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A responsivity-based criterion for accurate calibration of FTIR emission spectra: Identification of in-band low-responsivity wavenumbers

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A responsivity-based criterion for accurate calibration of FTIR emission spectra: Identification of in-band low-responsivity wavenumbers

Abstract and Figures

Spectra measured by remote-sensing Fourier transform infrared spectrometers are often calibrated using two calibration sources. At wavenumbers where the absorption coefficient is large, air within the optical path of the instrument can absorb most calibration-source signal, resulting in extreme errors. In this paper, a criterion in terms of the instrument responsivity is used to identify such wavenumbers within the instrument bandwidth of two remote-sensing Fourier transform infrared spectrometers. Wavenumbers identified by the criterion are found to be correlated with strong absorption line-centers of water vapor. Advantages of using a responsivity-based criterion are demonstrated.
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A responsivity-based criterion for acc urate
calibration of F T IR emission spectra:
identification of in-band low-responsivity
wavenumbers
Penny M. Rowe,
1,* Steven P. Neshy ba,
2 Ch ristopher J. C ox,
1 and Von P. Walden1
1Department*of*Geography,*University*of*Idaho,*875*Perimeter*Drive,*Moscow,*Idaho*83843,*USA*
2Department*of*Chemistry,*University*of*Puget*Sound,*1500*N.*Warner,*Tacoma,*Washington*98416,*USA*
*prowe@ harbornet.co
m
Abstr act : Spectra measured by remote-sensing Fourier transform infrared
spectrometers are often calibrated using two calibration sources. At
wavenumbers where the absorption coefficient is large, air within the
optical path of the instrument can absorb most calibration-source signal,
resulting in extreme errors. In this paper, a criterion in terms of the
instrument responsivity is used to identify such wavenumbers within the
instrument bandwidth of two remote-sensing Fourier transform infrared
spectrometers. Wavenumbers identified by the criterion are found to be
correlated with strong absorption line-centers of water vapor. Advantages of
using a responsivity-based criterion are demonstrated.
©2011 Optical Society of America
O CIS codes: (120.0280) Remote sensing and sensors; (120.3180) Interferometry; (120.5630
Radiometry; (120.6200) Spectrometers and spectroscopic instrumentation; (280.4991) Passive
remote sensing.
References and links
1. R. O. Knuteson, H. E. Revercomb, F. A. Best, N. C. Ciganovich, R. G. Dedecker, T. P. Dirkx, S. C. Ellington,
W. F. Feltz, R. K. Garcia, H. B. Howell, W. L. Smith, J. F. Short, and D. C. Tobin³Atmospheric emitted
radiance interferometer. Part I: instrument design´J. Atmos. Ocean. Technol. 21(12), 1763±1776 (2004a).
2. H. E. Revercomb, H. Buijs, H. B. Howell, D. D. Laporte, W. L. Smith, and L. A. Sromovsky³Radiometric
calibration of IR Fourier transform spectrometers: solution to a problem with the high-resolution interferometer
sounder´Appl. Opt. 27(15), 3210±3218 (1988).
3. L. A. Sromovsky³Radiometric errors in complex Fourier transform spectrometry´Appl. Opt. 42(10), 1779±
1787 (2003).
4. P. M. Rowe, S. P. Neshyba, and V. P. Walden³A responsivity-based criterion for accurate calibration of FTIR
emission spectra: theoretical development and bandwidth estimation´Opt. Express 19(6) 5451-5463 (2011)
5. G. Lesins, L. Bourdages, T. Duck, J. Drummond, E. Eloranta, and V. Walden³Large surface radiative forcing
from topographic blowing snow residuals measured in the high arctic at eureka´Atmos. Chem. Phys. 9(6),
1847±1862 (2009).
6. A. Shimota, H. Kobayashi, and S. Kadokura³radiometric calibration for the airborne interferometric monitor for
greenhouse gases simulator´Appl. Opt. 38(3), 5 71±576 (1999).
7. P. J. Minnett, R. O. Knuteson, F. A. Best, B. J. Osborne, J. A. Hanafin, and O. B. Brown³the marine-
atmospheric emitted radiance interferometer: a high-accuracy, seagoing infrared spectroradiometer´J. Atmos.
Ocean. Technol. 18(6), 994±1013 (2001).
8. S. Chandrasekhar,
Radi ative Transfer
. (Dover, 1960).
9. L. S. Rothman, D. Jacquemart, A. Barbe, D. Chrisbenner, M. Birk, L. Brown, M. Carleer, C. Chackerianjr, K.
Chance, and L. Coudert³The 2004 molecular spectroscopic database´J. Quant. Spectrosc. Radiat. Transf.
96(2), 139±204 (2005).
10. S. A. Clough, M. W. Shephard, E. J. Mlawer, J. S. Delamere, M. J. Iacono, K. Cady-Pereira, S. Boukabara, and
P. D. Brown³Atmospheric radiative transfer modeling: a summary of the AER codes´J. Quant. Spectrosc.
Radiat. Transf. 91(2), 233±244 (2005).
11. R. Knuteson, Cooperative Institute for Meteorological Satellite Studies ± SSEC, University of Wisconsin-
Madison, 1225 W. Dayton St., Madison, WI 53706 (personal communication, 2010).
12. R. O. Knuteson, H. E. Revercomb, F. A. Best, N. C. Ciganovich, R. G. Dedecker, T. P. Dirkx, S. C. Ellington,
W. F. Feltz, R. K. Garcia, H. B. Howell, W. L. Smith, J. F. Short, and D. C. Tobin³Atmospheric emitted
radiance interferometer. Part II: instrument performance´J. Atmos. Ocean. Technol. 21(12), 1777±1789
(2004b).
#138323 - $15.00 USD
Received 18 Nov 2010; revised 25 Feb 2011; accepted 8 Mar 2011; published 16 Mar 2011
(C) 2011 OSA
28 March 2011 / Vol. 19, No. 7 / OPTICS EXPRESS 5930
13. P. Rowe, L. Miloshevich, D. Turner, and V. Walden³Dry bias in Vaisala RS90 radiosonde humidity profiles
over antarctica´J. Atmos. Ocean. Technol. 25(9), 1529±1541 (2008).
14. Data from the AERI operated at the North Slope of Alaska is available at
http: //www.archive.ar
m
.gov
.
