American Institute of Aeronautics and Astronautics

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Accuracy, Precision, and Scatter in TDLAS Measurements

Michael S. Brown1

Innovative Scientific Solutions, Inc., Dayton, OH, 45440

Dominic L. Barone2

Air Force Research Laboratory, Propulsion Directorate, WPAFB, OH, 45433

William F. Terry3

Innovative Scientific Solutions, Inc., Dayton, OH, 45440

Todd Barhorst4, and Skip Williams

5

Air Force Research Laboratory, Propulsion Directorate, WPAFB, OH, 45433

By taking advantage of impressive advances in diode laser and associated technology,

tunable diode laser absorption spectroscopy (TDLAS) has become a widespread tool for

optical diagnostics in reacting flows. Measurements of target species concentration,

temperature, pressure, and velocity have been demonstrated in a wide variety of

environments by several research groups. Commercialization of niche measurement

instruments based on TDLAS has been achieved. The current maturity of the technique

enables close scrutiny of collected signals to ascertain expectations of accuracy and precision

in measurements coupled to reacting flows. In this paper we examine some TDLAS data sets

in this context. We also discuss the propagation of uncertainties from primary measured

values of signal amplitude versus time to secondary values of temperature, number density,

and pressure.

I. Nomenclature

a y intercept of straight line

b slope of straight line

x dependent variable

y independent variable

A integrated area of an absorption transition

E energy of lower level in an absorption transition (cm-1

)

k unit conversion constant, 0.69503 cm-1

/K

L pathlength (cm)

LS line strength of absorption transition (cm2 molecule

-1 cm

-1)

N number density of target species (cm-3

)

P pressure (atm)

Q partition function

T temperature (K)

S see equation A6)

Sx see equation A6)

Sy see equation A6)

Sxx see equation A6)

Sxy see equation A7)

1 Research Physicist, 2766 Indian Ripple Rd., AIAA Associate Fellow.

2 Masters Student, AIAA Member.

3 Senior Technician, 2766 Indian Ripple Rd., AIAA Member.

4 Research Engineer, AFRL/RZAS, AIAA Member.

5 Principal Research Physicist, AFRL/RZAS, AIAA Senior Member.

48th AIAA Aerospace Sciences Meeting Including the New Horizons Forum and Aerospace Exposition4 - 7 January 2010, Orlando, Florida

AIAA 2010-302

This material is declared a work of the U.S. Government and is not subject to copyright protection in the United States.

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V velocity (m/s)

σx uncertainty in the quantity x (standard deviation)

Δ see equation A7)

ΔνD Doppler line width associated with Voigt spectral profile (cm-1

)

ΔνL Lorentzian line width associated with Voigt spectral profile (cm-1

)

ΔνV total line width (full width a half maximum) of Voigt spectral profile (cm-1

)

II. Introduction

pplication of tunable diode laser absorption spectroscopy (TDLAS) to reacting flows has been repeatedly

demonstrated in various environments primarily for temperature determination.1 At least one study

demonstrated system control using TDLAS-based measurement input in real time. In this work, closed-loop

temperature-based control of a pulsed incinerator was achieved by maximizing the measured time-averaged

temperature through adjustment of the phase between primary and secondary air forcing.2 Overall, the TDLAS

technique has matured to the point that several companies including Zolo Technologies Inc., Southwest Sciences,

Inc., Los Gatos Research, Inc., and Physical Sciences Inc. have delivered TDLAS-based commercial products for

niche measurement applications.

