i s composi ion on ice elec ical p ope ies,” P oc. A c . An a c .
Ins ., in Russian, ol. 295, pp. 89–95, 1970.
Radiome ic Sensi i i y Compu a ion in
Ape u e Syn hesis In e e ome ic Radiome y
Ad iano Camps, Ignasi Co bella, Ja ie Ba ´
a, and F ancesc To es
Abs ac — This pape is conce ned wi h he adiome ic sensi i i y
compu a ion o an ape u e syn hesis in e e ome ic adiome e de o ed
o ea h obse a ion. The impac o sys em pa ame e s and he use
o simul aneous edundan measu emen s a e analyzed. The In e e o-
me ic Radiome e Unce ain y P inciple is p esen ed; i quan i ies he
ela ionship be ween adiome ic sensi i i y and angula esolu ion.
Index Te ms—In e e ome y, adiome y, emo e sensing, sensi i i y.
I. INTRODUCTION
An in e e ome ic adiome e measu es he co ela ion be ween he
analy ic signals collec ed by di e en an ennas [
S
1
(
)
and
S
2
(
)
].
These co ela ions p o ide he samples o he so-called isibili y
Manusc ip ecei ed Decembe 27, 1995; e ised July 2, 1997. This wo k
was suppo ed by he Eu opean Space Agency. wi hin he amewo k o ESA
MIRAS CCN 2, Wo k O de 10, ESTEC Con ac 9777/92/NL/PB ac i i ies,
wi h MATRA MARCONI SPACE as main con ac o .
The au ho s a e wi h he Depa men o Signal Theo y and Communica-
ions, Uni e si a Poli `
ecnica de Ca alunya, 08034 Ba celona, Spain (e-mail:
[email p o ec ed]).
Publishe I em Iden i ie S 0196-2892(98)00737-2.
0196–2892/98$10.00 1998 IEEE
IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 36, NO. 2, MARCH 1998 681
unc ion, which has dimensions o Kel in.
V
1
;
2
(
u;
)=
1
2
E
[
S
1
(
)
S
3
2
(
)]
=

+


1
T
(
; 
)~
12
0
u
+

c
1
e
0
j
2

(
u
+

)
d d
(1)
whe e
(
u;
)
is he baseline and is equal o he di e ence be ween he
an enna posi ions o e he
XY
plane no malized o he wa eleng h;
T
(
; 
)
K is he so-called modi ied b igh ness empe a u e [1].
T
(
; 
)=
T
B
(
; 
)
1
0

2
0

2
F
n
1
(
; 
)
F
3
n
2
(
; 
)
(2)
whe e
T
B
(
; 
)
, dimensions o Kel in, is he b igh ness empe a u e;
(
;
)
a e he di ec o cosines, wi h espec he
(
X; Y
)
axes, equal
(sin

cos
;
sin

sin

)
;
F
n
1
;
2
(
; 
)
a e he no malized an enna
ol age pa e n; and
~
12
(

)
, he inge-wash unc ion, (wi hou uni s)
akes in o accoun spa ial deco ela ion e ec s [2].
In he ideal case, no deco ela ion e ec s
~
12
(

)

