Coordinate Rotation–Amplification in the Uncertainty and Bias in Non-orthogonal Sonic Anemometer Vertical Wind Speeds

Abstract

Recent research indicates that non-orthogonal sonic anemometers underestimate vertical wind velocity and consequently eddy-covariance fluxes of mass and energy. Whether this is a general problem among all non-orthogonal sonic anemometers, including those calibrated for flow-shadowing effects, is unknown. To investigate this, we test two sonic anemometer designs, orthogonal (3Vx-probe, Applied Technologies, Inc.) and non-orthogonal (R3-50, Gill Instruments, Ltd.), in a series of field manipulation experiments featuring replicate instruments mounted in various orientations, and use a Bayesian analysis to determine the most likely posterior correction to produce equivalent measurements. The 3Vx-probe experiment was conducted on a 24-m scaffold at the Glacier Lakes Ecosystem Experiments Site (GLEES), Wyoming, USA AmeriFlux site while R3-50 anemometer experiments were conducted at the GLEES field site and on a 2.9-m scaffold at the Pawnee National Grassland, Colorado, USA. Without applying a shadowing correction to the 3Vx-probe, the posterior correction significantly increases the standard deviation of the horizontal velocity component by 5–15% (95% Bayesian credible interval) but without a significant change in the horizontal temperature flux; with the shadowing correction applied neither of these have significant changes. Similarly, for the R3-50 GLEES experiment, the standard deviation of the vertical velocity and vertical temperature flux significantly increase by 13–18% and 6–10% (95% credible intervals); results from the Pawnee experiment are contradictory and inconclusive. The reason for the underestimated vertical velocity is undetermined, though a mathematical by-product of the non-orthogonal geometry is that small systematic measurement biases can become large uncertainties in the vertical velocity. This could affect all non-orthogonal designs.

Appendix: Non-orthogonal Vertical Velocity Measurements are Unpredictable for Near-Horizontal Angles of Attack

If any systematic measurement errors occur with non-orthogonal sonic anemometer transducers, then the vertical wind velocity will have an unpredictable measurement error, or conversely an unpredictable correction, for near-horizontal flows. While transducer shadowing is a well-documented source of systematic measurement error (Kaimal 1979), this is generalizable for any cause of bias. The following derivation is an adaptation from the open-access discussion of Mauder and Zeeman (2018) (https://www.atmos-meas-tech.net/11/249/2018/amt-11-249-2018-discussion.html) and applies to non-orthogonal anemometers with transducers tilted 45° from the horizontal plane, including the Gill R3-50. Assuming systematic transducer measurement errors can be corrected with a set of arbitrary functions α, the corrected vertical wind component in sonic coordinates, $\hat{w}_{s}$, is evaluated by substituting the corrected transducer velocities u₁α₁, u₂α₂, and u₃α₃ into Eq. 1

$$ \hat{w}_{s} = \frac{\sqrt 2 }{3}\left[ {u_{1} \alpha_{1} \left( {\lambda ,\varphi } \right) + u_{2} \alpha_{2} \left( {\lambda ,\varphi } \right) + u_{3} \alpha_{3} \left( {\lambda ,\varphi } \right)} \right]. $$

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This can be expressed as a relative correction by dividing by the uncorrected w_s

$$ \frac{{\hat{w}_{s} }}{{w_{s} }} = \frac{\sqrt 2 }{3}\left[ {\frac{{u_{1} }}{{w_{s} }}\alpha_{1} \left( {\lambda ,\varphi } \right) + \frac{{u_{2} }}{{w_{s} }}\alpha_{2} \left( {\lambda ,\varphi } \right) + \frac{{u_{3} }}{{w_{s} }}\alpha_{3} \left( {\lambda ,\varphi } \right)} \right]. $$

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The relative correction can be expressed using only sonic coordinates by substituting with Eq. 2,

$$ \frac{{\hat{w}_{s} }}{{w_{s} }} = \frac{\sqrt 2 }{3}\left[ {\frac{{\frac{\sqrt 3 }{2\sqrt 2 }u_{s} + \frac{1}{2\sqrt 2 }v_{s} + \frac{1}{\sqrt 2 }w_{s} }}{{w_{s} }}\alpha_{1} \left( {\lambda ,\varphi } \right) + \frac{{ - \frac{1}{\sqrt 2 }v_{s} + \frac{1}{\sqrt 2 }w_{s} }}{{w_{s} }}\alpha_{2} \left( {\lambda ,\varphi } \right) + \frac{{ - \frac{\sqrt 3 }{2\sqrt 2 }u_{s} + \frac{1}{2\sqrt 2 }v_{s} + \frac{1}{\sqrt 2 }w_{s} }}{{w_{s} }}\alpha_{3} \left( {\lambda ,\varphi } \right)} \right], $$