In-stream measurements of temperature, pressure, species concentration and velocity are possible in hypersonic

reacting flows using TDLAS.3 Specifically, demonstration measurements of temperature and water concentration

have been made at the exit of a direct-connect model scramjet engine,4 and air velocity has been measured in a

direct-connect isolator using water and oxygen absorption.5 Measurements of temperature and water concentration

have also been reported in wind-tunnel testing of model scramjet engines.6 More recently, spatial information about

the shock train in a scramjet isolator has been captured using multiple TDLAS lines of sight and a cluster math

algorithm approach to tomography.7 These results are all highly encouraging and point toward successful integration

of the technique into ground and flight tests. The Hypersonic International Flight Research Experimentation

(HIFiRE) program has embraced TDLAS technology. The Flight 1 payload includes two oxygen-based instruments

for mass capture determination. And, water-based measurements of concentration, temperature, and pressure, will be

executed on Flight 2 using 8 lines of sight at the exit plane of the combustor.8

III. Discussion

The significant TDLAS measurement achievements noted above point to a general need for understanding, in a

detailed way, the accuracy and precision to which TDLAS measurements can be made in reacting flows. While the

relative accuracy and precision may vary from one particular application to another, the limits of both will be

dictated by general considerations of signal-to-noise ratios and signal processing algorithms. In applications

associated with

reacting flows, the

largest source of

uncertainty is

typically the signal-

to-noise ratio (SNR).

The SNR in turn

depends on intrinsic

detector noise, A/D

bit depth, parasitic

etalons, gross beam

steering, beam

obscuration, etc. Limits of accuracy and precision will also be determined by the accuracy of the fundamental

spectroscopic constants associated with the target species and the specific transitions accessed during signal

collection. Some of this thinking has been addressed as researchers have sought resonant features that exhibit

temperature sensitivity over the anticipated measurement temperature range. We are currently examining such issues

in a detailed manner that is independent of hardware details using TDLAS data acquired in a number of flowing

environments. The following discussion is focused on single-scan (un-averaged) water absorption spectra acquired

in a time-multiplexed manner over the transitions shown in Table 1. The spectra were acquired using direct

absorption and individually processed to determine the integrated area under each spectral feature. Temperature was

A

Table 1 Near-infrared (~1.39 m) water transitions used in study.

Transition

Label

Transition

Frequency

(cm-1

)

Lower Energy

Level

(cm-1

)

Line Strength at 296 K

(cm2 molecule

-1 cm

-1)

A 7161.410 224.838 1.168E-20

B 7179.752 1216.195 2.438E-22

C 7181.156 136.762 1.488E-20

D 7185.597 1045.058 8.335E-22

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then derived from the integrated areas using two approaches. In the first, ratios of integrated areas were calculated

and equated to the corresponding ratio of the line strengths as shown here

𝐴1

𝐴2 =

𝐿𝑆1

𝐿𝑆2 (1)

The line strengths themselves are known functions of temperature and the ratio of two is simply given by

𝐿𝑆1

𝐿𝑆2 =

𝐿𝑆1 𝑇𝑜

𝐿𝑆2 𝑇𝑜 𝑒

−1

𝑘 𝐸1 − 𝐸2

1

𝑇 −

1

𝑇𝑜

(2)

where Ei denotes the lower energy of the ith

transition and To a reference temperature (296 K).

Extracting a temperature value from the line ratio

method is simply a matter of finding the unique

value for T that satisfies Eqn. 1 and 2. (The line

strengths at the reference temperature are found in

the open literature.) The second approach for

determining the temperature is based on the well-

known Boltzmann plot. In this prescription, the

logarithm of the integrated areas normalized by

their corresponding lines strengths at the reference

temperature are graphed as a function of the

corresponding values of the lower-state energy

level for the transitions. A linear, least-squares fit

to the plotted data yields both the temperature and

density as seen from Eqn. A5 in the Appendix.

The derived density (N) associated with the first

approach is found directly from A1 = LS1(T) N L

using the temperature as found from Eqn. 1 and

Eqn. 2. The anticipated numerical uncertainties

associated with the derived temperatures and

densities associated with both approaches are written explicitly in the Appendix. Both approaches presume local

thermodynamic equilibrium of the target species over the time duration of the acquisition of a single spectrum. This

assumption is sound under the conditions that the data presented here were acquired.

