Applications of Kalman filtering to realtime trace gas concentration measurements.
ABSTRACT A Kalman filtering technique is applied to the simultaneous detection of NH3 and CO2 with a diodelaserbased sensor operating at 1.53 micrometers. This technique is developed for improving the sensitivity and precision of trace gas concentration levels based on direct overtone laser absorption spectroscopy in the presence of various sensor noise sources. Filter performance is demonstrated to be adaptive to realtime noise and data statistics. Additionally, filter operation is successfully performed with dynamic ranges differing by three orders of magnitude. Details of Kalman filter theory applied to the acquired spectroscopic data are discussed. The effectiveness of this technique is evaluated by performing NH3 and CO2 concentration measurements and utilizing it to monitor varying ammonia and carbon dioxide levels in a bioreactor for water reprocessing, located at the NASAJohnson Space Center. Results indicate a sensitivity enhancement of six times, in terms of improved minimum detectable absorption by the gas sensor.

Article: Pulsed airborne lidar measurements of atmospheric optical depth using the Oxygen Aband at 765 nm.
Haris Riris, Michael Rodriguez, Graham R Allan, William Hasselbrack, Jianping Mao, Mark Stephen, James Abshire[Show abstract] [Hide abstract]
ABSTRACT: We report on an airborne demonstration of atmospheric oxygen optical depth measurements with an IPDA lidar using a fiberbased laser system and a photon counting detector. Accurate knowledge of atmospheric temperature and pressure is required for NASA's Active Sensing of CO<sub>2</sub> Emissions over Nights, Days, and Seasons (ASCENDS) space mission, and climate modeling studies. The lidar uses a doubled erbiumdoped fiber amplifier and single photoncounting detector to measure oxygen absorption at 765 nm. Our results show good agreement between the experimentally derived differential optical depth measurements with the theoretical predictions for aircraft altitudes from 3 to 13 km.Applied Optics 09/2013; 52(25):63696382. · 1.69 Impact Factor  Antonio Castrillo, Hemanth Dinesan, Giovanni Casa, Gianluca Galzerano, Paolo Laporta, Livio Gianfrani[Show abstract] [Hide abstract]
ABSTRACT: We propose a method for the measurements of the 17O/16O isotope amount ratio in water, based upon the use of a pair of offsetfrequency locked extendedcavity diode lasers at 1.39 μm. This method enables one to acquire absorption spectra with an extremely high fidelity, exploiting the highly accurate, absolute, and repeatable frequency axis. One of the two lasers, namely the socalled slave laser, is continuously scanned across a pair of H216O and H217O lines at 7183.5 cm−1 and it interacts with a water vapor sample inside a multiple reflections cell, thus producing absorption spectra with a signaltonoise ratio of the order of 4000 for a detection bandwidth of 1 kHz. The determination of the isotope amount ratio is performed through a careful analysis of the acquired spectra, by using semiclassical line profiles. In this respect, the influence of the choice of the line shape model is investigated. The experimental reproducibility of the spectrometer has been carefully assessed by means of an Allan variance analysis. Finally, the application of the Kalman filtering technique has shown that a precision of 0.6‰ can be achieved, from repeated spectral acquisitions over a time span of 6000 s.Physical Review A 11/2012; 86(5). · 3.04 Impact Factor  SourceAvailable from: Yingjian WangTao Wu, Weidong Chen, Eric Fertein, Pascal Masselin, Xiaoming Gao, Weijun Zhang, Yingjian Wang, Johannes Koeth, Daniela Brückner, Xingdao He[Show abstract] [Hide abstract]
ABSTRACT: A compact isotope ratio laser spectrometry (IRLS) instrument was developed for simultaneous measurements of the D/H, 18O/16O and 17O/16O isotope ratios in water by laser absorption spectroscopy at 2.73 μm. Special attention is paid to the spectral data processing and implementation of a Kalman adaptive filtering to improve the measurement precision. Reduction of up to 3fold in standard deviation in isotope ratio determination was obtained by the use of a Fourier filtering to remove undulation structure from spectrum baseline. Application of Kalman filtering enables isotope ratio measurement at 1 s time intervals with a precision (<1‰) better than that obtained by conventional 30 s averaging, while maintaining a fast system response. The implementation of the filter is described in detail and its effects on the accuracy and the precision of the isotope ratio measurements are investigated.Sensors (Basel, Switzerland). 01/2014; 14(5):902745.
Page 1
DOI:10.1007/s003400100751
Appl. Phys. B 74, 85–93 (2002)
Lasers and Optics
AppliedPhysicsB
d.p.leleux1,✉
r.claps1
w.chen1
f.k.tittel1
t.l.harman2
Applications of Kalman filtering to realtime
trace gas concentration measurements
1Rice Quantum Institute, Rice University, Houston, TX 772511892, USA
2Computer Engineering Department, University of Houston Clear Lake, Houston, TX 770581098, USA
Received: 13 July 2001/Revisedversion: 11 October 2001
Published online: 29 November 2001• © SpringerVerlag 2001
ABSTRACT A Kalman filtering technique is applied to the sim
ultaneous detection of NH3and CO2with a diodelaserbased
sensoroperatingat1.53 µm.Thistechniqueisdevelopedforim
proving the sensitivity and precision of trace gas concentration
levels based on direct overtone laser absorption spectroscopy in
thepresenceof various sensor noise sources. Filter performance
is demonstrated to beadaptive to realtimenoise and data statis
tics. Additionally, filter operation is successfully performed
with dynamic ranges differing by three orders of magnitude.