1. I nt rod uction
Atmospheric infrared radiation is routinely measured using Fourier transform infrared (FTIR)
spectrometers [1]. Raw measurement spectra are calibrated using spectra of hot and cold
calibration sources of known emission (e.g [2]). The variance of a calibrated spectrum due to
noise in raw spectra can be expressed analytically if approximations are made that are valid in
the limit of low noise [3]. Outside the low-noise limit, however, the variance must be
calculated numerically as described in a companion paper by three of the authors (Rowe,
Neshyba, and Walden [4]; hereafter RNW). RNW present a general expression for the
variance (that is, for any noise level), showing that it is (formally) infinite. The variance can
be approximated as finite in the limit of low noise, or more precisely, when the uncertainty in
WKHPHDVXUHPHQWRI WKHLQVWUXPHQWUHVSRQVLYLW\ı
r
) divided by the true system responsivity
(
r
LVVPDOOı
r
/
r
<<1). However, outside the low-noise limit, the infinite variance corresponds
to arbitrarily large noise spikes in calibrated radiances. Furthermore, within the low-noise
limit, errors in the calibrated radiance due to random noise in raw spectra are also random, but
outside this limit errors are not generally random. Instead, large noise causes a bias in
calibrated radiances. For these reasons, it is important to set a criterion for accurate
calibration 7KH FULWHULRQ ı
r
/
r
<0.3 was shown by RNW to result in a negligible bias and a
variance for 99.999% of measurements that agrees with the low-noise approximation to
within 20%.
$V DQ DSSOLFDWLRQ RI WKH FULWHULRQ 51: FDOFXODWHG ı
r
/
r
for upwelling and downwelling
)7,5 PHDVXUHPHQWV WR DVVHVV ZKHWKHU WKH FRQGLWLRQ ı
r
/
r
< 0.3 is met within the bandwidth
specified. For an Atmospheric Emitted Radiance Interferometer (AERI) operated at Eureka,
Canada [5@ ı
r
/
r
was found to be generally less than 0.3 within the specified bandwidth,
indicating that most in-EDQGUDGLDQFHVDUHDFFXUDWHO\FDOLEUDWHG+RZHYHU ı
r
/
r
was found to
EH JUHDWHU WKDQ  DW SDUWLFXODU ZDYHQXPEHUV ZLWKLQ WKH LQVWU XPHQW¶V VSHFLILHG EDQGZLGWK
where
r
was small. At the same wavenumbers, extremely large errors were sometimes
observed in calibrated spectra. For example, on 1 July 2008 errors were as large as 2800 RU
[radiance unit; 1 RU = 1 mW m2 sr1 (cm1)1]. Such large errors correspond to spectral
features in raw uncalibrated spectra that have been attributed to absorption by CO2 and H2O
inside the instrument [2]. At these wavenumbers, the absorption coefficients of CO2 and H2O
are strong enough to absorb almost all of the calibration-source radiation, making accurate
calibration impossible. This problem only occurs when air is present inside the instrument and
thus is not observed in space-borne instrumentation, such as the Interferometric Monitor for
Greenhouse Gases (IMG [4,6]).
Identifying measured radiances with large errors and replacing them with reasonable
values is particularly important since further processing involving Fourier transforms (e.g.
FRUUHFWLQJ WKH LQVWUXPHQW¶V ILQLWH ILHOG RI YLHZ DQG ³]HUR-SDGGLQJ´ WKH VSHFWUXP RQWR D
VWDQGDUG ZDYHQXPEHU JULG FDQ FDXVH WKH HUURU WR ³ULQJ´ Lnto radiances at neighboring
wavenumbers.
,QWKLVZRUNZHVXJJHVWXVLQJDFULWHULRQLQWHUPVRIı
r
/
r
to identify wavenumbers where
absorption by trace gases within the instrument prevents accurate calibration. Because the
responsivity is low at such wavenumbers, as will be shown, we refer to these hereafter as
³ORZ-UHVSRQVLYLW\ ZDYHQXPEHUV´ ,Q FXUUHQW $(5, SURFHVVLQJ VRIWZDUH > 7], quality control
includes identification of large spikes in calibrated radiances using a criterion we term the
³UDWLR FULWHULRQ´ %RWK WKH UHVSRQVLYLW\ FULWHULRQ DQG WKH ratio criterion are applied to
measurements made with AERI instruments to show that the responsivity criterion is better
suited for identifying low-responsivity wavenumbers and is widely applicable to FTIR
spectrometers. We suggest replacing radiances at ZDYHQXPEHUVZKHUHı
r
/
r
> 0.3, based on the
work of RNW, but this threshold can be modified as needed for specific instruments and
experiments. Wavenumbers are identified over a 21-month field season using an AERI at
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Received 18 Nov 2010; revised 25 Feb 2011; accepted 8 Mar 2011; published 16 Mar 2011
(C) 2011 OSA
28 March 2011 / Vol. 19, No. 7 / OPTICS EXPRESS 5931
Eureka, Canada, and the likelihood of identification is compared to the strength of the water-
vapor absorption coefficient.
The organization of the paper is as follows. In section 2, we give a theoretical overview,
ILUVW VKRZLQJ WKDW WKH WUDQVPLVVLRQ RI DLU LQVLGH WKH LQVWUXPHQW¶V RSWLFDO SDWK L s implicitly
included in the instrument responsivity. We then summarize the theoretical development of
RNW, in which expressions for the variance and bias in a calibrated radiance, valid at all
noise levels, were derived. In section 3, we describe the ratio criterion for identifying highly
erroneous radiances and apply both the ratio criterion and the responsivity criterion to
measured AERI spectra. We also argue that radiances at identified wavenumbers should be
replaced with surrogates based on the ambient surface temperature, in contrast to existing
practice. Finally, we suggest a method to avoid biases that occur when corrected radiances are
averaged, showing that the method is well suited for the responsivity criterion. Section 4
presents conclusions.
2. Theoretical over view
The calibrated spectral radiance of a scene is given by Revercomb et al. [2] as
> @
Re ,
sc
sL h c c
hc
V V
L L L L
V V
H
ªº
«»
¬¼
(1)
where we have explicitly included the error in
Ls
due to noise, İ
L
, and where
Vs
,
Vh
and
Vc
are
raw spectra of the scene and hot and cold calibration source of known radiances,
Lh
and
Lc
. Re
represents taking the real part, and all terms are functions of wavenumber. At wavenumbers
where the absorption coefficient of a trace gas is large, we expect most scene radiation to
originate from near the instrument. If the cold calibration source is kept at ambient
temperature, then, at these wavenumbers, in the absence of errors,
Vs
-
Vc
= 0, and
Ls
=
Lc
.