The accuracy of TDLAS-derived quantities such as temperature can be ascertained via measurements made in

steady-state environments with known controlled

values or in similar environments in which the

desired parameter is measured using a second

known method. We have acquired water-based

TDLAS measurements (using the transitions in

Table 1) above a flat-flame burner along with

corrected thermocouple measurements in the same

methane/air flames at a fixed height above the

burner surface. (This work will be discussed in

detail in a future paper.) For the particular case of

an equivalence ratio of 0.9, the TDLAS

measurements yielded a path-integrated

temperature across the burner of 1825 K using

Boltzmann-plot data analysis. A sequence of

thermocouple measurements made with three

different bead diameters along the same path

yielded a path-integrated temperature value of

1785 K. These two values agree with each other at

the 2% level. Similar accuracy assessments appear

in the open literature. For example, in preparation

Figure 1. Single-shot TDLAS-based temperature

measurements (T in K) acquired in a Mach 1.8 flow stream at

the entrance to the isolator.

Figure 2. Single-shot TDLAS-based temperature

measurements (T in K); same conditions as Fig. 1.

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for wind-tunnel testing of a model scramjet engine, researchers compared water-based TDLAS measurements of

temperature with those of a thermocouple in a premixed flame.6 Over the temperature range of 1000 K to 1600 K

they found the TDLAS-derived temperature to be a few percent greater than the thermocouple temperature with the

difference increasing with flame temperature.

For measurements made in reacting flows – particularly those set up to test hardware – it is frequently not

possible to obtain a secondary temperature measurement. Indeed, optical diagnostics are typically employed for

temperature measurements in environments where other thermometry approaches are not possible or practical. It is

prudent therefore to examine TDLAS data reduction strategies with regards to anticipated numerical scatter given

finite signal-to-noise ratios in the context of apparent measurement scatter (i.e. precision). Such close examination

can lead to conclusions that may be data-set dependent and care must be used in transferring such considerations

between data sets.

Using transitions A, B and D of Table 1, the temperature of ambient laboratory air was measured following the

Boltzmann plot and two-line approaches for extracting temperature. The data set included 1000 single scans of the

employed diode lasers acquired at 1 kHz. The three derived average temperatures agreed to within 6%. (We note

that the transitions of Table 1 are better suited for elevated temperature environments than for room temperature

environments.) The Boltzmann plot average temperature of 293 K agrees with the two-line temperature (288 K)

using the B/A line ratio at the 1.7% level. These three approaches yielded similar average values for the SNR-driven

average uncertainty of ~33 K. However, the observed standard deviations of the derived temperatures were smaller,

~21 K. The larger SNR-driven uncertainty value is certainly due the relatively small signal associated with

transitions B and D at room temperature.

As an example of a measurement in a reacting flow, we examine a sequence of single-scan (single-shot)

temperature acquired at the entrance to the isolator in a direct-connect hypersonic test facility. The measurements

were made using a single line-of-sight across the mid-plane of the circular hardware just downstream of the Mach

1.8 facility nozzle. The incoming air stream consisted of heated (non-vitiated) air with an average water molecule

number density of 6 × 1016

cm-3

. The data was collected at 1 kHz using the transitions noted in Table 1 and

processed to yield values of the frequency-integrated absorbance for each shot. Temperatures were derived from

these integrated areas using the area ratio approach (Eqn. 1 and 2) and the Boltzmann plot approach (Eqn. A5). The

four transitions provide for six unique area ratios. The derived temperatures for an arbitrarily chosen sequence of 40

shots are shown in Fig. 1 and 2 for the Boltzmann plot approach and five of the ratio-based approaches. As seen in

Fig. 1, the results for the Boltzmann plot approach and the ratios of transition combinations B/A and B/C (see Table

1 for transition labeling convention) follow each other quite closely. For example they each exhibit local maxima

near shot numbers 23, 37, and 52 and local minima near 32 and 47. The apparent oscillation in temperature persists

for all times recorded and probably reflects facility air supply characteristics such as compressor surge.