Details of Kalman filter theory applied to the acquired spectro
scopic data are discussed. The effectiveness of this technique is
evaluated by performing NH3and CO2concentration measure
ments and utilizing it to monitor varying ammonia and carbon
dioxide levels in a bioreactor for water reprocessing, located at
the NASA–Johnson Space Center. Results indicate a sensitiv
ity enhancement of six times, in terms of improved minimum
detectable absorption by the gas sensor.
PACS 42.68.Ca; 42.62.Fi; 95.75.z
1Introduction
The primary focus of this work is the development
of an adaptive filtering technique, known as Kalman filtering,
to the sensitive, selective and realtime detection of NH3and
CO2usingatunablediodelaserbased tracegassensor[1–3].
The motivation for the application of this technique is the
fact that these trace gas sensors have inherent laser, electri
calandopticalnoiseassociatedwiththemthatmanifestsitself
as shortterm variations in gas concentration measurements.
Examplesoftheserealtimenoisesourcesincludethermalex
pansionorcontractionofopticalcomponents,laserfrequency
drift and etalons. These fluctuations in concentration occur
with every measurement, compared to true gas concentration
variations thatresultfromchanges inthesampleenvironment
which evolve over the course of a series of measurements
as reported in Sect. 4. This noise defines the minimum de
tection limit of the gas sensor, which can be characterized
by calculating an Allan variance for the specific optical sen
sor configuration [4]. The Allan variance defines the optimal
✉ Fax: +1713/3485686, Email: leleux@rice.edu
averagingtimeforthegasanalyzer,whichminimizestheshot
toshot variability of concentration measurements. Once this
averaging time has been optimized, further detection sensi
tivity improvements can be achieved through the use of the
Kalman filtering technique[5].
While numerous filtering techniques can be applied to
postprocessing of gas concentration measurements, only
afewsuccessfultechniqueshavebeenidentifiedthatallowef
ficient online filtering of concentration measurements. One
direct technique, which can be applied in real time, is simple
averaging of the previous n measurements. The advantages
of the Kalman filtering technique over simple averaging are
threefold:
– theKalmanfilterincorporatesallavailablemeasurements,
regardless of their precision, and is not limited to a nar
row window of n measurements to estimate the actual gas
concentration;
– the Kalman filter (although the equations are more com
plex than an averaging technique) is computationally effi
cient because it is recursive and only requires two sets of
information instead of n to be transferred from measure
mentto measurementforfilter calculations; and
– the Kalman filter is adaptive and can adjust to changes in
signalstatistics anddynamic rangeduringoperation.
AcomparisonbetweenaKalmanfiltersolutionandamov
ing average technique is presented in Sect. 4.3. A Kalman fil
tercombinesallavailablemeasurementdata,pluspriorknow
ledge about the system and measuring devices, to produce an
estimate of theactual concentration suchthattheerroris min
imized statistically.Theuseofthisalgorithmenablesthefilter
to solve the Gauss’ leastsquares technique in real time with
little or nodelay in processingtime.
The filtering algorithm was first developed in 1960 by
Kalman [5] and applied to aerospace navigation problems.
In 1993, Werle et al. employed adaptive filtering techniques
to tunable diodelaser absorption spectroscopy (TDLAS) in
the infrared molecular “fingerprint”region [6]. Subsequently,
Kalmanfilteringappliedtotunablediodelaserbasedgassen
sors was studied by Riris et al. in 1994 [7] and 1996 [8]. This
paper describes the effective approach of applying Kalman
filtering to realtime trace gas concentration measurements
using diodelaser overtonespectroscopy.
Page 2
86Applied Physics B – Lasers and Optics
InSect. 2,thetheoretical considerationsofaKalmanfilter
willbedescribed.Sect.3willexplainsensorconfigurationand
discussdata acquisition and processing. In Sect. 4, the results
of the experiments will be analyzed and a comparison will be
made between the Kalman filter and a moving average tech
nique.The resultswill besummarized in Sect. 5.
2
2.1
Theoretical considerations
Kalman filter model for trace gas concentration
measurements
In this section, we report on the application of
Kalmanfilteringtoaddresstheproblemofdetermininganop
timal estimate of the true gas concentration, xk, of a sample
gas with a nearinfrared diodelaserbased sensor in the pres
ence of two sources of uncertainty (measurement noise in
troduced by the sensor, vk, and true concentration variability,
wk). This scalar optimal estimation problem can be modeled
bythelinear stochastic differenceequation [9,10]:
xk+1= xk+wk,
withmeasurements fromthetracegassensor, zk,modeled as
(1)
zk= xk+νk,
where the expected value, E[wk] = E[vk] = 0, and wkand vk
are uncorrelated random variables with white noise variance
of σ2
valid for trace gas concentration measurements [9], since the
noise variances are not correlated in time, i.e. knowing the
noise variance at time x does not aid in predicting what its
valuewillbeatanyothertime.Thevarianceσ2
true concentration variability while σ2
urementnoiseintroduced by thesensor.