However, if air is present within the instrument, the absorption coefficients of CO2 and H2O
can be so large that these trace gases absorb and re-emit almost all calibration-source
radiation, so that
Vh
=
Vc
, and the denominator in Eq. (1) approaches zero. In this section, we
first show that the transmission of air inside the instrument is included implicitly in the
instrument responsivity,
r
, so that identifying wavenumbers where the instrument does not
transmit enough source radiation for accurate calibration can be accomplished with a criterion
LQWHUPVRIı
r
/
r.
Following this, we summarize the work of RNW, showing the effect of large
ı
r
/
r
on the variance and bias of calibrated spectra.
2.1 Trans
m
ission of air inside the instru
m
ent
The equation of radiative transfer through an atmosphere in the absence of scattering and in
local thermodynamic equilibrium is
 
0
00
(',)
() ( )( ,) B (') '
'
z
z
dtz z
I z I z tz z T z dz
dz
ªº
«»
¬¼
³
(2)
[8], where is position,
I
(
z0
) is the source radiation,
t
is transmittance, and B is the Planck
function at temperature
T
. The parenthesis in Eq. (2) represent functionality.
I
,
t
, and B are
functions of wavenumber as well as position. For FTIR spectrometers, this equation governs
the transfer of radiation through the air inside the instrument, that is, from the calibration
source at position
z0
to the detector at position
z
. For the view of the hot calibration source,
I
(
z0
) is
Lh
and for the cold calibration source,
I
(
z0
) is
Lc
. The transmittance
t
(
zo
,
z
) is the
transmittance through the air inside the instrument; we define
tAI
{
t
(
zo
,
z
). The integral term in
Eq. (2) represents the radiance contribution from within the instrument, given the symbol
O
.
Thus the radiation transferred from the hot calibration source through the air inside the
instrument is
LhtAI
+
O
.
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Received 18 Nov 2010; revised 25 Feb 2011; accepted 8 Mar 2011; published 16 Mar 2011
(C) 2011 OSA
28 March 2011 / Vol. 19, No. 7 / OPTICS EXPRESS 5932
The signal measureGE\WKHGHWHFWRUDOVR GHSHQGVRQWKHRYHUDOOV\VWHP HIILFLHQF\ȘWKH
instrument phase,
·
, and the detector response,
Rd
. Including these terms as dictated by
Sromovsky [3], the signal measured when viewing the hot calibration source is given by
(3)
where this equation differs from Eq. (18) of Sromovsky [3] in two respects: the prime on
Vh
indicates that errors are not taken into account, and the transmittance term
tAI
is included here
(but is not in [3]). As pointed out by Sromovsky (first paragraph of section 6C of [3]) a term
needs to be added to account for error. We express the error as
nh
exp(
i
I
) instead of
Rd
Șİ
x
(as in [3]), for reasons given in RNW, so that
0.5 exp( ) exp( ).
h d AI h d h
V R tL i R O n i
KI I
(4)
For views of the cold calibration source and sky, subscripts
h
are replaced with subcripts
c
and
s
, respectively.
Sromovsky defines the system responsivity,
r
, to be the proportionality constant 0.5Ș5
d
. In
the present treatment, we show that
r
also includes
tAI
. Since the detector response and optical
efficiency vary slowly with wavenumber [2], fine spectral structure in the instrument
responsivity is expected to be due to
tAI
.
The system responsivity can be estimated from measurements as
 
/
m
hchc
r V V L L
{ 
(5)
(as in [2], but note that we have not taken the magnitude of the right hand side). The standard
deviation of
r
m
ı
r
, can be estimated from sequential measurements of
r
m
or from knowledge
of the statistical properties of
nh
and
nc
, since
2
2
r r
e
V
(RNW), where the brackets
indicate the mean and the double bars indicate the magnitude, and where
er
{ (
nh
-
nc
)/(
Lh
-
Lc
)
(ignoring variations in
Lh
,
Lc
,
·
and
O
).
Figure 1 shows ||
r
m
|| and an estimate of
tAI
for an AERI operated at Eureka, Canada. The
transmittance was calculated using absorption coefficients from the HITRAN database [9] as
input into the Line By Line Radiative Transfer Model (LBLRTM v. 10.3 [10]) for a 1 meter
path, a temperature of 283 K, and 100% relative humidity. The transmittance and the
responsivity both vary slowly with wavenumber for much of the spectrum, but drop
precipitously at the center of the strong CO2 band at 667 cm1 and at the centers of strong
OLQHV LQ WKH Ȟ2 band of water vapor (1300 to 1900 cm1; circles). As discussed below,
wavenumbers where
r
IDOOV EHORZ WKH WKUHVKROG ı
r
(also shown in the figure) should be
rejected as corresponding to calibrated radiances having errors that may be arbitrarily large.
Since
r
is proportional to
tAI
,
r
will decrease with
tAI
as the humidity increases. Thus we
expect to identify more wavenumbers for rejection in warmer, wetter ambient conditions.
2.2 Variance and bias of cal ibrated spec tra
Here we briefly review the theoretical development of RNW, leading to expressions for the
YDULDQFHDQGELDVLQD FDOLEUDWHGUDGLDQFHH[SUHVVHGLQWHUPVRIı
r
/
r
, which are valid for any
noise level. As shown in RNW, error in the calibration radiance is
 
,
Lschcrcs
sf f L L fL L
H
(6)
where
fs
,
fc
, and
fr
are given by replacing the subscript
x
with subscripts
s
,
c
, or
r
in
/
Re ,
1( /)
x
x
r
er
f
er
ªº
{«»
¬¼
(7)
and where
ec
{
nc
/(
Lh
-
Lc
) and
es
{ (
ns
/
s
) /(
Lh
-
Lc
), by analogy with
er
.
#138323 - $15.00 USD
Received 18 Nov 2010; revised 25 Feb 2011; accepted 8 Mar 2011; published 16 Mar 2011
(C) 2011 OSA
28 March 2011 / Vol. 19, No. 7 / OPTICS EXPRESS 5933
Fig. 1. Estimate of system responsivity (||
r
m
||) and the standard deviation of
r
m
ı
r
) for an
Atmospheric Emitted Radiance Interferometer (AERI), and an estimate of the transmittance of
air through the instrument (
tAI
). Circles indicate points with large errors.
