«GEORGE TREVIÑO CHIRES Incorporated, PO Box 201481, San Antonio, TX 78220-8481, U.S.A. EDGAR L. ANDREAS U. S. Army Cold Regions Research and ...»
AVERAGING INTERVALS FOR SPECTRAL ANALYSIS OF
CHIRES Incorporated, PO Box 201481, San Antonio, TX 78220-8481, U.S.A.
EDGAR L. ANDREAS
U. S. Army Cold Regions Research and Engineering Laboratory, Hanover, NH 03755-1290, U.S.A.
(Received in ﬁnal form 13 December 1999) Abstract. We formulate a method for determining the smallest time interval T over which a turbulence time series can be averaged to decompose it into instantaneous mean and random components.
From the random part the method deﬁnes the optimal interval (or averaging window) AW over which this part should be averaged to obtain the instantaneous spectrum. Both T and AW vary randomly with time and depend on physical properties of the turbulence. T also depends on the accuracy of the measurements and is thus independent of AW. Interesting features of the method are its real-time capability and the non-equality between AW and T.
Keywords: Turbulence spectrum, Nonstationarity, Real-time, Averaging windows, Intermittency.
1. Introduction The atmospheric boundary layer (ABL) is by nature nonstationary at all time scales. Another name for nonstationarity is intermittency, but, for the ABL, intermittency generally refers to short-term events. A key to understanding ABL intermittency is the ability to say something meaningful about the turbulence during such events.
Many researchers have reported results of investigations into requisite averaging intervals for estimating turbulence statistics (e.g., Bradshaw, 1971; Lumley and Panofsky, 1964, p. 36ff.; Gupta et al., 1971; Wyngaard, 1973; Blackwelder and Kaplan, 1976; Wyngaard and Clifford, 1978; Sreenivasan et al., 1978; Andreas, 1988; Kaimal et al., 1989; Lenschow et al., 1994; Gluhovsky and Agee, 1994;
Gupta, 1996). None, however, has adequately considered the intervals necessary to determine turbulence statistics within intermittent ABL events.
Here we extend the idea of minimum averaging to its lowest limit, one that approaches the duration of intermittent events – tens of seconds. Keep in mind that spectral analysis of the ABL requires quantifying not only the instantaneous spectrum but also changes in it from one intermittent event to the next. Such changes typically appear in frequency content, distribution, domain, and bandwidth.
To accomplish this, we formulate a method for identifying the smallest interval T over which an ABL time series can be averaged to deﬁne its instantaneous Boundary-Layer Meteorology 95: 231–247, 2000.
© 2000 Kluwer Academic Publishers. Printed in the Netherlands.
232 GEORGE TREVIÑO AND EDGAR L. ANDREASmean and accompanying deviations from the mean. In some circles, mean is referred to as the systematic part and deviations are called the random part (see Bendat and Piersol, 1971). From the random part, the method determines the optimal interval (or window) AW over which this part should be averaged to estimate.
The interesting feature of these determinations is not only that AW and T are determined in real time and depend on instantaneous properties of the ABL, but also that AW does not always equal T. The difference between AW and T is due to dynamical effects and not to kinematical scale changes. That is, it is not possible to re-scale the turbulence to make AW and T equal to one another. This difference is consistent with the fact that mean and random components represent different degrees of freedom. In determining we treat the mean as an instantaneous DC component; from emerges all the parameters necessary to assess the instantaneous character of ABL turbulence.
The method localizes by assigning uniform weight to the random part of the time series within AW and zero weight to anything outside. The location of AW in time is designated by t, the present time. The length of AW is proportional to a t-dependent parameter called the memory. As t advances, the memory changes; and, thus, the magnitude of AW varies. This has the effect of varying the ‘block length’ of data used in computing (cf. Henjes, 1997). The magnitude and variability of the block length here, though, are deﬁned by the random part of the ABL time series rather than by the analyst. In this way, the method adapts to instantaneous ABL properties and reduces the likelihood that short-lived events are lost or masked due to user-deﬁned averaging intervals that are too short or too long.