1
and iden ical
an enna pa e ns
F
n
1
=
F
n
2
=
F
n
, he modi ied b igh ness empe -
a u e can be eco e ed by means o a disc e e Fou ie T ans o m o
he isibili y samples
T
(
; 
)=
F
0
1
[
V
(
u;
)]
:
(3)
In la ge in e e ome e s, in o de o simpli y he signal dis ibu ion
ne wo k, he c oss-co ela ions a e usually pe o med a baseband by
means o eal co ela o s a e in-phase and quad a u e demodula ion
V
1
;
2
/
E
[
i
1
(
)
i
2
(
)] +
jE
[
q
1
(
)
i
2
(
)]
:
(4)
II. RADIOMETRIC SENSITIVITY COMPUTATION
Radiome ic sensi i i y is de ined as he minimum inpu change ha
can be de ec ed a he ou pu [2]–[4]. In a in e e ome ic adiome e ,
i is limi ed by he disc e iza ion and he ini e co e age o he spa ial
equencies plane
(
u;
)
and he SNR, which can be imp o ed by
inc easing he in eg a ion ime and/o he p ede ec ion bandwid h
[3]. The ini e
(
u;
)
co e age and he disc e iza ion e o s se he
sa u a ion limi ha is eached o high SNR’s.
A. Disc e iza ion and Fini e
(
u;
)
Co e age
In a o al-powe o Dicke adiome e , he measu ed an enna em-
pe a u e is gi en by equa ions 4.55–4.60 o [4]. The e o commi ed
depends on he pa icula b igh ness empe a u e dis ibu ion being
obse ed and can be minimized by maximizing he an enna main-
beam e iciency (MBE), which equi es he use o an ennas wi h
a ape ed illumina ion ha , in u n, educe he achie able spa ial
esolu ion.
On he o he hand, an in e e ome ic adiome e o ms he b igh -
ness empe a u e map by a disc e e-in e se Fou ie ans o m o he
isibili ies measu ed by he a ay (3). I has been shown [1], [5], [6]
ha , as p oposed in [7], he op imum shape o a wo-dimensional
(2-D) in e e ome ic a ay is a
Y
.
Y
-a ays gene a e he la ges
egula
(
u;
)
co e age o e an hexagonal g id o a gi en numbe
o an ennas, hus maximizing he angula esolu ion o , con e sely,
minimizing he ha dwa e equi emen s [1].
The impulse esponse o he in e e ome e in he di ec ion
(

0
;
0
)
can be in e p e ed as he beam syn hesized by he a ay, and i is
called he equi alen a ay ac o (
AF
eq
) [3] because o i s simila i ies
wi h phased a ays
AF
eq
(
; ; 
0
;
0
)=
A
n
W
(
u
n
;
n
)~
n
u
+

o
1
e
+
j
2

(
u
(

0

)+
(

0

))
(5)
whe e
A
is he pixel’s a ea in he
(
u;
)
plane;
A
=
d
2
o
T
-a ays;
A
=
p
3
d
2
=
2
o
Y
-a ays;
A
=
d
o one-dimensional (1-D) a ays;
and
d
is he spacing be ween adjacen an ennas no malized o he
wa eleng h o he minimum baseline. The unc ion
W
(
u;
)
is a
window used o weigh he isibili y samples.
In a simila way, he MBE can be de ined as
MBE
=
main lobe
j
AF
eq
(
; 
)
j
d

4

j
AF
eq
(
; 
)
j
d

(6)
whe e he
AF
eq
is no squa ed because i e e s o b igh ness
empe a u es, a powe measu emen , howe e , om (5), he
AF
eq
may ha e nega i e lobes. The MBE can be op imized by a p ope
selec ion o he window unc ion. Table I shows he sidelobe le el
(SLL) and he MBE a he SLL o i e di e en windows o
an
Y
-a ay wi h
N
EL
=43
an ennas pe a m spaced
d
=0
:
89
wa eleng hs, as p oposed o MIRAS [7]. Deco ela ion e ec s ha e
been neglec ed since
B=
0