Furthermore, the factor
s
before
fs
in Eq. (6) is defined such that
ns
/
s
has the same standard
deviation as
nh
(as noted in RNW, we have assumed that the standard deviations of
nh
and
nc
are the same, but that the standard deviation of
ns
may differ due to acquiring raw spectra with
different numbers of spectral averages, or coadditions).
In the low-noise limit that
er
/
r
<< 1, <İ
L
> = 0, and thus the variance (<İ
L
2>-<İ
L
>2) is <İ
L
2>.
Sromovsky shows that the variance in the low-noise limit is (in the notation of RNW)
22
22 2
2
22 2 .
sch
n n n
hscs
L
hchc
L L L L
L L L L
r r r
VV V
H
§· §·
§· §·

| 
¨¸ ¨¸
¨¸ ¨¸
¨¸ ¨¸

©¹ ©¹
©¹ ©¹
(8)
$PRUHJHQHUDOH[SUHVVLRQIRUİL
2> is derived in RNW, given as
 
      
22
222 2 2 2,
Lschcrcscr h c c s
s f f L L fL L f f L L L L
H

(9)
Reviewing Eq. (7), we see that the
f
2 terms in Eq. (9) approach infinity as
er
/
r
approaches
 $OWKRXJK WKH YDULDQFH LV WKXV IRUPDOO\L QILQLWH IRUı
r
/
r
<< 1, |
er
/
r
| is unlikely to ever
approach values as large as one, and the variance can be approximated as finite [Eq. (8)].
+RZHYHU DV ı
r
/
r
gets larger, the probability of infinite
f
2 increases; this corresponds to a
probability of arbitrarily large errors in calibrated radiances.
Outside the low-noise limit, RNW find that <İ
L
> z 0, but rather,
 
0.5 .
sL r h c
LfL L
H
(10)
In RNW, <
fr
! LV FDOFXODWHG QXPHULFDOO\ DQG IRXQG WR YDU\ IURP WR  DV ı
r
/
r
increases
from 0 to infinity (see Fig. 2 of RNW). Figure 3 of RNW shows that averages of calibrated
radiances approach 0.5(
Lh
+
Lc
) outside the instrument bandwidth and at specific points in-
EDQGZKHUHı
r
/
r
is very large.
In RNW, numerical calculations were used to determine that the threshold ´
r
/
r
d 0.3
corresponds to a bias [d ~105(
Lh
+
Lc
)] that is negligible compared to most error budgets, and
a variance for 99.999% of measurements that is within 20% of the variance in the low-noise
approximation. This threshold was used to check the bandwidth of an AERI and a satellite-
borne instrument.
#138323 - $15.00 USD
Received 18 Nov 2010; revised 25 Feb 2011; accepted 8 Mar 2011; published 16 Mar 2011
(C) 2011 OSA
28 March 2011 / Vol. 19, No. 7 / OPTICS EXPRESS 5934
The opposite condition, ´
r
/
r
> 0.3, can be used to identify calibrated radiances at
wavenumbers in-EDQG ZKHUH DEVRUSWLRQ E\ WUDFH JDVHV ZLWKLQ WKH LQVWUXPHQW¶V RSWLFDO SDWK
makes accurate calibration impossible. The remainder of this paper focuses on these in-band
wavenumbers for specific instruments.
3. Applications
In this section, we show that the responsivity criterion can accurately identify low-
responsivity wavenumbers. To put the responsivity criterion into context, we first discuss a
PHWKRG WHUPHG WKH ³UDWLR FULWHULRQ´ FXUUHQWO\ LQFOXGHG DV TXDOity-control in the standard
AERI processing [7], whose purpose is to identify large spikes in calibrated radiances (that
lead to ringing when Fourier transforms are performed in further processing). [ 11] Here, we
apply both criteria to measurements from two AERI instruments, one with low noise and one
with relatively high noise. We also provide a method for avoiding biases that can occur when
corrected radiances are averaged. We then show how Fourier transforms can cause errors to
ring into neighboring wavenumbers, making it important to replace radiances that have large
errors. Finally, we apply the responsivity criterion to a year-long field experiment at Eureka,
Canada, showing that the number of times a given wavenumber is identified correlates well
with the strength and proximity of the nearest strong line center in the absorption spectrum of
water vapor.
3.1 C ase study: low noise
Figure 2a shows the calibrated downwelling radiance measured by an AERI at Eureka,
Canada at 0357 UTC on 1 July 2008 [5]. The AERI instrument is described in general by
Knuteson et al. [12]. This AERI has a standard detector and is sensitive from about 500 to
1800 cm1, exhibiting relatively low noise [13]. Absorption features due to CO2, O3, and H2O
are labeled in the figure. IQWKH³DWPRVSKHULFZLQGRZ´IURP WR FP1, there is little
emission from trace gases, except O3. At the center of the CO2 band (667 cm1DQGLQWKHȞ2
band of water vapor (1300 to 1900 cm1), the absorption coefficient can be strong enough that
almost all source radiation is absorbed and re-emitted within the optical path of the
instrument. A few radiances in the water-YDSRU Ȟ2 band having large errors (t10 RU) are
indicated (circles); these correspond to the extremely low responsivities indicated previously
in Fig. 1. Since the emission is known to be strong at these wavenumbers, we expect that if
accurate calibration were possible, then
Ls
would be |B(
Ta
), where B indicates the Planck
function and
Ta
is the ambient temperature. An estimate of B(
Ta
) is shown on the figure,
derived from the radiance observed between 672 and 682 cm1
, a region known to saturate
close to (but outside) the instrument. Also shown is the Planck function of the temperature of
the cold calibration blackbody, B(
Tc
), which is (approximately) the radiance emitted by the
cold calibration-source. B(
Tc
) is greater than B(
Ta
) because the cold calibration source was
warmed to slightly above-ambient temperatures.
