Atmospheric Acoustic Remote Sensing - Chapter 6 pdf

The worst-case position of the spectral peak with respect to the frequency bins then gives frequency estimates which are higher because they are on a wider curve.. Averaging can-not be d

In previous chapters we have described the atmospheric properties accessible to

SODARs, elements of SODAR design, and instrument calibration In a number of

instances we have also discussed signal-to-noise ratio in general terms In practice,

separating valid signals from the noise background is a major part of SODAR

hard-ware and softhard-ware design We consider these features in the current chapter

6.1 SIGNAL ACQUISITION

6.1.1 S AMPLING

Although already discussed in Chapter 2, sampling will be briefly revisited In the

simplest case, a SODAR transmits a signal

Asin(2 Qf T t)

gen-eral is Doppler shifted and has modified phase

having frequency 960 + 960/3 = 1280 Hz gives the same digitized values as if it had

frequency 960/3 = 320 Hz The same is true for negative E This means that higher

frequency components can add into the lower frequency spectrum This is called

aliasing This means that all frequency components outside of nf s ± f s/2 should be

excluded from the signal before digitizing This is called the Nyquist criterion

Usu-ally this is interpreted as using anti-aliasing low-pass filters to remove all frequency

components outside of ±f s/2, but in fact the criterion is satisfied if band-pass filters

6.1.3 M IXING

For a SODAR, the bandwidth of the Doppler spectrum is generally much smaller

2 × 10 sin(π/10)×960/340 = 82 Hz So typically a filter need only have a bandwidth

of, say, 200 Hz It is usual to implement this filter as a low-pass filter, but this means

that the signal frequencies of interest must lie below say 100 Hz, rather than be

cen-tered around, say, 4500 Hz This is achieved by demodulation or mixing down the

signal to be centered around 0 Hz

The signal p(t) is multiplied by a mixing waveform

M t I 2sin2Pf t m

(6.2)giving

M t p t I  A [cos2P§©¨f T f m$f t·¹¸ F cos2P§©¨f T f m$f t·¹¸ F ]

filter can give just

I t A cos 2P §©¨f T f m$f t·¹¸ J

(6.3)

nega-tive Doppler shifts are then easily identified by looking at only the posinega-tive frequency

part of the spectrum, as shown in Figure 6.1

output is switched to the speaker during the transmitted pulse This is a very

con-venient signal to use as the mixing signal, so that f m = f0 However, since cos(2π∆ft)

= cos(2π[−∆f]t), positive and negative Doppler shifts cannot be distinguished This

means that, say, easterly and westerly winds will give the same result To overcome

this limitation, a quadrature, or 90° phase, signal is also mixed with the echo signal

M Q t 2cos2Pf t T

giving

pler shift to be determined since

I t jQ t A e j2P$f tJ

and the Fourier spectrum of this combination has either a single positive peak (for ∆f

positive, or a single negative peak (for ∆f negative) Generation of a quadrature, cosine,

to be passed through two mixing circuits and sampled with two ADC channels

SODAR, 180 Hz for a 4.5-kHz system, and 230 Hz for a 6-kHz system It is

nec-essary to sample at least twice the highest frequency, and depending on BP filter

characteristics, perhaps three or four times the highest frequency For example, the

= 1130 samples for a height range of 200 m In practice SODAR systems will

usu-ally sample a little longer than for the range displayed or recorded, to avoid

combin-ing echoes from more than one pulse This also affords the opportunity to measure

the background noise during the period at the top of the range in which no echoes

are being returned The total number of samples per pulse is not large, and so can

be stored pending Fourier transforming The fast Fourier transform (FFT) can be

Negative Doppler shift

FIGURE 6.1 Positive and negative Doppler shifts are readily distinguished providing

completed on sequential groups of samples, corresponding to the displayed range

gate length, or from overlapped groups of samples so that more spectra can be

dis-played (although not with additional information) FFTs are most conveniently

gate of 64 × 340/(2 × 960) = 11 m, but the AeroVironment reports Doppler spectra at