Temperatures derived from area ratio combinations involving transition D (B/D, D/C, and D/A) exhibit large

fluctuations (see Fig. 2) and don’t reproduce the periodic structure evident in Fig. 1. Transition D exhibited the

lowest signal-to-noise ratio observed for this data set and is the likely reason behind the excessive scatter seen in

Fig. 2.

Using a full second worth of data (1000 shots), the average temperature was calculated for each approach along

with the standard deviation of the measurements. These values are shown in Table 2. Similarly, the table includes

the average of the uncertainties of the derived temperature values based on measured signal-to-noise ratios and the

procedures detailed in the Appendix. Since the signal-to-noise ratio can include systematic error sources we do not

add the individual uncertainties in quadrature but rather report the simple average. Note that the Boltzmann plot

derived temperature using all four transitions exhibits larger observed scatter than that of the same plot with

transition D eliminated. The temperature derived from the three-transition Boltzmann analysis is plotted in Fig. 1.

As anticipated from Fig. 1 and Fig. 2, the observed experimental scatter (standard deviation) as well as the

anticipated numerical scatter are least for the three-transition Boltzmann approach and the ratio combinations of B/C

Table 2. Extracted temperatures with associated uncertainties using Boltzmann analysis and area ratios

for transitions of Table 1.

Boltzmann

Plot for

A,B,C,D

Boltzmann

Plot for

A,B,C

B / C B / A B / D A / C D / C D / A

avg. T (K) 488 425 427 437 300 371 479 503

std expt. (K) 72 49 35 38 104 122 58 68

avg. uncert. (K) 70 84 16 30 215 198 73 146

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and B/A. The mean temperatures derived via these three approaches agree within 2.8%. The aforementioned

oscillation of the temperature accounts for the measured scatter being larger than the anticipated numerical scatter.

The two area ratio approaches exhibit less measured scatter as well as less anticipated numerical uncertainty.

Likewise, as anticipated from Fig. 2, temperatures derived from area ratios involving transition D exhibit widely

varying mean values along with large measured scatter and large anticipated scatter.

The primary quantities derived from a TDLAS measurement are the temperature and number density of the

target species. Numerous other quantities can be derived as well; for example, the pressure which is derivable from

the line width of an absorption transition. The relevant expressions are shown in Appendix Eqn. A12 – A17 where

use has been made of empirical relationships between the full width at half maximum of the transition, the

temperature, and the pressure. The uncertainty in the derived pressure is then dependent upon the uncertainty in the

derived temperature and measured line width. We take, as an example, the measured temperature of Table 2 (430 K)

and the Voigt line width of the 7185.6 cm-1

transition (0.067 cm-1

) along with constant values of To=296 K, n =

0.65, and o = 0.041 cm-1

/atm taken from the literature9 to find a measured pressure of 0.9 atm. Assuming no

uncertainty in the empirical constants, the uncertainty in the derived value of the pressure is 0.07 atm for a 10%

temperature uncertainty and a 5% line width uncertainty. That is, the combined effects of temperature and line width

uncertainty lead to an 8% pressure uncertainty.

For large data sets with thousands of single-shot spectra to be analyzed it is helpful to develop routines that

afford some means of rejecting compromised spectra. Degradation can occur for multiple reasons such as gross

beam steering or obscuration or transient electronic noise. One such approach that we have found valuable is close

examination of the fitting associated with the Boltzmann plot. The scatter about the linear fit can be quantified using

the standard rules for calculating χ2. For a single data set at constant flow condition we calculate the individual

values of χ2, average them, and then reject outliers that show large deviation from this average.