The ratio criterion identifies wavenumbers where the real part of the ratio of uncalibrated
spectral differences, Re[(
Vs
-
Vc
)/(
Vh
-
Vc
)], falls outside a set of pre-determined bounds. The
bounds are intended to be generous enough to filter out large spikes in calibrated radiances
but not noisy data generally, and to be widely applicable geographically and with variations in
instruments [11]. However, the uncalibrated sky spectrum (
Vs
) is highly variable both
spectrally and with atmospheric conditions. Figure 2b demonstrates how the ratio criterion is
applied. A set of bounds is chosen, in this case at +1.5 and 1.5 (red dashed lines). The ratio
is calculated for each scene spectrum. The blue curve in the figure indicates the ratio for the
case study on 1 July 2008 at 3.94 UTC. Wavenumbers are identified where Re[(
Vs
-
Vc
)/(
Vh
-
Vc
)] > 1.5 or Re[(
Vs
-
Vc
)/(
Vh
-
Vc
)] < 1.5. For this case, a single point is identified above 1.5,
while many points are identified below 1.5. In current AERI processing, calibrated radiances
at these wavenumbers are then replaced with the radiance from the cold calibration source,
Bc
.
Over the course of the day, the ratio will change slightly at most wavenumbers as
Vs
changes
with atmospheric conditions and
Vc
changes with the temperature of the cold calibration-
#138323 - $15.00 USD
Received 18 Nov 2010; revised 25 Feb 2011; accepted 8 Mar 2011; published 16 Mar 2011
(C) 2011 OSA
28 March 2011 / Vol. 19, No. 7 / OPTICS EXPRESS 5935
source (
Vh
is generally constant because the hot calibration-source is generally maintained at a
constant temperature). At low-responsivity wavenumbers, the ratio will change dramatically
with noise fluctuations, causing different ratio values to fall outside the bounds and be
identified.
Fig. 2. a) The calibrated downwelling radiance (
Ls
+ İ
L
) measured with an Atmospheric
Emitted Radiance Interferometer (AERI) at Eureka, Canada on 1 July 2008 at 0357 UTC, and
the Planck functions of the temperature of the cold calibration-source, B(
Tc
), and ambient
temperature, B(
Ta
). A few radiances having significant errors are circled. b) The real part of the
ratio of uncalibrated difference spectra for the measurement shown in a). Bounds used to
identify spectral data points with large errors are shown. Dots indicate points identified with a
criteriRQLQWHUPVRIWKHUHODWLYHHUURULQWKHLQVWUXPHQWUHVSRQVLYLW\ı
r
/
r
.
Wavenumbers identified as corresponding to high radiance errors and low responsivity in
Fig. 2a (circles) are also circled in Fig. 2b. Of these wavenumbers, only one is identified by
the ratio criterion. Furthermore, most wavenumbers between 800 and 1000 cm1 are
incorrectly identified as having large errors because most ratio values fall below the lower
bound (1.5). At these wavenumbers, replacing the calibrated radiance with
Lc
would result in
errors of up to 100 RU (as evident in Fig. 2a). The ratio values fall outside the bounds not
because of radiance errors, but rather because the bounds were set for a warmer location,
where
Vs
is larger in the atmospheric window.
Vs
changes with geographic location, season,
the diurnal cycle, and weather conditions. The lower bound needs to be reduced so these
wavenumbers are not identified. Thus the ratio criterion, designed to identify large error
spikes in
Vs
, is not well-suited to identify low-responsivity wavenumbers since the bounds are
largely determined by the variability in
Vs
, rather than the responsivity.
7RLGHQWLI\ZDYHQXPEHUVXVLQJWKHUHVSRQVLYLW\FULWHULRQı
r
/
r
was calculated for this case
study. Since instrument temperatures and tKHLQVWUXPHQWSKDVHZHUHIDLUO\VWDEOHZLWKWLPHı
r
was calculated as the standard deviation of 20 sequential measurements of
r
m
taken over about
15 minutes, and
r
was estimated as ||<
r
m
!__ 7KH FULWHULRQ WKDW ı
r
/
r
> 0.3 was then used to
identify low-responsivity wavenumbers.
In contrast to the ratio criterion, the responsivity criterion identified all but one of the
circled points, in addition to 21 others (points identified using the responsivity criterion are
shown in Fig. 2b as small black dots). Visual examination confirms that at many of the
wavenumbers identified by the responsivity criterion there are noise spikes in the ratio. At a
few wavenumbers identified there do not appear to be noise spikes. This is not surprising,
since the criterion for selection is not based on knowledge that errors are large, but rather on
the likelihood that errors might be arbitrarily large. Since these wavenumbers are all near the
centers of extremely strong lines in the water-vapor absorption coefficient, the true radiance is
known to be close to the Planck function of ambient temperature [B(
Ta
) in Fig. 2a], so
replacing measured radiances with B(
Ta
) will not cause large errors.
The responsivity criterion has several advantages for identifying low-responsivity
wavenumbers. It accurately identifies low-responsivity wavenumbers if the humidity at the
#138323 - $15.00 USD
Received 18 Nov 2010; revised 25 Feb 2011; accepted 8 Mar 2011; published 16 Mar 2011
(C) 2011 OSA
28 March 2011 / Vol. 19, No. 7 / OPTICS EXPRESS 5936
surface changes, without adjusting any parameters, since there will be a corresponding
decrease in
r
. It is not affected by atmospheric conditions above the surface, since
r
does not
depend on
Vs
 )XUWKHUPRUH WKH WKUHVKROG IRU ı
r
/
r
is associated with a bias magnitude and
variance that have been quantified by RNW. Thus the threshold can be set to achieve the
desired statistics (RNW recommend a threshold of 0.3) regardless of atmospheric conditions.
Fig. 3. a) The calibrated downwelling sky radiance (
Ls
İL) measured with an Atmospheric
Emitted Radiance Interferometer (AERI) at the North Slope of Alaska at 0102 UTC on 1 Feb.
2010. Radiances appearing to have large errors are circled. b) The corresponding ratio of
uncalibrated difference spectra as well as symmetric and asymmetric bounds used in quality
control and points identified using a responsivity criterion (dots). c) The average of calibrated
radiances measured over the course of the day (<
Ls
+İL>, blue) and the average of spectra
corrected using the asymmetric bounds (green dashed). d) <
Ls
İL> (blue) and the averages of
spectra corrected using symmetric bounds (red dashed) and the responsivity criterion (dots).
