Tilt of a sonic anemometer from true vertical can introduce errors in the measured sensible heat and momentum fluxes. The tilt error in u^* or in a scalar flux is on the order of 5% per degree of tilt in the vertical plane aligned with the mean wind direction. Consequently, it is often necessary to rotate the coordinates of three-dimensional sonic wind data to correct for this tilt. NCAR estimates the sonic tilt angles from the archived wind data using the planar fit technique described in Wilczak, Oncley, and Stage, 2001, "Sonic Anemometer Tilt Correction Algorithms," Boundary Layer Meteor., 99, pp. 127-150.

Briefly, this technique assumes that the time-averaged wind field is confined to a plane surface, nominally parallel to the ground. It is assumed that the fluxes of interest are those normal to this wind-field plane and thus the vertical velocity axis is re-oriented normal to that plane. The orientation of the wind-field plane relative to the measurement axes of the sonic anemometer is determined by a least-squares fit of the wind data to the equation

w = a + bu + cv

where u, v, w are averaged horizontal and vertical wind components measured by the sonic anemometer.

Confidence in the fit to this model requires data with as wide a range of wind directions as possible; in practice, 90° or greater. Thus confidence can best be achieved with data extending over many hours or days. If the data are available, this requirement is acceptable because the sonic tilt angles are expected to be unchanged unless the sonic anemometer is moved (or the surrounding surface itself changes!). Thus the tilt angles are calculated for periods delimited by known physical changes in the sonic orientation, e.g. re-leveling the mast or swapping the sonic anemometer.

Correcting Wind Vectors for Instrument Tilt

To rotate wind vectors from instrument coordinates to the new coordinate system where the W axis is normal to the plane of mean flow, first determine the unit axes of the new flow coordinates, Uf, Vf, Wf, expressed in instrument coordinates.

Using the equation of the plane:

a = -bu - cv + w

the normal to the plane, W_f is:

W_f = (-b, -c, 1) / sqrt(b² + c² + 1)

Wf can be defined in terms of two angles:

tilt = atan(sqrt(b² + c²))

tiltaz = atan(-c/-b)

W_f = ( sin(tilt)*cos(tiltaz), sin(tilt)*sin(tiltaz), cos(tilt) )

tilt is the angle between the sonic W axis (W_s) and W_f, and is positive, from 0 to 90 degrees. tiltaz is the azimuth of the projection of the flow normal vector in the sonic UV plane, measured counter-clockwise from U_s.

Note: use the two argument arctangent function, atan2(-c,-b) in programming languages such as C, Java and R when computing tiltaz in order to return the full range of -pi to pi.The y and x arguments of atan2 are switched in spreadsheets such as Excel and Google Docs, so, in spreadsheets, tiltaz = atan2(-b,-c).

You have a choice of where to put the U_f axis in the plane of mean flow.

U_f can be defined as the intersection of the plane of mean flow with the plane defined by the  sonic U and W axes, U_s=(1,0,0), W_s=(0,0,1), using the cross-product:

U^'_f = V_s x W_f

In this case, since V_s = (0,1,0):

U^'_f = ((cos(tilt), 0, -sin(tilt)*cos(tiltaz))

Normalizing to a unit vector:

U_f = U^'_f / |U^'_f|

Or U_f can be defined as the intersection of the plane of mean flow with the plane defined by U_s and W_f, and then normalizing:

U^''_f = (W_f x U_s) x W_f

U_f = U^''_f / |U^''_f|

This last definitiion of U_f is used in the ISFS NIDAS code.

In either case, V_f is defined from the cross-product:

V_f = W_f x U_f

For a wind vector (u,v,w) in sonic coordinates, with a w offset "a", the corrected vector in flow coordinates (u_f,v_f, w_f) is computed from dot products with the new axes:

u_f = U_f dot (u,v,w-a)

v_f = V_f dot (u,v,w-a)

w_f = W_f dot (u,v,w-a)