Vertical resolution is a specification that is typically given for a Doppler sodar system by the various sodar manufacturers.  This specification is often of interest to those who either use or contemplate using a Doppler sodar system to collect wind profile data.  Amongst most sodar users, there is a desire to have the finest vertical resolution possible, particularly in the wind energy community.  Unfortunately, there appears to be a lack of consistency and information amongst and from sodar manufacturers in defining the vertical resolution of their systems.  Further, many sodar users have a lack of understanding of this issue and may not fully appreciate some of the implications involving the vertical resolution of sodar systems.  Although it might be expected that a very fine vertical resolution would be the most desirable, configuring or operating a sodar system with a very fine vertical resolution may actually degrade overall system performance.  The following paragraphs shall attempt to explain why this is so.

The vertical resolution of a sodar system is the smallest distance interval over which the height of a sodar signal can be resolved.  It is NOT simply the altitude interval over which a sodar system is configured to output data.  Sodar output intervals, which are usually termed range gates, may not be independent of one another, i.e., they may include overlapping data, depending on the sodar configuration.  To understand how the true vertical resolution can be ascertained, we can first talk in general about how a sodar system functions.  The following discussion will be limited to single-frequency sodar systems only, but the extension to multi-frequency sodar systems is similar.

Very basically, a sodar operates by transmitting a high-energy acoustic pulse in a vertical or near vertical direction and then switching to a receive mode to sample the small amount of energy that is backscattered toward the sodar as the transmit pulse propagates through the atmosphere.  In the receive mode, a sodar samples for a period that corresponds to the amount of time needed for sound to travel the round-trip distance from the sodar up to the maximum altitude range and then return back to the sodar.  A series of samples is obtained during this interval and each sample represents an instant in time.  At each sampling instant, the sodar "sees" a signal that is due to the total backscattered energy from the transmit pulse.  Longer transmit pulses will provide for a stronger signal and higher altitude performance because there will be more backscattered energy arriving back at the sodar at any instant in time.  Each sample due to the energy backscattered by the transmit pulse has a "signal depth" that is proportional to the duration of the transmit pulse.  Hence, longer transmit pulses result in less vertical resolution since the sodar, in effect, integrates across the total amount of energy backscattered by the entire length of the transmit pulse on each sample.

Even without going into any mathematics or specifics, it can be seen that the vertical resolution of a sodar system is related, at least in part, to the duration of the transmit pulse.  Most, if not all, commercial sodar systems, however, are not capable of using just a single sample to derive a signal.  The signal in this case is a measurement of the frequency of the backscattered energy.  The frequency of the signal is generally determined by obtaining a series of signals over time and then performing a Fast Fourier Transform (FFT) on the signal data.  This is done at each sodar range gate (reporting height).  Because the series of data points is collected over time and the data arrive from higher altitudes with each time step, the FFT of the signal data at each range gate results in a "sampling depth".  The sampling depth is a function of the round-trip speed of sound, the sample rate and the number of data points included in the FFT.

The vertical resolution of a Doppler sodar system is then the signal depth plus the sampling depth, which we shall call the "effective sampling depth".  The effective sampling depth is probably the truest measure of the vertical resolution of a sodar system.  It is a function of the transmit pulse duration, sodar sample rate, FFT size and the speed of sound (which is dependent on ambient temperature).

Looking at this more specifically, let us first examine the issue of signal depth.  The physical length of the transmit pulse as it propagates through the atmosphere is simply:

P = (S) (d)

where,

P = transmit pulse length (m)

S = speed of sound (m/s)

d = transmit pulse duration (s)

The signal depth, however, is actually one-half this value, due to the fact that the sodar signal is the result of backscattering.  At any particular height from which the signal arrives at the sodar, it will have traveled twice this distance.  Thus, the speed of the sodar return signal as it propagates up to any particular height and then returns to the sodar is effectively one-half the speed of sound when it is measured as the height that the signal has reached.  The backscattered signal that arrives at the sodar is, in effect, folded in half.  Another way to look at this is to consider that at the instant the sodar stops transmitting and begins receiving the backscattered signal, the backscattered energy has already traveled a maximum round-trip distance corresponding to the pulse length, or up to a height corresponding to one-half the pulse length.  All other backscattered energy at that instant will have traveled a distance ranging from 0 up to that height.  At any point later in time, the difference in the range of heights of the backscattered signal (i.e., the signal depth) will always be one-half the transmit pulse length.