ˆ x−
for a step k, with knowledge of the gas concentration prior to
step k,and ˆ xkto be thea posterioriconcentration estimate for
step k after the measurement value zkis incorporated. Note
that a hat above a variable such as ˆ xkindicates an estimated
or predicted quantity, and a superscript negative sign above
avariable such as ˆ x−
orianda posterioriestimate errorscan bedefinedas:
(2)
wand σ2
v, respectively. The assumption of “whiteness” is
wrepresentsthe
vrepresents the meas
kcanbedefinedtobetheaprioriconcentration estimate
kindicates an a prioriquantity. Thea pri
e−
k≡ xk− ˆ x−
and
k,
(3)
ek≡ xk− ˆ xk.
The a priori estimate error variance is then given by the ex
pected value
(4)
P−
k= E?e−
and the a posteriori estimate error variance is given by the
expected value
ke−
k
?,
(5)
Pk= E [ekek] .
InderivingtheKalman filterequations,theobjectiveistofind
an equation that computes an a posteriori concentration es
timate ˆ xkas a linear combination of an a priori estimate ˆ x−
(6)
k
and a weighted difference between an actual concentration
measurement zkand a measurement prediction ˆ x−
below:
kas shown
ˆ xk= ˆ x−
The difference (zk− ˆ x−
residual. The residual reflects the discrepancy between the
predicted measurement ˆ x−
A residual of zero means that the predicted and actual meas
urements are in complete agreement. The value of Kk in
(7) is called the Kalman gain and is chosen to minimize
the a posteriori error variance from (6). This minimiza
tion procedure can be accomplished by substituting (7) into
(4), which defines ek. Next, this result is substituted in (6),
followed by performing the indicated expectations, taking
the derivative of the result with respect to Kk, and set
ting this value equal to zero. Solving for Kk yields the
definition of the Kalman gain, which minimizes (6), given
by:
k+ Kk
?zk− ˆ x−
k
?.
(7)
k) in (7) is called the measurement
kand the actual measurement zk.
Kk= P−
k
?P−
k+σ2
v
?−1=
P−
k
k+σ2
?P−
v
? .
(8)
From (8), it can be seen that the gain Kkweights the residual
(zk− ˆ x−
approaches zero. In practical terms, this can be interpreted as
a sensor that is reliable and provides measurements with low
variability. Specifically, from(8),
k) more in (7) as the measurement error variance σ2
v
lim
σ2
v→0Kk= 1.
However,asthemeasurement errorvariance σ2
gain Kkweights the residual (zk− ˆ x−
tical terms, this can be interpreted as a sensor that provides
measurements with significant variability. In the limit, as σ2
approaches infinity, from(8),
vincreases,the
k) less. Again, in prac
v
lim
σ2
v→∞Kk= 0.
Another way of interpreting the weighting by Kk is that
as the measurement error variance σ2
actual measurement zk is “trusted” more because the sen
sor becomes more reliable and the predicted measurement
ˆ x−
ance σ2
because the sensor is considered less reliable in terms of
variability, while the predicted measurement ˆ x−
more.
By substituting (8) into (7) and using (3) through (6) to
solve for Pk, the solution provides the a posteriori estimate
error varianceas:
vapproaches zero, the
kis trusted less. However, as the measurement error vari
vincreases, the actual measurement zkis trusted less
kis trusted
Pk= (1− Kk) P−
With (8) and (9), the complete a posteriori estimate and error
variance are defined as a function of the weighting factor Kk
that is conditioned based on both the previous uncertainty of
theestimate and theuncertainty ofthemeasuring device.
k.
(9)
Page 3
LELEUXet al. Applications of Kalman filtering to realtime trace gas concentration measurements87
2.2
The Kalman filteralgorithm
The Kalman filter estimates the gas concentration
by using a form of feedback control. The filter estimates the
concentrationatsomegiventimeandthenobtainsfeedbackin
the form of measurements possessing a noise component. As
such,theKalmanfilterequationsfallintotwogroups:timeup
date equations and measurement update equations. The time
update equations are responsible for projecting forward in
timecurrentconcentration and error varianceestimates to ob
tainaprioriestimatesforthenexttimestep.Table 1showsthe
equationsused inthetime updateportion ofthealgorithm.
If the gas concentration is determined by a process that
can be modeled directly or inferred (e.g. from temperature),
this information can be used to obtain a better predictive ca
pability instead of assuming that the next concentration will
be equal to the previous concentration plus some noise, as in
(1). If a control input can be modeled by a variable, uk(e.g.
thegasconcentrationisdeterminedbythetemperatureinare
action chamber at a chemical processing plant), then the time
update (10) can be modified to include this information, as
shownin (12).