3.2 C ase study: high noise
Figure 3a shows an AERI measurement made at the North Slope of Alaska (NSA)
Atmospheric Radiation Measurement (ARM) site, near Barrow, Alaska, at 0102 UTC on 1
February 2010, when a thin cloud was overhead. The NSA AERI uses a detector that is
sensitive at longer wavelengths than the AERI at Eureka (i.e. below 500 cm1), but has much
larger detector noise from 500 to 1800 cm1 [14]. A few radiances with large errors at low-
responsivity wavenumbers are circled; these are less readily apparent in this figure due to the
higher level of instrument noise than for the instrument used at Eureka. As shown in this
subsection, identification of low-responsivity wavenumbers becomes more challenging when
noise is high. In addition, it is shown that correction methods based on both the ratio criterion
and the responsivity criterion can introduce biases in averages of corrected calibrated
radiances; a simple procedure to avoid these biases is suggested.
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Received 18 Nov 2010; revised 25 Feb 2011; accepted 8 Mar 2011; published 16 Mar 2011
(C) 2011 OSA
28 March 2011 / Vol. 19, No. 7 / OPTICS EXPRESS 5937
In addition to asymmetric bounds, Fig. 3b also shows a set of symmetric bounds (at ±0.5)
developed especially for this case study. These bounds are only used to identify wavenumbers
from 1500 to 1800 cm1. Figure 3d (red dashed line) shows the average of calibrated
radiances after replacing radiances at wavenumbers where the ratio falls outside the
symmetric bounds. We see that using symmetric bounds eliminates most of the bias (the plot
of the average of uncorrected spectra falls directly on top of the red dashed line). However, at
a few wavenumbers the red dashed line differs from the average of uncorrected spectra. Thus,
even bounds that are symmetric about zero can induce biases at particular wavenumbers. To
avoid biases the bounds need to be symmetric not about zero, but about the value the ratio
would have in the absence of errors. The ratio is only expected to be zero when
Vs
=
Vc
, which
in turn occurs only near strong line centers when
Ta
|
Tc
. On this day,
Tc
lagged
Ta
and was on
average 1.2 K colder, making the ratio slightly negative near strong line centers, so that
negative errors were more likely to be identified.
Low-responsivity wavenumbers were also identified using the responsivity criterion. To
estimate
r
DQG ı
r
, the average and standard deviation of 20 sequential measurements of
r
m
were used (taken before, during, and after the spectrum of interest). For each spectrum,
rDGLDQFHVZHUHUHSODFHGDWZDYHQXPEHUVZKHUHı
r
/
r
> 0.3. For example, black dots in Fig. 3b
LQGLFDWHUDWLRYDOXHVIRUZKLFKı
r
/
r
>0.3 at 0102 UTC. As for the ratio criterion, the corrected
radiances for the full day were averaged. Slightly different wavenumbers were identified for
each spectrum so that over the course of the day 66 wavenumbers were identified at least once
(Fig. 3d, which thus has more dots than for the single case shown in Fig. 3b). Many of the
wavenumbers identified (43) fall between 1750 and 1800 cm1, where the instrument is less
sensitive. The black dots shown in Fig. 3d indicate not only the wavenumbers identified, but
also the averages of the corrected radiances. Comparing this average to the average of
uncorrected spectra, we see that correcting spectra using the responsivity criterion can also
cause biases in averaged spectra. These biases are smaller than when wavenumbers are
identified using the ratio with bounds of ±0.5 and occur at fewer wavenumbers. The bias
occurs because wavenumbers where the errors in
r
m
are negative (
nh
-
nc
< 0) are more likely to
be identified than errors that are positive, since negative errors reduce
r
m
, bringing it closer to
WKHWKUHVKROGı
r
. The bias is mitigated by averaging
r
m
to decrease noise in the estimate of
r
.
To avoid biases in averaged calibrated radiances, we suggest a simple procedure. For the
time period chosen for averaging, the lists of wavenumbers identified for each spectrum
should be combined. Radiances should then be replaced in all spectra at all wavenumbers
identified over the entire time period. A responsivity criterion is better suited than a ratio
criterion for implementing this procedure for the following reasons. If noise is large,
occasionally errors in calibrated spectra will be large enough to be identified by the ratio
criterion. However, as long as errors are random, they will average out, and replacing these
radiances is not needed. Indeed, accurate radiance-averages may be replaced with incorrect
values if the responsivity is not low (i.e. if the radiance does not originate from close to the
instrument). Thus the goal is not to identify large random errors, but rather to i dentify
wavenumbers where errors are systematic. As discussed by RNW, high noise and/or low
UHVSRQVLYLW\FDXVHV\VWHPDWLFHUURUVLQFDOLEUDWHGVSHFWUD7KXVDFULWHULRQLQWHUPVı
r
/
r
is best
suited to identify wavenumbers where averaged radiances are inaccurate.
3.3 Ringing
If radiances at low-responsivity wavenumbers are not replaced with reasonable values, further
processing may corrupt results at neighboring wavenumbers. For example, correction of the
instrument lineshape for the effects of the instrumenW¶VILQLWHILHOGRIYLHZDQGUHVDPSOLQJWKH
spectrum onto a standard wavenumber grid both involve taking the fast Fourier transform
))7 DQG LQYHUVH ))7 ,))7 FDXVLQJ HUURU WR ³ULQJ´ LQWR QHLJKERULQJ ZDYHQXPEHUV DQG
corrupting them as well. If the error is moderate, the ringing may be acceptable compared to
other errors. For example, a noise spike of 40 RU results in error due to ringing of 0.6 RU
approximately 2 wavenumbers from the spike. However, larger errors lead to unacceptable
corruption of nearby wavenumbers. Figure 4 shows the ringing due to resampling onto a
#138323 - $15.00 USD
Received 18 Nov 2010; revised 25 Feb 2011; accepted 8 Mar 2011; published 16 Mar 2011
(C) 2011 OSA
28 March 2011 / Vol. 19, No. 7 / OPTICS EXPRESS 5938
standard wavenumber grid by zero-filling (taking the IFFT, zero-padding, and taking the FFT)
for an AERI measurement at Eureka at 0648 UTC on 1 July 2008. The plot was enlarged to
show the ringing about a noise spike of 1410 RU (the noise spike falls outside the upper plot
boundary). The dashed curve shows the original radiance without resampling, the solid curve
shows the radiance after resampling if the noise spike is not corrected, and the red dots show
the radiance after resampling if the noise spike is replaced with the radiance shown at 1521
cm1. If the erroneous point is not replaced with a reasonable value, the error due to ringing is
still 4 RU as far as ±10 wavenumbers from the line center.