IV. Conclusion

While issues associated with the accuracy, precision, and scatter associated with flow field parameters derived

from TDLAS measurements have been discussed here using specific data sets, the methodology is quite general and

can be extended to other data sets. We recommend using a combination of Boltzmann plot analysis combined with

integrated area ratio analysis to extract the local path-integrated temperature. For a set of transitions it is best to

exclude a transition or transitions that exhibit poor signal-to-noise ratios as compared to other transitions in the set.

As indicated in Table 2, calculation of the anticipated numerical scatter provides guidance in selecting line pairs for

careful analysis.

V. Appendix

The following indicates the manner in which the numerical uncertainties in the derived values of temperature,

target species number density and pressure discussed above are found using standard statistical analysis for both the

line ratio approach and the Boltzmann plot approach.

The uncertainties in the measured absorption signals (A) denoted by A are determined from signal-to-noise

conditions as noted above. For any function f that is dependent upon A or a set of Ais, the uncertainty in f is

determined from the following

𝜎𝑓2 = 𝜎𝑧𝑖2𝑛

𝑖=1 𝜕𝑓

𝜕𝑧𝑖

2

A1)

where zi denotes Ai or a function of Ai. Note, we are presuming independence of the variables zi. Herein, the

uncertainty () is expressed as a standard deviation. The 95% confidence interval can be found from through

multiplication by 1.96.

We first consider the uncertainty in the temperature derived from a two-line ratio. Defining the ratio of the

integrated absorbances as R = A1/A2, the fractional temperature uncertainty can be written from Eqn. 1 and Eqn. 2 as

𝜎𝑇

𝑇 =

𝑘∆𝐸

𝑘∆𝐸 𝑙𝑛 𝑅

𝐿𝑆2 𝑇𝑜 𝐿𝑆1 𝑇𝑜 + 1 𝑇𝑜

𝜎𝑅 A2)

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where ΔE = E1 – E2. Similarly, the uncertainty in the extracted number density can be expressed via

𝜎𝑁2 = 1

𝐿𝑆 𝐿

2

𝜎𝐴2 + 𝜕𝑁

𝜕 𝐿𝑆

𝜕 𝐿𝑆

𝜕𝑇

2

𝜎𝑇2 2

A3)

𝜕𝑁

𝜕 𝐿𝑆 =

−𝐴

𝐿𝑆2 𝐿

𝜕 𝐿𝑆

𝜕𝑇 = 𝐿𝑆 𝑇𝑜

𝑄 𝑇𝑜

𝑄 𝑇 𝑒−

𝐸

𝑘

1

𝑇−

1

𝑇𝑜 A4)

A Boltzmann analysis of absorption spectra stems from the following expression for each (ith

) transition in the

spectrum (we omit factors associated with stimulated emission that are insignificant in this context)

𝑙𝑛 𝐴𝑖

𝑆𝑖 𝑇𝑜 𝑒

−𝐸𝑖

𝑘 𝑇𝑜 = −𝐸𝑖

𝑘 𝑇 + 𝑙𝑛

𝑄 𝑇𝑜

𝑄 𝑇 𝑁 𝐿 A5)

where To denotes a reference temperature (usually 296 K). A Boltzmann plot is generated by graphing the left hand

side of Eqn. A5 as a function of Ei for two or more transitions. A linear fit to the plotted points yields the

temperature (T) via the slope and the number density (N) of the target species via the y intercept. To extract a

measure of the uncertainty in the derived values for T and N, we write the linear fit to the Boltzmann plot in the form

y = a + b x. Uncertainties in a and b are derivable from the uncertainties in the values of y that in turn reflect the

uncertainties in the set of Ai. A solution for a and b is found using well-established routines that we reproduce here

following the notation of Press et al.10

Each value of yi has an associated uncertainty noted as σi.