6.1.4 W INDOWING AND S IGNAL M ODULATION

Sampling a finite length of the time series record, for the purposes of doing an FFT,

is equivalent to sampling the entire time series and then multiplying the series of

in the FFT The effect of this is to convolve the power spectrum of the time series

with a sin(πNf/f s )/(πNf/f s) function The spectral peak level from a single sine

compo-nent will vary in value depending on what frequency the peak is at

The four plots in Figure 6.2 show part of the positive half of a spectrum which

contained 64 points in the FFT and was sampled at 960 Hz The top plot shows the

result (solid diamond points) for a sine wave at 97.5 Hz Because of where the

sam-pled points fall in relation to the peak of the sin(πNf/f s )/(πNf/f s) function, the

result-ing estimate of the peak is only 0.4 instead of 1.0 The second and third plots show

results for sine waves at frequencies of 100 and 105.5 Hz The bottom plot shows the

result when the sampled time series has been multiplied by the Hanning window

so that the sampled values always are small at the start of the sampled group and at

the end The result of this “windowing” is that the spectrum for a pure sine wave is

wider (as shown in the ideal curve on the bottom plot) The worst-case position of

the spectral peak with respect to the frequency bins then gives frequency estimates

which are higher because they are on a wider curve They are still only 0.7 instead

of 1.0, however

Other windows can be used: all give better estimation of peak value but poorer

frequency resolution, when compared to the no-window case

6.1.5 D YNAMIC R ANGE

The amplified, filtered, and demodulated signal is an analog time series This is fed

to an ADC The digital bit pattern is then stored as a representation of the sampled

voltage of the SODAR signal If the circuit has ramp gain to offset the spherical

spreading loss, and has a band-pass filter to limit the noise bandwidth, then a 10-bit

0.1% In practice, this is far more accurate than the generally noisy input signals

However, if no ramp gain is used, a SODAR signal could be expected to vary by at

is required at the upper height, then 20 bits are required Thus to have a simpler

0.0 0.2 0.4 0.6 0.8 1.0

Frequency (Hz)

0.0 0.2 0.4 0.6 0.8 1.0

Frequency (Hz)

0.0 0.2 0.4 0.6 0.8 1.0

Frequency (Hz)

0.0 0.2 0.4 0.6 0.8 1.0

preamplifier circuit, the ADC bit width should be preferably 24 bits so as to have

sufficient dynamic range.

Once the FFTs have been performed, spectral peak detection methods are used

to determine velocity components and the raw samples are usually discarded Note

that sampling at, say, 960 samples per second gives turbulence samples every 0.18 m,

which is much smaller than the real spatial resolution for turbulence Consequently,

some averaging, say to 5 m (~30 samples) is usual, and only the averages are stored

Such averaging will normally be done in log space (dB values are averaged)

6.2 DETECTING SIGNALS IN NOISE

Reasonable wind estimates can be made in noisy conditions in which the power SNR

is less than 1 The signal peak needs to be detected, however, by some characteristic

which distinguishes it from the noise Such characteristics include the following

6.2.1 H EIGHT OF THE P EAK ABOVE A N OISE T HRESHOLD

Background noise can be estimated within a power spectrum from the highest

fre-quency parts of the spectrum, since the spectrum is usually considerably wider than

necessary for typical winds For example, the noisy spectrum in Figure 6.3 has a

sig-nal peak at 100 Hz, and the peak at that frequency is a likely candidate because of its

width and height The noise threshold might have been set at say 1.0 based on noise

levels from 300 to 480 Hz, but in this example this still leaves two possible peaks

6.2.2 C ONSTANCY OVER S EVERAL S PECTRA

Most commonly, averaging of power spectra is used to improve SNR Averaging

can-not be done on the time series, since this has positive and negative voltages and the

phase is random, so any averaging reduces the signal component as well as the noise

But the power series is the square of the absolute value of the Fourier spectrum, and

all phase information is therefore removed Averaging the signal component does not

Frequency (Hz) 0.0

0.5 1.0 1.5 2.0

FIGURE 6.3 Threshold detection of possible signal peaks.

change it, but averaging the noise component, which is random, reduces its

fluctua-tions by the square root of the number of spectra in the average (see Figure 6.4)

For example, the AeroVironment 4000 typically records spectra at a particular

range gate every 4 s, but displays data every five minutes This means that 75 spectra

are averaged Taking the above example, and averaging successive spectra, gives the

solid curve in Figure 6.4

The peak position is often estimated from the average frequency in the spectrum

but this should only be applied to the full, double-sided, spectrum

6.2.3 N OT G ENERALLY B EING AT Z ERO F REQUENCY

In many circumstances it is known that there is some wind, and therefore any peak at zero

frequency must be from a fixed echo This part of the spectrum can then be ignored

