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#### Correction of Sodar Tilted Beams for Vertical Speed

Most sodar systems operate by emitting a sequence of three or more pulses. One of the pulses propagates vertically (which we shall call W) and at least two pulses propagate at a small angle tilted from the vertical. The two tilted pulses (which we shall call U and V) have components that are orthogonal in the horizontal plane.  U and V pertain to the orientation of the sodar antenna and not necessarily to the traditional east/west and north/south directions.

As a transmitted pulse propagates through the atmosphere, a small portion of the energy is backscattered by the air and Doppler shifted by the movement of the air relative to the sodar. It is this Doppler-shifted backscattered energy that represents the signal in a monostatic sodar system (one in which the transmit and receive antennas are collocated). The size of the Doppler shift is proportional to the radial wind speed along the beam. In the case of the vertical beam, because the beam is directed vertically and there is no horizontal component, the radial wind speed is the same as the vertical wind speed. However, in the case of the tilted beams, the Doppler-shifted return signal (or radial wind speed) is a function of the tilt angle and both the vertical wind speed component and the horizontal wind speed component. If the vertical wind speed is zero, then the measured radial speed will be proportional to the horizontal wind speed component and the tilt angle only.

The W, V and U components can be expressed mathematically as follows:

W = -f S / (2F)

V  = -f S / [2 F sin(θ)] - W / tan(θ)

U  = -f S / [2 F sin(θ)] - W / tan(θ)

where,

W = vertical component speed (m/s)

V = horizontal component speed (m/s)

U = horizontal component speed (m/s)

f = Doppler shift (Hz)

F = transmit frequency (Hz)

θ = beam tilt angle (degrees)

S = speed of sound (approximately 340 m/s)

The above expressions pertain to the three-beam Model VT-1 sodar. In other sodar systems, the components are calculated similarly, but the signs may be different depending on the beam geometry.

If the vertical component (W) is small enough to be negligible, the computations simplify, and the horizontal speed and direction of the wind can be derived from the return signals generated by the two tilted transmit pulses (U and V) only. The U and V components uncorrected for vertical speed are then:

V  = -f S / [2 F sin(θ)]

U  = -f S / [2 F sin(θ)]

In operation, typically the mean U and V components will first be calculated uncorrected for vertical speed. As an example, let us assume that we have a situation where the mean uncorrected U and V components for an averaging period are both +5 m/s. Then, assuming that the vertical speed is zero, the speed of the horizontal wind is:

Speed = Sqrt((U * U) + (V * V)) = Sqrt((5 * 5) + (5 * 5)) = 7.1 m/s

and the horizontal wind direction is:

Direction = Atan(U / V) = Atan(5 / 5) = 45 degrees

In actual operation, the direction would be rotated based on the orientation of the sodar antenna and the directions of the U and V beams.

Let us assume now that the vertical speed is non-zero. In typical applications, the sodar averaging time is configured for 10 to 15 minutes. Although on the long-term average of the vertical component may be zero or near zero, during short averaging periods, vertical speeds of ±0.5 m/s or higher may occur. This is especially so in complex terrain areas or even in non-complex terrain where thermals may occur during sunny, unstable daytime conditions. If we assume again that the uncorrected U and V components are each +5 m/s and the vertical speed is +0.5 m/s, then the corrected components (Uc and Vc) are:

Uc = U - W / tan(θ)

Vc = V - W / tan(θ)

The tilt angle of the Model VT-1 is approximately 18 degrees. Hence, the corrected horizontal components for this example are:

Uc = 5 – 0.5 / tan(18) = 3.5 m/s

Vc = 5 – 0.5 / tan(18) = 3.5 m/s

After correcting the tilted components for vertical speed, the horizontal wind speed is:

Speed = Sqrt((Uc * Uc) + (Vc * Vc))

= Sqrt((3.5 * 3.5) + (3.5 * 3.5)) = 5.0 m/s

and the horizontal direction after the tilted component correction is:

Direction = Atan(U / V) = Atan(3.5 / 3.5) = 45 degrees

In this particular example, the horizontal wind speed changed rather substantially (by 2.1 m/s) while there was no change in the horizontal wind direction. This is because the U and V components were equal in value and had the same sign, causing the error to compound for wind speed but cancel for wind direction).