Themeasurementupdateequationsareresponsibleforthe
feedback, i.e. for incorporating a new measurement into the
a priori concentration estimate to obtain an improved a pos
terioriestimate, and aregivenin Table 2.
The time update equations act then as predictor equa
tions, while the measurement update equations act as correc
tor equations. The final estimation algorithm is a predictor–
corrector algorithm for solving the leastsquares method in
realtimeas shownin Fig. 1.
The first step in the sequence involves selecting initial
a priori values for the concentration estimate, ˆ x−
error variance, P−
critical, since the filter will converge to an appropriate value.
However, they should be chosen to be within the dynamic
range of the expected gas concentrations and errors to speed
convergence. The next step involves taking a measurement
update from the sensor by first computing the Kalman gain,
Kk, using (13). Subsequently, the sensor performs a concen
trationmeasurementtoobtainzk.Thisisfollowedbycalculat
ing an a posteriori concentration estimate using (14), and the
final step is to obtain an a posteriori error variance estimate
from(15).
Once a posteriori estimates of the concentration and vari
ance are obtained, a time update is processed using (10) and
(11), or (10) and (12) if a control input is modeled. This step
k, and the
k. The selection of these quantities is not
ˆ x−
P−
ˆ x−
TABLE 1
k+1= ˆ xk
k+1= Pk+σ2
k+1= ˆ xk+uk
(10)
(11)
(12)
w
Kalman filter time update equations
Kk= P−
ˆ xk= ˆ x−
Pk= (1−Kk)P−
TABLE 2
k(P−
k+Kk(zk− ˆ x−
k+σ2
v)−1
(13)
(14)
(15)
k)
k
Kalman filter measurement update equations
FIGURE 1
rent concentration estimate ahead in time. The measurement update adjusts
the projected concentration estimate by an actual measurement at that time.
Note: σ2
ware constants
Complete Kalman filter cycle. The time update projects the cur
vand σ2
projects the concentration and variance estimates from time
step k to stepk+1.
After each time and measurement update pair, the pro
cess is repeated with the previous a posteriori estimates used
to project or predict the new a priori estimates. This recur
sive nature is one of the appealing features of the Kalman
filter, because it recursively conditions the current concentra
tion estimate on all of the past measurements. Furthermore,
the selection of σ2
tuning the Kalman filter to provide the optimum sensor per
formance, asdiscussedin Sect. 4.
wand σ2
vis one of the central features in
3
3.1
Experimental details
Diodelaserbased gas sensor
The diodelaserbased sensor shown in Fig. 2 for
gas detection and quantification using vibrational overtone
spectroscopy at 1.53µm was used to investigate the Kalman
FIGURE 2
NH3and CO2concentration measurements at 1.53 µm
Diodelaserbased trace gas sensor configuration for continuous
Page 4
88 Applied Physics B – Lasers and Optics
Linestrength Frequency
cm−1
Wavelength
nm
Ref.
cm−1/(molecule/cm2) cm−2atm−1(@ 296 K)
A
B
C
2.33×10−21
1.24×10−21
5.19×10−25
0.0578
0.0307
1.29×10−5
6528.76
6528.89
6528.895
1531.68
1531.65
1532.65
[11]
[11]
[12]
TABLE 3
sorption lines used in this work
NH3and CO2overtone ab
filtering technique. This sensor has been described in detail
previously [1–3]. An ammonia (or any other gas) concentra
tion measurement for narrowlinewidth radiation sources is
obtainedbyusingtheBeer–Lambertabsorptionlaw,whichre
latestheintensityoflightenteringintoanabsorptionmedium,
I0,to thetransmitted intensity, I, asfollows:
I
I0
whereSνisthetemperaturedependentabsorptionlinestrength
for a specific transition denoted by ν[cm−2atm−1], φνis the
line shape function (cm), χjis the fractional concentration of
the species j, P is pressure (atm), and l is the optical path
length throughthemedium(cm).
Table 3 depicts the three NH3and CO2absorption lines
thatareusedinthiswork.LinesAandBarethetwoammonia
lines used: A is an isolated line useful for measuring ammo
nia concentrations while line B overlaps with line C, which
isthe(00001)← (30011), R(36),overtonetransition ofCO2.
These three lines are accessible using a 1.53µm telecommu
nications diodelaser.
The sensor consists of three main components: a single
frequency, fiber pigtailed, tunable, distributed feedback
(DFB) diodelaser (NTTNEL Electronics), a multipass ab
sorption cell and a dualbeam autobalanced InGaAsdetector
(Nirvana 2017, New Focus, Inc). The laser diode is driven
by a compact current supply (MPL250, Wavelength Elec
tronics) with ripple noise < 1 µA, so that the frequency fluc
tuations of the laser line due to current noise are negligible
(< 1 MHz). The current supply is scanned at 20Hz about an
averagecurrentof67mAwithasawtoothrampfunction.The
laser temperature was controlled to within 0.003◦C, close to
32◦C,byacurrentmodule(HTC1500,WavelengthElectron
ics). The scan range of the laser under these conditions was
0.3cm−1, which allowed all the spectral lines of interest to be
accessedon everyscan.