𝑆 = 1

𝜎𝑖2

𝑁𝑖=1 𝑆𝑥 =

𝑥𝑖

𝜎𝑖2 𝑆𝑦 =

𝑦𝑖

𝜎𝑖2 𝑆𝑥𝑥 = 𝑁

𝑖=1𝑁𝑖=1

𝑥𝑖2

𝜎𝑖2

𝑁𝑖=1 A6)

𝑆𝑥𝑦 = 𝑥𝑖𝑦𝑖

𝜎𝑖2

𝑁𝑖=1 ∆ = 𝑆 𝑆𝑥𝑥 − 𝑆𝑥 2 A7)

The fitted values of a and b along with their uncertainties are then given by

𝑎 = 𝑆𝑥𝑥 𝑆𝑦 − 𝑆𝑥 𝑆𝑥𝑦

∆ 𝑏 =

𝑆 𝑆𝑥𝑦 − 𝑆𝑥 𝑆𝑦

∆ A8)

𝜎𝑎2 = 𝑆𝑥𝑥

∆ 𝜎𝑏2 =

𝑆

∆ A9)

Following a Boltzmann analysis, the uncertainty in the derived value for the temperature and target species number

density are given from the above by

𝜎𝑇 = 𝜎𝑏

𝑘 𝑏2 = 1

𝑘

𝑆 ∆1.5

𝑆 𝑆𝑥𝑦 − 𝑆𝑥 𝑆𝑦 2 A10)

𝜎𝑁 = 𝑄 𝑇

𝑄 𝑇𝑜

𝑒𝑎

𝐿 𝜎𝑎 A11)

TDLAS measurements can be used to determine the local pressure through the line widths (full width at half

maximum) of the recorded spectral transitions. Under the physical environments associated with hydrocarbon-based

reacting flows, the spectral line shapes are well characterized by the Voigt spectral profile. This profile includes line

broadening due to temperature (Doppler broadening) as well as pressure (collision broadening). The collision

broadening contribution to the line width can be separated from the total line width and Doppler contribution as

follows

∆𝜈𝐿 = Δ𝜐𝑉

2 − 𝛽2 𝑇

∆𝜈𝑉 A12)

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where denotes constants associated with Doppler width expression (ΔνD = T½). Careful experimental work has

revealed scaling laws that empirically show the relationship between the collision width and the pressure that are

often written in the following form

𝑃 = Δ𝜈𝐿

2 𝛾 𝛾 = 𝛾𝑜

𝑇𝑜

𝑇 𝑛

A13)

where n and o are transition-specific constants and To the reference temperature. The above can be combined into a

single expression

𝑃 = 1

2 𝛾𝑜 𝑇𝑜𝑛 𝑇𝑛 Δ𝜈𝑉 −

𝛽2

Δ𝜈𝑉 𝑇 A14)

The statistical uncertainty in the pressure can then be found from

𝜎𝑃2 = 𝜕𝑃

𝜕𝑇

2

𝜎𝑇2 + 𝜕𝑃

𝜕 ∆𝜈𝑉

2

𝜎Δ𝜈𝑉2 A15)

𝜕𝑃

𝜕∆𝜈𝑉 =

1

2 𝛾𝑜 𝑇𝑜𝑛 𝑇𝑛 +

𝛽2 𝑇𝑛+1

∆𝜈𝑉2 A16)

𝜕𝑃

𝜕𝑇 =

1

2 𝛾𝑜 𝑇𝑜𝑛 𝑇𝑛−1 ∆𝜈𝑉 𝑛 −

𝛽2 𝑇𝑛 𝑛+1

∆𝜈𝑉 A17)

Acknowledgments

This work is supported in part by the Advanced Propulsion Test Technology (APTT) Focus Area, Tom

Fetterhoff, Executing Agent, and the Air Force Office of Scientific Research (Dr. Michael Berman, Program

Manager).

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103, 2007, 565-577. 10 Press, W. H., Teukolsky, S. A., Vetterling, W. T., Flannery, B. P., Numerical Recipes in C++, Chap. 15, (Cambridge

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