2%. The sa u a ion o he adiome ic
sensi i i y shown in Fig. 1 is due o he disc e iza ion o MBE e o
compu ed a he cen e o he ins an aneous ield o iew (FOV) [1].
The e o dec eases wi h he a ay size and he window smoo hness.
B. SNR
I he eal and imagina y pa s o he isibili y unc ion a e
ob ained by c oss-co ela ing he in-phase and quad a u e componen s
o he signals collec ed by he an ennas once digi alized [7], sligh ly
di e en esul s a e ob ained om hose p esen ed in [2], [3], and
[8] o he 1-D in e e ome e ESTAR, o adioas onomy. Th ee
e ec s ha now ha e been aken in o accoun a e p ede ec ion il e s’
shape ( ec angula o Gaussian), single sideband (SSB) o double
sideband (DSB) ecei e s wi h he same p ede ec ion bandwid h, and
co ela o ’s ype.
The MIRAS inge-wash unc ion was compu ed in [5] and [6],
aking in o accoun he o e all equency esponse o he ecei ing
chain [7]. I was ound ha he inge-wash unc ion is be e
app oxima ed by a Gaussian il e (7a) han by a ec angula il e
(7b) wi h he same noise bandwid h
B
(7c).
j
H
(
)
j
=
e
0
((
0
)
=B
)
(7a)
j
H
(
)
j
=5
0
o
B
(7b)
B
1
=
+
1
01
j
H
(
)
j
2
d
(7c)
whe e
5(
x
)=1
o
j
x
j
1
=
2
and
0
elsewhe e. Consequen ly, i is
expec ed ha mo e accu a e esul s a e ob ained wi h he Gaussian
model.
Following he p ocedu e used in [3], he s anda d de ia ion o he
eal and imagina y pa s o he isibili y unc ion can be compu ed,
aking in o accoun ha I/Q demodula ion is pe o med p io o he
co ela ion [6, App. 1]. The main esul s a e lis ed below o Gaussian
(8a) and o ec angula p ede ec ion il e s (8b)

2
; i
=1
2
p
2
B
e
(
T
A
+
T
R
)
2
1+
e
0

(21
=
p
2
B
)
+
V
2
; i
(
u;
)1+
e
0

(21
=
p
2
B
)
0
V
2
i;
(
u;
) 1
0
e
0

(21
=
p
2
B
)
(8a)
682 IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 36, NO. 2, MARCH 1998
TABLE I
SLL [dB] AND MBE FOR THE MIRAS SPACEBORNE-INSTRUMENT EQUIVALENT-ARRAY FACTOR (

2
mn
=
u
2
mn
+
2
mn
,

max
=
p
3
N
EL
d
)
Fig. 1. Radiome ic sensi i i y dBK (10 log
1
T
) e sus SNR [10 log (
T
A
=
V
)].
T
A
=
200 K, MIRAS ins umen . Radiome ic sensi i i y sa u a ion
is due o he disc e iza ion and ini e (
u;
) co e age e o s.

2
; i
=1
2
B
e
(
T
A
+
T
R
)
2
1+3 21
B
+
V
2
; i
(
u;
)1+3 21
B
0
V
2
i;
(
u;
) 1
0
321
B
(8b)
whe e
3(
x
)=1
0
j
x
j
o
j
x
j
1
and
0
elsewhe e;
V
and
V
i
a e he
eal and imagina y pa s o he isibili y unc ion;
T
A
is he an enna
empe a u e;
T
R
=
T
R
1
=
T
R
2
is he ecei e s’ noise empe a u e;
1
=
o
0
lo is he di e ence be ween he il e ’s cen al equency
o
and he local oscilla o ’s equency
lo; and

e is he e ec i e
in eg a ion ime ha depends on co ela o ’s ype, i.e.,

e
=

o an
analog co ela o , and

e
=

/2.46 o 1-bi
2
1-bi co ela o wi h
IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 36, NO. 2, MARCH 1998 683
sampling equencies
s
=2
B
[9]. A his poin , i is in e es ing o
no e he ollowing.
1) The a iance

2
V
=

2
+

2
i
, compu ed wi h ec angula
p ede ec ion il e s, is
p
2
imes la ge han wi h Gaussian il e s
because noise is mo e compac ed in equency and su e s less
om deco ela ion e ec s.
2) Fo any kind o il e , an imp o emen by a ac o o wo is
achie ed in SSB ecei e s (
j
1
j
>B=
2
), as compa ed o
DSB ecei e s (
1
=0
), a he expense o highe speed
co ela o s and highe powe consump ion. In p ac ice, he
use o SSB demodula ion simpli ies ecei e ’s design when
e adia ion om local oscilla o o he an enna mus be kep
below he h eshold o he signals ecei ed in he p o ec ed
band, i.e., 1.400–1.427 MHz [7]. I also educes local oscilla o
AM noise and o se s can be easily emo ed by high-pass
il e ing he signals p io o co ela ion.
3) The use o digi al co ela o s educes he in eg a ion ime

by a ac o ha depends on quan iza ion le els and sampling
equency [9].
Since he b igh ness empe a u e map is ob ained by means
o a disc e e Fou ie ans o m o he isibili y samples, he
isibili y e o s a e ansla ed in o he empe a u e map
^
T
(
; 
)=
A
m n
W
mn
1
[
V
(
u
mn
;
mn
)+
e
V
(
u
mn
;
mn
)
+
je
V
(
u
mn
;
mn
)]
1
e
j
2