As another example, let us assume that the uncorrected U component is -3 m/s and the uncorrected V component is +6 m/s. The uncorrected horizontal wind speed and direction for this example are:

Speed = Sqrt((U * U) + (V * V)) = Sqrt((-3 * -3) + (6 * 6)) = 6.7 m/s

Direction = Atan(U / V) = Atan(-3 / 6) = 333 degrees

If we assume again a vertical speed of +0.5 m/s and a tilt angle of 18 degrees, the corrected U and V components are:

Uc = (-3) – 0.5 / tan(18) = -4.5 m/s

Vc = 6 – 0.5 / tan(18) = 4.5 m/s

After correcting the tilted components for vertical wind speed, the horizontal wind speed and direction are:

Speed = Sqrt((Uc * Uc) + (Vc * Vc))

= Sqrt((-4.5 * -4.5) + (4.5 * 4.5)) = 6.4 m/s

Direction = Atan(U / V) = Atan(-4.5 / 4.5) = 315 degrees

In this case, where the uncorrected U and V components had opposite signs, the vertical correction made only a small difference (0.3 m/s) in the horizontal wind speed but a large difference (18 degrees) in the horizontal wind direction.

Some sodar systems have a tilt angle as large as 30 degrees, which reduces the vertical correction term by about one half compared to the 18-degree tilt angle used by the Model VT-1. Even so, in many instances, particularly in complex terrain areas, more accurate measurements of the horizontal wind speed and direction will be obtained by correcting the tilted components for vertical speed. It is worth noting, too, that although a larger tilt angle will reduce the calculated error in the uncorrected tilted components, it introduces other issues. Ideally, the beam pattern in the horizontal should be kept as small as possible when it is desired to make the measurements site specific, which is often the case. While a larger tilt angle reduces the vertical speed error in the tilted components, it enlarges the beam pattern. With an 18-degree tilt, the horizontal distance between the vertical and the tilted beams is only about 30 m at a height of 100 m. With a 30-degree tilt, the return signal from the tilted beams at 100 m above ground is actually coming from a point nearly 60 m in the horizontal from the sodar location.

In the Model VT-1, the user has the option of turning the vertical speed correction of the tilted beams on or off. The default setting is on, and in most situations, this will yield the most accurate data. At this point, one might ask why not always correct for vertical speed? In some cases, it may be undesirable to correct for vertical speed. If the vertical speeds are always zero or near zero, correcting for vertical speed could actually increase rather than decrease the error in the measurement of the horizontal wind speed and direction. This is because there is a certain amount of error in the measurement of each of the W, V and U components. And when the vertical speed correction to the tilted components is made, any error in W is multiplied by a factor of about 2 to 3, depending on the tilt angle, (due to the division of W by tan(θ) in the vertical term) and added to both the U and V components. This is why it is essential to have accurate measurements of the vertical component. One important element of accuracy is spectral resolution.  The Model VT-1 has a spectral resolution of 1 Hz, which corresponds to vertical speed resolution of 0.04 m/s at the transmit frequency of 4504 Hz. Some sodar systems operate at half this frequency, which means they must have a spectral resolution of at least 0.5 Hz to achieve the same vertical speed resolution.

In sodar systems that operate with five or nine beams, one vertical and either four or eight tilted beams are emitted. The tilted beams are transmitted such that each one has a counterpart in the opposite direction. The magnitude of the Doppler shift is determined by averaging the signals from both directions. Because of the difference in sign, in one direction any non-zero vertical wind speed will tend to add to the Doppler shift and in the other subtract. Thus, any effect due to a non-zero vertical wind speed will tend to cancel. Hence, in a five- or nine-beam system, it is unnecessary to make a specific correction for vertical speed. One of the tradeoffs, though, is that a five- or nine-beam system substantially enlarges the beam pattern in the horizontal plane, making the measurements less site specific. At a height of 100 m and with a tilt angle of 30 degrees, there will be a horizontal distance of about 120 m between the opposing beams.

The bottom line to this discussion is that in most cases three-beam sodar systems should correct the tilted components for vertical speed. If this is not done, errors of a few meters per second or more may occur in the measurement of horizontal wind speed and errors of about 10 to 20 degrees could occur in the horizontal wind direction.