Sonic anemometers are the principal instruments in
micrometeorological studies of turbulence and ecosystem fluxes. Common
designs underestimate vertical wind measurements because they lack a
correction for transducer shadowing, with no consensus on a suitable
correction. We reanalyze a subset of data collected during field experiments
in 2011 and 2013 featuring two or four CSAT3 sonic anemometers. We introduce
a Bayesian analysis to resolve the three-dimensional correction by optimizing
differences between anemometers mounted both vertically and horizontally. A
grid of 512 points (

This paper was written and prepared by a US Government employee on official time, and therefore it is in the public domain and not subject to copyright.

The eddy-covariance technique has become the most commonly used method for measuring the ecosystem exchange of mass and energy with the atmosphere. It is fundamental to the global network of flux towers that are central to quantifying terrestrial carbon sinks and sources (Baldocchi, 2003), to hydrological studies accounting for evapotranspiration and sublimation (Biederman et al., 2014; Reba et al., 2012), and to the energy balance through the turbulent fluxes of sensible and latent heat (Welch et al., 2015; Anderson and Wang, 2014). There is a growing consensus within the micrometeorology and ecosystem flux communities that many sonic anemometers, the core instrument for all modern eddy-covariance systems, systematically underestimate the vertical wind component (Frank et al., 2016; Horst et al., 2015; Kochendorfer et al., 2012). The ramifications for this are that all vertical fluxes (i.e., carbon dioxide, water vapor, latent heat, sensible heat, momentum) are similarly underestimated for any ecosystem. This underestimate is roughly consistent with the persistent energy balance closure problem across flux sites (Leuning et al., 2012; Stoy et al., 2013; Wilson et al., 2002), where a vast majority are assumed to be systematic biased towards low turbulent fluxes of sensible and latent heat.

Photograph of the 2011 experiment with two CSAT3 sonic anemometers
mounted vertically and two horizontally. The cardinal

Horst et al. (2015) and Frank et al. (2016) have shown that the error in at least two non-orthogonal sonic anemometer designs can be traced to transducer shadowing that remains uncorrected in the anemometer's firmware. In both studies, shadowing was described a priori by theoretical formulations based on the wind tunnel tests of Kaimal (1979), yet there was no consensus on a correction. A shortcoming in the use of formulations derived for single transducer pairs in laminar flow to describe turbulent flow distortions around more complex geometries (Fig. 1) is that shadowing between all transducers and structures cannot be accurately represented or incorporated. A second problem is that in turbulent flow fields there are few standards available to use as a calibration reference. Advancements in Bayesian techniques (Gelman et al., 2014) have created the potential to resolve both of these issues by incorporating prior knowledge of transducer flow distortions with a model that evaluates the omnidirectionality of a sonic anemometer to produce a posterior 3-D correction.

To quantify a 3-D correction of the CSAT3 sonic anemometer, we reanalyze data
from field experiments conducted by Frank et al. (2013, 2016), where wind
measurements from non-orthogonal anemometers mounted vertically and
horizontally were significantly different. We develop a Bayesian hierarchical
model to evaluate three hypotheses:

A 3-D shadowing correction based solely on wind location can make a non-orthogonal sonic anemometer omnidirectional.

This correction increases vertical wind measurements more than expected from single transducer shadowing because it accurately represents all shadowing between transducers.

In ecosystems where these instruments are deployed, the application of this correction will result in significantly higher Bayesian credible intervals for the turbulent components of the energy budget and improved surface energy budget closure.

Summary of the subset of data from Frank et al. (2013, 2016)
reanalyzed in this study listing the four CSAT3 anemometers
(A–D), their location within the five-position horizontal array, and whether
they are mounted horizontally (

We reanalyze data from field campaigns conducted by Frank et al. (2013,
2016). To summarize them, experiments were conducted in 2011 and 2013 where
multiple sonic anemometers were deployed in a horizontal array at 24.5 m
height on the Glacier Lakes Ecosystem Experiments Site (GLEES) AmeriFlux
scaffold above a subalpine forest in southeastern Wyoming, USA (Frank et al.,
2014). The anemometers were initially mounted vertically, oriented west,
arranged south to north, staggered up and down, and located 0.50 m
center-to-center from each other (Fig. 1). Periodically, some of the
anemometers were rotated 90