(
u 
+

)
(9)
whe e (
e
V
,
e
Vi
) a e he e o s in he eal and imagina y
pa s o he isibili y unc ion. P io o compu a ion o he
adiome ic sensi i i y some conside a ions abou edundancy
and he mi ici y mus be poin ed ou .
1) He mi ici y o he Visibili y Samples: Only hal o he base-
lines mus be measu ed (
u
≥0,
≥0 and
u
<0,
> 0). The o he
hal is ob ained by conjuga ing he measu ed baselines. In doing so,
he noise is He mi ian oo.
V
3
(
u;
)=
1
2
E
[
S
1
(
)
S
3
2
(
)]
3
=
1
2
E
[
S
2
(
)
S
3
1
(
)] =
V
(
0
u;
0
)
:
(10)
2) Redundancy and Co ela ion Be ween E o s: In [5], [6], [10]
and [11], i is shown ha he c oss-co ela ion o he e o s o wo
iden ical baselines 1–2 and 3–4 (excep o he an enna posi ions)
wi h he same in eg a ion ime

, one o hem delayed

d
, is gi en
by ( ec angula p ede ec ion il e s)
E
[1
V
12
(
+

d
)1
V
3
34
(
)] =
R
^
V
^
V
(

d
)
0
V
12
V
3
34
=
V
13
V
3
24
B
e sinc

d
:
(11)
being
R
^
V
^
V
(

d
)
, he c oss-co ela ion be ween he measu ed is-
ibili ies
V
12
and
V
34
a
=

d
. No e ha , in an ac ual onboa d
in e e ome e like MIRAS [7], all baselines a e measu ed in he same
ime in e al and

d
=0
. Gi en i s impo ance, we explici ly show
ha he noise o a isibili y sample (8b) can be ob ained om (11).

2
V
=

2
V
+

2
V
=
E
[
j
1
V
12
j
2
]=
R
1
V
1
V
(0)
=
V
11
V
3
22
B
e
=
j
V
(0
;
0)
j
2
B
e
=(
T
A
+
T
R
)
2
B
e (12)
F om (11), i can be seen ha , wi h ideal noise- ee ecei e s, e o s
be ween simul aneous measu emen s (

d
=
0) o di e en isibili y
samples a e s ongly co ela ed i he spacing be ween he an enna
pai s 1–2 and 3–4 is much smalle han he down all o he ampli ude
o he isibili y unc ion. This si ua ion holds o scenes consis ing
on poin sou ces [2], [10], and a e aging simul aneous measu emen s
does no imp o e SNR signi ican ly. On he con a y, o a smoo h
empe a u e dis ibu ion, as in he case o ea h obse a ion, he
isibili y unc ion decays apidly, e o s a e only pa ially co ela ed,
and a e aging educes noise powe .
On he o he hand, i he ecei e ’s noise empe a u e is much
highe han he b igh ness empe a u e o be measu ed, he a e -
aging o simul aneous measu emen s imp o es he SNR, due o
he educ ion o ecei e ’s noise. This is no he case wi h ea h
obse a ion a low mic owa e equencies, in which ecei e ’s noise
empe a u e (
T
R

