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**Atmospheric Measurement Techniques**
An interactive open-access journal of the European Geosciences Union

Journal topic

- About
- Editorial board
- Articles
- Special issues
- Highlight articles
- Manuscript tracking
- Subscribe to alerts
- Peer review
- For authors
- For reviewers
- EGU publications
- Imprint
- Data protection

- About
- Editorial board
- Articles
- Special issues
- Highlight articles
- Manuscript tracking
- Subscribe to alerts
- Peer review
- For authors
- For reviewers
- EGU publications
- Imprint
- Data protection

- Abstract
- Introduction
- Site, instrumentation, and data
- Data check and instrument diagnosis
- Algorithm to recover the data of 3-D wind and sonic temperature
- Application
- Verification
- Adjustment
- Discussion
- Conclusion remarks
- Data availability
- Appendix A: Transform matrixes
- Appendix B: Iteration algorithm for sonic transducer-shadow corrections
- Appendix C: MATLAB code
- Appendix D: Sonic temperature from air temperature, relative humidity, and atmospheric pressure
- Author contributions
- Competing interests
- Acknowledgements
- References
- Supplement

**Research article**
30 Oct 2018

**Research article** | 30 Oct 2018

Recovery of the three-dimensional wind and sonic temperature data from a physically deformed sonic anemometer

^{1}Guangdong Province Key Laboratory for Climate Change and Natural Disaster Studies, School of Atmospheric Sciences, Sun Yat-sen University, Zhuhai 519082, China^{2}CAS-CSI Joint Laboratory of Research and Development for Monitoring Forest Fluxes of Trace Gases and Isotope Elements, Institute of Applied Ecology, Chinese Academy of Sciences, Shenyang 110016, China^{3}Campbell Scientific Incorporation, Logan, Utah 84321, USA^{4}Beijing Techno Solutions Ltd., Beijing 100089, China^{5}Nanjing University of Information Science and Technology, Nanjing 210044, China^{6}National Marine Environmental Forecasting Center, Beijing 100081, China

^{1}Guangdong Province Key Laboratory for Climate Change and Natural Disaster Studies, School of Atmospheric Sciences, Sun Yat-sen University, Zhuhai 519082, China^{2}CAS-CSI Joint Laboratory of Research and Development for Monitoring Forest Fluxes of Trace Gases and Isotope Elements, Institute of Applied Ecology, Chinese Academy of Sciences, Shenyang 110016, China^{3}Campbell Scientific Incorporation, Logan, Utah 84321, USA^{4}Beijing Techno Solutions Ltd., Beijing 100089, China^{5}Nanjing University of Information Science and Technology, Nanjing 210044, China^{6}National Marine Environmental Forecasting Center, Beijing 100081, China

**Correspondence**: Qinghua Yang (yangqh25@mail.sysu.edu.cn) and Ning Zheng (ning.zheng@campbellsci.com.cn)

**Correspondence**: Qinghua Yang (yangqh25@mail.sysu.edu.cn) and Ning Zheng (ning.zheng@campbellsci.com.cn)

Abstract

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A sonic anemometer reports three-dimensional (3-D) wind and sonic temperature
(*T*_{s}) by measuring the time of ultrasonic signals transmitting along each
of its three sonic paths, whose geometry of lengths and angles in the
anemometer
coordinate system was precisely determined through production calibrations
and the geometry data were embedded into the sonic anemometer operating
system (OS) for internal computations. If this geometry is deformed, although
correctly measuring the time, the sonic anemometer continues to use its
embedded geometry data for internal computations, resulting in incorrect
output of 3-D wind and *T*_{s} data. However, if the geometry is remeasured
(i.e., recalibrated) and to update the OS, the sonic anemometer can resume
outputting correct data. In some cases, where immediate recalibration is not
possible, a deformed sonic anemometer can be used because the ultrasonic
signal-transmitting time is still correctly measured and the correct time can
be used to recover the data through post processing. For example, in 2015, a
sonic anemometer was geometrically deformed during transportation to
Antarctica. Immediate deployment was critical, so the deformed sonic
anemometer was used until a replacement arrived in 2016. Equations and
algorithms were developed and implemented into the post-processing software
to recover wind data with and without transducer-shadow correction and *T*_{s}
data with crosswind correction. Post-processing used two geometric datasets,
production calibration and recalibration, to recover the wind and *T*_{s}
data from May 2015 to January 2016. The recovery reduced the difference of
9.60 to 8.93 ^{∘}C between measured and calculated *T*_{s} to 0.81 to
−0.45 ^{∘}C, which is within the expected range, due to normal
measurement errors. The recovered data were further processed to derive
fluxes. As data reacquisition is time-consuming and expensive, this
data-recovery approach is a cost-effective and time-saving option for similar
cases. The equation development can be a reference for related topics.

How to cite

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How to cite.

Zhou, X., Yang, Q., Zhen, X., Li, Y., Hao, G., Shen, H., Gao, T., Sun, Y., and Zheng, N.: Recovery of the three-dimensional wind and sonic temperature data from a physically deformed sonic anemometer, Atmos. Meas. Tech., 11, 5981–6002, https://doi.org/10.5194/amt-11-5981-2018, 2018.

1 Introduction

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The three-dimensional (3-D) sonic anemometer is commonly used for both
micrometeorological research and applied meteorology (Horst et al., 2015).
It directly measures boundary-layer flows at high measurement rates (10 to
50 Hz) and outputs wind speeds expressed in the 3-D right-handed orthogonal
anemometer coordinate system relative to its structure frame (see Appendix
A, hereafter, referred as 3-D anemometer coordinate system) and sonic
temperature calculated from the speed of sound (Hanafusa et al., 1982). Its
outputs are commonly used to estimate the fluxes of momentum and sonic
temperature and, when combined with fast-response scalar sensors, the fluxes
of CO_{2}, H_{2}O, and other atmospheric constituents.

It has three pairs of sonic transducers forming three sonic paths (Fig. 1), each of which is between paired sonic transducers. The three paths are situated as optimized angles for wind measurements in the 3-D anemometer coordinate system, structuring the geometry of sonic anemometer. This geometry is quantitatively defined by the path lengths and path angles that are precisely measured during production calibration. A sonic anemometer measures the time of ultrasonic signals transmitting along each path (hereafter, referred as transmitting time). In reference to the sonic path length, the transmitting time is used to calculate the speeds of flow and sound along the path, which will be detailed in Sect. 4 as follows. According to the angles of three sonic paths, the speeds from the three paths are expressed in the 3-D anemometer coordinate system for wind and as sonic temperature for air heat property.

A sonic anemometer has geometry information embedded into its operating system (OS) for internal data processing (see Appendix A), allowing output of 3-D wind and sonic temperature. However, if it is geometrically deformed from the manufacturer's setting at millimeter scales, or even smaller, due to an unexpected physical impact in transportation, installation, or other handling, the geometry embedded in the OS is not representative of the current geometry of this sonic anemometer. As a result, the anemometer no longer outputs correct wind speeds and sonic temperatures because the deformation in geometry changes the relative spatial relationship among its six sonic transducers. If, an impact displaces a transducer relative to the others, the displacement must change at least one of the sonic path lengths and one of the sonic path angles. Fortunately, if geometrical deformation is the only problem, rather than physical damage to the transducers, the sonic anemometer can, according to its working physics (Schotland, 1955), correctly perform its transmitting-time measurements. Due to the change in a sonic path length, the speeds of air flow and sound along the path are incorrectly computed because the sonic path length embedded in the OS does not match the true length when the transmitting time was measured. As a result, the incorrect speeds along with the change in any sonic path angle might cause all 3-D wind speeds as well as sonic temperature outputs to be incorrect. These incorrect outputs are recoverable because the transmitting time was correctly measured and the deformed geometry can be remeasured (i.e., recalibrated) by the manufacturer to whom the anemometer can be shipped back with care. However, the equations and algorithms for the recovery are needed if a sonic anemometer is found to be geometrically deformed in a remote site where its use has to be continued. From such a site, it could take months, seasons, or even longer for a deformed anemometer to be transported back to the manufacturer for geometry remeasurements, recalibration, and shipped back to the site. In this case, if the measurements were not continued, a measurement season or year could be easily missed.

This study demonstrates data recovery from such a case when a sonic
anemometer as a component of the IRGASON (integrated CO_{2}/H_{2}O open-path
gas analyzer and 3-D sonic anemometer, Campbell Scientific Inc., 2018) was
geometrically deformed during transportation to the Antarctic Zhongshan Station
from China in early 2015 and had to be used until its replacement arrived at
the site early the next year. If the deformed sonic anemometer was not used,
one measurement-year would have been missed because the only transportation
of R/V *Xue Long* (i.e., Snow Dragon in English) from China to the Zhongshan
Station served a round-trip to the site on an annual basis. More
importantly, the 2015 data were also needed by related projects for
collaborations. Therefore, the geometrically deformed sonic anemometer was
used to acquire the 2015 data. In early 2016, the deformed anemometer was
shipped, with a pair of buffer bumpers for protection, to the manufacturer
of Campbell Scientific Inc. in the US for remeasurements of its geometry to
update its OS (i.e., recalibration).

Using the measurements of sonic path lengths and sonic path angles for this sonic anemometer from production calibration in April 2014 before its transportation and from recalibration in March 2016 after the field use in the Zhongshan Station, this study aims to develop and verify the equations and algorithms to recover the 2015 data measured using this geometrically deformed sonic anemometer to data as if measured with the this anemometer after recalibration although actually measured before the recalibration, providing a reference to similar cases and/or related topics.

2 Site, instrumentation, and data

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The observation site was in the coastal landfast sea ice area of the
Zhongshan Station (69^{∘}22^{′} S and 76^{∘}22^{′} E), East Antarctica (Yang et al., 2016; Yu et al., 2017; Zhao et al., 2017).
In this area, as influenced by the unique solar cycles, the climate is
characterized by the polar night from late March to mid-July and the polar
day from mid-November to January. The polar day and the polar night are
inhabitable to human life, but drive atmospheric dynamics in a way that is of
interest to human beings (Valkonen et al., 2008); therefore, this region has
attracted scientists to measure its surface heat balance; However, these
measurements are not an easy task in terms of financial support, technical
infrastructure, and administrative management. As such, only a few studies
on such measurements have been conducted in this region (e.g., Vihma et al.,
2009; Liu et al., 2017).

The fluxes of CO_{2}/H_{2}O, heat, radiation, momentum, and
atmospheric variables were measured so that the sea ice and snow surface energy
budget during both melting and frozen periods can be quantified. For these
measurements, the project established two open-path eddy-covariance (OPEC)
flux stations in May 2015. One station (see Fig. 2) was configured with
the IRGASON (SN: 1131) for the fluxes, four-component net radiometer (model:
CNR4, Kipp & Zonen, Delft, the Netherlands) for net radiation and
radiation fluxes; one temperature and relative humidity probe (model:
HMP155A, SN: H5140031, Vaisala, Helsinki, Finland) inside a 14-plate
naturally aspirated radiation shield of model 41005 for air temperature and
air relative humidity; and one infrared radiometer (model: SI-111, SN: 2962,
Apogee, UT, USA) for surface temperature. In early 2016, a CSAT3B (Campbell
Scientific Inc., UT, USA) was added for additional data of 3-D wind and sonic
temperature. This OPEC station was also equipped with a built-in barometer
(model: MPXAZ6115A, Freescale Semiconductor, TX, USA) for atmospheric
pressure and a built-in 107 temperature probe (model: 100K6A1A, BetaTherm,
Finland) inside a 6-plate naturally aspirated radiation shield of model
41303-5A for air temperature, the IRGASON was connected to and controlled by
an EC100 electronic module (SN: 1542, OS: EC100.04.10) that, in turn, was
connected to and instructed by a central CR3000 Measurement and Control
Datalogger (SN: 7720, OS 25) for these sensor measurements, data processing,
and data output. While receiving the data output from EC100 at 10 Hz, the
CR3000 also controlled and measured slow response sensors at 0.1 Hz such as
the CNR4, HMP155A, and others in support to this study.
EasyFlux_CR3OP (version 1.00, Campbell Scientific Inc., 2016)
was used inside CR3000. The data of 3-D wind, sonic temperature, CO_{2} and
H_{2}O amounts, atmospheric pressure, diagnosis codes for the 3-D sonic
anemometer and open-path infrared gas analyzer, air temperature, and
relative humidity were stored 10 records per second (i.e., 10 Hz). The data
from all sensors were computed and stored by the CR3000 at every half-hour
interval.