Bayesian statistics is based on Bayes' theorem (Bayes and Price, 1763), which
in modern applications relates the posterior probability of a model parameter
conditioned on data to the product of the likelihood of the data and the
prior probability of that parameter (Gelman et al., 2014). In essence, the
prior represents an initial educated guess or belief in the value of a model
parameter; the likelihood is the probability of observing the data if they
were deterministically generated from a model; and the posterior is an
updated belief in the model parameter considering each the prior, the model,
and the data. Analytical evaluation of the posterior is rarely possible, as
is in our case; thus the posterior is commonly estimate through the Markov
chain Monte Carlo (MCMC) method, Gibbs sampling (Appendix A1), and the
Metropolis–Hastings algorithm (Kruschke, 2010). The framework of our Bayesian
model is to divide the sphere around the sonic anemometer into approximately
equal grid points and to define a prior probability distribution of the 3-D
shadowing correction for each transducer pair at each location. Then, the
model proposes new corrections for each grid point, recalculates the
fast-response data set, summarizes new 5 min wind statistics, determines the
probability that the updated measurements from vertical and horizontal
anemometers are more equivalent using the proposed correction versus the old
one (i.e., the Metropolis–Hastings ratio, which is Eq. A13 evaluated for the
proposed versus old correction), and finally accepts/rejects the proposal
probabilistically from this ratio to construct the posterior correction. The
model recursively adjusts the distribution that generates the proposals to
achieve between 25 and 50 % acceptance rates, which are theoretically
optimal (Gelman et al., 2014). We define a grid of 512 points (

In our mathematical notation, we use uppercase and lowercase subscripts to
distinguish variables as scalars, vectors, or matrices. Uppercase subscripts
are part of the variable name, denote the dimensionality of the variable,
and describe the coordinate system. For example,

We test three prior corrections: no shadow correction, the Kaimal correction
(Kaimal, 1979; Frank et al., 2016; Horst et al., 2015), and a doubling of the
Kaimal correction (Frank et al., 2016). The Kaimal correction is defined as

The model predicts the standard deviation of the data in cardinal
coordinates,

The non-orthogonal data are transformed via matrix multiplication into
orthogonal sonic coordinates,

The matrix,

For the model to predict data simultaneously from both vertical and
horizontal anemometers, a final corrected time series data set is produced in
cardinal coordinates,

Using the corrected time series data in cardinal coordinates, the model
calculates the average correction along the three dimensions,

Equation (8) is equivalent to the ratio of the standard deviation of

Increase in

SD: standard deviation;

Finally, the model predicts the mean for the standard deviation data as the
reference divided by the average correction (Eq. 10).

To complete the Bayesian model definition, the model error is a state
variable which is assigned a prior probability distribution with a gamma
distribution (Eq. 11).

Distributions are defined where normal distributions are

Our Bayesian analysis was conducted using R (version 3.2.2, R Core Team,
2015) within RStudio (version 0.99.486, RStudio Team, 2015). We constructed
an MCMC chain of 10 000 steps for each of the three priors. Because the
Bayesian model estimates are relative and not an absolute correction (see
discussion in Sect. 4.1), we normalized each chain. This was done in
post-processing by dividing

Because each MCMC chain was based on a different prior, they are not
replicate chains from the same Bayesian analysis. Instead, these are three
separate solutions for the posterior correction. But after considering the
results (see Sect. 3.3) and recognizing that, apart from normalization, the
prior had minimal influence on the solution, we combined the three priors to
create a single chain containing 204 independent samples of the posterior
correction. We rescaled the correction to be absolute by forcing the
condition that the correction will not change, on average, equatorial wind
measurements (i.e., (

Computation of the Bayesian model was extremely intensive: completion of the
three chains took upwards of 2 months of continuous computer processing
(Windows 7, Intel^{®}
Core^{™} i7-3630QM CPU @ 2.40 GHz processor,
1 TB solid-state hard drive, 20 GB RAM). During beta testing we attempted
to estimate the 3-D correction independently for all grid points and all
transducer pairs, with a single MCMC chain requiring a half year to complete.
Likewise, we investigated increasing the number of grid points to obtain
better resolution around the sphere as well as increasing the amount of sonic
anemometer data used from the Frank et al. (2013, 2016) data sets. In both
cases we desired an order-of-magnitude better resolution or more data, but
the time required to complete a single MCMC chain quickly made these
improvements impractical. Instead, we determined that 512 grid points and
5 % of the original data were optimal considering these processing
constraints.