80 K) is usually lowe han he a e age b igh ness
empe a u e (
T
A

250 K). In any case, he imp o emen shown by
(13), ep oduced om [3], will always be lowe han he uppe bound
ound o a linea a ay, which akes in o accoun unco ela ed e o s
1
T
no edundancy
=
T
B
+
T
R
p
B
p
N
V
!
1
T
edundancy
unco e o s
=
T
B
+
T
R
p
B
p
c
+ln
N
V
(13)
whe e
N
V
s ands o he o al numbe o isibili ies and
c
is he
Eule ’s cons an .
A de ailed analysis o edundancy and i s imp o emen on adio-
me ic sensi i i y equi es a speci ic a ay con igu a ion and scene
unde obse a ion. Howe e , o
Y
-a ays, which p o ide a e y
low deg ee o edundancy [5], [7], [10], [11], his imp o emen can
be app oxima ely ound i we ealize ha only baselines ela ing
an ennas on he same a m can be edundan . By he ze o baseline
i is unde s ood ha he one co esponding o
u
=
=0
, which
in MIRAS is non edundan , since i is measu ed by a dedica ed
Dicke adiome e . Recall also ha when he He mi ian p ope y is
conside ed e e y
(
u;
)
-poin is ac ually duplica ed. Fo he
Y
-a ay
wi h h ee a ms, each wi h
N
EL
=43
elemen s, plus a cen al
elemen , he e a e
3
N
EL
(3
N
EL
+1)
/
2 + 1 = 8386
baselines
[ he ex a one co esponding o
V
(0
;
0)
],
3
N
2
EL
+3
N
EL
+1=
5551
non edundan baselines o non edundan
(
u;
)
poin s, and
3(
N
EL
0
1) = 126
edundan
(
u;
)
poin s wi h di e en deg ees o
edundancy. I means ha
8386
0
5551 = 2709
edundan complex
co ela ions ( isibili ies) lead o only 126 edundan
(
u;
)
poin s.
This leads o an imp o emen o a 1% o a 43 an ennas pe a m
Y
-a ay [10], [11], e en in he case in which e o s be ween hese
isibili y samples a e assumed o be comple ely unco ela ed.
3) Snapsho Radiome ic Sensi i i y: As shown in he p e ious
sec ion, isibili y e o s a e He mi ian and, o compu a ional pu -
poses, unco ela ed om sample o sample. Wi h hese conside a ions,
he snapsho adiome ic sensi i i y, ha is, he a e age e o in each
b igh ness empe a u e map ob ained a e an in eg a ion ime o

seconds, is
T
(
; 
)=
A
m n
W
mn
1
[
e
V
(
u
mn
;
mn
)+
je
V
(
u
mn
;
mn
)]
1
e
j
2

(
u 
+

)
1
T
(
; 
)=
E
[
T
(
; 
)
T
(
; 
)
3
]
=
A
2
m n
W
2
mn
(

2
mn
+

2
imn
)
+
u >
0
;

0
u

0
; >
0
W
2
mn
(

2
mn
+

2
imn
)
1
cos[4

(
u
mn

+
mn

)]
(14)
684 IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 36, NO. 2, MARCH 1998
TABLE II
BASIC PARAMETERS OF MIRAS INSTRUMENT
which can be app oxima ed by
1
T
(
; 
)
'
AT
A
+
T
R
B
e

w

ol

p
N
V
(15)
whe e
A
=
p
3
d
2
=
2
, he local oscilla o ac o is gi en by

ol
=
p
2=1
:
41
o DSB ecei e s and

ol
=1
o SSB ecei e s;
he il e ac o is gi en

=
4
p
2=1
:
19
o Gaussian il e s
and

=1
o ec angula il e s; and he windowing ac o

w
is de ined as

w
=
m n
W
2
mn
=N
V
(16)
whe e he numbe o isibili y samples, including he He mi ian
ones, is
N
V
=6
N
2
EL
+6
N
EL
+1
o
Y
-a ays. In he
MIRAS case,
N
EL
=43
and he windowing ac o