3 Data check and instrument diagnosis

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Immediately after the station started to run, all measured values were
checked. Unfortunately, the sonic temperature from the 3-D sonic anemometer
was incorrect because it was around 10 ^{∘}C higher than the air
temperature from HMP155A or 100K6A1A. Given a H_{2}O density of about 1.00 g m^{−3} and air temperature about −20 ^{∘}C, sonic temperature
should be around 0.13 ^{∘}C higher than air temperature (see Eq. 5
in Schotanus et al., 1983) if the sonic temperature was measured, although
impossible, without an error. Further diagnosis for sonic anemometer
measurements found that the sonic temperature values from the three sonic
paths unexpectedly deviated around −12, 5, and −7 ^{∘}C,
respectively, as shown by device configuration (Campbell Scientific Inc.,
UT, USA) connected to EC100 through a notebook computer while the station
was running. Apparently, the largest absolute difference in sonic
temperature among the three paths reached 17 ^{∘}C, although the
difference from an IRGASON sonic anemometer was expected to be < 1 ^{∘}C. Such a large unexpected absolute difference
(e.g., 17 ^{∘}C) among the three values from the three sonic paths might be
caused by the geometrical deformation of sonic anemometer. To confirm the
diagnosis, the body of the IRGASON was visually examined and painting on the
knuckle of side one (i.e., first sonic path) among the top three claws was
found removed as it was apparently impacted (Fig. 3). Therefore, with confidence, it
was concluded that the incorrect outputs of sonic temperature were caused by
the geometrical deformation of sonic anemometer while being transported to
Antarctica from China. The deformation also might cause the incorrect
outputs of 3-D wind. Therefore, this IRGASON should have been shipped back to
the manufacturer for remeasurements of its geometry to update its OS
(recalibration). However, as addressed in the introduction, the 2015 data would
have been missed if it was shipped back to the manufacturer at that point. To make
measurements as planned, this IRGASON continued its field duty until the next
round-trip of R/V *Xue Long* to Antarctica from China until the end of 2015 when
its replacement from the manufacturer arrived at the site.

In early 2016, it was replaced in the field and was shipped back to the manufacturer, where it was remeasured for sonic geometry in the recalibration process in March. The remeasurements verified our diagnosis conclusion that the IRGASON sonic anemometer was geometrically deformed (see Table A1 in Appendix A). Therefore, the 2015 data from this sonic anemometer needed to be recovered as if measured by the same anemometer after recalibration, although the data were acquired from the measurements before the recalibration.

4 Algorithm to recover the data of 3-D wind and sonic temperature

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An IRGASON sonic anemometer measures wind flows along its three
non-orthogonal sonic paths (i.e., the three sonic paths non-orthogonally
situated in relation to each other, see Fig. 1), each of which is between a pair of sonic
transducers. Sensing each other in each sonic path, the pair separately
pulse two ultrasonic signals in opposite directions at the same time. The
signal pulsed by the transducer facing the air flow direction along the sonic
path takes less time to be sensed by its paired one than the one pulsed by
the transducer against the air flow direction. In a path, the transmitting
time of the ultrasonic signal upward [*t*_{ui} where subscript *i* can be 1, 2, or
3, denoting the sequential order of sonic path (Fig. 1). This subscript
denotes the same variable throughout] and downward (*t*_{di}) are measured by the
sonic anemometer (Hanafusa, 1982; Foken, 2017). In the case shown in Fig. 1 for the third sonic path, or *i*=3, the transmitting time of ultrasonic
signal upward in the path is given by the following equation:

$$\begin{array}{}\text{(1)}& {t}_{u\mathrm{3}}={\displaystyle \frac{{d}_{\mathrm{3}}}{{c}_{\mathrm{3}}+{u}_{\mathrm{3}}}},\end{array}$$

where, along the third sonic path, *d*_{3} is its length precisely measured
during production or recalibration process using a coordinate measurement
machine (CMM), *c*_{3} is the speed of sound, and *u*_{3} is the speed of air
flow (Fig. 1); and the transmitting time of ultrasonic signal downward is
given by the following equation:

$$\begin{array}{}\text{(2)}& {t}_{d\mathrm{3}}={\displaystyle \frac{{d}_{\mathrm{3}}}{{c}_{\mathrm{3}}-{u}_{\mathrm{3}}}}.\end{array}$$

Equations (1) and (2) lead to

$$\begin{array}{}\text{(3)}& {u}_{\mathrm{3}}={\displaystyle \frac{{d}_{\mathrm{3}}}{\mathrm{2}}}\left[{\displaystyle \frac{\mathrm{1}}{{t}_{u\mathrm{3}}}}-{\displaystyle \frac{\mathrm{1}}{{t}_{d\mathrm{3}}}}\right].\end{array}$$

Using the same procedure, *u*_{1} and *u*_{2} (see Fig. 1) can be derived as
the same form. In reference to Eq. (3), the equation for *u*_{i}; where $i=\mathrm{1},\mathrm{2}$, or 3; can be expressed as follows:

$$\begin{array}{}\text{(4)}& {u}_{i}={\displaystyle \frac{{d}_{i}}{\mathrm{2}}}\left[{\displaystyle \frac{\mathrm{1}}{{t}_{\mathrm{u}i}}}-{\displaystyle \frac{\mathrm{1}}{{t}_{\mathrm{d}i}}}\right].\end{array}$$

Similar to *d*_{3}, *d*_{1} and *d*_{2} are also precisely measured using CMM.
The three flow speeds of *u*_{i} (*i*=1, 2, or 3) from the three
non-orthogonal paths are expressed in the 3-D anemometer coordinate system of
*x*, *y*, and *z*; where *x* and *y* are the horizontal coordinate axes and *z* is the vertical
axis; and through a transform matrix **A** as the 3-D wind speeds (*u*_{x}, *u*_{y},
and *u*_{z}) commonly used in practical applications:

$$\begin{array}{}\text{(5)}& \left[\begin{array}{l}{u}_{x}\\ {u}_{y}\\ {u}_{z}\end{array}\right]=\mathbf{A}\left[\begin{array}{l}{u}_{\mathrm{1}}\\ {u}_{\mathrm{2}}\\ {u}_{\mathrm{3}}\end{array}\right],\end{array}$$

where the 3-D anemometer coordinate system (see Figs. 1 and A1) is defined by
its origin at the center of sonic measurement volume, the
*u*_{x}−*u*_{y} plane, parallel to the imagery plane, leveled by a built-in bubble
in the anemometer structure, and the *u*_{y}−*u*_{z} plane through the first
sonic path and **A** is a 3×3 matrix constructed using precisely
measured geometry of the sonic paths in angles relative to the 3-D anemometer
coordinate system (see its derivations in Appendix A). Matrix **A** is unique
for each sonic anemometer and is embedded in its OS; therefore, the 3-D wind
data outputted from the anemometer are the three components of *u*_{x},
*u*_{y} and *u*_{z} in the 3-D anemometer coordinate system.

Due to shadowing from the sonic transducer itself (transducer shadowing),
the measured *u*_{i} is assumed to be lower than its true value in magnitude
(Wyngaard and Zhang, 1985; Kaimal and Finnigan, 1994). As denoted by
*u*_{Ti_n} where subscript *T* indicates “True” and subscript
*n* indicates that *u*_{Ti_n} was estimated from
*n* counts of iterations of transducer-shadow correction as shown in Appendix
B, this true value is assumed to be approached through the transducer-shadow
correction from *u*_{i}. Now, the shadow correction was implemented as an
option if the OS of EC100 for the IRGASON sonic anemometer is version 5 or
newer. Therefore, depending on the option, Eq. (5) alternatively can be
expressed as follows:

$$\begin{array}{}\text{(6)}& \left[\begin{array}{l}{u}_{x}\\ {u}_{y}\\ {u}_{z}\end{array}\right]=\mathbf{A}\left[\begin{array}{l}{u}_{\mathrm{T}\mathrm{1}\mathit{\_}n}\\ {u}_{\mathrm{T}\mathrm{2}\mathit{\_}n}\\ {u}_{\mathrm{T}\mathrm{3}\mathit{\_}n}\end{array}\right].\end{array}$$

Following Host et al. (2015), based on Wyngaard and Zhang (1985), the correction equation for the sonic transducer size and sonic path geometry of the IRGASON sonic anemometer is given by

$$\begin{array}{}\text{(7)}& {u}_{Ti\mathit{\_}\mathrm{1}}={\displaystyle \frac{{u}_{i}}{\mathrm{0.84}+\mathrm{0.16}\mathrm{sin}{\mathit{\alpha}}_{i}}},\end{array}$$

where *α*_{i} is the angle of the total wind vector to the wind
vector along sonic path *i* and is unknown before the two vectors are
estimated, but, referencing Figs. 1 and 4, the sin*α*_{i} in Eq. (7)
can be alternatively expressed as a function of flow speed values to lead
Eq. (7) as follows:

$$\begin{array}{}\text{(8)}& {u}_{Ti}={\displaystyle \frac{{u}_{i}}{\mathrm{0.84}+\mathrm{0.16}\frac{\sqrt{{U}_{\mathrm{T}}^{\mathrm{2}}-{u}_{Ti}^{\mathrm{2}}}}{{U}_{\mathrm{T}}}}},\end{array}$$

where *U*_{T} is the magnitude of total true wind vector, given by

$$\begin{array}{}\text{(9)}& {U}_{\mathrm{T}}=\sqrt{{u}_{x}^{\mathrm{2}}+{u}_{y}^{\mathrm{2}}+{u}_{z}^{\mathrm{2}}}.\end{array}$$

In Eq. (8), all independent variables are actually related to the variables
in Eq. (5). As such, using this equation, *u*_{Ti} can be computed; however,
there are two inconvenient issues in this equation application to
transducer-shadow corrections: (1) an analytical solution for *u*_{Ti} is not
easily available because *u*_{Ti} is in a second order term under a square
root in the right side of Eq. (8), although *u*_{Ti} is analytically expressed
in its left side and (2) *U*_{T} is not available either because *u*_{x},
*u*_{y}, and *u*_{z} are derived from *u*_{1}, *u*_{2}, and *u*_{3} before the
transducer-shadow corrections. Fortunately, the corrections are small in
magnitude, as shown in Eq. (8); therefore, *u*_{i} is closed to *u*_{Ti}. As a
result, *u*_{x}, *u*_{y}, and *u*_{z} from Eq. (5) are close to those from
Eq. (6). Accordingly, an iteration algorithm may be the right approach to the
corrections using Eq. (8), or for the estimation of *u*_{Ti}.

For the first iteration, *u*_{Ti} in the right side of Eq. (8) could be
replaced with *u*_{i} as its estimation. Given that *U*_{T} should be
calculated using *u*_{x}, *u*_{y}, and *u*_{z} from Eq. (6), before the
transducer-shadow corrections, *U*_{T} can be estimated using *u*_{x},
*u*_{y}, and *u*_{z} from Eq. (5); see Appendix B: Iteration algorithm for
sonic transducer-shadow corrections. The iterations ensure that the
difference in *u*_{x}, *u*_{y}, or *u*_{z} between the last and previous iterations
are <1 mm s${}^{-\mathrm{1}}\approx \mathrm{1.96}\mathit{\sigma}$ < 1, where *σ* is the maximum precision (i.e., standard
deviation at constant wind) among *u*_{x}, *u*_{y}, and *u*_{z} (Campbell
Scientific Inc., 2018). The *u*_{T1_n},
*u*_{T2_n}, and *u*_{T3_n} from the last
interaction are finally used for Eq. (6) to compute the 3-D wind of
*u*_{x}, *u*_{y}, and *u*_{z} as sonic anemometer output.

As addressed in Eqs. (4) to (6), a sonic anemometer measures *t*_{ui} and
*t*_{di} to calculate the 3-D wind of *u*_{x}, *u*_{y}, and *u*_{z}; therefore,
sonic path lengths (*d*_{i}) in Eq. (4) and transform matrix **A** in Eqs. (5)
and (6) are embedded into the OS of sonic anemometer in the manufacture
processes (see the embedded data for our study sonic anemometer in Appendix
A). If the anemometer was physically deformed in transportation,
installation, or other handling; the sonic path lengths and sonic path
angles must be changed from what they were at the time when *d*_{i} and **A**
were embedded into its OS; therefore, *d*_{i} in Eq. (4) and sonic path
angles reflected by **A** in Eqs. (5) and (6) are no longer valid for this
anemometer. Consequently; the output of *u*_{x}, *u*_{y}, and *u*_{z} still
based on embedded *d*_{i} and **A** from production calibration or recalibration
process are erroneous. To correct the erroneous output *u*_{x}, *u*_{y}, and
*u*_{z} need to be transformed back into *t*_{ui} and *t*_{di} and be
recalculated using *t*_{ui} and *t*_{di} based on the true sonic path lengths
and true sonic path angles at the time when *t*_{ui} and *t*_{di} were
measured in the field by the sonic anemometer physically deformed away from
the
manufacturer's geometrical settings before its field deployment.

For the true sonic path lengths and true sonic path angles, the IRGASON (SN:
1131) was returned to the manufacturer in the way described in Sect. 3.
In the same way as in the manufacture process, the lengths and angles were
remeasured using CMM. The remeasured lengths are denoted by *d*_{Ti} (*i*=1, 2, or 3) and the remeasured angles were used to reconstruct the
transform matrix **A** as **A**_{T} (see Appendix A). Both *d*_{Ti} and **A**_{T} are
used to update the OS of this IRGASON for future field uses and to correct
*u*_{x}, *u*_{y}, *u*_{z} and *T*_{s} (sonic temperature, see Sect. 4.2) that
were outputted in the field before the remeasurements. The correction
procedures are different for the output of *u*_{x}, *u*_{y}, *u*_{z} with or
without transducer-shadow corrections.