There is a slight distinction to be made between the prior corrections – which
are defined as a function,

We quantified the impact of shadowing on measurements of the standard
deviations of winds in the three dimensions and the sensible heat flux
(

Finally, we quantified the impact of the 3-D correction on the sum of the
turbulent components of the energy balance (i.e., sensible and latent
(LE) heat flux) at various sites across North America (Table 2).
Each site featured a CSAT3, a fast-response hygrometer, and ancillary
meteorological data. Measurements of LE were calculated similar to

Uncorrected measurements of the 5 min standard deviation of wind
(

The Kaimal correction, one of three priors tested in this study, for
the

We conducted a validation experiment of the posterior 3-D correction at the
Colorado State University, Agricultural Research Development and Education
Center (ARDEC), Fort Collins, CO, USA (40

Kaimal-corrected measurements (i.e., one of three priors tested) of
the 5 min standard deviation of wind (

Without any shadow correction applied, measurements between a vertically and
a horizontally mounted anemometer were different, which becomes clear when
the variance between two vertical anemometers is taken into account
(Fig. 2b, d, f vs. a, c, e). The RMSE in the 5 min
standard deviation of wind along all cardinal dimensions (

The Kaimal correction, one of three priors tested in this study,
evaluated among 512 cells for the

The Kaimal correction is symmetrical with respect to each sonic transducer
path (Fig. 3a, c, e). Yet the same correction when viewed in sonic
coordinates reveals unique responses for

Figure 5 illustrates the approach of the Bayesian model. The model
initializes the 512 grid points with a prior, in this case the Kaimal
correction. No matter the transducer pair or vertical versus horizontal
mounting, the 3-D corrections for all cases are identical but rotated versions
of a common correction based on 138 unique state variables. For a single
instantaneous wind, the simultaneous corrections for all six combinations of
transducer pairs and mounting orientations will be different. As the MCMC
chains progress, the Bayesian model will continuously adjust each of the 138
unique state variables so that measurements from the vertically and
horizontally mounted anemometers are most similar based on the univariate
conditional posterior probability distribution (Eq. A13). Much of the
predictive power of the model comes from resolving the inconsistencies along
the cardinal

Each MCMC chain was initialized with the mean of each prior, yet after
convergence their posterior corrections were remarkably similar regardless of
the choice of prior correction, with one peculiarity (Fig. 6). There was a
clear linear relationship between the prior correction averaged across all
512 grid points (1.000 for no correction, 1.040 for the Kaimal correction,
and 1.080 for the double-Kaimal correction) and the magnitude of the
posterior correction (1.030, 1.064, and 1.098, respectively) that relates to
the Bayesian model estimating a relative and not absolute correction (see
discussion in Sect. 4.1). The posterior correction is more than an estimate
of the optimal solution, as it intrinsically accounts for the uncertainty of
the correction at each of the 512 grid points (Fig. 7). Whereas each prior
was defined with 10 % uncertainty (Eq. 2), much of the posterior
correction has much lower standard deviations, especially around the
transducers where values were as low as 2.5 % (Fig. 7a). These
uncertainties can be expressed in sonic coordinates for the

The A transducer pair correction evaluated among 512 cells for the
three prior corrections tested in this study:

Standard deviations of the posterior correction for

The posterior correction evaluated for the

Figure 8 illustrates the completion of the Bayesian model where the same
posterior correction is applied to all transducer pairs and both mounting
orientations. For every instantaneous wind, application of these six
different corrections ultimately results in the 5 min standard deviations
of wind along the cardinal