The fiberpigtailed DFB laser diode delivers 15mW at
1531.7nm with a specified linewidth of < 10MHz. The fiber
was fusionspliced to a 70/30% directional coupler with an
insertion loss of less than 2%. The 70% power arm of the di
rectional coupler was sent to the multipass cell, using a lens
( f = 7mm, 0.5NA) mounted in a precision holder with 5
degrees of freedom (Optics For Research) to set the beam
waist at the midpoint of the multipass cell. The 30% power
arm was used as the reference beam for the balancing detec
tor. Such an integrated laser and optical fiber configuration
delivers 10mW of laser light at the input of the multipass
cell. The output power of the beam obtained from the cell
after 182 passes was 200µW, resulting in a throughput ef
ficiency of 2.0% (a factor of 10 better than for the cell de
scribed in [1]). The output beam from the cell is focused on
the signal and reference photodiodes of the dual beam de
tector by a goldcoated, parabolic mirror of 2.54cm diam
= exp?−SνφνPχjl?
(16)
eter ( fnumber= 2)and alens ( f = 7mm, 0.5NA)mounted
on a precision holder, respectively. Since the reference beam
power (Pref) was much greater than the power of the sig
nal beam coming from the cell (Psignal), it was attenuated
using a variable fiber attenuator. For optimum performance
of the autobalanced detector, the reference power was set as
Pref= 2.2× Psignal, at the center frequency of the laser scan.
This in turn reduces the attenuation level required at the ref
erence input of the detector to ∼ 3dB, a range where the
operation ofthefiberattenuator isreliableandfreeoffringing
effects.
All elements, except the input and output of the multi
pass cell, are coupled to optical fibers in order to make the
system suitable for field applications. A twostage, micro
mechanical diaphragm pump mounted next to the multipass
cell provides a continuous flow of sample gas at a rate that
is controlled by two needle valves at the input and output
of the multipass cell, which has a volume of 0.3liters. For
these experiments, an optimal flow rate of about 100sccm
allows for a fast response of the instrument to true con
centration changes, while keeping the concentration rates
of the gas sample stable. The multipass cell was heated to
a temperature of 40◦C to minimize ammonia adsorption on
its glass walls. All the measurements presented here were
obtained with the sample gas inside the multipass cell at
a fixed pressure of 100Torr. For the measurements, a cylin
der containing certified 99.99% pure CO2(Scott Specialty
Gases, Inc.) was connected in parallel with an ammonia gas
dilution system [1]. The ammonia sample was obtained by
diluting a certified, 100ppm NH3 mixture in N2 (Mathe
son), with a pure N2 sample. The fraction of NH3 intro
duced in the mixture was controlled to within ±0.2ppm
by using a mass flow controller [MKS Instruments, Inc.],
whereas the CO2 concentration was regulated to within
±1000ppm by adjusting the outlet pressure of the cylinder
regulator.
Other modifications were introduced in order to improve
the operation of the instrument that was reported in [1–3]:
A pure ammonia cell (1 Torr) was used for realtime fre
quency calibration. This cell was attached to a motorized
translational stage (New Focus, Model No. 8892) that inserts
the reference cell periodically into the optical path of the sig
nal beam. This is accomplished using a softwaregenerated
TTLpulsefromthedataacquisition routine.Toautomatically
account for operational fluctuations in the gas handling sys
tem, pressure and temperature are measured in real time and
their values are updated in the data analysis routine for each
concentration measurement. By implementing these modifi
cations, the system is now a standalone instrument that can
be used for online remote operation over extended periods
of time, monitoring the concentration of trace gases such as
NH3 and CO2. This is particularly useful for demonstrat
ing the capabilities of the Kalman filtering technique, which
Page 5
LELEUXet al. Applications of Kalman filtering to realtime trace gas concentration measurements 89
provides an optimal estimate of the true gas concentration
continuously.
3.2
Data processing: acquisition and analysis
A laptop PC running LabView 5.0 software was
usedfordataacquisitionandprocessing.Withthephotodetec
tor operating in autobalance mode, 500 spectral scans were
averaged for each single concentration measurement. The
total data collection time, averaging and processing to obtain
a single concentration measurement was less than 30s. For
convenience,thesensorwasconfiguredtotakemeasurements
every minute in order to correlate ammonia measurements to
external daily events. The capability to remotely monitor the
sensor performance and concentration data was implemented
byusing“PCAnywhere”software.
Therealtime fitting routine implemented in LabView has
been reported elsewhere [13], and only minor software mod
ifications were required for this work. Initially a direct trans
mission spectrum from the logarithmic output of the Nirvana
detector,operatinginautobalancemode,isobtainedanda3rd
order polynomial is fitted to the baseline of the absorption
scan. If a voltage, V∗(ν), is assigned to the baseline values,
and the logarithmic signal values are assigned a voltage V(ν),
then the corresponding optical transmission for the given fre
quencyvalue, T(ν),in percent, is givenby [14]:
T(ν) = 100e−A
TV∗(ν)+1
e−A
TV(ν)+1
(17)
where T is the temperature of the photodetector (taken to be
300K), and A is a constant determined by the gain in the
amplifier circuit of the detector (A = 273K/V). This proced
ure provides a balanced transmission spectrum and simultan
eously corrects for the baseline of the absorption signal. To
thisspectrum(afterdividingby100andtakingthenaturallog
arithm), an absorption lineshape function is fitted to obtain
thegas concentration as described below. Theconvenience of
this procedure is that only one detection channel is needed in
this configuration to accountfor both thesignal and reference
outputsin thephotodetector.