6.2.4 S HAPE

The spectrum shape for the signal component is often known from considerations of

pulse length, etc One way of discriminating against noise is to successively fit this

shape with its peak at each spectral bin, and accept the position giving the best fit A

good approximation is a Gaussian, or even a parabola of the right width

An even simpler variant is to take a weighted sum of several spectral bin values,

and accept the position giving the highest sum The weights can be all unity

(search-ing for maximum power in a given signal BW), or reflect the expected shape of the

signal peak

Frequency (Hz) 0.0

0.5 1.0 1.5 2.0 2.5 3.0

FIGURE 6.4 The effect of averaging the power spectrum shown in Figure 6.3.

6.2.5 S CALING WITH T RANSMIT F REQUENCY

A much more sophisticated method is to use two or more transmit frequencies The

Doppler shift scales with the transmit frequency, so peaks at the correct position in

the spectra from different transmit frequencies indicate a true signal This method is

probably used by Scintec

6.3 CONSISTENCY METHODS

performed on small blocks of samples, perhaps equivalent to 5 m vertically A

spec-tral peak detection algorithm then finds the individual Doppler shifts at each range

gate Velocity components are combined to give speed and direction This results in

individual and independent estimates of velocities at a series of vertical points.

Consistency checks and smoothing algorithms are then applied This step makes

a connection between the independent estimates (or assumes a connection)

Combin-ing velocity components may be interleaved with this check/smooth process

Is it possible to come up with a systematic algorithm for smoothing, allowing for

poor data points, and combining several profiles and points within a profile as

con-sistency checks? The following method has been described by Bradley and

Hüner-bein (2004)

A typical plot of spectra versus height shows generally higher spectral peaks

near the ground, and increasing spectral noise at higher altitudes Examination of

plots such as Figures 6.5 and 6.6 can indicate the most likely velocity profile by

fol-lowing the progression of spectral peaks with height

At height z m (m = 1, 2, …, M), power spectral estimates P im = P( f i , z m) are

compo-FIGURE 6.5 Typical raw power spectra versus height.

400

200 150 100

0 0

300 200 100 0

nents u i Higher values of P im are more likely associated with the echo signal rather

than with noise The quantity

therefore represents the relative uncertainty of a particular f i being at the signal peak

for height z m We therefore treat the f i , or equivalently the corresponding u i, as

2

Assume that the u are a linear function of basis functions K(z) with unknown

coefficients x as follows.

This puts the problem into the context of the solution of a set of linear equations

In particular, use of constraints, such as smoothness, profile rate of change, limiting

the deviation from other data points, etc., can be applied by calling upon the huge

constrained linear inversion literature

There are still a number of arbitrary decisions required, however These include

1 The relationship between the power spectral estimates and the variance,

2 The choice of basis functions, and

3 How to include other profile data as constraints

Other possible relationships between P im and Sim

2 include

80 60 40 20

The center of a wider peak is a good estimator:

S

N N

im

m i

i

P

2

2 2

  1 1 

2 2

2

SS

(6.8)Figure 6.7 shows a typical fit using this method, but without any constraints from

other profiles The method appears to show promise

6.4 TURBULENT INTENSITIES

There are two basic requirements in obtaining meaningful turbulent intensities:

1 Calibration of the system variable part of the SODAR equation and

2 Allowing for the background noise

Calibration is actually quite difficult One can try putting some well-defined

scattering object above the SODAR, but this must be above the reverberation part

200 180 160 140 120 100

80 60 40 20

Along-beam Velocity (m/s)

FIGURE 6.7 The fit through the spectra (white line) to give the spectral peak at each height

A Gaussian constraint is used for smoothness of velocity variations in the vertical.

of the SODAR range (i.e., above 20 m or so) and must be in the main beam of the

SODAR (i.e., at 20 m the object must be located to within ±1 or 2 m horizontally)

This is quite difficult with a tethered balloon, for example, but it might be possible to

use an object on an overhead wire Alternatively, a sonic anemometer can be used,

providing one can work out how to extract meaningful records from it, and then

allow for the extra vertical distance to the first usable SODAR range gate

lev-els recorded from the highest one or two range gates, or from receiving without

transmitting for a while, or from the wings of the power spectra Then

If calibrated turbulence levels are required, care must also be exercised that fixed

echoes are not contaminating the time series record Gross fixed echoes are always

evident on the SODAR facsimile display, but there is a problem with part

2 estimate, to see if there is a significant peak at zero frequency The true signal spectral peak is