w
=
1
;
0
:
5212
;
0
:
5717
;
0
:
5446
;
and
0
:
4517
o he ec angula , iangula ,
Hamming, Hanning, and Blackmann windows, espec i ely (Table I).
No e ha he weighing unc ion a enua es isibili y samples be ween
dis an an ennas, whe e SNR is wo se; hus, he adiome ic sensi i i y
is imp o ed a he expense o a loss in he angula esolu ion. MIRAS
spacebo ne snapsho adiome ic sensi i i y can be compu ed om
(15) and (16) wi h he pa ame e s lis ed in Table II [7]. Fig. 1 shows
he snapsho adiome ic sensibili y in decibels
10 log (
1
T
) [dBK]
g
e sus he SNR. Fo an SNR in he MIRAS ange 31.6–33.2 dB, he
adiome ic sensi i i y is bounded by 7.1–15.0 K and 3.2–6.8 K o
he ec angula and Blackmann windows, espec i ely.
4) Radiome ic Sensi i i y Imp o emen by Pixel A e aging:
Radiome ic sensi i i y can be imp o ed in a 2-D in e e ome ic
adiome e by means o “pixel a e aging.” Tha is, since a pixel
emains in he FOV o a long ime, he eco e ed alues can be
a e aged a e p ope co ec ion o he dependence wi h he angle
o incidence. In he MIRAS case, a pixel emains in he FOV o
abou 22 s (

FOV
=
FOV wid h/pla o m eloci y
= 165
Km/7
Km/s
=22
s), om which 11 s co espond o each pola iza ion.
The imp o emen on he adiome ic sensi i i y in each pola iza ion
is hen
1
T
pixel a g.
=
1
T
snap-sho
= (11
s=
0
:
3
s
)=6
o
6
p
2
in a single pola iza ion ins umen . This imp o emen is achie ed
because unsimul aneous measu emen s a e independen and he e o
is educed by he squa e oo o he numbe o measu emen s,
o equi alen ly, he in eg a ion ime is inc eased o he o al ime
he pixel emains in he FOV

FOV. A e pixel a e aging, o
he MIRAS ins umen (dual pola iza ion ins umen ), he expec ed
adiome ic sensi i i y alues a e hen
1
T
MIRAS

2
:
5
and
1
:
1
K o he ec angula and Blackmann windows, espec i ely, and
T
A

200
K.
III. RADIOMETRIC SENSITIVITY IN INTERFEROMETRIC
RADIOMETERS AND TOTAL POWER RADIOMETERS:
THE INTERFEROMETRIC RADIOMETER UNCERTAINTY PRINCIPLE
In o de o compa e in a homogeneous way he adiome ic
sensi i i ies o in e e ome ic adiome e s and ideal o al-powe
adiome e s, we mus ake in o accoun all he a ailable in eg a ion
ime. No e ha a 2-D in e e ome ic adiome e images all he
space simul aneously, while a o al-powe adiome e images only he
pixel poin ed by he an enna beam. Tha is, he MIRAS spacebo ne
ins umen will image (
3
N
EL
+1)
2
= 130
2
=16
:
900
pixels
simul aneously [1] e e y

=
0.3 s, om which he e a e 8.689
in he alias- ee FOV. An ideal o al-powe adiome e imaging
only he alias- ee FOV pixels wi h he same angula esolu ion
would ha e a maximum in eg a ion ime o

pixel
=
=
8
:
689 =
0
:
3
s
=
8
:
689 = 34
:
5

s, leading o a wo s -case adiome ic sensi i i y
o
1
T
TPRad
pixel
=
T
sys
=
(
B
pixel
)

14
:
5
K, which is e y close o
he snapsho adiome ic sensi i i y o he in e e ome e adiome e
when he ec angula window is used (Sec ion II-B3).
The adiome ic sensi i i y imp o emen achie ed by windowing
can be now unde s ood as he spa ial a e aging o he pixel’s alue
wi h i s neighbo s. In ac , he sensi i i y imp o emen by windowing
is app oxima ely ela ed o he hal -powe syn hesized beamwid hs
gi en in [6] and [12] by
1
T
In Rad
ec angula
1
T
In Rad
W
=1

W
'
1

0
3dB
W
1

0
3dB
ec angula
:
(17)

IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 36, NO. 2, MARCH 1998 685
In [6] and [12], he angula esolu ion o
Y
-a ays is analyzed,
and o he ec angula window
1

0
3dB
ec .