Transfer *u*_{x}, *u*_{y}, and *u*_{z} in the 3-D anemometer coordinate system
to the flow speeds along the sonic paths after transducer-shadow
corrections.

$$\begin{array}{}\text{(10)}& \left[\begin{array}{l}{u}_{\mathrm{T}\mathrm{1}\mathit{\_}n}\\ {u}_{\mathrm{T}\mathrm{2}\mathit{\_}n}\\ {u}_{\mathrm{T}\mathrm{3}\mathit{\_}n}\end{array}\right]={\mathbf{A}}^{-\mathrm{1}}\left[\begin{array}{l}{u}_{x}\\ {u}_{y}\\ {u}_{z}\end{array}\right]\end{array}$$

Using Eq. (B5), flow speed along the *i*th sonic path before
transducer-shadow
correction (*u*_{i}) can be expressed as follows:

$$\begin{array}{}\text{(11)}& {u}_{i}={u}_{\mathrm{T}i\mathit{\_}n}\left(\mathrm{0.84}+\mathrm{0.16}{\displaystyle \frac{\sqrt{{U}_{\mathrm{T}}^{\mathrm{2}}-{u}_{Ti\mathit{\_}m}^{\mathrm{2}}}}{{U}_{\mathrm{T}}}}\right),\end{array}$$

where *U*_{T} can be calculated using Eq. (9) and *u*_{Ti_m}
can be reasonably approximated using *u*_{Ti_n} because
*u*_{Ti_m} and *u*_{Ti_n} are close enough to
ensure *u*_{x}, *u*_{y}, and *u*_{z} to converge at their measurement
precision (see Appendix B). Using *u*_{i} and *d*_{i}, the time term inside
the square bracket in Eq. (4) can be recovered as follows:

$$\begin{array}{}\text{(12)}& \left[{\displaystyle \frac{\mathrm{1}}{{t}_{\mathrm{u}i}}}-{\displaystyle \frac{\mathrm{1}}{{t}_{\mathrm{d}i}}}\right]={\displaystyle \frac{\mathrm{2}{u}_{i}}{{d}_{i}}},\end{array}$$

Additionally, according to Eq. (4) and using *d*_{Ti}, the speed of air flow along the
*i*th sonic path can be recalculated as *u*_{ci}:

$$\begin{array}{}\text{(13)}& {u}_{ci}={\displaystyle \frac{{d}_{Ti}}{\mathrm{2}}}\left[{\displaystyle \frac{\mathrm{1}}{{t}_{\mathrm{u}i}}}-{\displaystyle \frac{\mathrm{1}}{{t}_{\mathrm{d}i}}}\right].\end{array}$$

Further replacing *u*_{i} with *u*_{ci} in the iteration algorithm for sonic
transducer-shadow corrections in Appendix B, *u*_{ci} is corrected for
transducer-shadowing as *u*_{cTi_n}. Using Eq. (6), the
recovered vector of 3-D wind in the 3-D anemometer coordinate system
[*u*_{cx} *u*_{cy} *u*_{cz}]^{′} can be expressed as follows:

$$\begin{array}{}\text{(14)}& \left[\begin{array}{l}{u}_{cx}\\ {u}_{cy}\\ {u}_{cz}\end{array}\right]={\mathbf{A}}_{\mathrm{T}}\left[\begin{array}{l}{u}_{cT\mathrm{1}\mathit{\_}n}\\ {u}_{cT\mathrm{2}\mathit{\_}n}\\ {u}_{cT\mathrm{3}\mathit{\_}n}\end{array}\right].\end{array}$$

Transfer *u*_{x}, *u*_{y}, and *u*_{z} in the 3-D anemometer coordinate system
to the flow speeds along individual sonic paths.

$$\begin{array}{}\text{(15)}& \left[\begin{array}{l}{u}_{\mathrm{1}}\\ {u}_{\mathrm{2}}\\ {u}_{\mathrm{3}}\end{array}\right]={\mathbf{A}}^{-\mathrm{1}}\left[\begin{array}{l}{u}_{x}\\ {u}_{y}\\ {u}_{z}\end{array}\right]\end{array}$$

Using Eqs. (12) and (13), the speed of flow along the *i*th sonic path
(*u*_{ci}) is recalculated (i.e., recovered). Based on Eq. (5), the recovered
speeds of flow along the three sonic paths can be expressed in the 3-D
anemometer coordinate system as follows:

$$\begin{array}{}\text{(16)}& \left[\begin{array}{l}{u}_{cx}\\ {u}_{cy}\\ {u}_{cz}\end{array}\right]={\mathbf{A}}_{\mathrm{T}}\left[\begin{array}{l}{u}_{c\mathrm{1}}\\ {u}_{c\mathrm{2}}\\ {u}_{c\mathrm{3}}\end{array}\right].\end{array}$$

Equations (1) and (2) also lead to

$$\begin{array}{}\text{(17)}& {c}_{\mathrm{3}}={\displaystyle \frac{{d}_{\mathrm{3}}}{\mathrm{2}}}\left[{\displaystyle \frac{\mathrm{1}}{{t}_{u\mathrm{3}}}}+{\displaystyle \frac{\mathrm{1}}{{t}_{d\mathrm{3}}}}\right].\end{array}$$

Using the same procedure, *c*_{1} and *c*_{2} (see Figs. 1 and 5) can be
derived as the same form. In reference to Eq. (17), the equation for *c*_{i},
where subscript *i*=1, 2, or 3; can be expressed as follows:

$$\begin{array}{}\text{(18)}& {c}_{i}={\displaystyle \frac{{d}_{i}}{\mathrm{2}}}\left[{\displaystyle \frac{\mathrm{1}}{{t}_{\mathrm{u}i}}}+{\displaystyle \frac{\mathrm{1}}{{t}_{\mathrm{d}i}}}\right]\end{array}$$

Here, *c*_{i} is the measured speed of sound along the sonic path *i* (see Fig. 5). When the crosswind (*u*_{⊥i}), or wind normal to the sonic path *i*, is
zero; *c*_{i} is the true speed of sound (*c*_{0i} where subscript 0 indicates
the speed of sound at crosswind speed equal to zero). Unfortunately,
crosswind is rarely zero and *c*_{i} needs to be corrected to *c*_{0i}.
According to Figs. 1 and 5, the true speed of sound is given by

$$\begin{array}{}\text{(19)}& {c}_{\mathrm{0}i}={\displaystyle \frac{{c}_{i}}{\mathrm{cos}{\mathit{\alpha}}_{i}}}={\displaystyle \frac{{c}_{i}}{{c}_{i}/\sqrt{{c}_{i}^{\mathrm{2}}+{u}_{\perp i}^{\mathrm{2}}}}}=\sqrt{{c}_{i}^{\mathrm{2}}+{u}_{\perp i}^{\mathrm{2}}}.\end{array}$$

Referencing the diagram for wind vectors in the left side of Fig. 5, this equation can be expressed as follows:

$$\begin{array}{}\text{(20)}& {c}_{\mathrm{0}i}^{\mathrm{2}}={c}_{i}^{\mathrm{2}}+{U}_{\mathrm{T}}^{\mathrm{2}}-{u}_{Ti}^{\mathrm{2}},\end{array}$$

According to the definition of sonic temperature (Kaimal and Finnigan,
1994), the sonic temperature (K) along the *i*th sonic path (*T*_{si})
should be expressed as follows:

$$\begin{array}{}\text{(21)}& {T}_{\mathrm{s}i}={\displaystyle \frac{{c}_{\mathrm{0}i}^{\mathrm{2}}}{{\mathit{\gamma}}_{\mathrm{d}}{R}_{\mathrm{d}}}},\end{array}$$

where *γ*_{d} (1.4003) is the ratio of dry-air-specific heat at
constant pressure (1004 J K^{−1} kg^{−1}) to dry-air-specific heat at
constant volume (717 J K^{−1} kg^{−1}) and *R*_{d} is gas constant for
dry air (287.04 J K^{−1} kg^{−1}). The sonic temperature outputted
from the
sonic anemometer (*T*_{s} in ^{∘}C) is the average from the three
sonic paths (van Dijk, 2002), given by

$$\begin{array}{}\text{(22)}& {T}_{\mathrm{s}}={\displaystyle \frac{\mathrm{1}}{\mathrm{3}}}\sum _{i=\mathrm{1}}^{\mathrm{3}}{T}_{\mathrm{s}i}-\mathrm{273.15}={\displaystyle \frac{\mathrm{1}}{\mathrm{3}{\mathit{\gamma}}_{d}{R}_{\mathrm{d}}}}\sum _{i=\mathrm{1}}^{\mathrm{3}}{c}_{\mathrm{0}i}^{\mathrm{2}}-\mathrm{273.15}.\end{array}$$

Substituting *c*_{0i} with Eq. (20) and then substituting *c*_{i} with Eq. (18), *T*_{s} can be expressed as follows:

$$\begin{array}{ll}{\displaystyle}{T}_{\mathrm{s}}& {\displaystyle}={\displaystyle \frac{\mathrm{1}}{\mathrm{3}{\mathit{\gamma}}_{\mathrm{d}}{R}_{\mathrm{d}}}}\left\{\sum _{i=\mathrm{1}}^{\mathrm{3}}\left[{\displaystyle \frac{{d}_{i}^{\mathrm{2}}}{\mathrm{4}}}{\left({\displaystyle \frac{\mathrm{1}}{{t}_{\mathrm{u}i}}}+{\displaystyle \frac{\mathrm{1}}{{t}_{\mathrm{d}i}}}\right)}^{\mathrm{2}}-{u}_{Ti}^{\mathrm{2}}\right]\right.\\ \text{(23)}& {\displaystyle}& {\displaystyle}\left.+\mathrm{3}{U}_{\mathrm{T}}^{\mathrm{2}}\right\}-\mathrm{273.15}.\end{array}$$

Equation (23) indicates that, given *d*_{i}, a sonic anemometer estimates
sonic temperature using its measured transmitting time of *t*_{ui} and
*t*_{di}, the flow speeds along the sonic paths (*u*_{i} or *u*_{Ti} if
corrected for transducer shadowing) that are also calculated from *t*_{ui}
and *t*_{di} (see Eq. 4), and the resultant wind speed (*U*_{T}, i.e., the
total wind) computed using Eq. (9), inside which the three wind components in
the 3-D anemometer coordinate system are transformed from *u*_{i} using
**A**, as explained by Eq. (5), without transducer-shadow corrections or from
*u*_{Ti} also using **A** as explained by Eq. (6), with transducer-shadow
corrections. As discussed in Sect. 4.1.2, when a sonic anemometer is
geometrically deformed in an incident, the sonic path lengths and sonic path
angles may be changed from what they were at the time when *d*_{i} and **A** were
embedded into its OS; therefore, *d*_{i} in Eq. (23) and **A** in Eqs. (5) and (6) for *u*_{i}∕*u*_{Ti} and
*U*_{T} in Eq. (23) are no longer valid for this
sonic anemometer. As a result, its output of *u*_{x}, *u*_{y}, *u*_{z}, and
*T*_{s} still based on embedded *d*_{i} and **A** must not be representative to
the field wind and sonic temperature to be measured. In Sect. 4.1, the
procedure to recover 3-D wind data was developed using remeasured sonic path
lengths (*d*_{Ti}) and redetermined sonic path angles for **A**_{T}. The
procedure to recover sonic temperature data also needs to be developed using
*d*_{Ti} and recovered 3-D wind data in this section.

Based on Eq. (20), the recovered speed of sound from the sonic path *i* after
crosswind corrections (*c*_{c0i}) can be expressed as follows:

$$\begin{array}{}\text{(24)}& {c}_{c\mathrm{0}i}^{\mathrm{2}}={c}_{ci}^{\mathrm{2}}+{U}_{cT}^{\mathrm{2}}-{u}_{cTi}^{\mathrm{2}},\end{array}$$

where *c*_{ci} is the recovered speed of sound along sonic path *i* and ${U}_{cT}=\sqrt{{u}_{cx}^{\mathrm{2}}+{u}_{cy}^{\mathrm{2}}+{u}_{cz}^{\mathrm{2}}}$. After replacement
of ${c}_{\mathrm{0}i}^{\mathrm{2}}$ with ${c}_{c\mathrm{0}i}^{\mathrm{2}}$ in Eq. (22), the recovered sonic
temperature (*T*_{cs} in ^{∘}C) can be written as follows:

$$\begin{array}{}\text{(25)}& {T}_{cs}={\displaystyle \frac{\mathrm{1}}{\mathrm{3}{\mathit{\gamma}}_{\mathrm{d}}{R}_{\mathrm{d}}}}\sum _{i=\mathrm{1}}^{\mathrm{3}}{c}_{c\mathrm{0}i}^{\mathrm{2}}-\mathrm{273.15}.\end{array}$$

Now, the term of ${c}_{c\mathrm{0}i}^{\mathrm{2}}$ needs to be derived. Subtracting Eq. (20) from (24) leads to

$$\begin{array}{}\text{(26)}& {c}_{c\mathrm{0}i}^{\mathrm{2}}={c}_{\mathrm{0}i}^{\mathrm{2}}+\left({c}_{ci}^{\mathrm{2}}-{c}_{i}^{\mathrm{2}}\right)+\left({U}_{cT}^{\mathrm{2}}-{U}_{\mathrm{T}}^{\mathrm{2}}\right)-\left({u}_{cTi}^{\mathrm{2}}-{u}_{Ti}^{\mathrm{2}}\right).\end{array}$$

