Fundamental uncertainties that arise in instantaneous measurements made by lidar of wake vortices.

RI Young

Laser Doppler Velocimetry, LDV (a branch of coherent laser radar or lidar) has proven a particularly useful tool for characterising the wake vortices trailing behind landing aircraft[1]. There have now been many papers published in the open scientific literature demonstrating the success of lidar in measuring in-situ these vortices. It is now well accepted that lidar complements other techniques (sodar, ground anemoters or instrument towers) and is the only approach that can provide multiple high-resolution images throughout the lifetime of vortices.

Lidar measurements can contribute to a better understanding of the hazard presented to other landing aircraft in two ways. First it allows a detailed picture to be built of the vortex in terms of its strength, persistence and trajectory. This data can then be used to aid the further development of theoretical models. Secondly by gathering bulk information on the persistence of vortices and correlating this to aircraft type and meteorological conditions a database can be compiled that can be used to predict the vortex hazard on subsequent occasions.

Most of the published papers have concentrated upon demonstrating that the lidar has been used to accurately measure the vortices. That was because the primary goal was to show the strength of vortices for different aircraft types. From the high resolution of these measurements it is argued how lidar would be invaluable for further studies of the structure of the vortices and the mechanisms governing their behaviour.

What is often neglected from the papers is the fact that all the relevant characteristics of the vortex (circulation, peak tangential velocity and core location) can never be measured simultaneously to the maximum resolution possible. That is, from the nature of the measurements there will always be an uncertainty in at least one of the key vortex parameters measured, the lidar uncertainty principle. As lidar moves away from the demonstration concept to a useful tool to aid in aircraft wing design the reasons for this uncertainty need to be well documented and understood. It is the intention of this paper to describe the key fundamental trade offs in measuring vortices and why they arise.

Lidar measures wind velocities by scattering coherent laser radiation from dust particles, aerosols, present in the atmosphere and detecting the shift in frequency of the scattered light. This Doppler shift is proportional to the line of sight component of velocity for the aerosols illuminated by the beam. For the application of lidar to wake vortex characterisation the probe beam needs to be passed through the rotational flow of the wake vortex. Subsequent analysis of the Doppler shifts enables the creation of a detailed description of the air motion along the beam, that is a high resolution mapping of the vortex flow.

The general technique is that the lidar beam is aimed at the vortex and the beam brought to a focus as it passes through the rotational flow. The diameter of the laser beam in the focal range is about 10 mm so in order to build a picture of the complete vortex the probe beam is either scanned or held stationary while the vortex descends or drifts through the beam. The returned signal is strongest at the chosen focal range and falls off gradually on either side of the focal region. The sensitive region along the beam is conventionally defined as the section where the signal strength is within 3 dB of the peak. For the DERA wake vortex lidar the length of this region is given by (6(F/100)² m^-1, where F is the focal range. For actual application the depth of the probe beam extends over several meters and there is a returned signal from scattering along the length of the probe region. Hence the bulk of this scattering will be away from the region of peak rotational velocity and so most of back-scattered signals power resides at lower Doppler frequencies. As the depth of field is proportional to the square of the focal range, measurements made at greater altitude will probe a larger volume of space and as a consequence produce measurements with more of the signal at the lower end of the Doppler frequency range in comparison to those made at low altitude.

Every 50 ms a surface acoustic wave (SAW) analyser converts the signal into a frequency spectrum with a bandwidth covering the Doppler shifts from 0 to 32 m/s or between ± 16 m/s. This information is then digitised with a velocity resolution of 0.08 m/s.

When selecting the scanning pattern and signal interrogation period there is a trade off between overall coverage, spatial resolution, velocity resolution and signal to noise ratio, SNR. Each individual 50 ms Doppler spectrum has a low signal to noise ratio, which can be improved by integrating adjacent spectra, though this causes a blurring due to evolution during the integration time. Alternatively a larger region can be covered or more frequent intersections of the core achieved by using a faster scanning rate but this results is a quicker sweep through the vortex so either blurring will increase or the SNR is lowered. As such there is little opportunity to maximise the resolution of the spectra gathered.

Due to the making of the measurements there is an impact on the precision of the spectra in three ways. There will be inaccuracies in the axial and lateral positions, plus velocity precision, where axial position refers to the scanning pattern and lateral position to distance along the LOS of the probe beam. In practice it is difficult to distinguish the uncertainties caused by one error source from the impact it has one the other two. Hence for the rest of the paper degraded spectrum resolution implies induced inaccuracies in the axial, lateral and velocity components without endeavouring to specify which term is most affected.

Typically for the DERA system 512 adjacent spectra are integrated. This gives sufficient signal to noise ratio, SNR, to determine the peak tangential velocity under most conditions of atmospheric scattering.

In practical terms this three-way trade off reappears when it is decided what to be measured. Multiple intersections of the cores are needed to precisely define vortex location or in order to observe decay in circulation strength with time for a single lidar. To do this requires a fast scanning rate which increases the blurring and hence lowers the spatial resolution of the spectrum. To measure the cores through their full life, the cores need to be probed whilst they are at altitude and tracked during descent until they disperse. The greater the height the greater the probe depth and so the smaller the part of the spectral power in the high frequency region of the spectrum, corresponding to the vortex tangential velocity. Thus to achieve accurate core following, resolution in the velocity profile across the core is sacrificed. The accuracy of these measurements will improve as the vortices descend and the probe beam focal length reduced as it tracks the descent.

To measure the peak tangential velocity of the vortex core requires maximum spectral resolution. For this the probe beam is held stationary (or a slow scan is used if there is no crosswind) and the vortex core allowed to drift through it. Thus maximum velocity resolution is achieved at the expense of a single intersection.