The posterior correction for each transducer pair is presented in Fig. 9.
These results take into account the recursive adjustment to the wind
locations and have been smoothed with a spherical spline. Significantly more
self-shadowing and cross-shadowing around the transducers are visible than
for the Kaimal prior (Fig. 9a, c, e vs. Fig. 3a, c, e, in locations
near all transducers). These results are more certain (i.e., low standard
deviations when compared to the original 10 % assigned to the prior) near
the transducers, poorly constrained near the Equator (Fig. 7a), and
independent of the choice of prior correction (Fig. 6). Transforming the
posterior correction into sonic coordinates reveals that, similar to the
Kaimal prior, minimal

Results of a validation experiment of CSAT3 sonic anemometers at
ARDEC, CO, showing the relative error in 5 min standard deviation of wind
(

RMSE: root mean square error; SD: standard deviation;

We applied the posterior correction to various sites across North America
that deploy the CSAT3 in their eddy-covariance instrumentation (Table 2). The
estimated increase in

The posterior correction for the

Posterior-corrected measurements of the 5 min standard deviation of
wind (

Though application of the Kaimal (dashed lines) and posterior (solid
lines) corrections results in similar changes to the 5 min standard deviations
of wind (

The validation experiment was conducted during excellent fall weather with no
precipitation, where winds averaged 2.0

Perhaps the most important shortcoming in almost every sonic anemometer study is the lack of a standard wind measurement to compare against. A fundamental problem is that the principle of sonic measurements (Barrett and Suomi, 1949; Kaimal and Businger, 1963) involves the observer effect; i.e., it is virtually impossible for sonic transducers to observe air parcels without influencing them (Buks et al., 1998). Thus, any method that relies on a sonic anemometer measurement as an absolute standard is flawed to an extent. And while we are justified to believe that some sonic anemometer measurements are more accurate that others (Frank et al., 2016), it is tenuous to choose any sonic anemometer measurement as a standard. Then, what are the alternatives? Wind tunnels are extremely useful (Horst et al., 2015; van der Molen et al., 2004), yet it is debatable that such laminar or quasi-laminar calibrations are transferrable to turbulent field conditions (Hogstrom and Smedman, 2004). And, while other new technologies such as Doppler Lidar exist (Sathe et al., 2011; Dellwik et al., 2015), their application as a field reference standard has been limited.

What we address is the general problem of determining a calibration given an unknown standard or nothing to compare against. Whether this problem exists in medicine (Lu et al., 1997), acoustics (MacLean, 1940; Monnier et al., 2012), or micrometeorology with respect to calibrating sonic anemometry in turbulent flow fields, all approaches have a commonality of testing the relative consistency of a response to unknown signals. In our situation, we hold the 3-D sonic anemometer to an omnidirectional standard of relative consistency and contend that the correction that best achieves this standard is statistically the most likely 3-D calibration. A CSAT3 without any 3-D shadow correction is clearly not omnidirectional (Fig. 2) as measurements depend on the instrument's orientation. A CSAT3 with the Kaimal transducer shadow correction is better at meeting this standard (Fig. 4). However, the posterior 3-D correction is remarkably effective in making the CSAT3 omnidirectional (Fig. 10). Because the posterior correction closely achieves the omnidirectional standard, we support our first hypothesis and argue that it is the most accurate correction, in general, for the three dimensions of the CSAT3. Whether or not the posterior correction reveals meaningful information regarding vertical winds and turbulent fluxes is another matter discussed below.

A consequence of the omnidirectional standard is that implicitly this
produces only relative results. Indeed, our Bayesian posterior has no meaning
in an absolute sense without the additional constraint that equatorial winds
should be unchanged by the correction. We do not specify the 3-D correction
at any of the grid points, nor we do we specify a reference or true
condition for the standard deviation of wind during any 5 min period. Because
of this, the parameter estimates for

There is a clear need to specify an absolute standard to reference our
results. Without one, our normalized posterior correction reduced the 5 min
standard deviations for equatorial winds (i.e., the

Recent studies have questioned the accuracy of CSAT3 vertical wind velocity
measurements (Frank et al., 2013; Kochendorfer et al., 2012), culminating with
Horst et al. (2015) and Frank et al. (2016), who identified the anemometer's
lack of transducer shadowing correction as the root cause. Quantifying the
inaccuracy and determining how to fix this problem have been a challenge.
While each of these studies estimated different errors in