2, but this will be only available at the vertical spatial resolution of the winds, rather than the vertical spatial resolution of the turbulence:

this reduced resolution may be adequate in many cases however

C V2

measures derived from SODAR winds should be treated with caution: they will usually be only an approximation to the true values since assumptions are nec-

essary on homogeneity and Taylor’s “frozen field” hypothesis

6.4.1 S ECOND M OMENT D ATA

These standard deviations are useful as

2 Statistic variables to obtain other quantities such as wind energy, and

3 Input into similarity relationships to derive other quantities

The latter is useful in, for example, obtaining estimates of surface heat flux, H,

in convective conditions through (Weill et al 1980)

Sw

z M

H T

3

(6.10)

where M is a constant and T is absolute temperature Also, the mixing layer height,

3 2

6.5 PEAK DETECTION METHODS OF

AEROVIRONMENT AND METEK

The SODAR incorporates signal-processing software to determine

1 The position in the spectrum of the signal peak (corresponding to Doppler

shift) and

2 The averages over a number of profiles (to improve SNR)

The methods for achieving these tasks vary a little between manufacturers

Some examples follow

6.5.1 A ERO V IRONMENT

The AeroVironment system performs peak detection on each individual 64-point

spectrum (128-point spectra can also be user-selected) This is done by finding the

highest power in any contiguous 5-spectral-point group (or 7-point for a 128-point

spectrum) across the frequency spectrum The SNR is then defined as the 5-point

power divided by the power in the remaining 59 points normalized by multiplying

estimated Doppler shift for the particular range gate and beam Note that if the user

selects the option to use beam 3 data, then a rejected beam 3 spectrum causes the

beam 1 and beam 2 peak estimates to also be rejected at that range gate for that

pro-file (i.e., the system does not default to a 2-beam configuration which might give

aver-ages of mixed 2-beam and 3-beam calculations) Numbers of accepted beam 1, beam

2, and beam 3 peak estimates in each averaging interval are output for the user

The system also employs an adaptive noise threshold as part of the decision to

accept/reject a spectrum This threshold is determined by sampling the background

noise prior to the transmit pulse, and appropriately scaling this threshold to account

for spherical beam divergence with altitude This option can be disabled or enabled

by the user If this option is disabled, the system uses a fixed noise threshold which

is applied at every altitude

Statistical analysis shows that the uncertainty in each estimate of the position of

the spectral peak in this scheme depends on

6.5.2 M ETEK

spectral estimate is

SNR P A

P

1S

If N s spectra are averaged, the average spectral estimate becomes P

A s N

N s

1 ,

60 profiles, giving 32 averaged spectral intensities Two noise spectra measurements

are made shortly before each pulse is transmitted and these are averaged to obtain

intensities are subtracted from the averaged intensities received after the pulse, to

give residual power spectra at each range gate It is assumed that the noise-free signal

power spectrum has a Gaussian shape

P f e

f

f f f

0

1 22

2

$PS

12

(6.13)

is a quadratic in f Using least-squares, the moments P0, ˆf , and T f can be estimated

In practice, only n spectral points within 1/4 height (6 dB) of the main peak are

included in the least-squares fit Simulations based on this scheme show that, for

are rejected which have SNR below a certain critical threshold, then this accuracy

com-ponent is Sv rSfˆ$z0 1 m s 1 and the error in the estimate of the width of the

horizontal velocity components are generally comparable and dominate over the

5 m s–1

6.6 ROBUST ESTIMATION OF DOPPLER

SHIFT FROM SODAR SPECTRA

6.6.1 F ITTING TO THE S PECTRAL P EAK

Assume that a sinusoidal signal s(t) of duration U is transmitted The amplitude

spec-trum of the received voltage is

i i

E

V i V i E

2

1 2PS

2

(6.16)From this probability distribution, the mean power spectral value at frequency

f i is

P i P p P dP i i i V e

E i

V i V i E

2

2 2

d V i V i SE

(6.17)

In other words, there is a systematic overestimate of the power spectral value by

the noise power quantity

N SE2

(6.18)

level when no signal is present (i.e., from the highest range gates) giving a reduced

power spectrum

P i P i Nˆ

(6.19)

The moments of are

(6.20)