Using this equation to substitute ${c}_{c\mathrm{0}i}^{\mathrm{2}}$ in Eq. (25), denoting ${U}_{cT}^{\mathrm{2}}-{U}_{\mathrm{T}}^{\mathrm{2}}$ with $\mathrm{\Delta}{U}_{cT}^{\mathrm{2}}$ and denoting ${u}_{cTi}^{\mathrm{2}}-{u}_{Ti}^{\mathrm{2}}$ with $\mathrm{\Delta}{u}_{cTi}^{\mathrm{2}}$ leads to

$$\begin{array}{}\text{(27)}& {T}_{cs}={T}_{\mathrm{s}}+{\displaystyle \frac{\mathrm{1}}{\mathrm{3}{\mathit{\gamma}}_{\mathrm{d}}{R}_{\mathrm{d}}}}\sum _{i=\mathrm{1}}^{\mathrm{3}}\left[\left({c}_{ci}^{\mathrm{2}}-{c}_{i}^{\mathrm{2}}\right)+\mathrm{\Delta}{U}_{cT}^{\mathrm{2}}-\mathrm{\Delta}{u}_{cTi}^{\mathrm{2}}\right].\end{array}$$

In this equation, the term of ${c}_{ci}^{\mathrm{2}}-{c}_{i}^{\mathrm{2}}$ is still unknown. Based on Eq. (18), ${c}_{ci}^{\mathrm{2}}$is given by

$$\begin{array}{}\text{(28)}& {c}_{ci}^{\mathrm{2}}={\displaystyle \frac{{d}_{Ti}^{\mathrm{2}}}{\mathrm{4}}}{\left[{\displaystyle \frac{\mathrm{1}}{{t}_{\mathrm{u}i}}}+{\displaystyle \frac{\mathrm{1}}{{t}_{\mathrm{d}i}}}\right]}^{\mathrm{2}}.\end{array}$$

Accordingly, the unknown term is given by

$$\begin{array}{ll}{\displaystyle}{c}_{ci}^{\mathrm{2}}-{c}_{i}^{\mathrm{2}}& {\displaystyle}={\displaystyle \frac{{d}_{Ti}^{\mathrm{2}}}{\mathrm{4}}}{\left[{\displaystyle \frac{\mathrm{1}}{{t}_{\mathrm{u}i}}}+{\displaystyle \frac{\mathrm{1}}{{t}_{\mathrm{d}i}}}\right]}^{\mathrm{2}}-{\displaystyle \frac{{d}_{i}^{\mathrm{2}}}{\mathrm{4}}}{\left[{\displaystyle \frac{\mathrm{1}}{{t}_{\mathrm{u}i}}}+{\displaystyle \frac{\mathrm{1}}{{t}_{\mathrm{d}i}}}\right]}^{\mathrm{2}}\\ \text{(29)}& {\displaystyle}& {\displaystyle}={\displaystyle \frac{\mathrm{1}}{\mathrm{4}}}{\left[{\displaystyle \frac{\mathrm{1}}{{t}_{\mathrm{u}i}}}+{\displaystyle \frac{\mathrm{1}}{{t}_{\mathrm{d}i}}}\right]}^{\mathrm{2}}\left({d}_{Ti}^{\mathrm{2}}-{d}_{i}^{\mathrm{2}}\right)={c}_{i}^{\mathrm{2}}{\displaystyle \frac{\mathrm{\Delta}{d}_{Ti}^{\mathrm{2}}}{{d}_{i}^{\mathrm{2}}}}.\end{array}$$

In this equation, the only unknown variable is ${c}_{i}^{\mathrm{2}}$. Based on Eq. (20), this equation can be expressed as follows:

$$\begin{array}{}\text{(30)}& {c}_{ci}^{\mathrm{2}}-{c}_{i}^{\mathrm{2}}=\left({c}_{\mathrm{0}i}^{\mathrm{2}}-{U}_{\mathrm{T}}^{\mathrm{2}}+{u}_{Ti}^{\mathrm{2}}\right){\displaystyle \frac{\mathrm{\Delta}{d}_{Ti}^{\mathrm{2}}}{{d}_{i}^{\mathrm{2}}}}.\end{array}$$

In the right side of this equation, ${c}_{\mathrm{0}i}^{\mathrm{2}}$ is the only unknown. However,
the whole term in the right side of Eq. (30) mathematically is a
differential term in which ${c}_{\mathrm{0}i}^{\mathrm{2}}$can be reasonably approximated using
its neighbor value, as close as possible to ${c}_{\mathrm{0}i}^{\mathrm{2}}$. The average of
${c}_{\mathrm{01}}^{\mathrm{2}},{c}_{\mathrm{02}}^{\mathrm{2}}$, and ${c}_{\mathrm{03}}^{\mathrm{2}}$ can be calculated from Eq. (22) because *T*_{s} is an output variable
of the
sonic anemometer. Without a measurement error and random error, the three
*c*_{0i} should be the same, independent of flow speed, because they are
the true speed of sound instead of measured speed of sound along an
individual sonic path (Schotanus et al., 1983; Liu et al., 2001); Therefore,
${c}_{\mathrm{0}i}^{\mathrm{2}}$ can be reasonably approximated using the average of three
${c}_{\mathrm{0}i}^{\mathrm{2}}$ as ${c}_{\mathrm{0}}^{\mathrm{2}}$, given by

$$\begin{array}{}\text{(31)}& {c}_{ci}^{\mathrm{2}}-{c}_{i}^{\mathrm{2}}=\left({c}_{\mathrm{0}}^{\mathrm{2}}-{U}_{\mathrm{T}}^{\mathrm{2}}+{u}_{Ti}^{\mathrm{2}}\right){\displaystyle \frac{\mathrm{\Delta}{d}_{Ti}^{\mathrm{2}}}{{d}_{i}^{\mathrm{2}}}},\end{array}$$

where ${c}_{\mathrm{0}}^{\mathrm{2}}$ can be computed from Eq. (22) as follows:

$$\begin{array}{}\text{(32)}& {c}_{\mathrm{0}}^{\mathrm{2}}={\mathit{\gamma}}_{\mathrm{d}}{R}_{\mathrm{d}}\left({T}_{\mathrm{s}}+\mathrm{273.15}\right).\end{array}$$

Due to the replacement of ${c}_{\mathrm{0}i}^{\mathrm{2}}$ with ${c}_{\mathrm{0}}^{\mathrm{2}}$, the relative error
of the whole term in the right side of Eq. (31) would be < 4 %, even if
the variability in sonic temperature due to the difference among ${c}_{\mathrm{0}i}^{\mathrm{2}}$ values reaches 10 ^{∘}C at an air temperature of −30 ^{∘}C
without wind (i.e., *U*_{T}=0 and *u*_{Ti}=0), which would be the worst
case. Substituting the term of ${c}_{ci}^{\mathrm{2}}-{c}_{i}^{\mathrm{2}}$ in Eq. (27) with Eq. (31) leads to

$$\begin{array}{ll}{\displaystyle}{T}_{cs}& {\displaystyle}={T}_{\mathrm{s}}+{\displaystyle \frac{\mathrm{1}}{\mathrm{3}{\mathit{\gamma}}_{\mathrm{d}}{R}_{\mathrm{d}}}}\sum _{i=\mathrm{1}}^{\mathrm{3}}\left[\left({c}_{\mathrm{0}}^{\mathrm{2}}-{U}_{\mathrm{T}}^{\mathrm{2}}+{u}_{Ti}^{\mathrm{2}}\right){\displaystyle \frac{\mathrm{\Delta}{d}_{Ti}^{\mathrm{2}}}{{d}_{i}^{\mathrm{2}}}}\right.\\ \text{(33)}& {\displaystyle}& {\displaystyle}\left.+\mathrm{\Delta}{U}_{cT}^{\mathrm{2}}-\mathrm{\Delta}{u}_{cTi}^{\mathrm{2}}\right].\end{array}$$

In the right side of this equation, the whole term after *T*_{s} is the sonic
temperature recovery term.

5 Application

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For our case without a transducer-shadow correction, Eqs. (15), (12), (13), and (16) were sequentially used to recover the 3-D wind data. In a case of transducer-shadow correction in option, Eqs. (10) to (16) are used. Based on the data of 3-D wind from the recovery process, Eqs. (9), (32), and (33) were used to recover the sonic temperature data. The whole recovery processes large data files (10 records per second), not only using these equations, but also operating the matrixes (A3) to (A5) (see Appendix A) for Eqs. (15) and (16) along with the data of sonic paths lengths in Table A1 for Eqs. (12) and (13). Apparently, the recovery process is a huge work load in computation. As such, these equations, matrixes, and data were implemented into a software package: “Sonic Data Recovery for IRGASON/CSAT3/A/B Used in Geometrical Deformation after Production/Calibration” (Appendix C and Fig. 6). As long as the path lengths and matrixes from production/calibration and from recalibration are input into the software as instructed by the interface (Appendix C), the software automatically recover the data in batches.

6 Verification

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In our station, an additional anemometer for wind was not under deployment when this study IRGASON was used in its deformed state; therefore, no data were available to verify the recovered 3-D wind data. However, the algorithms as addressed using Eqs. (10) to (16) to recover the 3-D wind data are solid without any estimation and the recovered 3-D wind data are not necessary for verification.

Fortunately, the data to verify sonic temperature are available in this
station. Air temperature, relative humidity, and atmospheric pressure were
measured using research grade sensors of the HMP155A and IRGASON built-in
barometer and the data of these variables also stored at 10 Hz (10 records
per second). These data can be used to estimate the sonic temperature (see
Appendix D: Sonic temperature from air temperature, relative humidity, and
atmospheric pressure). The recovered data of sonic temperature using Eq. (33) were compared to the calculated sonic temperature over the range of
sonic temperature for three representative values: $-\mathrm{20.01}\pm \mathrm{0.14}$ ^{∘}C in Fig. 7a, $-\mathrm{9.06}\pm \mathrm{0.13}$ ^{∘}C in Fig. 7b, and
$-\mathrm{1.90}\pm \mathrm{0.22}$ ^{∘}C in Fig. 7c. The difference between measured
(i.e., unrecovered) and calculated sonic temperature values of 9.60±0.14 K in Fig. 7a, 9.53±0.17 K in Fig. 7b, and 8.93±0.24 K
in Fig. 7c was narrowed to 0.99±0.14 K, 0.57±0.17 K, and
$-\mathrm{0.25}\pm \mathrm{0.24}$ K, respectively, as the difference between recovered and
calculated sonic temperature values. Given the accuracy of ±0.5 K in
sonic temperature from the IRGASON sonic anemometer (Personal communication with
Larry Jacobsen, the designer of the sonic anemometer, 2017) and the accuracy of
±0.2 ∼0.3 K in air temperature below 0 ^{∘}C
and 1.2 % in relative humidity from HMP155A (Vaisala Corp., 2017), from which
the calculated sonic temperature was derived (see Appendix D), recovered
sonic temperature data can be reasonably judged as satisfactory if the
difference in mean sonic temperature between recovered and calculated ranges
within ±0.80 K or even wider, which could be considered a
likelihood range of possible difference between correctly measured and
calculated sonic temperature. As shown in Fig. 7, Eq. (33) apparently did an
excellent job in recovering the sonic temperature data measured using sonic
anemometer in its deformed state, but was less satisfactory in the case of Fig. 7a (i.e., 0.99±0.14 K, the difference in sonic
temperature between recovered and calculated) although the range of 0.99±0.14 K was not significantly different from
±0.80 K. The less satisfactory recovery might be caused by the
approximation of *c*_{0i} from *c*_{0} that is fully valid if all *c*_{0i} are
not measured by a sonic anemometer in its deformed state, but this is not the case in
this study.

According to Eq. (22), it is impossible to have an individual *c*_{0i} from
*T*_{s}, which is the sole output for sonic temperature from any sonic
anemometer. Now, the average of ${c}_{\mathrm{01}}^{\mathrm{2}},{c}_{\mathrm{02}}^{\mathrm{2}}$, and ${c}_{\mathrm{03}}^{\mathrm{2}}$ is known and the changes in sonic
path lengths are known. It is possible to estimate the difference among the
three speeds of sound and to adjust their average (${c}_{\mathrm{0}}^{\mathrm{2}})$
to ${c}_{\mathrm{01}}^{\mathrm{2}},{c}_{\mathrm{02}}^{\mathrm{2}}$, and ${c}_{\mathrm{03}}^{\mathrm{2}}$ in approximation, although the exact values are
impossible to determine. The adjusted values can reflect the variability among
${c}_{\mathrm{0}i}^{\mathrm{2}}$ to some degree and are reasonably expected to improve the data
recovery.