Solely the fact that the posterior correction makes the CSAT3 more omnidirectional
does not imply that field measurements of vertical wind and turbulent fluxes
are impacted, nor does this assure that these impacts would be due to
anything more than chance. Even with the uncertainty in the posterior

Also of note, there are instabilities in the prior and posterior

Energy balance is a fundamental ecosystem concept where the flow of available
energy into an ecosystem influences the microclimate, drives photosynthesis,
and establishes trophic levels among the biota (Odum, 1957; Fisher and
Likens, 1973; Teal, 1962). Yet eddy-covariance studies of ecosystem fluxes
seldom delve into details of energy flow beyond the generation of sensible
and latent heat. It is often stated that most eddy-covariance sites
underestimate these turbulent components of the energy balance by
10–20 % when compared to the available energy (Wilson et al., 2002;
Foken, 2008; Stoy et al., 2013; Leuning et al., 2012; Franssen et al., 2010).
Even when sites thoroughly account for lesser components such as energy
stored in the biomass or canopy air, the turbulent energy can still be
1–14 % underestimated (Heilman et al., 2009; Oliphant et al., 2004; Barr
et al., 2006; Wang et al., 2012). It is common for sites to deal with this
problem by forcing energy balance closure by increasing

After applying the posterior correction to the CSAT3 at our site,
measurements of one of the energy balance components,

Are the results from this study applicable to the non-orthogonal sonic
anemometers produced by other manufacturers? Possibly. Frank et al. (2016)
showed that the Applied Technologies, Inc. A-probe shares a similar
transducer geometry, a lack of a shadow correction algorithm, and similar
differences between vertically and horizontally mounted anemometers, so it
would be reasonable to expect a similar 3-D correction for that instrument.
But other manufacturers do apply wake corrections in their firmware that are
traceable to wind tunnel calibrations. Are these adequate? Maybe not, as
non-orthogonal anemometers from other manufacturers have been implicated to
erroneously measure the vertical wind (Kochendorfer et al., 2012; Nakai et
al., 2014; Nakai and Shimoyama, 2012). Without details of the calibrations or
the wake corrections it is difficult to know. Regardless, for any
non-orthogonal sonic anemometer with vertically oriented transducers,
equatorial instabilities are likely to exist (Appendix A2) that would be
extremely difficult to characterize with only a series of wind tunnel
calibrations. One benefit of our methodology is that it allows an independent
check on the sufficiency of these wake corrections. If such an instrument
failed to consistently measure three-dimensional winds (i.e., it responds like
Fig. 2), then our methodology would estimate a posterior correction that
could correct a wake-corrected anemometer. Because

While these results reveal much about the nature of shadowing in a
non-orthogonal sonic anemometer, there is much more to be done. First, due to
the intense computational burden of this analysis we never fully utilized our
data. While we only analyzed 5 % of the available data, limited the 3-D
correction to approximately

Our results draw extensively on the symmetry of the CSAT3, which fails to
account for the upper and lower mounting arms that extend back into the
electronics housing and support block. We beta-tested our model to solve for
the 3-D correction independently for each transducer and for all grid points
around the sphere. We abandoned this because winds at GLEES are fairly
unidirectional, causing many of the grid points to be poorly characterized.
Plus with an order-of-magnitude more unique grid points to solve, the
computation took over 5 months to complete just one MCMC chain! There is a
middle ground between assuming symmetry and pooling data; i.e., the
correction for the A transducer pair could be considered symmetrical along
the

Sonic anemometer corrections should be verified and validated. There is an
opportunity to statistically cross-validate the posterior 3-D correction with
subsets of the other 95 % of available data; we decided against this
because the 5 % used was already partitioned equally throughout the full
data set; plus, analyzing multiple rounds of training and validation data sets
would take additional months of computation. Instead of a statistical
cross-validation analysis, we conducted a validation field experiment to
determine (1) if our results are reproducible and (2) if they can explain
other manipulations. From this, we first conclude that our results are
reproducible. In both our main experiments at GLEES and the validation
experiment at ARDEC, there was improved agreement between vertically and
horizontally mounted anemometers when using the posterior correction versus
the Kaimal correction or no correction (Table 3). The largest differences
between anemometers was for

Our results using the posterior correction (Fig. 10) show that there is still unexplained residual error, though we expect some of this to be reduced with our suggestions above. While Horst et al. (2015) showed that to a first order that transducer shadowing is a function of the longitude and latitude of the instantaneous wind, the impact of other covariates such as wind velocity and turbulence may need to be considered. An advantage of performing our analysis in a Bayesian framework is that the model can be expanded to incorporate the effects of these covariates.