This results in moments

2 2

f f

D f i f

the reduced power spectrum has a quadratic dependence on frequency

We find the nearest spectral frequency to the peak position, and write the index

i relative to this, so the nearest spectral frequency to the peak is labeled f0

Least-squares is used to estimate the three coefficients of the quadratic using 2Q+1 points

because in the case of unweighted least-squares this leads to simplification

We now apply the above methods to raw spectral data recorded from a Metek

SODAR/RASS The relevant parameters are given in Table 6.1 The time-series

echo strength is recorded for 3.2 s and each range gate (region over which each

spec-trum is valid) is 16 m in vertical extent The atmospheric conditions were low wind

and fairly neutral conditions (so relatively weak reflections) but with low levels of

external background acoustic noise

Figure 6.8 shows a typical Hanning-windowed time series for one range gate

Figure 6.9 shows the corresponding amplitude spectrum Note the signal peak near

the transmitting frequency From such spectra more localized spectra are selected,

so that only possible Doppler shifts are included in the analysis For a beam tilt angle

TABLE 6.1

List of parameters for the Metek SODAR

–0.10 –0.08 –0.06 –0.04 –0.02 0.00 0.02 0.04 0.06 0.08 0.10

Frequency (Hz)

Doppler shift of 67 Hz, so considering 16 spectral frequencies over the range 1593

values) from the range gate centered at 197 m Estimation of SNR using the wings

of the spectrum around the peak value gives SNR = 8 dB In Figure 6.11 the data

0 10 20 30 40 50 60 70 80 90

FIGURE 6.10 A local spectrum taken from range gate 13 (height 197 m) and for which the

estimated SNR is 8 dB.

FIGURE 6.11 Plot of log-corrected power spectrum from data in Figure 6.10 and with

Q = 3 Data are shown with dark dots and the fit with a solid line.

–6 –4 –2 0 2 4 6

6.6.2 E STIMATION OFTW

In practice reflections are from an ensemble of scatterers which provide a continuum

of Doppler shifts This gives spread to the Doppler spectrum which is particularly

, in vertical velocity is an important boundary layer parameter

Assume that the Doppler frequency from the ensemble has a Gaussian probability

centered on f D and with standard deviation TD This range of Doppler frequencies will

cause spectral broadening of the signal, and estimation of this extra broadening from

a vertical beam provides useful insights into turbulent eddy dissipation rates through

the standard deviation in vertical velocity, Tw A typical value for Tw is 0.3 m s–1,

Each scatterer in the ensemble contributes a power spectrum which may be

approximated by a Gaussian, so that the total spectrum is

12

1 2

2

P S

S max

D

1 2

2

S c

°

T

f f D T

maxSS

S 1 2

2

(6.24)which has a variance, ST2 S2fSD2 , equal to the sum of the contributing variances,

as expected There is now an extra variability (in addition to the background noise

discussed above) given by

2

1 21

2 1 2

2

f f D

D

S c

or

S

SSSS

P

D f D f

2 2

2 2

21

The regression methods discussed above can be used to estimate ST2 S2fSD2

and hence TD since Sf

2

is known from the system design The relative error in

SS

S

S

SS

T T

2

11

(6.27)

If SD2 S2f there is a large relative error multiplication factor in (6.27)

and

f T

f

S

A value of

f T

f

The time series from successive profiles should not be averaged, since they are

inco-herent and will average toward zero

Averaging of power spectra from successive profiles is useful, since phase

information has been removed The noise power fluctuates more than the signal,

FIGURE 6.12 Relative error in sigma-w value Five-point fits with peak at a spectrum

providing the averaging time is not too long (say no longer than 20 minutes, but this

the ith profile, at a particular range gate, are summed in the averaging process

(6.29)

so the standard deviation of the noise goes down as the square root of the number

of averages

6.7.1 V ARIANCE IN W IND S PEED AND D IRECTION OVER O NE A VERAGING P ERIOD

Generally wind data from a number of profiles are averaged In the following we will

restrict attention to the horizontal wind components The ith profile may contain an

The means and variances from a single averaging period are

N

u i i

N

v i i

N

v v i i N

and v i are available so N V ≤ N u , N V ≤ N v Also

V u v

u v

1

(6.31)Note that the wind direction needs to be calculated using four quadrants The

average wind speed and variance in wind speed are just found in the usual way

V

N V i u i v i N u v

N

V V

i i i