7 Adjustment

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The measured speed of sound after crosswind correction (*c*_{0i}) is
independent of wind speed (Schotanus et al., 1983; Liu et al., 2001) while
depending on moist air density and atmospheric pressure (Barrett and Suomi,
1949). Without wind, *c*_{0i} is equal to the measured speed of sound
(*c*_{i}) from sonic path *i* (see Eq. 19). In this case, again without wind,
*t*_{ui} and *t*_{di} in Eq. (18) are the same and can be denoted by
*t*_{i}. Accordingly, Eq. (18) in this case is equivalent to

$$\begin{array}{}\text{(34)}& {c}_{\mathrm{0}i}\equiv {\displaystyle \frac{{d}_{i}}{{t}_{i}}}.\end{array}$$

In Eq. (33), ${c}_{\mathrm{0}}^{\mathrm{2}}$ is the average of three squared *c*_{0i} (see Eqs. 22 and 32), but an individual *c*_{0i} is unknown; therefore, for
recovery improvement, it has to be estimated from ${c}_{\mathrm{0}}^{\mathrm{2}}$ through a
reasonable adjustment. The difference in magnitude between ${c}_{\mathrm{0}}^{\mathrm{2}}$ and
${c}_{\mathrm{0}i}^{\mathrm{2}}$ must be related to the ${c}_{\mathrm{0}i}^{\mathrm{2}}$ error due to the
geometrical deformation of sonic anemometer. Squaring both sides of Eq. (34)
leads to

$$\begin{array}{}\text{(35)}& {c}_{\mathrm{0}i}^{\mathrm{2}}={\displaystyle \frac{{d}_{i}^{\mathrm{2}}}{{t}_{i}^{\mathrm{2}}}}.\end{array}$$

The total differentiation of ${c}_{\mathrm{0}i}^{\mathrm{2}}$ is given by

$$\begin{array}{}\text{(36)}& \mathrm{\Delta}{c}_{\mathrm{0}i}^{\mathrm{2}}={\displaystyle \frac{\mathrm{2}{d}_{i}}{{t}_{i}^{\mathrm{2}}}}\mathrm{\Delta}{d}_{i}-{\displaystyle \frac{\mathrm{2}{d}_{i}^{\mathrm{2}}}{{t}_{i}^{\mathrm{3}}}}\mathrm{\Delta}{t}_{i}.\end{array}$$

Given the transmitting time is correctly measured by a sonic anemometer
(i.e., Δ*t*_{i}=0) even in its geometrical deformation, this equation
becomes

$$\begin{array}{}\text{(37)}& \mathrm{\Delta}{c}_{\mathrm{0}i}^{\mathrm{2}}={\displaystyle \frac{\mathrm{2}{d}_{i}}{{t}_{i}^{\mathrm{2}}}}\mathrm{\Delta}{d}_{i}={c}_{\mathrm{0}i}^{\mathrm{2}}{\displaystyle \frac{\mathrm{2}\mathrm{\Delta}{d}_{i}}{{d}_{i}}}={c}_{\mathrm{0}i}^{\mathrm{2}}{\displaystyle \frac{\mathrm{2}\left({d}_{i}-{d}_{Ti}\right)}{{d}_{i}}}.\end{array}$$

Mathematically in differentiation, ${c}_{\mathrm{0}i}^{\mathrm{2}}$ can be reasonably
approximated by *c*_{0}, given by

$$\begin{array}{}\text{(38)}& \mathrm{\Delta}{c}_{\mathrm{0}i}^{\mathrm{2}}\approx \mathrm{2}{c}_{\mathrm{0}}^{\mathrm{2}}\left(\mathrm{1}-{\displaystyle \frac{{d}_{Ti}}{{d}_{i}}}\right)\end{array}$$

This is the error of ${c}_{\mathrm{0}i}^{\mathrm{2}}$ away from ${c}_{\mathrm{0}}^{\mathrm{2}}$. This error can be reasonably used to represent the deviation of ${c}_{\mathrm{0}i}^{\mathrm{2}}$ away from ${c}_{\mathrm{0}}^{\mathrm{2}}$. The deviations of three ${c}_{\mathrm{0}i}^{\mathrm{2}}$ values away from ${c}_{\mathrm{0}}^{\mathrm{2}}$ are the measures of variability among three ${c}_{\mathrm{0}i}^{\mathrm{2}}$ away from ${c}_{\mathrm{0}}^{\mathrm{2}}$.

Although an individual ${c}_{\mathrm{0}i}^{\mathrm{2}}$ is unknown, the average of three ${c}_{\mathrm{0}i}^{\mathrm{2}}$ is known as ${c}_{\mathrm{0}}^{\mathrm{2}}$. This average should be unchanged after adjustments because of the adjustment within the variability among ${c}_{\mathrm{0}i}^{\mathrm{2}}$ away from ${c}_{\mathrm{0}}^{\mathrm{2}}$. If the average of adjusted ${c}_{\mathrm{0}i}^{\mathrm{2}}$ is not equal to ${c}_{\mathrm{0}}^{\mathrm{2}}$, all adjusted ${c}_{\mathrm{0}i}^{\mathrm{2}}$ should be added or subtracted with the same constant to make the average of three adjusted ${c}_{\mathrm{0}i}^{\mathrm{2}}$ values as ${c}_{\mathrm{0}}^{\mathrm{2}}$, but the variability among ${c}_{\mathrm{0}i}^{\mathrm{2}}$ values is kept the same. This constant must be the mean of three $\mathrm{\Delta}{c}_{\mathrm{0}i}^{\mathrm{2}}$ values. Based on these analyses, the adjustment of ${c}_{\mathrm{0}}^{\mathrm{2}}$ to ${c}_{\mathrm{0}i}^{\mathrm{2}}$ can be constructed as follows:

$$\begin{array}{}\text{(39)}& {c}_{\mathrm{0}i}^{\mathrm{2}}\equiv {c}_{\mathrm{0}}^{\mathrm{2}}+\left(\mathrm{\Delta}{c}_{\mathrm{0}i}^{\mathrm{2}}-{\displaystyle \frac{\mathrm{1}}{\mathrm{3}}}\sum _{i=\mathrm{1}}^{\mathrm{3}}\mathrm{\Delta}{c}_{\mathrm{0}i}^{\mathrm{2}}\right).\end{array}$$

Using this equation to replace ${c}_{\mathrm{0}i}^{\mathrm{2}}$ in Eq. (30) and the resultant equation with this replacement then is used for ${c}_{ci}^{\mathrm{2}}-{c}_{i}^{\mathrm{2}}$ in Eq. (27) as follows:

$$\begin{array}{ll}{\displaystyle}{T}_{cs}& {\displaystyle}={T}_{\mathrm{s}}+{\displaystyle \frac{\mathrm{1}}{\mathrm{3}{\mathit{\gamma}}_{\mathrm{d}}{R}_{\mathrm{d}}}}\sum _{i=\mathrm{1}}^{\mathrm{3}}\left\{\left[{c}_{\mathrm{0}}^{\mathrm{2}}+\left(\mathrm{\Delta}{c}_{\mathrm{0}i}^{\mathrm{2}}-{\displaystyle \frac{\mathrm{1}}{\mathrm{3}}}\sum _{j=\mathrm{1}}^{\mathrm{3}}\mathrm{\Delta}{c}_{\mathrm{0}j}^{\mathrm{2}}\right)\right.\right.\\ \text{(40)}& {\displaystyle}& {\displaystyle}\left.\left.-{U}_{\mathrm{T}}^{\mathrm{2}}+{u}_{Ti}^{\mathrm{2}}\right]{\displaystyle \frac{\mathrm{\Delta}{d}_{Ti}^{\mathrm{2}}}{{d}_{i}^{\mathrm{2}}}}+\mathrm{\Delta}{U}_{cT}^{\mathrm{2}}-\mathrm{\Delta}{u}_{cTi}^{\mathrm{2}}\right\}.\end{array}$$

In the right side of this equation, the whole term after *T*_{s} is the
adjusted sonic temperature recovery term.

The data recovered using Eq. (33) were recovered again using Eq. (40). Apparently, this equation did a better job than Eq. (33). The difference in sonic temperature between the recovered and calculated values was reduced to 0.81±0.14, 0.38±0.17, and $-\mathrm{0.45}\pm \mathrm{0.24}$ K, respectively, as shown from panels a to c in Fig. 7. These values for the difference fell into the range of ±0.80 K in a statistical sense. Eventually, Eq. (40) was used for data recovery and was incorporated into the software (Appendix C).

8 Discussion

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Although not explicitly verified, the recovered 3-D wind data were implicitly verified through the verification of recovered sonic temperature data because (1) sonic temperature is more sensitive than wind speeds in ultrasonic sonic measurements (Thomas Foken, 2018, review comment for this publication) and (2) the recovery of sonic temperature data must rely on recovered 3-D wind data (Eqs. 33 and 40). According to Eqs. (3), (17), and (21), it is apparent that sonic temperature is sensitive to one order higher than wind speed to the errors in measurements of sonic path lengths and ultrasonic signal transmitting time values. If the recovered sonic temperature is within the accuracy limits of sensors, this should be realized for the wind data recovery as well (Thomas Foken 2018, review comment for this publication). Additionally, the cross wind correction for sonic temperature needs 3-D wind data (Liu et al., 2001). If 3-D wind had not been well recovered, sonic temperature data could not have been recovered satisfactorily. Therefore, the satisfactory recovery of sonic temperature data in this study implicitly verified the satisfactory recovery of 3-D wind data.

The recovered sonic temperature was sourced from the measurements of a fast response sonic anemometer, and the calculated sonic temperature was sourced from the measurements of a slow response air temperature and relative humidity probe as well as a barometer built into the IRGASON (see Appendix D). Therefore, the former reflected the fluctuations in the sonic temperature at high frequency, and the latter reflects the same fluctuations at lower frequency. As such, a pair of recovered and calculated sonic temperature values from simultaneous measurements (i.e., the same records in a time series data file) were not comparable. The difference between the pair is meaningless; therefore, the mean difference between recovered and calculated sonic temperature values over a half-hour period was used for their data comparison.

See Fig. 7. Compared to calculated sonic temperature, the recovered sonic
temperature from Eq. (40) is 0.81±0.14 K higher at −20.01 ^{∘}C (Fig. 7a) and
0.38±0.17 K higher at −9.06 ^{∘}C (Fig. 7b); however, at −1.90 ^{∘}C, even $-\mathrm{0.45}\pm \mathrm{0.24}$ K lower
(Fig. 7c). This trend of difference with temperature may be related to the
performance of sonic anemometer at different temperature and the lower
accuracy of temperature and humidity probe in a lower temperature range
(Vaisala Corp., 2017).

The sonic path lengths and geometry of the sonic anemometer were measured in the
manufacture environment of an air temperature around 20 ^{∘}C (i.e.,
manufacture temperature) and embedded into its OS for field applications.
However, above or below the manufacture temperature, the sonic path lengths
must become, due to thermo-expansion or -contraction of sonic anemometer
structure, longer or shorter than those at the manufacture temperature while the
length values of sonic paths inside the OS are unchanged. As a result, the
sonic anemometer could under- or overestimate the speed of sound, thus
sonic temperature. The under- or overestimation may be insignificant when
temperature is not much above or below the manufacture temperature while the
anemometer must work best around the manufacturer temperature. In this
study, the working air temperature for the sonic anemometer was as low as
−20 ^{∘}C, within which the sonic paths become shorter to some
degree so that its measurement performance was possibly impacted. Although
an assessment on the measurement performance of sonic anemometer at low or
high air temperature could not be found in literature, overestimation of the
speed of sound from a sonic anemometer at temperatures dozens of degrees below the
manufacture temperature and thus sonic temperature is anticipated as shown
in Fig. 7a to c.

However, at different air temperature the performance of the temperature
and relative humidity probe and barometer built into the IRGASON, whose
measurements are used to calculate the sonic temperature (see Appendix D),
more stable than a sonic anemometer while their accuracies are the best
at 20 ^{∘}C and become lower with temperature away from 20 ^{∘}C (Vaisala Corp., 2017). For example, HMP155A has an accuracy in air
temperature to be ±0.1 ^{∘}C at 20 ^{∘}C and
±0.25 ^{∘}C at −20 ^{∘}C, as well as an accuracy in
relative humidity (RH) of $\pm (\mathrm{1.0}+\mathrm{0.008}$RH) % at 20 ^{∘}C
and to be $\pm (\mathrm{1.2}+\mathrm{0.012}$RH) % at −20 ^{∘}C. The greater
disagreement between recovered and calculated sonic temperature values at
lower temperature in Fig. 7a may also be due to the fact that the
lower the air temperature, the lower the accuracies of HMP155A and the
barometer.

Compared to the recovered sonic temperature using Eq. (40), the calculated
sonic temperature was 0.45±0.24 ^{∘}C higher over a whole
period of 12:00 to 12:30 and even 0.65±0.19 ^{∘}C higher
over a partial period of 12:15 to 12:27, which may be contributed to in part by
higher incoming solar radiation of 750 W m^{−2} in short-wave on the
radiation shield of HMP155A (Fig. 7c). As addressed in Appendix D, the
calculated sonic temperature was sourced from the measurements of air
temperature and relative humidity from HMP155A, as well as atmospheric
pressure from the barometer built into the IRGASON. The HMP155A housed inside a
radiation shield (Fig. 2) was subject to contamination from solar radiation.
A radiation shield was used to shade HMP155A from sunlight, when such a
shield was used, any heat generated from the shield under sunlight and the
sensor under electronic power was dissipated inefficiently (Lin et al.,
2001). As a result, the air and HMP155A sensing elements inside the shield
were warmer than ambient air of interest. How warm the air is inside the
radiation shield depended on shield structure, ambient wind speed, and other
environmental conditions (Blonquist et al., 2009). In the case of Fig. 7c at
750 W m^{−2} of incoming short-wave radiation, air being a degree warmer
inside the radiation shield was not unusual (Lin et al., 2001). In our
study, this higher air temperature could directly cause the overestimation
of calculated sonic temperature (Eq. D1 in Appendix D).