One potential solution to this is to hold the beam stationary, await the passage of the vortex cores through the beam and then move the beam to another stationary position. There are a number of problems with this, firstly it would require some modifications to the lidar station. However once the current upgrades to the optical scanner are complete it would be possible to undertake this operation. However there is a further complication with this approach, to obtain maximum resolution of the core profile the focal range is set short so that the probe depth is short. After the vortex core has been probed it is likely to go into ground effect and be destroyed so only a single intersection is likely to be made.

Even then the measured peak tangential radial velocity is unlikely to be the real one. The peak tangential radial velocity only occurs over an extremely small part of the vortex core profile. Therefore there will be very few aerosols moving at the peak velocity to scatter a signal back to the lidar. Hence the actual peak tangential velocity vortex parameter is likely to be lost in system noise, though a good estimate of it can sometimes be made.

It is the turning torque of the vortex rather than the peak tangential velocity of the vortex that is the hazard. Circulation is the parameter that represents the turning torque and can be measured without needing to know the maximum tangential velocity. Currently measuring precisely this maximum radial velocity is only really an issue for the dynamic fluid flow fundamentalists. Peak radial velocity may become a more key parameter if a relationship between itself and vortex persistence is proven. For now it is sufficient to note the uncertainty in making the measurement and perhaps begin developing an extrapolation technique to estimate the peak velocity from what is measured.

DERA’s solution to these problems is to run most of the lidar trial with slow scanning speeds to allow a few vortex core intersections with good resolution. DERA can achieve this because with direction sensing we are not looking for the actual maximum tangential radial velocity of the vortex. Thus we can build up multiple high resolution core profiles adequate for trajectory predictions and circulation estimation. During suitable conditions during the measurement campaign a few fixed probe beam measurements will be taken to ensure spectra are obtained of the highest resolution.

Direction sensing allows accurate determination of core profile. This is obtained at the expense of the maximum velocity that it is possible to measure. In the peak velocity profile mode measurements of airflow up to 32 m/s can be made whilst directional sensing only allows peak velocities of ± 16 m/s to be detected. This is not viewed as a problem, most of the aircraft core velocities will be in the ± 16m/s region. Furthermore peak velocity is not needed to derive an estimate of vortex circulation.

In addition to the above there are further subtleties with the interpretation of the lidar data that need to be clearly understood if this technique for vortex characterisation is to become effective. The data is gathered as a time series of spectra to build up a complex picture of the vortices radial velocities as a function of time and position. It is tempting to believe that much information can be gleaned from these spectra. There is a limit to what vortex characteristics can be accurately determined from these spectra as is explained below. For instance the impact on line of sight, LOS, velocity determination from the spectral spreading due to the finite instrumental linewidth of lidar needs to be considered.

The spectra can be thought of as the histogram of the LOS velocity profile weighted by the lidar response function. The maximum velocity in the spectrum is representative of the vortex velocity perpendicular to the LOS of the lidar beam. In practice the measured spectrum is modified from the theoretical spectrum by the addition of shot and speckle noise added and then convolved with the system spectral point spread function. Without a-priori knowledge of actual back scatter strength, aerosol concentration and noise factors this cannot be deconvolved from the spectra to give the actual aerosol distribution along the LOS for the measurement. Without detailed knowledge of the distribution of aerosol concentration along LOS the actual amount of air moving at a particular radial velocity remains uncertain.

However it has been shown that there will always be a peak in the lidar spectrum corresponding to the tangential LOS velocity[2]. This forms a robust feature that can be accurately measured though if the lidar focus is so far from the vortex then the peak loses its clarity and eventually it too is lost in the back ground noise. This is shown in fig 6 of reference 2. In this case the maximum velocity is defined in terms of the noise threshold and will be an overestimate if the true tangential velocity due to the system spectral spreading, due to the Hamming amplitude window[3]. This error can be compensated for by prior knowledge of the lidar instrumentation linewidth, something which is known

Therefore the position of the maximum spectra signal represents the tangential velocity from which the core profile and hence the circulation can be derived. Thus the only feature of the spectra that can be accurately measured is the peak tangential velocity.

A further consequence of the above discussion is that it is misleading to look for other meaningful data in the lidar spectra. To do so would assume that the particulates causing the back scatter in the vortex were uniformly distributed and this is unlikely. Whilst uniformity of particulate distribution is a useful assumption to prepare a first-order mathematical model it is unlikely to be true in nature. The situation is further complicated by engine exhaust entrainment or condensation within the vortex core. Looking at the spectra we seldom witness the slow, uniform variation in signal around the lower velocities expected for uniform scattering. This implies the scattering is non-uniform.

For lidar to become a useful tool it has to be accepted by the aircraft industry that there will always be some uncertainty in at least one of the key parameters of the vortex pair being measured. Once the fundamental trade off between quality and quantity have been accepted then work can begin on countering the uncertainty in the measurements. For a successful experiment two key requirements need to be met: The desired data can be measured and that the results are meaningful.

This can only be achieved through informed discussion between lidar scientists and aircraft design people. Hopefully the relevant European parties will come together under the C-wake banner to resolve these difficulties and realise all the potential benefits of wake vortex characterisation by lidar measurement.

1) Measurements of aircraft wake vortices at Heathrow by Laser Doppler Velocimetry. JS Greenwood & JM Vaughan. Air Traffic Quarterly Vol 6(3) Pg 179-203 98.

2) Coherent laser radar and the problem of aircraft wake vortices G Constant et al. Journal of Modern Optics 41(11) 2153-2173 1994.

3) Analysis of circulation data from a wake vortex lidar. RM Heinrichs & J Dasey. AIAAI. 1997.