The fit employs a nonlinear, leastsquares Levenberg–
Marquardtalgorithm and uses a given absorption profileto fit
thedata.Sincethepressureregimeofoperation(100 Torr)lies
between the predominantly Doppler and pressurebroadened
regimes, a triple Voigt profile with fixed pressure broadening
and Doppler width was used. Typical absorption spectra ob
tained were fitted to a thirdorder polynomial and three Voigt
profiles [15,16], with a linewidth of 0.027cm−1and a pres
sure broadening contribution of 0.020cm−1. Small pressure
fluctuations in thesystemof±1Torrareaccounted forin real
time by the data collection software. Therefore, the linewidth
of the ammonia and carbon dioxide absorption peaks was ex
pected to fluctuate by less than ±0.001cm−1[17–19]. This
introduces an error for the ammonia and carbon dioxide con
centration measurements in the fitting routine of 2%. The
residual of the Voigt fit in typical 2.8ppm NH3spectra (peak
absorption of 0.14%) yields an uncertainty level for a single
measurement (with 500 averages) of ±0.01% – absorption
– (note that 100×ln(I/I0) is equivalent to % absorption for
small absorptions). The rms uncertainty level is then 0.014%
absorption and a peak absorption and corresponds to 2.8ppm
of NH3. This is the minimum concentration that can be meas
ured reliably with the reported gassensor design, withoutany
further datafiltering. Opticalfringesfrominterference effects
introduced by the multipass cell limit the sensitivity and are
the primary source of uncertainty in the ammonia and carbon
dioxide concentration values reported.
4
4.1
Experimental results and discussion
Kalman filter parameters and tuning
The development of a Kalman filter for realtime
measurements that may vary by an order of magnitude re
quires the selection of appropriate values for σ2
centration variability) and σ2
by the sensor). Both of these variances represent the concen
tration variability from one measurement to the next. For in
stance, σ2
be expected to vary from one measurement to the next, and
σ2
introduceinto themeasured concentration fromonemeasure
ment tothenext.
The σ2
sor at a constant gas concentration to determine the variance
of the measured concentrations. This value may then be used
as a constant in a realtime filter operation. However, such
a calculation is not valid over all dynamic ranges, since the
concentration variability changes in direct proportion to the
concentration range. This is shown in Fig. 3, where the vari
w(true con
v(measurement noise introduced
wrepresents how much the true concentration would
vrepresents how much noise the sensor may be expected to
v, value may be measured while operating the sen
a
b
FIGURE 3
ments for a16hperiod. Kalman filter results are indicated by athick bold line
while raw data measurements appear as thin lines depicting system noise
Simultaneous NH3 (a) and CO2 (b) concentration measure
Page 6
90Applied Physics B – Lasers and Optics
ability of the ammonia is 0.1ppm for values of ∼ 4ppm, but
the variability of the carbon dioxide is about 158ppm for
valuesof∼ 2000ppm.Underanygivenexperimentalcircum
stances, once the measurement system has reached equilib
rium,then:
lim
t→∞
σ2
σ2
v
w
= ?,
(18)
where t is the time the system has been operating and ? is
a constant. Therefore, if the true concentration changes by
orders of magnitude, the variance will also change by orders
ofmagnitude. Themeasurementuncertainty may alsochange
as a result of different operating conditions resulting from
temperatureorhumidityvariationsaswellasagingofthesen
sor. These effects can be particularly obvious in the case of
field experiments where controlling the environmental con
ditions for the sensor is difficult. Recalculation of the vari
ance of a set of measurements periodically during the oper
ation of the sensor and using that variance to recalculate the
valuesofσ2
valuesforfilteroperationcanmitigatetheeffectoflargerange
changes.
Sincevariability scalesroughlylinearly astrueconcentra
tionchanges,thisobservationcanbeusedtodevelopascheme
for automating the selection of σ2
volvesfirstapplyingasmallwindow(e.g.theprevious10raw
measurements) to calculate the sample concentration vari
ance. This value is then used as the choice for σ2
the measurement variability (σ2
centration changes (σ2
musthold trueaccording to (18):
vandσ2
wwhilesubsequentlyusingtherecalculated
vand σ2
w. This method in
v. Since
v) scales linearly as true con
w), the following steadystate condition
σ2
σ2
v
w
= ?,
(19)
where ? is a constant. σ2
ingσ2
to control how rapidly the filter will respond to measurement
changes in calculating the Kalman filter output. The larger
the ratio, the longer it will take for radical changes in meas
ured concentrations to be incorporated into the filter output.