A geometrically deformed sonic anemometer outputs erroneous data. These data may be recoverable or unrecoverable, depending on the degree of deformation. If the degree is too large, the sonic anemometer cannot perform its normal measurements for the transmitting time. In this case, a Campbell sonic anemometer sets high for one to six of its first six measurement warning flags (low amplitude, high amplitude, poor signal lock, large sonic temperature difference, ultrasonic signal loss, and calibration signature error; see Table 10-2 in Campbell Scientific Inc., 2018). The geometrical deformation in sonic paths could trigger one or two flags high that indicate poor signal lock and/or ultrasonic signal loss. Regardless, in the case that any of the six warning flags from a deformed sonic anemometer were frequently, regularly, or continuously high, the erroneous data must not be recoverable (i.e., data recovery is not possible). While all six warning flags are low under normal measurement conditions, the transmitting time of ultrasonic signals along each sonic path is correctly measured and the data should be recoverable. The 3-D wind data can be recovered without uncertainty although there is little uncertainty in sonic temperature (see Eqs. 33 and 40). The subsequent question is the necessity to recover the recoverable data.

A sonic anemometer is used primarily for the fluxes of momentum and heat from the fluctuations in 3-D wind speeds and sonic temperature. If the fluctuations are not significantly influenced by the geometric deformation of the sonic anemometer, the data from this anemometer may not need recovering although the data are recoverable. The fluctuations in a wind speed component or sonic temperature are measured by variance. Therefore, this influence of sonic anemometer deformation on fluctuations in wind speed and sonic temperature can be tested through analyzing the homogeneity in variance of each wind component and sonic temperature between unrecovered and recovered data.

For this study case, the 2-day data, without a missing record or any high warning flag from 10 and 11 May 2015, were used for the analyses. After data recovery processing (Fig. 6), two datasets, unrecovered and recovered, were acquired. In the unrecovered dataset, for each wind speed component or sonic temperature, the data of 18 000 values from each half-hour were used to compute its variance (${s}_{k}^{\mathrm{2}})$, given by

$$\begin{array}{}\text{(41)}& {s}_{k}^{\mathrm{2}}={\displaystyle \frac{\mathrm{1}}{\mathrm{18}\phantom{\rule{0.125em}{0ex}}\mathrm{000}}}\sum _{j=\mathrm{1}}^{\mathrm{18}\phantom{\rule{0.125em}{0ex}}\mathrm{000}}{\left({x}_{kj}-{\overline{x}}_{k}\right)}^{\mathrm{2}},\end{array}$$

where *x* represents *u*_{x}, *u*_{y}, *u*_{z}, or *T*_{s}; subscript *j* denotes the
*j*th values in *k*th half-hour interval, and the upper bar indicates the
average over the interval. In the recovered dataset, this variance was
similarly computed and denoted by ${s}_{Rk}^{\mathrm{2}}$, where subscript *R* indicates
that this variance was computed from recovered dataset. For each wind
component or sonic temperature, 96 variance values were available in each
datasets and 192 variance values were available in both dataset. The 192
variance values for each wind component or sonic temperature can be used to
construct an F-statistic (Snedecor and Cochran, 1989) to analyze the
homogeneity in variance of each wind component or sonic temperature between
unrecovered and recovered data, given by

$$\begin{array}{}\text{(42)}& \sum _{k=\mathrm{1}}^{\mathrm{96}}{s}_{k}^{\mathrm{2}}/\sum _{k=\mathrm{1}}^{\mathrm{96}}{s}_{Rk}^{\mathrm{2}}\sim F(\mathrm{1}\phantom{\rule{0.125em}{0ex}}\mathrm{727}\phantom{\rule{0.125em}{0ex}}\mathrm{904},\phantom{\rule{0.125em}{0ex}}\phantom{\rule{0.125em}{0ex}}\mathrm{1}\phantom{\rule{0.125em}{0ex}}\mathrm{727}\phantom{\rule{0.125em}{0ex}}\mathrm{904}).\end{array}$$

From this statistic, four *F* values were acquired for three wind components
and sonic temperature. The four *F* values were either > 1.00 or
< 1.00, showing the inhomogeneity in variance between unrecovered
and recovered data (*P* < 0.001), which indicates that the geometrical
deformation of the sonic anemometer did significantly influence the fluctuations
in each of its measured variables.

Further, using EddyPro (LI-COR Biosciences, 2016), the same datasets were
used to compute two sets of sensible heat flux, latent heat flux, and
CO_{2} flux for each half-hour interval. One set was computed using
unrecovered data and the other set from recovered data. The two sets of flux
data were shown in Fig. 8. Compared to the flux from unrecovered data, the
flux from recovered data was 1.5 W m^{−2} lower for sensible heat (*P*=0.031), 0.14 W m^{−2} higher for latent heat (*P*=0.001), and
0.08 µmol m^{−2} s^{−1} higher for CO_{2} (*P*=0.000). These values were
small in magnitude, but significant in comparison to these flux values over
the ice surface in Antarctica.

Analyses of the *F* tests and Fig. 8 show that the data measured from a
geometrically deformed sonic anemometer need to be recovered; otherwise,
there were significant uncertainties in the wind speed and sonic temperature
fluctuations for flux estimations.

Any sonic anemometer is slender (e.g., < 1.00 cm in each diameter of six claws to hold individual sonic transducers) and as light as possible to minimize its aerodynamic resistance to air flows and to maximize its stability on supporting infrastructure (e.g., tripod) to wind momentum load, which sacrifices its durability in keeping its geometrical shape. Therefore, a sonic anemometer is easily deformed if not well cared for during transportation (e.g., the case in this study), installation, or other handling. As shown in this study, a slight geometrical deformation of sonic path length as small as millimeters or less (see Table A1 in Appendix A) could cause significant errors in 3-D wind and especially in sonic temperature. According to our recalibration experience with 3-D sonic anemometers at Campbell Scientific Inc., these cases as addressed in this study have been not unusual, but the equations and algorithms to recover the data measured by a deformed 3-D sonic anemometer were not available. As requisitions of these datasets are expensive, their recovery would be a cost-effective and time-saving option.

The equations and algorithms in this study were developed based on the
measurement working physics and sonic path geometry of the IRGASON sonic
anemometer. The physics is the same as those for other models of Campbell
Scientific 3-D sonic anemometers in use, such as CSAT3, CSAT3A, and CSAT3B (Campbell
Scientific Inc., UT, USA; Horst et al., 2015).
However, the sonic path geometry of the IRGASON sonic anemometer is different
from other models in the assigned azimuth angle of the first sonic path in
the 3-D anemometer coordinate system. This angle was assigned as
90^{∘} in the IRGASON sonic anemometer, but as 0^{∘} in other
models (e.g., CSAT3, CSAT3A, and CSAT3B). Even so, given the sonic path
lengths and transfer matrixes of sonic anemometer that were measured and
determined in the manufacture or calibration process (*d*_{i} in Eq. 12
and **A** in Eq. 15) and in the recalibration process after use in the
geometrical deformation state (*d*_{Ti} in Eqs. 13, 33, and 40 and
**A**_{T} in Eqs. 14 and 16), the equations and algorithms from this study
are applicable to all models of Campbell Scientific 3-D sonic anemometers
(Fig. 6) except for CSAT3 if its bugged OS version 4 is used (Burns et al., 2012). The derivation procedures and even equations based on the
measurement working physics are applicable as a reference to the development
of the equations and algorithms to recover the data measured using other
brands of 3-D sonic anemometers that incurred deformations or to studies on
similar topics.

9 Conclusion remarks

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An IRGASON 3-D sonic anemometer (SN: 1131) was geometrically deformed by an impact during transportation to Antarctica from China in early 2015. To fulfill the field measurement plans for the year, it had to be deployed there in the Zhongshan Station until early 2016 when it was replaced in the field with another IRGASON provided by the manufacturer and was returned to the manufacturer, Campbell Scientific Inc., for recalibration through the remeasurement of its sonic path geometry (lengths and angles), redetermination of its transfer matrix, and an update of its operating system (OS). To recover the 3-D wind and sonic temperature data measured by this sonic anemometer in its deformed state before the recalibration, equations and algorithms were developed and implemented into a software package: “Sonic Data Recovery for IRGASON/CSAT3/A/B Used in Geometrical Deformation after Production/Calibration” (Fig. 6 and Appendix C). Given two sets of sonic path lengths and two transfer matrixes of sonic anemometer that were measured and determined in the manufacture and calibration process and also in recalibration process after the use in its deformed state, the data measured by the IRGASON 3-D sonic anemometer, even in its deformed state, were recovered as if measured by the same anemometer recalibrated immediately after its deformation.

Inside a Campbell Scientific sonic anemometer, the transducer-shadow correction for 3-D wind (Wyngaard and Zhang, 1985) is an available, programmable option for a user. However, the crosswind correction for sonic temperature (Liu et al., 2001) is internally applied as default by its OS. In a case of transducer-shadow correction in option, the 3-D wind data are recovered using Eqs. (10) to (16). If not, Eqs. (15), (12), (13), and (16) are sequentially used. Based on the data from the recovery process of 3-D wind, the sonic temperature data are recovered using Eqs. (9), (32), (38), and (40); therefore, the satisfactory recovery for both 3-D wind data and sonic temperature can be eventually reflected by the satisfactory of sonic temperature data recovery.

The software based on the equations and algorithms from this study can
recover the 3-D wind data with or without the transducer-shadow correction
inside the sonic anemometer and sonic temperature data with crosswind
correction also inside the sonic anemometer. It was verified by comparing
the recovered to calculated sonic temperature data (Appendix D). As shown in
Fig. 7, the recovered data of sonic temperature using Eqs. (33) and (40)
were compared to the calculated sonic temperature of three representative
values over the range of measured sonic temperature from −20.01 to −1.90 ^{∘}C. The difference of 9.60 to 8.93 K between unrecovered and
calculated sonic temperature (i.e., unrecovered minus calculated) was
narrowed by Eq. (40) to 0.81 to −0.45 K (i.e., recovered minus calculated),
which was satisfactory for measurements of sonic anemometer below 0 to −20 ^{∘}C. After verification, the software was used to recover the data
measured by the IRGSON (SN: 1131) 3-D sonic anemometer in its deformed state
from May 2015 to January 2016. The 8-month data were recovered using
3 days of one engineer's time. Further, using EddyPro 6.2.0 (LI-COR Inc.,
2016), the recovered data were processed for the fluxes of
CO_{2}/H_{2}O, sensible heat, and momentum. The data quality (Foken et
al., 2012) mostly ranged in 1 to 3 and the energy closure without
considering surface heat flux into ice were > 83 % when
friction velocity was > 0.2 m s^{−1}. Although energy balance
closure is not a good indicator for data quality (Foken et al., 2012), this
closure rate is fair.

The use of a deformed 3-D sonic anemometer is a practical case. The analyses of our study case indicated that the measured fluctuations in wind speeds and sonic temperature as well as fluxes were significantly influenced by the deformation. If the data from such a use cannot be recovered, the requisition of these data is expensive and their recovery would be a cost-effective and time-saving option. The equations, algorithms, and software are applicable to all models of Campbell Scientific 3-D sonic anemometers such as CSAT3, CSAT3A, and CSAT3B that are used around the world. The derivation procedures and even equations based on the measurement working physics of sonic anemometers are applicable as a reference to the development of the equations and algorithms to manage the data measured using other brands of 3-D sonic anemometers or recover the data measured by an anemometer in its deformed state.

Data availability

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Data availability.

The data in this paper can be accessed via connecting the Supplement.