And finally, our posterior correction and methodology should be compared to other independent analysis of sonic anemometer shadowing such as wind tunnel data (Horst et al., 2015) or an independent Doppler lidar system (Sathe et al., 2011). Care should be taken when incorporating these results, as anemometers could respond differently under laminar flow in a wind tunnel versus under turbulent field conditions. Regardless, a key to resolving this problem will be to embrace new technologies, new experimental designs, and new analyses.

The non-orthogonal CSAT3 sonic anemometer produces different results (Fig. 2) when it is mounted horizontally instead of vertically (Fig. 1). Assuming that the primary source of this error is shadowing across the various transducers, a Bayesian model can estimate a posterior correction (Fig. 8) that ultimately makes measurements from vertically and horizontally mounted anemometers most similar (Fig. 10). Even when taking into account the uncertainty of the posterior correction (Fig. 7), the increases in vertical wind velocity and sensible heat flux measurements are significantly larger and are approximately twice the magnitude of the Kaimal correction (Fig. 11). When this posterior correction is applied to various eddy-covariance sites across North America, the turbulent components of the ecosystem energy balance (sensible plus latent heat flux) increased between 8.1 and 11.6 %, with an average 95 % confidence that this increase was between 6.1 and 13.8 % (Table 2). Considering this is the most common sonic anemometer in the AmeriFlux network and is found in all the regional networks that comprise FLUXNET, these results have major implications for countless studies that use the eddy-covariance technique to measure terrestrial–atmospheric exchange of mass and energy.

The data and source code are available at:

For the univariate conditional posterior distribution functions there is a
distinction between independent grid points and those linked together
through symmetry. In the case of the former, these functions can be
evaluated for each unique grid point,

First, using Bayes' theorem, the joint posterior distribution of the model
parameters can be expressed as being proportional to the product of the
likelihood of the data and the joint prior distribution of the model
parameters (Eq. A1).

Because the prior distributions for three model parameters are independent,
the joint prior distribution can be written as the product of the individual
probabilities (Eq. A2).

The likelihood of the data is normally distributed (Eq. A3).

Because

Gibbs sampling for each model parameter is based on the univariate
conditional posterior distribution, which assumes that all other model
parameters plus the data are given (in the case of sampling within a
multidimensional array, all other parameters within that array are given
except the one at the index being evaluated). For

The underbar denotes all elements within a multidimensional array, while the
notation

Assuming that all but

Substituting in the likelihood from Eq. (A3) and simplifying gives the
univariate conditional posterior distribution for

The univariate conditional posterior distribution for

Again, only the first term in the numerator must be evaluated while assuming
that all but

Substituting in both the likelihood of the data (Eq. A3) and the prior
distribution for

An important issue is that

Only the first term in the numerator must be evaluated while assuming that
all but

Substituting in the likelihood from Eq. (A3) and simplifying yields the
univariate conditional posterior distribution for

For a CSAT3, the amount of correction applied to the vertical wind velocity
– expressed as the individual corrections

If the individual corrections for the three transducer pairs never approach 0
or

This is satisfied by

We thank Jorge Ramirez, Susan Howe, Mario Bretfeld, Kelly Elder, Banning Starr, Bill Kustas, and Joe Alfieri for providing data from their unique field sites. We especially thank Ben Bird for his comments and countless hours of statistical advice in developing the Bayesian model. We thank Jay Ham for his generous assistance in conducting the field validation experiment. We thank Bob Hall and the two anonymous reviewers, whose thoughts and comments improved this manuscript. This study was funded by the US Forest Service, the Wyoming Water Development Commission, the USGS, the NSF (awards EPS-1208909 and EAR-0444053), and the DOD Army Research Office (W911NF-05-1-0558 and W911NF-05-1-0126). Edited by: L. Bianco Reviewed by: three anonymous referees