The smaller the ratio, the more susceptible the filter will be
to accept large changes in measured concentrations. In other
words, the higher frequency components will have less af
fect on the filter output as ? is increased. This method was
used for filter calculations reported in this work. The selec
tion of a value for ? was made empirically as described in
Sect. 4.2. The use of the variance ratio technique allows for
a completely automated selection process for σ2
sumingaconstantvaluefor?.Notethatthevalueof?depends
on the specific circumstances of the concentration measure
ment, which includes both the sensor and the system that is
beingmeasured.Therefore?isnotanintrinsicpropertyofthe
gassensor.
wcan then be calculated by divid
vby?.Thevalue?canthenbeusedasatuningparameter
vand σ2
was
4.2
Kalman filter resultsand sensor performance
The Kalman filter using the variance ratio tech
niquewasapplied during theoperation of thetrace gas sensor
for a period of approximately 16h. For this period, 1000 sim
ultaneous concentration measurements of NH3and CO2were
acquired. The results of such a test are shown in Fig. 3. Am
moniaandcarbondioxideconcentrationsareshowninFig.3a
and 3b, respectively. A thin line indicates raw data meas
urements, while Kalman filter calculations are designated by
thick lines. As expected, the raw data is significantly nois
ier than the filtered data. The detection sensitivity for the
NH3was 0.12ppm without Kalman filtering. After applying
Kalmanfiltering,thedatavariabilitywasreducedtoaconcen
tration uncertainty of 0.02ppm. By dividing the uncertainty
before the Kalman filter was applied by the uncertainty after
employing the Kalman filter, a detection sensitivity improve
ment by a factor of six was indicated. Detection sensitivity
improvementisequaltotheratioofthemeasurementstandard
deviation(σ)withoutaKalmanfiltertothemeasurementstan
a
b
c
FIGURE 4
lection run indicating good filter performance for a factor of 4 change in the
CO2dynamic range (a and b); a typical 1h zoomin of the NH3concentra
tion data with ? set to 50 is depicted in c
NH3and CO2concentration measurements for a 39h data col
Page 7
LELEUXet al. Applications of Kalman filtering to realtime trace gas concentration measurements91
darddeviationwithaKalmanfilter.Animprovementbyafac
tor of two was observed for CO2measurements. The Kalman
filter output clearly follows true concentration changes mod
eled in terms of σ2
sensornoiseandmodeled byσ2
Inafurtherevaluationtest,theperformanceoftheKalman
filter was investigated over a significant dynamic range by
allowing the CO2 concentration to change by a factor of
4, from 8000ppm to 2000ppm. The NH3 concentration
was held constant at 4 ppm. This test was allowed to run
for approximately 39h. During this time, 2500 concentra
tion measurements were obtained. The result of this experi
ment is shown in Fig. 4. As in the previous test, ammonia
and carbon dioxide concentrations are shown in Fig. 4a
and 4b. The detection sensitivity for the NH3measurement
was 0.12ppm without the Kalman filtering. After apply
ing the Kalman filtering, the data variability was reduced
to a concentration uncertainty of 0.02ppm. This indicates
a detection sensitivity improvement by a factor of six. Sim
ilarly, an improvement by a factor of 2.5 was observed for
CO2 measurements. Again, the Kalman filter output fol
lowed true concentration changes over a large dynamic range
and the variability caused by sensor noise is minimized. As
previously, a value of 50was used for ?. Figure 4c, a 1h
zoomin on Fig. 4a data, clearly illustrates the effective
ness of applying the Kalman filter to improve the detection
sensitivity.
A subsequent experiment involved performing several
data collection runs for different values of ? in order to as
certain the effects of varying this parameter. A result of this
experiment, shown in Fig. 5, indicates that as ? increases, the
filter solution lags behind thetrueconcentration changes. A ?
valueof250wasusedinFig. 5.Thiseffectisoneofthelimita
tionsofthevarianceratiotechnique.If?ismadetoolarge,the
filter solution will lag behind true changes in concentration.
As?isincreased, thiseffect becomesmorepronounced. Data
collection wasperformedusingfivevaluesfor?including50,
100, 150, 200, and 250. The best overall sensor performance
was observed with a value of 50, although reasonable filter
performancewasobservedwithall ofthesevalues.
AfieldtestofthesensorwasperformedattheNASAJohn
son Space Center in Houston, TX as described in [1–3]. The
measurement involved the monitoring of a packedbed bio
logical water processor (BWP) that is part of a NASA water
recovery system (WRS). This system produces potable wa
ter from waste water using aerobic and anaerobic microbial
processes. Two of the byproducts of the chemical reaction in
the bioreactor are ammonia and carbon dioxide. During one
datacollectionrun,theKalmanfilterwasoperatedinrealtime
as data was being collected. Figure 6 shows an excerpt of the
CO2data collected together with the corresponding Kalman
filterresults processedafter therun.