=
(4
p
3
N
EL
d
)
. Fo an
a bi a y window
W
, he p oduc
1
T
W
1

0
3dB
;W
is ound o be
1
T
W
1

2
0
3dB
;W
'
p
3
2
d
2
T
A
+
T
R
B
e

w

ol

F
p
N
V


w
4
p
3
N
EL
d
'
p
3

24
T
A
+
T
R
B
e

ol

F
;
(
N
EL
>
1)
(18)
which can be app oxima ed by
1
T
1

2
0
3dB
;W
'
T
A
+
T
R
B
e

ol

F
d:
(19)
Equa ion (19) is he new In e e ome ic Radiome e Unce ain y
P inciple. I s a es ha he p oduc o he adiome ic sensi i i y
1
T
by he 2-D angula esolu ion
1

2
0
3dB
is a cons an ha depends only
on ecei e s and co ela o s pa ame e s, and i is independen on he
window used o p ocess he isibili y samples. I can be iewed as
he in e e ome ic adiome e e sion o he o al-powe adiome e
unce ain y equa ion gi en in (6.149) o [4].
IV. CONCLUSIONS
The adiome ic sensi i i y o a gene al 2-D in e e ome ic a-
diome e has been compu ed in his pape . The impac o he il e s’
shape has been analyzed and quan i ied as well as he ype o demod-
ula ion (SSB o DSB) and he kind o co ela o . The imp o emen
achie ed by means o pixel a e aging has been discussed and esul s
ha e been pa icula ized o he MIRAS ins umen , a
Y
-shaped
in e e ome ic adiome e wi h 43 an ennas pe a m, cu en ly unde
s udy a he Eu opean Space Agency. I has been shown ha a e
pixel a e aging, adiome ic sensi i i ies a e expec ed o be abou
2.5 o 1.1 K, depending on he weighing unc ion used o ape
he isibili y samples. Finally, The new in e e ome ic adiome e
unce ain y p inciple has been s a ed: i es ablishes ha he p oduc
o he adiome ic sensi i i y by he angula esolu ion is a cons an
ha depends only on he kind o ecei e s, co ela o s, and minimum
baselines.
REFERENCES
[1] A. Camps, J. Ba ´a, I. Co bella, and F. To es, “The p ocessing o hexago-
nally sampled signals wi h s anda d ec angula echniques: Applica ion
o ape u e syn hesis in e e ome e adiome e s,” IEEE T ans. Geosci.
Remo e Sensing, ol. 35, pp. 183–190, Jan. 1997.
[2] R. Thompson, J. Mo an, and G. Swenson, In e e ome y and Syn hesis
in Radio As onomy. New Yo k: Wiley, 1986.
[3] C. S. Ru , C. T. Swi , A. B. Tanne , and D. M. LeVine, “In e e ome ic
syn he ic ape u e adiome y o he emo e sensing o he Ea h,” IEEE
T ans. Geosci. Remo e Sensing, ol. 26, pp. 597–611, Sep . 1988.
[4] F. T. Ulaby, R. K. Moo e, and A. K. Fung, Mic owa e Remo e Sensing,
ol. I. No wood, MA: A ech House, 1981.
[5] J. Ba ´a, I. Co bella, F. To es, and A. Camps, “Two-dimensional disc e e
o mula ion o ape u e syn hesis adiome e s,” ESA-ESTEC, Final
Rep., CNN 2 o Wo k O de 10 o ESTEC Con ac 9777/92/NL/PB,
Jan. 1996.
[6] A. Camps, “Applica ion o in e e ome ic adiome y o Ea h ob-
se a ion,” Ph.D. disse a ion, Uni e si a Poli `
ecnica de Ca alunya,
Ba celona, Spain, No . 1996.
[7] Ma a Ma coni Space, “MIRAS: Mic owa e imaging adiome e wi h
ape u e syn hesis. Mic owa e adiome y c i ical echnical de elop-
men ,” ESA-ESTEC, Final Rep., ESTEC Con ac 9777/92/NL/PB, Jan.
1995.
[8] M. E. Tiu i, “Radio as onomy ecei e s,” IEEE T ans. An ennas P op-
aga ., ol. AP-11, pp. 930–938, Dec. 1964.
[9] J. B. Hagen and D. T. Fa ley, “Digi al co ela ion echniques in adio