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which can be further simplified as

$$ \frac{{\hat{w}_{s} }}{{w_{s} }} = \frac{1}{3}\left[ {\left( {\frac{\sqrt 3 }{2}\frac{{u_{s} }}{{w_{s} }} + \frac{1}{2}\frac{{v_{s} }}{{w_{s} }} + 1} \right)\alpha_{1} \left( {\lambda ,\varphi } \right) + \left( { - \frac{{v_{s} }}{{w_{s} }} + 1} \right)\alpha_{2} \left( {\lambda ,\varphi } \right) + \left( { - \frac{\sqrt 3 }{2}\frac{{u_{s} }}{{w_{s} }} + \frac{1}{2}\frac{{v_{s} }}{{w_{s} }} + 1} \right)\alpha_{3} \left( {\lambda ,\varphi } \right)} \right]. $$

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By definition, tan(λ) = v_s/u_s and tan(φ) = w_s/$\sqrt {u_{s}^{2} + v_{s}^{2} }$, such that the ratios u_s/w_s and v_s/w_s can be expressed as

$$ \frac{{u_{s} }}{{w_{s} }} = \frac{\cos \lambda }{\tan \varphi } $$

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and

$$ \frac{{v_{s} }}{{w_{s} }} = \frac{\sin \lambda }{\tan \varphi }. $$

(11)

Equations 10 and 11 can be substituted into Eq. 9

$$ \frac{{\hat{w}_{s} }}{{w_{s} }} = \frac{1}{3}\left[ {\left( {\frac{\sqrt 3 }{2}\frac{\cos \lambda }{\tan \varphi } + \frac{1}{2}\frac{\sin \lambda }{\tan \varphi } + 1} \right)\alpha_{1} \left( {\lambda ,\varphi } \right) + \left( { - \frac{\sin \lambda }{\tan \varphi } + 1} \right)\alpha_{2} \left( {\lambda ,\varphi } \right) + \left( { - \frac{\sqrt 3 }{2}\frac{\cos \lambda }{\tan \varphi } + \frac{1}{2}\frac{\sin \lambda }{\tan \varphi } + 1} \right)\alpha_{3} \left( {\lambda ,\varphi } \right)} \right]. $$

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It is important to distinguish that λ and φ are the original/uncorrected wind direction and angle of attack based on original/uncorrected transducer measurements. An equation similar to Eq. 12 can be expressed with respect to the corrected angles $\hat{\lambda }$ and $\hat{\varphi }$,

$$ \frac{{w_{s} }}{{\hat{w}_{s} }} = \frac{1}{3}\left[ {\left( {\frac{\sqrt 3 }{2}\frac{{\cos \hat{\lambda }}}{{\tan \hat{\varphi }}} + \frac{1}{2}\frac{{\sin \hat{\lambda }}}{{\tan \hat{\varphi }}} + 1} \right)\alpha_{1} \left( {\hat{\lambda },\hat{\varphi }} \right) + \left( { - \frac{{\sin \hat{\lambda }}}{{\tan \hat{\varphi }}} + 1} \right)\alpha_{2} \left( {\hat{\lambda },\hat{\varphi }} \right) + \left( { - \frac{\sqrt 3 }{2}\frac{{\cos \hat{\lambda }}}{{\tan \hat{\varphi }}} + \frac{1}{2}\frac{{\sin \hat{\lambda }}}{{\tan \hat{\varphi }}} + 1} \right)\alpha_{3} \left( {\hat{\lambda },\hat{\varphi }} \right)} \right]. $$

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In Eq. 12, the left-hand side is the relative correction while in Eq. 13 it is the relative error. Finally, the limit of the relative correction as the angle of attack becomes near-horizontal is

$$ \mathop {\lim }\limits_{\varphi \to 0} \frac{{\hat{w}_{s} }}{{w_{s} }} = \frac{1}{3}\left( {\alpha_{1} \left( {\lambda ,\varphi } \right) + \alpha_{2} \left( {\lambda ,\varphi } \right) + \alpha_{3} \left( {\lambda ,\varphi } \right)} \right) + \frac{1}{3}\left[ {\left( {\frac{\sqrt 3 }{2}\cos \lambda + \frac{1}{2}\sin \lambda } \right)\alpha_{1} \left( {\lambda ,\varphi } \right) + \left( { - \sin \lambda } \right)\alpha_{2} \left( {\lambda ,\varphi } \right) + \left( { - \frac{\sqrt 3 }{2}\cos \lambda + \frac{1}{2}\sin \lambda } \right)\alpha_{3} \left( {\lambda ,\varphi } \right)} \right]\mathop {\lim }\limits_{\varphi \to 0} \frac{1}{\tan \varphi }, $$

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which approaches ± ∞ for most α functions and leads directly to Eq. 5.