Fig. 4. An expanded region of a downwelling radiance spectrum centered on the strong water-
vapor absorption line at 1521.3 cm1, where an error of 1410 RU occurs (blue dashed curve;
the error is outside the plot limits). Resampling using Fourier transforms causes the error to
³ULQJ´LQWR QHLJKERULQJZDYHQXPEHUVJUHHQ VROLGOLQH XQOHVVWKHHUURU LVILUst removed (red
dots). This spectrum was obtained at Eureka, Canada at 0648 UTC on 1 July 2008.
To reduce errors due to ringing as much as possible, it is important to replace radiances
with the best estimate possible. Since the problem occurs because absorption and emission is
so strong that most radiance originates from within the instrument, we know that if air were
not present in the instrument, most radiance would originate from only a short distance away.
Thus the radiance should be close to that of a blackbody at ambient temperature (
Ta
). If the
temperature of the cold calibration source (
Tc
) is very close to
Ta
, it is convenient to replace
erroneous radiances with the (known) cold calibration-source radiance. However, the
temperature of the cold calibration source is sometimW 4es different than ambient
temperature. The temperature of the cold calibration source often lags ambient temperature
and, in cold conditions the cold source may be heated slightly to prevent the formation of frost
on its surface. For example, in one case the cold blackbody temperature and the ambient
temperature differed by 2.4K, corresponding to a radiance difference of 1 RU at 1600 cm1.
Instead of using
Tc
,
Ta
can be estimated from the brightness temperature between 672 and 682
cm1, a region known to saturate close to (but outside) the instrument.
We suggest that the responsivity criterion be used to identify low-responsivity
wavenumbers, so that erroneous radiances can be replaced.
3.4 Field season
:DYHQXPEHUV ZKHUH ı
r
/
r
> ~0.3 were identified for AERI spectra measured at Eureka,
Canada for a 21-PRQWK ILHOG VHDVRQ IURP 0DUFK  WR 'HFHPEHU  %HFDXVH ı
r
is
relatively low for this data set, it is relatively easy to identify points where
r
~0, due to the
dramatic difference VHHQ LQ ı
r
/
r
DW WKHVH SRLQWV 7KXV ERWK ı
r
and
r
were calculated very
roughly, as follows. It was assumed that the noise was low enough that
r
could be
approximated as ||
r
m
|| for each measurement (i.e. averaging sequential values of
r
m
was
unnecessary). TRHVWLPDWHı
r
, the standard deviation of
r
m
with wavenumber was calculated in
regions where the instrument is not sensitive (between 400 and 460 cm1 and between 1845
#138323 - $15.00 USD
Received 18 Nov 2010; revised 25 Feb 2011; accepted 8 Mar 2011; published 16 Mar 2011
(C) 2011 OSA
28 March 2011 / Vol. 19, No. 7 / OPTICS EXPRESS 5939
and 1920 cm1) and a linear variation across the spectrum was assumed. (A more accurate
aSSURDFKZRXOGWDNHLQWRDFFRXQWWKDWı
r
is proportional to 1/[
Lh
-
Lc
]).
The most commonly identified wavenumbers for 934,544 spectra measured using the
channel 1 detector (500-1800 cm1) are shown in Table 1 (excluding wavenumbers near the
center of the CO2 band at 667 cm1). The columns of the table show the frequency with which
the wavenumbers were identified (as a percentage of all spectra) and the corresponding
wavenumber. Also shown are line strengths and positions of the closest strong water-vapor
absorption lines. Line strengths were obtained from the HITRAN database [9]. Columns are
ordered by frequency of identification to show the correlation between frequency of
identification and line strength; this correlation supports the robustness of the criterion. For
example, wavenumbers were identified near all lines having strengths greater than 0.80 x
1019 cm1/(molecule cm2). If the humidity increased, the same wavenumbers would be
identified, but the frequency of identifications would increase, and additional wavenumbers
would be identified. As an alternative to identifying wavenumbers for every spectrum, lists of
wavenumbers such as this may be compiled and used for all spectra.
Table 1. Low -Responsivi ty Wav enumbers Identif ied fo r an Atmospher ic Emitted
Ra diance I nterf erometer (A ER I) O perating from April 2006 to De cember 2007 at
Eurek a, C anada
Na
Ȟobsb
Ȟ0c
Sd
N
Ȟobs
Ȟ0
S
N
Ȟobs
Ȟ0
S
(%)
(cm1)
(cm1)
(SU1)
(%)
(cm1)
(cm1)
(SU)
(%)
(cm1)
(cm1)
(SU)
43
1653.3
1653.267
2.42
10
1684.7
1684.835
2.99
0.3
1473.5
1473.514
1.01
41
1695.8
1695.928
2.63
8
1541.9
1542.160
1.10
0.2
1696.2
1695.928
2.63
40
1539.1
1539.061
2.07
8
1734.8
1734.650
1.31
0.1
1772.4
1772.714
1.43
39
1576.2
1576.185
2.59
8
1506.8
1507.058
1.90
0.1
1739.6
1739.839
0.90
37
1652.4
1652.400
2.41
7
1792.7
1792.659
0.97
0.1
1683.2
1683.178
0.67
37
1616.7
1616.711
2.32
6
1652.9
1653.267
2.61
0.1
1522.7
1522.686
0.71
34
1717.5
1717.405
2.09
6
1685.2
1684.835
2.99
0.1
1675.0
1675.173
0.93
31
1700.6
1700.776
0.90
5
1505.8
1505.604
1.90
0.1
1557.9
1557.609
0.58
29
1700.1
1699.934
1.77
2
1472.0
1472.051
1.06
0.1
1768.1
1768.120
0.47
29
1521.2
1521.309
0.85
2
1540.5
1540.300
1.63
0.1
1654.3
1654.511
0.88
28
1436.8
1436.818
1.04
2
1554.5
1554.352
1.29
0.04
1771.5
1771.287
0.48
25
1646.1
1645.969
1.60
2
1734.3
1734.650
1.31
0.04
1663.0
1662.809
0.79
24
1559.8
1560.257
2.17
1
1772.9
1772.714
1.43
0.04
1623.4
1623.559
0.83
23
1635.5
1635.652
1.79
1
1751.2
1751.423
0.86
0.03
1706.4
1706.349
0.50
21
1560.3
1560.257
2.17
0.9
1496.1
1496.249
1.45
0.03
1540.0
1540.300
1.63
18
1675.5
1675.515
0.54
0.8
1558.8
1558.531
2.33
0.03
1636.0
1635.652
1.79
17
1456.6
1456.887
1.69
0.7
1761.8
1761.828
0.76
0.02
1715.0
1715.155
0.69
14
1419.5
1419.508
1.02
0.7
1701.1
1700.776
0.90
0.02
1718.9
1718.612
0.52
13
1733.4
1733.391
1.42
0.6
1569.9
1569.789
0.98
0.01
1718.4
1718.612
0.52
11
1507.2
1507.058
1.90
0.5
1699.6
1669.393
0.96
0.01
1464.8
1464.905
0.88
11
1669.2
1669.393
0.96
0.5
1557.4
1557.486
0.51
0.01
1740.1
1739.839
0.90
10
1558.3
1558.531
2.33
0.3
1635.0
1634.967
0.59
0.01
1704.4
1704.453
0.50
10
1457.1
1456.887
1.69
0.3
1684.2
1683.984
0.47
0.01
1545.3
1545.157
0.79
aIdentifications, as percent of all spectra measured. b7KH LGHQWLILHGZDYHQXPEHUȞobs). cPosition of the strongest line
within 0.48 cm1. dCorresponding line strength. eSU = 1019 cm1/(molecule cm2) at 296K.