science,” Radio Sci. ol. 8, pp. 775–784, Aug./Sep . 1973.
[10] J. Ba ´a, A. Camps, F. To es, and I. Co bella, “Baseline edundancy and
adiome ic sensi i i y: A c i ical e iew,” in Soil Mois u e and Ocean
Salini y Measu emen s and Radiome e Techniques Consul a i e Mee -
ing. Noo dwijk, The Ne he lands: ESA-ESTEC, Ap . 20–22, 1995.
[11] , “The co ela ion o isibili y e o s and i s impac on he
adiome ic esolu ion o an ape u e syn hesis adiome e ,” submi ed
o publica ion.
[12] , “Angula esolu ion o wo-dimensional hexagonally sampled
in e e ome ic adiome e s,” Radio Sci., o be published.
Recip oci y o he Bidi ec ional Re lec ance
Dis ibu ion Func ion (BRDF) in Measu emen s
and Models o S uc u ed Su aces
William C. Snyde
Abs ac —The bidi ec ional e lec ance dis ibu ion unc ion (BRDF)
is one o he mos impo an su ace p ope ies o e es ial emo e
sensing, bu i s de ini ion o s uc u ed su aces is no ully unde s ood.
The BRDF o la su aces has a s aigh o wa d de ini ion and is usually
conside ed o be ecip ocal, which means he alue is he same when
he sou ce and de ec o angles a e swi ched. S uc u ed su aces, such
as o es canopies and g asslands, equi e an ex ension o he de ini ion
o BRDF and some addi ional measu emen condi ions. In his pape ,
a de ini ion o he BRDF o s uc u ed su aces is p oposed, and i
is shown ha wi h his de ini ion, he BRDF is ecip ocal. In addi ion,
some o he ela ed geome ical measu emen equi emen s a e discussed.
I is concluded ha ecip oci y should apply o bo h measu emen s
and models o s uc u ed su aces and ha ield measu emen s iola e
ecip oci y no because he BRDF i sel is non ecip ocal, bu because o
unco ec ed geome ic and adiome ic ac o s.
Index Te ms—Elec omagne ic sca e ing by ough su aces, adia i e
ans e , adiome y, emo e sensing.
I. INTRODUCTION
Land-co e bidi ec ional e lec ance is o p ime impo ance in e -
es ial emo e sensing. In he sola - e lec i e egion o he spec um,
he bidi ec ional p ope ies a e applied o no malize he e ec s o
di e en sun-senso geome ies o p o ide consis en su ace ea u es
o classi ica ion and change de ec ion [1]. In he he mal in a ed
egion, he bidi ec ional cha ac e is ics a e applied o accoun o he
e lec ed downwelling i adiance and o compu e he angula emis-
si i y [2]. The bidi ec ional e lec ance dis ibu ion unc ion (BRDF)
cha ac e izes su ace bidi ec ional e lec ance o all combina ions
o inciden and e lec ed zeni h and azimu h angles. BRDF is an
op ical p ope y o a ma e ial ha does no depend on ex e nal
ac o s, such as illumina ion o a mosphe ic ansmission. In p ac ice,
BRDF can be modeled, bu canno be measu ed o applied di ec ly
Manusc ip ecei ed Feb ua y 19, 1997; e ised June 4, 1997. This wo k
was pe o med a he Ins i u e o Compu a ional Ea h Sys em Science,
Uni e si y o Cali o nia, San a Ba ba a, and suppo ed by Ea h Obse ing
Sys em P og am Con ac NAS5-31370 o he Na ional Ae onau ics and Space
Adminis a ion.
The au ho is wi h GDE Sys ems Inc., San Diego, CA 92150-9008 USA
(e-mail: [email p o ec ed]).
Publishe I em Iden i ie S 0196-2892(98)00547-6.
0196–2892/98$10.00 1998 IEEE