In a subsequent laboratory and field test on the same
BWP system described above, a pulsed, thermoelectrically
cooled, singlefrequency quantum cascade laser was used
to access two NH3absorption lines of the fundamental ν2
band at 10microns [20]. Comparable results to the over
tone data were obtained by measuring changes in ammo
nia concentration over a 15h time period. The NH3 con
centration was varied from 10ppm to 40ppm. The detec
wwhile ignoring the variability caused by
v.Avalueof50wasusedfor?.
a
b
FIGURE 5
time series with ? set to 250, indicating a filter delay in accepting large
changes to the measured data
NH3(a) and CO2(b) concentration measurements for a 9h
FIGURE 6
from a biological water processor located at the NASA Johnson Space
Center, Houston, TX. The Kalman filter data depicted was applied to the
concentration measurements after its acquisition
CO2 concentration measurements and Kalman filter results
tion sensitivity was 0.3ppm without Kalman filtering and
improved the precision to 0.04ppm after applying Kalman
filtering to the NH3concentration data. This indicates an en
hancement of the minimum detection sensitivity by a factor
of 7.5.
4.3
Comparison of Kalman filterresults to a moving
average
As discussed in the introduction, there are a num
ber of advantages of the Kalman filter over the moving
average technique. Using the raw CO2measurement data in
Page 8
92Applied Physics B – Lasers and Optics
a
b
c
FIGURE 7
tween a Kalman filter solution and two moving average solutions. Inset
a shows a Kalman filter solution versus a 10point moving average while
inset b displays the same Kalman filter solution against a 20point moving
average. The final inset c displays the raw data used in both the Kalman filter
as well as the moving average solutions
Time series of CO2measurements showing a comparison be
Fig. 7c, a Kalman filter solution was calculated as well as
a 10 and 20point moving average. These results are shown
in Figs. 7a and 7b respectively. The results show similar be
havior between the Kalman filter and the moving average;
however, the advantages of the Kalman filter show up in two
keyways.
The first place the Kalman filter shows superiority to
the moving average is when abnormally large spikes are
generated by the sensor. For instance at 3/3/01 5 : 52 in
Fig. 7c, a spike of 5500ppm was generated. Observing sur
rounding data, a “real” concentration of 5500ppm seems
highly unlikely. The Kalman filter is conditioned on all pre
vious data, so it is unaffected by this spike. The moving
average technique, however, is significantly impacted by the
spike and takes several measurements before it returns to
normal.
One way of dealing with these spikes is to increase the
number of points in the moving average. The 20point mov
ing average shows a diminished impact of the spike; how
ever, its overall performance is degraded. If the number of
points in the moving average window is made too large, a
“lagging” effect will be observed causing key features to be
missed, e.g. at 3/3/01 0 : 39. This feature is similar to the
behavior of an untuned Kalman filter. However, once ? is
optimized, both of the problems identified above are solved,
as described in Sect. 4.1. This is one of the key advantages
of the Kalman filter, which is its adaptivity to changing data
statistics.
5 Conclusions and outlook
In summary, an automated trace gas sensor that re
quired no operator intervention during data collection was
demonstrated. The use of a Kalman filter significantly im
proved the detection sensitivity of the diodelaserbased gas
sensor by a factor of two to six when using 1.5µm overtone
absorption lines and by a factor of 7.5 when performing
ammonia concentration measurements at 10µm with funda
mental absorption lines, by determining online an optimum
estimate of the true concentration to reduce the variability.
Both thetheory andimplementation ofapracticalKalman fil
ter were discussed and limitations of certain aspects of this
technique were analyzed. The operation of the Kalman filter
with two different dynamic ranges corresponding to the two
gases used was successfully demonstrated. Detection sensi
tivity factor improvements of six and two were observed at
concentration levels of 4 and 1500ppm for NH3and CO2,
respectively.
Potential applications of the Kalman filter technique in
clude its usein environmental, medical and industrialprocess
control. For example, identification and modeling of gas con
centrations in chemical manufacturing processeswillprovide
measurable indicators or indirect control of the gas concen
trations. This could be modeled in terms of a control input
ukas discussed in Sect. 2.2. This modeling can improve the
predictive capability and accuracy of the filter. This modeling
was not included in the experiments presented in this paper,
since no measurable indicators were available for improving
thepredictive capability of thefilter.
One limitation of the Kalman filtering technique as pre
sented in this work is the fact that it must operate on concen
tration values after a data processing algorithm has reduced
them. A potentially more accurate method would be to oper
ate the filter directly on the absorption profile output of the
detector prior to the nonlinear leastsquares fitting routine as
discussed in [1]. Significant variations in absorption profiles
often cause problems with applied nonlinear leastsquares
routines even after averaging, which can add additional noise
to the sensor output above the hardware noise due to white
noisevarianceof σ2
v.
ACKNOWLEDGEMENTS Funding for this project was pro
vided by the National Aeronautics and Space Administration (NASA), the
Institute for Space Systems Operations (ISSO), the Texas Advanced Technol
ogy Program, and the Welch Foundation. D. Leleux would like to acknowl
edge support by the NASAJSC Graduate Student Researcher’s Program
(GSRP).
Page 9
LELEUX et al. Applications of Kalman filtering to realtime trace gas concentration measurements93
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