Appendix A: Transform matrixes

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In micrometeorological applications, the wind speeds are expressed in a
three-dimensional (3-D) orthogonal coordinate system of anemometer or natural
wind, but a sonic anemometer measures flow velocities along its three
non-orthogonal sonic paths (i.e., situated non-orthogonally from each other, see
Figs. 1 and A1); therefore, for applications, the flow velocities along the
three sonic paths need to be transformed into a 3-D right-handed orthogonal
coordinate system in reference to the geometry of sonic anemometer, as shown
in Fig. A1 (i.e., the 3-D orthogonal anemometer coordinate system). Given
*u*_{x} and *u*_{y} are two horizontal velocities in the *x* and *y* direction,
respectively, and *u*_{z} is vertical velocity in the *z* direction (Fig. A1);
*x*, *y*, and *z* are the three coordinate axes in the 3-D orthogonal anemometer
coordinate system. This system is defined with the *x*–*y* plane, parallel to the
anemometer bubble-leveled plane, with the first sonic path on the *y*–*z* plane, and
with origin in the center of measurement volume. A flow speed along the
*i*th ($i=\mathrm{1},\mathrm{2}$, or 3) sonic path is a combination of component velocities of
*u*_{x}, *u*_{y}, and *u*_{z}; given by

$$\begin{array}{}\text{(A1)}& {\displaystyle}{u}_{i}=\left({u}_{x}\mathrm{cos}{\mathit{\varphi}}_{i}+{u}_{y}\mathrm{sin}{\mathit{\varphi}}_{i}\right)\mathrm{sin}{\mathit{\theta}}_{i}+{u}_{z}\mathrm{cos}{\mathit{\theta}}_{i},\end{array}$$

where *θ*_{i} and *ϕ*_{i} are the zenith and azimuth
angles of the *i*th sonic path in the 3-D orthogonal anemometer coordinate
system. In this system (see Fig. A1), given the first sonic path has an
azimuth angle of *ϕ*_{1} equal to 90^{∘} as fixed on the
*x*−*y* plane, Eq. (A1) can be expressed in a matrix form of

$$\begin{array}{l}{\displaystyle}\left[\begin{array}{c}{u}_{\mathrm{1}}\\ {u}_{\mathrm{2}}\\ {u}_{\mathrm{3}}\end{array}\right]=\left[\begin{array}{ccc}\mathrm{0}& \mathrm{sin}{\mathit{\theta}}_{\mathrm{1}}& \mathrm{cos}{\mathit{\theta}}_{\mathrm{1}}\\ \mathrm{sin}{\mathit{\theta}}_{\mathrm{2}}\mathrm{cos}{\mathit{\varphi}}_{\mathrm{2}}& \mathrm{sin}{\mathit{\theta}}_{\mathrm{2}}\mathrm{sin}{\mathit{\varphi}}_{\mathrm{2}}& \mathrm{cos}{\mathit{\theta}}_{\mathrm{2}}\\ \mathrm{sin}{\mathit{\theta}}_{\mathrm{3}}\mathrm{cos}{\mathit{\varphi}}_{\mathrm{3}}& \mathrm{sin}{\mathit{\theta}}_{\mathrm{3}}\mathrm{sin}{\mathit{\varphi}}_{\mathrm{3}}& \mathrm{cos}{\mathit{\theta}}_{\mathrm{3}}\end{array}\right]\\ \text{(A2)}& {\displaystyle}\left[\begin{array}{c}{u}_{x}\\ {u}_{y}\\ {u}_{z}\end{array}\right]={\mathrm{A}}^{-\mathrm{1}}\left[\begin{array}{c}{u}_{x}\\ {u}_{y}\\ {u}_{z}\end{array}\right],\end{array}$$

where **A** is a matrix expressing the flow speeds along the three
non-orthogonal sonic paths in the 3-D orthogonal anemometer coordinate
system. Nominally for the sonic paths of the IRGASON, *θ*_{1}, *θ*_{2}, and *θ*_{3} are all 30^{∘} and *ϕ*_{2} and
*ϕ*_{3} are 330 and 210^{∘}, respectively (see Fig. A1). Given
*ϕ*_{1}=90^{∘}, these angles are calculated using
measured data from a coordinate measurement machine and, along with the sonic
path lengths, are listed in Table A1 for the IRGASON serial no. of 1131
before and after its geometrical deformation.

Using the data in this table, matrix **A** in Eq. (A2) and its inversion
**A**^{−1} for this IRGASON before its geometric deformation (i.e., as used
in the
IRGASON OS but not valid in the field after deformation) are given as
follows:

$$\begin{array}{}\text{(A3)}& {\displaystyle}\mathbf{A}=\left[\begin{array}{ccc}\mathrm{0.034785}& \mathrm{1.142665}& -\mathrm{1.183914}\\ \mathrm{1.365505}& -\mathrm{0.696580}& -\mathrm{0.660515}\\ \mathrm{0.367627}& \mathrm{0.401124}& \mathrm{0.380356}\end{array}\right],\end{array}$$

and

$$\begin{array}{}\text{(A4)}& {\displaystyle}{\mathbf{A}}^{-\mathrm{1}}=\left[\begin{array}{ccc}\mathrm{0.00000}& \mathrm{0.499023}& \mathrm{0.866589}\\ \mathrm{0.418196}& -\mathrm{0.246062}& \mathrm{0.874394}\\ -\mathrm{0.441030}& -\mathrm{0.222826}& \mathrm{0.869391}\end{array}\right].\end{array}$$

After the IRGASON geometrical deformation, matrix **A** became

$$\begin{array}{}\text{(A5)}& {\displaystyle}{\mathbf{A}}_{\mathrm{T}}=\left[\begin{array}{ccc}\mathrm{0.006035}& \mathrm{1.276412}& -\mathrm{1.323287}\\ \mathrm{1.363991}& -\mathrm{0.724862}& -\mathrm{0.600545}\\ \mathrm{0.368690}& \mathrm{0.417250}& \mathrm{0.345690}\end{array}\right],\end{array}$$

where subscript T indicates “True” because, after the IRGASON deformation, it should be used in the field although it was not used. The inversion of this matrix is given as follows:

$$\begin{array}{}\text{(A6)}& {\displaystyle}{\mathbf{A}}_{\mathrm{T}}^{-\mathrm{1}}=\left[\begin{array}{ccc}\mathrm{0.000000}& \mathrm{0.498879}& \mathrm{0.866672}\\ \mathrm{0.347992}& -\mathrm{0.246063}& \mathrm{0.904629}\\ -\mathrm{0.420029}& -\mathrm{0.235072}& \mathrm{0.876537}\end{array}\right].\end{array}$$

Matrixes **A**^{−1}, **A**_{T}, and ${\mathbf{A}}_{\mathrm{T}}^{-\mathrm{1}}$ were used for our data recovery
and **A** was also used in the sonic anemometer OS.

Appendix B: Iteration algorithm for sonic transducer-shadow corrections

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Given transform matrix **A**, using Eq. (5), the measured wind vector
[*u*_{1} *u*_{2} *u*_{3}]^{′} along the sonic paths is transformed to the wind
vector in the 3-dimensional orthogonal anemometer coordinate system
[*u*_{x} *u*_{y} *u*_{z}]^{′}. Subsequently, *U*_{T}
is calculated using Eq. (9). Replace *u*_{Ti} with *u*_{i} under the square root in the right side of
Eq. (8), an approximate equation for the first iteration is given as follows:

$$\begin{array}{}\text{(B1)}& {\displaystyle}{u}_{Ti\mathit{\_}\mathrm{1}}\approx {\displaystyle \frac{{u}_{i}}{\mathrm{0.84}+\mathrm{0.16}\frac{\sqrt{{U}_{\mathrm{T}}^{\mathrm{2}}-{u}_{i}^{\mathrm{2}}}}{{U}_{\mathrm{T}}}}},\end{array}$$

where *i* is 1, 2, or 3 and subscript 1 of *u*_{Ti} indicates
that it is calculated from the first iteration.

Equation (B1) is used for sonic transducer-shadow corrections in the first iteration.

$$\begin{array}{}\text{(B2)}& {\displaystyle}\left[\begin{array}{l}{u}_{x}\\ {u}_{y}\\ {u}_{z}\end{array}\right]=\mathbf{A}\left[\begin{array}{l}{u}_{T\mathrm{1}\mathit{\_}\mathrm{1}}\\ {u}_{T\mathrm{2}\mathit{\_}\mathrm{1}}\\ {u}_{T\mathrm{3}\mathit{\_}\mathrm{1}}\end{array}\right]\end{array}$$

Using Eq. (9), *U*_{T} is recalculated. Replace *u*_{i} with
*u*_{Ti_1} under the square root in the right side of Eq. (B1), an approximate equation for the second iteration is given as follows:

$$\begin{array}{}\text{(B3)}& {\displaystyle}{u}_{Ti\mathit{\_}\mathrm{2}}={\displaystyle \frac{{u}_{i}}{\mathrm{0.84}+\mathrm{0.16}\frac{\sqrt{{U}_{\mathrm{T}}^{\mathrm{2}}-{u}_{Ti\mathit{\_}\mathrm{1}}^{\mathrm{2}}}}{{U}_{\mathrm{T}}}}}\end{array}$$

…

$$\begin{array}{}\text{(B4)}& {\displaystyle}\left[\begin{array}{l}{u}_{x}\\ {u}_{y}\\ {u}_{z}\end{array}\right]=\mathbf{A}\left[\begin{array}{l}{u}_{T\mathrm{1}\mathit{\_}m}\\ {u}_{T\mathrm{2}\mathit{\_}m}\\ {u}_{T\mathrm{3}\mathit{\_}m}\end{array}\right]\end{array}$$

where subscript $m=n-\mathrm{1}$. Using Eq. (9), *U*_{T} is also recalculated.
Similar to the calculation for *u*_{Ti_2},
*u*_{Ti_n} is calculated using the following equation:

$$\begin{array}{}\text{(B5)}& {\displaystyle}{u}_{\mathrm{T}i\mathit{\_}n}={\displaystyle \frac{{u}_{i}}{\mathrm{0.84}+\mathrm{0.16}\frac{\sqrt{{U}_{\mathrm{T}}^{\mathrm{2}}-{u}_{Ti\mathit{\_}m}^{\mathrm{2}}}}{{U}_{\mathrm{T}}}}},\end{array}$$

to ensure that the difference in *u*_{x}, *u*_{y}, or *u*_{z} between the last and
previous iterations is <1 mm s${}^{-\mathrm{1}}\approx \mathrm{1.96}\mathit{\sigma}$, where *σ* is the maximum precision (i.e., standard
deviation at constant wind) among *u*_{x}, *u*_{y}, and *u*_{z} (Campbell
Scientific Inc., 2018). Our numerical tests within the measurement ranges
in *u*_{x}, *u*_{y}, and *u*_{z} concluded that the iterations mostly converged
at *n*=2 and entirely at *n*≤3.

Appendix C: MATLAB code

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Sonic data recovery for the IRGASON/CSAT3/A/B used in geometrical deformation after production/calibration (Code lines were formatted for readability and the electronic version of this code is available from the corresponding authors).

% sonicdatarecovery Sonic Data Recovery for IRGASON/CSAT3/A/B Used in Geometrical Deformation after Production/Calibration

function [Ux,Uy,Uz,Ts,Ts1,Ts2,Raw] =

sonicdatarecovery(RAW)

% *um* Measured 3-D wind speeds in the orthogonal anemometer coordinate system
(OCS)

% *T*_{s} Measured sonic temperature

% **A** Matrix of sonic to OCS before geometrical deformation

% **A**_{T} Matrix of sonic to OCS after geometrical deformation

% *di* Sonic path length before geometrical deformation (*i*=1, 2, or 3)

% *dTi* Sonic path length after geometrical deformation (*i*=1, 2, or 3)

shadow_correction_flag =1; % %Shadow correction has been done (=1) or not (=0) inside OS

gama_d=1.4003; %% the ratio of dry air specific heat at constant pressure to that at constant volume

Rd =287.04; %% gas constant for dry air

RV $=\mathrm{4.61495}e-\mathrm{4}$; %% gas constant for water vapor

Av =60.064621; Bv =60.973392; Cv =60.387959;

Ah =0.000000; Bh =59.527953; Ch =63.195226;

Avt =60.074122; Bvt =64.773415; Cvt =61.227399;

Aht =0.000000; Bht =54.736084; Cht =60.766176;

hwait = waitbar(0,“Please select the file to be processed”);

pause(0.5)

[name,path] = uigetfile(“*.*”,“stabilitylect a folder”);

fname = [path name];

close(hwait);

RAW = dlmread(fname,',', 4, 1);

UX = RAW(:,2); UY = RAW(:,3); UZ = RAW(:,4);

TRAW = RAW(:,5); H2O = RAW(:,8);

Temp = RAW(:,10); P = RAW(:,11);

amb_e = RV.*H2O.*(Temp +273.15);

TS_emp = (Temp +273.15).*(1+0.32*amb_e./P)−273.15;

The1 = ((90-Av)/180)*pi; The2 = ((90-Bv)/180)*pi;

The3 = ((90-Cv)/180)*pi;

Phi1 = ((90-Ah)/180)*pi; Phi2 = ((270+Bh)/180)*pi;

Phi3 = ((270-Ch)/180)*pi;

A_inversion = [0 sin(The1) cos(The1);

sin(The2)*cos(Phi2)
sin(The2)*sin(Phi2) cos(The2);

sin(The3)*cos(Phi3) sin(The3)*sin(Phi3) cos(The3)];

**A**=A_inversion(−1);
$d=[\mathrm{11.6486};\mathrm{11.5240};\mathrm{11.4968}]$;

The1 = ((90-Avt)/180)*pi; The2 = ((90-Bvt)/180)*pi;

The3 = ((90-Cvt)/180)*pi;

Phi1 = ((90-Aht)/180)*pi; Phi2 = ((270+Bht)/180)*pi;

Phi3 = ((270-Cht)/180)*pi;

AT_inversion = [0 sin(The1) cos(The1);

sin(The2)*cos(Phi2)
sin(The2)*sin(Phi2) cos(The2);

sin(The3)*cos(Phi3) sin(The3)*sin(Phi3)
cos(The3)];

AT = AT_inversion^{∧}(-1);

dT =[11.6159;11.1245;11.3548];

hwait = waitbar(0,“Processing> > > > > > ”)

[mRaw,nRaw] = size(RAW);

for *i*=1:mRaw;

um = [UX(i);UY(i);UZ(i)];

UT = (um(1)^{2}+um(2)^{∧}2+um(3)^{∧}2)${}^{\wedge}(\mathrm{1}/\mathrm{2})$; %% Calculate the
total wind magnitude

if isequal(shadow_correction_flag, 1) %% TSC has been done (=1) inside firmware

*u*= A_inversion*um; %% Calculate the vector of the
three flow speeds using Eq. (10)

ut1(1) =*u* (1)/(0.84+0.16.*((UT^{∧}2-u

(1)^{∧}2)${}^{\wedge}(\mathrm{1}/\mathrm{2})$)./UT);

%% Eq. (11), recover flow speed along
sonic path 1 before TSC

ut2(1) =*u* (2)/(0.84+0.16.*((UT^{∧}2-u

(2)^{∧}2)${}^{\wedge}(\mathrm{1}/\mathrm{2})$)./UT);

%% Eq. (11), recover flow speed along
sonic path 2 before TSC

ut3(1) =*u* (3)/(0.84+0.16.*((UT^{∧}2-u

(3)^{∧}2)${}^{\wedge}(\mathrm{1}/\mathrm{2})$)./UT);

%% Eq. (11), recover flow speed along
sonic path 3 before TSC

uc = [ut1.*(dT (1)./d(1));ut2.*(dT(2)./d(2));ut3.