4. Conclusions
In a previous paper (RNW [4]), an equation for the variance of the calibrated radiance from
FTIR spectrometers, known for low errors, was generalized for all noise levels. In addition,
51:UHYHDOHGDELDVLQVSHFWUDDW KLJKQRLVHOHYHOV$FULWHULRQ ı
r
/
r
 ZKHUHı
r
/
r
is the
relative error in the system responsivity) was presented for identifying accurately calibrated
radiances. This criterion was used to identify the bandwidth of two representative FTIR
spectrometers, a ground-based Atmospheric Emitted Radiance Interferometer (AERI) and the
satellite-mounted Interferometric Monitor for Greenhouse Gases (IMG).
In this paper we have focused on in-band spectral regions, in an effort to identify
wavenumbers that, despite lying inside the nominal band of the instrument, nevertheless
cannot be accurately calibrated. Two representative FTIR spectrometers are considered, both
ground-based Atmospheric Emitted Radiance Interferometers (AERIs); these spectrometers
#138323 - $15.00 USD
Received 18 Nov 2010; revised 25 Feb 2011; accepted 8 Mar 2011; published 16 Mar 2011
(C) 2011 OSA
28 March 2011 / Vol. 19, No. 7 / OPTICS EXPRESS 5940
differ considerably in the sensitivity of their detectors and therefore afford different
opportunities to test the utility of the responsivity-based method developed by RNW.
Although most of the in-EDQGVSHFWUDOUHJLRQLVFKDUDFWHUL]HGE\KLJKUHVSRQVLYLW\KHQFHı
r
/
r
< 0.3), we find numerous wavenumbers ZKHUH WKHUHVSRQVLYLW\LV ORZ ı
r
/
r
> 0.3). We find
that the predominant reason for this poor in-band responsivity is absorption by trace gases
within the spectrometer, especially water vapor. Errors can be quite large (thousands of RU)
and further processing can cause them to corrupt neighboring wavenumbers (for example,
errors of 20 to 40 RU result in errors of 0.3 to 0.6 RU approximately 2 wavenumbers away).
A comprehensive comparison of the responsivity-based criterion to currently employed
³UDWLRFULWHULRQ´PHWKRGVIRULGHQWLI\LQJXQUHVSRQVLYHZDYHQXPEHUVLVDOVRXQGHUWDNHQ7KH
two methods have different goals: the ratio criterion identifies wavenumbers where errors
happen
WR EH ODUJH LQ D JLYHQ VSHFWUXP %\ FRQWUDVW VLQFH ı
r
is a statistical quantity, it
identifies wavenumbers where errors are
likely
to be large. The responsivity criterion is found
to be well-VXLWHGIRULGHQWLI\LQJORZUHVSRQVLYLW\ZDYHQXPEHUVVLQFHı
r
/
r
does not depend on
the uncalibrated sky spectrum,
Vs
, but only on instrument characteristics, and should apply in
all atmospheric conditions, including the polar regions and lower latitudes. Furthermore, since
ı
r
LV D VWDWLVWLFDO TXDQWLW\ WKH YDOXH RI ı
r
/
r
corresponds to known error characteristics (i.e.
variance and bias, as given in RNW).
We show that both the responsivity criterion and the ratio criterion can induce biases when
many corrected spectra are averaged. Biases can be large when asymmetric bounds are used
in the ratio method and are expected to be small for bounds that are symmetric and when the
responsivity criterion is used. Because such biases occur at identified wavenumbers, a
reasonable strategy for avoiding a bias is to combine the lists of identified wavenumbers for
the entire averaging period and replace radiances at these wavenumbers in all spectra.
The responsivity criterion is applied to 21 months of AERI spectra at Eureka, Canada, and
identified wavenumbers are found to correlate with strong line centers of water vapor. The
responsivity criterion is tunable; making the threshold less stringent (increasing it) generally
identifies wavenumbers near increasingly weaker lines, while making it more stringent
(decreasing it) generally identifies wavenumbers near only the strongest lines.
Acknowledgments
We are grateful to the Canadian Network for the Detection of Atmospheric Change
(CANDAC) technicians for operating the AERI instrument at Eureka, Canada from March
2006 through April 2009 and to the Environment Canada Weather Station at Eureka,
Nunavut, for their suppRUW :H WKDQN WKH 'HSDUWPHQW RI (QHUJ\¶V $WPRVSKHULF 5DGLDWLRQ
Measurement (ARM) program for providing the data for the AERI instrument operating at the
North Slope of Alaska (NSA) and Dave Turner for help obtaining the NSA AERI data. We
thank Dave Turner and Robert Knuteson for helpful discussions. Support for this research
came from the National Aeronautics and Space Administration (NASA) Research
Opportunities in Space and Earth Science program (Contract NNX08AF79G), from the
National Science Foundation (NSF) Idaho Experimental Program to Stimulate Competitive
Research (EPSCoR) and by the NSF under award number EPS-0814387. SPN acknowledges
support from the University of Puget Sound.
#138323 - $15.00 USD
Received 18 Nov 2010; revised 25 Feb 2011; accepted 8 Mar 2011; published 16 Mar 2011
(C) 2011 OSA
28 March 2011 / Vol. 19, No. 7 / OPTICS EXPRESS 5941
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