Confirming that the signal depth is actually one-half the length of the transmit pulse is perhaps best done by constructing a table of values showing the heights of the top and bottom of the transmit pulse and the heights of the top and bottom of the backscattered return signal as a function of time.  If we assume the speed of sound is 340 m/s, then the speed of the return signal is in effect one-half this value or 170 m/s (because the distance traveled by the signal is twice the distance to the signal height).  If we further assume that the sodar operates with a 50-ms transmit pulse, then the heights of the top and bottom of both the transmit pulse and the return signal can be calculated as a function of time from the beginning of the transmit pulse:

In the above table, the pulse length is simply the difference between the pulse top and the pulse bottom.  Similarly, the signal depth is the difference between the signal top and the signal bottom.  In this example, the transmit pulse length was 17.0 m and the signal depth was one-half this value or 8.5 m.  Signal sampling does not begin until after the transmit pulse is fully emitted, and the signal depth is constant with time after the transmit pulse is fully issued.

As indicated previously, the sodar vertical resolution is actually the sum of the signal depth and the sampling depth.  The sampling depth is given by:

SD = (S/2) (FFT size) / (SR)

where,

SD = sampling depth (m)

S = speed of sound (m/s)

FFT size = number of sample points

SR = sample rate (Hz)

Ideally, the sampling depth should be made as small as possible to obtain a fine vertical resolution.  The two parameters we have to work with here are the FFT size and the sample rate.  Thus, either the FFT size could be made small and/or the sample rate could be made large (fast) to achieve a minimum sampling depth.  However, in any sodar system that uses an FFT to derive the frequency of the signal data, the sampling depth is not an independent parameter in the sodar configuration.  That is because the sodar spectral resolution is inversely related to the sampling depth.  In a sodar system using an FFT to identify the signal frequency, the spectral resolution is given by:

Resolution (Hz) = (SR) / (FFT size)

Hence, a very fine sampling depth results in a very coarse spectral resolution, and vice versa.  (Note: The actual spectral resolution of sodar systems is typically enhanced by bin averaging of the FFT results, but the inverse relationship with sampling depth remains in effect.  Also, computational techniques are available, such as "zero padding", that improve the sampling depth without degrading the spectral resolution.  While zero padding may improve the sampling depth without degrading spectral resolution, the tradeoff is usually reduced signal-to-noise ratio and reduced altitude performance.)

 Ideally, the sodar spectral resolution should be as fine as possible to obtain the most accurate results with a sodar system.  But configuring the sodar to operate with a fine spectral resolution results in a coarse vertical resolution.

In practice, the vertical resolution of a Doppler sodar system must be balanced with spectral resolution and altitude performance to obtain optimum results.  If the vertical resolution is taken to be the effective sampling depth as defined above, then typically the finest vertical resolution that can be achieved while maintaining optimum accuracy and altitude performance will be about 10 to 20 m.  However, the actual performance will generally be better for typical wind profiles.  This is because data near the center of the effective sampling depth will tend to carry more weight than data from the upper and lower extremes which will often tend to cancel each other.  Hence, comparisons of sodar data with data from wind instruments at corresponding levels on a collocated tower will be found to be quite good even with larger effective sampling depths.

The ART Model VT-1 is configurable with a combination of settings for transmit pulse duration, FFT size and sample rate that result in an effective sampling depth ranging from about 10 to 40 m.