*(dT(3)./d(3))]; %%
Eq. (13)

uts1 = ut1; uts2 = ut2; uts3 = ut3;

%%Corrected 3-D wind speed

um_c=AT*uc; %% Eq. (16)

UT_C=(um_c (1)^{∧}
2+um_c ()^{∧}2+um_c
(3)^{∧}2)^{∧}(1/2); %% Total wind magnitude

% 1st iteration

uct1 = uc (1)/(0.84+0.16.*((UT^{∧}2-uc (1)^{∧}

2)^{∧}(1/2))./UT); % %flow speed 1

uct2 = uc (2)/(0.84+0.16.*((UT^{∧}2-uc (2)^{∧}

2)^{∧}(1/2))./UT); % %flow speed 2

uct3 = uc (3)/(0.84+0.16.*((UT^{∧}2-uc (3)^{∧}

2)^{∧}(1/2))./UT); % %flow speed 3

% 2nd iteration

for q = 2:5; %% 5 steps of iterations after 1st iteration are adequate

%TSC for flow speed 3

uct_m = [uct1(q-1);uct2(q-1);uct3(q-1)]; %% Vector of three path flow speeds

um_C = AT*uct_m; %%Vector in 3-D orthogonal system

UT_C = (um_C (1)^{∧}
2+um_C (2)^{∧}2+um_C

(3)^{∧}2)^{∧}(1/2);

%% Total wind magnitude,
again

uct3(q) = uc (3)/(0.84+0.16.*((UT_C^{∧}2-uct3

(q-1)^{∧}2)^{∧}(1/2))./UT_C);

%% TSC for flow speed 3

% TSC for flow speed 2

uct_mm = [uct1(q-1);uct2(q-1);uct3(q)];

%%Vector of
three flow speeds, again

um_C = AT*uct_mm; %% Vector in 3-D orthogonal system, again

UT_C = (um_C (1)^{∧}
2+um_C (2)^{∧}2+um_C

(3)^{∧}2)^{∧}(1/2); %% Recalculated the
total wind magnitude

uct2(q) = uc (2)/(0.84+0.16.*((UT_C^{∧}2-uct2

(q-1)^{∧}2)^{∧}(1/2))./UT_C);
%%% TSC for flow speed 2

%TSC for flow speed 1

uct_mm = [uct1(q-1);uct2(q);uct3(q)]; %%Vector of three flow speeds, again

um_C = AT*uct_mm; %% Vector in 3-D orthogonal system

UT_C = (um_C (1)^{∧}
2+um_C (2)^{∧}2+um_C

(3)^{∧}2)^{∧}(1/2); %% Total wind magnitude,
again

uct1(q) = u (1)/(0.84+0.16.*((UT_C^{∧}2-uct1

(q-1)^{∧}2)^{∧}(1/2))./UT_C);
%%%TSC for flow speed 1

% Judge the steps of iterations

uct_n = [uct1(q);uct2(q);uct3(q)]; %%Vector from current iteration

ABS_C = uct_n-uct_m; %%Difference between two iterations

% Exit condition

if(abs(ABS_C(1))<
= 0.001&&abs(ABS_C(2))<

= 0.001&&abs(ABS_C(3))< = 0.001);

ucm = AT*uct_n; %% Eq. (14)

ucts1 = uct1(q); ucts2 = uct2(q); ucts3 = uct3(q);

break; %% %Exit iterations

end

end

else

u = A_inversion*um; %% Acquire the flow speeds along 3 sonic paths, Eq. (10)

uc = [dT(1)./d(1).*u(1); dT(2)./d(2).*u(2);

dT(3)./d(3).*u(3)];
%%Correction

ucm = AT*uc; %%3-D orthogonal data after recovery

uts1 = uc(1); uts2 = uc(2); uts3 = uc(3);

ucts1 = ucm(1); ucts2 = ucm(2); ucts3 = ucm(3);

end

Ts = TRAW(i);

UcT = (ucm (1)^{∧}2 + ucm (2)^{∧}2 + ucm
(3)^{∧}2)^{∧}(1/2); %% Total wind

C02 = gama_d*Rd*(Ts + 273.15); %% Eq. (32)

DELTUcT2 = UcT^{∧}2 – UT^{∧}2;

DELTucT21 = ucts1^{∧}2 – uts1^{∧}2;
DELTucT22 = ucts2^{∧}2 – uts2^{∧}2;
DELTucT23 = ucts3^{∧}2 – uts3^{∧}2;

DELTC21 = (C02 - UT^{∧}2 + uts1^{∧}
2)*((dT(1)^{∧}2 –

d(1)^{∧}2)/d(1)^{∧}
2); %% Eq. (30)

DELTC22 = (C02 - UT^{∧}2 + uts2^{∧}
2)*((dT(2)^{∧}2 –

d(2)^{∧}2)/d(2)^{∧}
2); %% Eq. (30)

DELTC23 = (C02 – UT^{∧}2 + uts3^{∧}
2)*((dT(3)^{∧}2 –

d(3)^{∧}2)/d(3)^{∧}
2); %% Eq. (30)

AAA = (DELTC21 + DELTUcT2 – DELTucT21);

BBB = (DELTC22 + DELTUcT2 – DELTucT22);

CCC = (DELTC23 + DELTUcT2 – DELTucT23);

DDD = (AAA + BBB + CCC);

EEE = 3*gama_d*Rd;

Tcs = Ts+(DDD/EEE); %% Eq. (33)

DELTC021_ad = C02*2*(1-dT(1)/d(1)); %% Eq. (38)

DELTC022_ad = C02*2*(1-dT(2)/d(2)); %% Eq. (38)

DELTC023_ad = C02*2*(1-dT(3)/d(3)); %% Eq. (38)

AAA_ad = ((dT(1)^{∧}2-d(1)^{∧}
2)/d(1)^{∧}2)*(C02-

(DELTC021_
ad+((DELTC021_ad+DELTC022_
ad+

DELTC023_ad)/3))-UT^{∧}
2+uts1^{∧}2)

+DELTUcT2-DELTucT21;

BBB_ad = ((dT(2)^{∧}2-d(2)^{∧}
2)/d(2)^{∧}2)

*(C02-(DELTC022_ad+((DELTC021_ad+DELTC022_ad

+DELTC023_ad)/3))-UT^{∧}
2+uts2^{∧}2)

+DELTUcT2-DELTucT22;

CCC_ad = ((dT(3)^{∧}2-d(3)^{∧}
2)/d(3)^{∧}2)

*(C02-(DELTC023_ad+((DELTC021_ad+DELTC022_ad

+DELTC023_ad)/3))-UT^{∧}
2+uts3^{∧}2)

+DELTUcT2-DELTucT23;

DDD_ad = (AAA_ad + BBB_ad + CCC_ad);

Tcs_ad = Ts+(DDD_ad/EEE); %% Eq. (40)

Data_recovery(i,1) = ucm(1); %%Recovered 3-D wind speed in x-direction

Data_recovery(i,2) = ucm(2); %% Recovered 3-D wind speed in y-direction

Data_recovery(i,3) = ucm(3); %% Recovered 3-D wind speed in z-direction

Data_recovery(i,4) = Tcs; %% Recovered Ts from raw Ts, Eq. (33)

Data_recovery (i,5) = Tcs_ad; %% Recovered Ts from raw Ts, Eq. (40)

Data_recovery (i,6) = TS_emp(i); %% Recovered T, Eq. (D1)

Data_recovery (i,7) = TRAW(i); %% Raw Ts

End

title = *{*“Recovered *u*_{x}”,“Recovered *u*_{y}”,“Recovered
*u*_{z}”,“Recovered Tcs”,“Recovered Tcs_ad”,“Recovered
T”,“RAW *T*_{S}”*}*;

fname = [path “\ Data_recovery”];

xlswrite(fname,title,“sheet1”);

xlswrite(fname,Data_recovery,“sheet1”,“A2”);

waitbar(0,hwait,`Done”);

pause(2);

close(hwait);

Appendix D: Sonic temperature from air temperature, relative humidity, and atmospheric pressure

Back to toptop
In case that air temperature (*T* in ^{∘}C), relative humidity (RH in
percentage), and atmospheric pressure (*P* in kPa) are measured in the field, sonic
temperature (*T*_{s} in ^{∘}C) can be calculated using the well-known
equation (Kaimal and Gaynor, 1991):

$$\begin{array}{}\text{(D1)}& {\displaystyle}{T}_{\mathrm{s}}=\left(T+\mathrm{273.15}\right)\left(\mathrm{1}+\mathrm{0.32}{\displaystyle \frac{e}{P}}\right)-\mathrm{273.15},\end{array}$$

where *e* is air water vapor pressure (kPa) and can be computed from *T*, RH, and
*P*.

Given *T* and *P*, saturated water vapor pressure (*e*_{s} in kPa) can be calculated
using Buck (1981):

$$\begin{array}{}\text{(D2)}& {\displaystyle}{e}_{\mathrm{s}}=\left\{\begin{array}{l}\mathrm{0.61121}\mathrm{exp}\left({\displaystyle \frac{\mathrm{17.368}T}{T+\mathrm{238.88}}}\right){f}_{w}(T,P)\phantom{\rule{0.125em}{0ex}}\phantom{\rule{0.125em}{0ex}}T\ge \mathrm{0}\\ \\ \mathrm{0.61121}\mathrm{exp}\left({\displaystyle \frac{\mathrm{17.966}T}{T+\mathrm{247.15}}}\right){f}_{w}(T,P)\phantom{\rule{0.125em}{0ex}}\phantom{\rule{0.125em}{0ex}}T<\mathrm{0}\end{array}\right.,\end{array}$$

where *f*_{w}(*T*, *P*) is the enhancement factor:

$$\begin{array}{ll}{\displaystyle}{f}_{w}(T,P)& {\displaystyle}=\mathrm{1.00041}+P\left[\mathrm{3.48}\times {\mathrm{10}}^{-\mathrm{5}}+\mathrm{7.4}\times {\mathrm{10}}^{-\mathrm{9}}\right.\\ \text{(D3)}& {\displaystyle}& {\displaystyle}\left.{\left(T+\mathrm{30.6}-\mathrm{0.38}P\right)}^{\mathrm{2}}\right]\end{array}$$

Using the definition of air relative humidity, air water vapor pressure is given by

$$\begin{array}{}\text{(D4)}& {\displaystyle}e={e}_{\mathrm{s}}{\displaystyle \frac{\mathrm{RH}}{\mathrm{100}}},\end{array}$$

Submit the measured *T* and *P* as well as the calculated *e* into Eq. (D1) and the
sonic temperature can be calculated.

Supplement

Back to toptop
Supplement.

The supplement related to this article is available online at: https://doi.org/10.5194/amt-11-5981-2018-supplement.

Author contributions

Back to toptop
Author contributions.

QY, NZ, and XiaZ proposed this study and coordinated and led the teamwork. They also developed equations and algorithms, analyzed data and results, and finalized the manuscript. YL and TG reviewed the manuscript, provided constructive comments, and helped finalize the manuscript. GH, HS, and YS collected and analyzed data. XinZ integrated research results for publication.

Competing interests

Back to toptop
Competing interests.

The authors declare that they have no conflict of interest.

Acknowledgements

Back to toptop
Acknowledgements.

This study was supported by the National Key R&D Program of China (grant no.: 2016YFA0600704), and the National Natural Science Foundation of China
(grant nos.: 41376005, 41406218, 41505004, and 31200432). We thank the
Chinese Arctic and Antarctic Administration and the Polar Research Institute
of China for their field logistical support; Campbell Scientific and Beijing
Techno Solutions Limited for their customer support; Steve Harston and
Antoine Rousseau for technical graphic work; Carolyn Ivans, Bo Zhou, Mark Blonquist, and Hayden Mahan for their English polishing; and
Linda Worlton-Jones for her professional English checks and revisions. We greatly
thank Thomas Foken and the anonymous reviewer for their comments and
input that improved our presentation and interpretations in this
paper.

Edited by: Laura Bianco

Reviewed by: Thomas Foken and one anonymous referee

References

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Short summary

The three-dimensional wind and sonic temperature data from a physically deformed sonic anemometer was successfully recovered by developing equations, algorithms, and related software. Using two sets of geometry data from production calibration and return re-calibration, this algorithm can recover wind with/without transducer shadow correction and sonic temperature with crosswind correction, and then obtain fluxes at quality as expected. This study is applicable as a reference for related topics.

The three-dimensional wind and sonic temperature data from a physically deformed sonic...

Atmospheric Measurement Techniques

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