Its real-time capability makes our method more versatile than existing algorithms, such as the sliding FFT (Fast Fourier Transform) and variable interval time averaging (VITA) (Gupta et al., 1971; Blackwelder and Kaplan, 1976; Gupta, 1996). Real-time capability in sliding FFT and VITA is unachievable because (i) both use information from future ABL behaviour to determine present frequency information; and (ii) both require interactive user input, which cannot reliably emulate the changing properties of the ABL. Information from the future is used whenever an algorithm determines ABL behaviour at time t by using information at time t + T, T 0, and all intervening times. The method formulated here uses only information from the present and not-too-distant past to deﬁne ; no information from the future is required. The method formulated here requires no interactive user input. Lastly, sliding FFT and VITA, in contrast to our method, use the same averaging interval for mean and random components of the turbulence.
depends on t but not on AW. That is, in For an optimal AW, the resulting spite of inherent variability, AW is deﬁned in such a way that is strictly (t, ω), not (t, ω, AW), where ω is radian frequency. As t advances, and successive sets of T, AW, and are identiﬁed, (t, ω) estimates the instantaneous spectrum of
SPECTRAL ANALYSIS OF NONSTATIONARY TURBULENCEthe ABL. We demonstrate our method by applying it to wind speed data collected in the ABL.
Table I illustrates this procedure for a present time of 09:46 local time and averages of 1 s. Column 1 is actual time beginning at 09:45:57; column 2 is actual wind speed data recorded at 10 Hz; column 3 is the 1 s averages (10 data points) of the data in column 2, beginning at 09:46 (the present) and running backwards until 09:45:59.1. That is, all the elements in column 3 are averages deﬁned as in (1) for the succession of ‘present’ times 09:45:59.1, 09:45:59.2, 09:45:59.3,..., 09:46:00.
The quantity in column 3 corresponding to 09:45:59.1 is the average of the ten quantities in Column 2 from 09:45:59.1 back to 09:45:58.2.
Column 4 is the difference between column 2 and column 3. If 1 s was indeed the window that ‘matches’ the related time scale of the data, Column 3 would then be the time-dependent mean of the data for the succession of ‘present’ times 09:45:59.1, 09:45:59.2, 09:45:59.3,..., 09:46:00, and column 4 would be the corresponding time-dependent random part. An alternate designation for the totality of information in Column 4 is χ(γ ), t − T ≤ γ ≤ t. This notation will be used hereafter.
Next, evaluate the quantity t m.s.e. = χ(γ )dγ T t− T
(4) where both the scale factor ( T − |τ |)−1 and the limits of the integration in γ are adjusted to accommodate the fact that only a ﬁnite amount of data are available
SPECTRAL ANALYSIS OF NONSTATIONARY TURBULENCE
the computation. The expression within the outer ( ) is the autocorrelation. The τ -dependence of the scale factor can be eliminated if T is considerably larger than the largest value of τ for which the autocorrelation is non-zero; that is, if T is large enough that ( T − |τ |)−1 ≈ ( T )−1 for the largest meaningful value of τ in the autocorrelation. We show later that this relationship between T and τ will always be the case. The resulting autocorrelation then depends on t, T, and τ but is subsequently integrated over τ. Thus, for a nonstationary ABL, the real-time dependence of m.s.e. is never lost.
For large T, though, the integral within ( ) in (4) stabilizes at C(t, τ ). The T stability is such that the subsequent integration in (4), viz. − T C(t, τ ) dτ, is unchanged by further increasing T ; that is, this stability eliminates the integral’s dependence on T. Thus, by the ‘requisite T ’ above we mean a T long enough that
T C(t, τ )dτ ≈ 0. T −T
¯ As a result, S(t, T ) is a valid estimate of the mean at time t, and χ(γ ), t − T ≤ γ ≤ t, represents deviations from the mean. The right-hand side of (4) is then σ 2 (t) (t)/ T, where σ 2 (t) is the variance of χ(γ ), t − T ≤ γ ≤ t, and (t) is its integral scale. Because of the ‘requisite’ nature of T, σ 2 and do not depend on T.
In numerical work, though, m.s.e. is taken to be zero whenever it becomes smaller than the square of the accuracy (ACC) of the measurements. This is equivalent ¯ to saying that the absolute value of χ (t, T ) in (3) is less than the absolute value ¯ of ACC, which effectively makes χ (t, T ) equal zero to within the accuracy of the measurements. It also makes the expected value of χ(γ ) equal zero to within ¯ the same accuracy. This is what we mean above by ‘small’, viz. χ (t, T ) is not identically zero but is smaller than what can be detected with our sampling device.
The smallest T that yields this condition is then the value for which, (i) a viable estimate of the mean at time t can be deﬁned, and (ii) a valid time description of the related random part can be determined. For example, for the anemometer we use to measure wind speed, the accuracy of the measurements is speciﬁed by the manufacturer as ±1% of the wind speed. This we take to mean that at any instant, ¯ ACC = 0.01S(t). The magnitude of T for our analysis is therefore deﬁned in the ¯ m.s.e. sense through (σ 2 / T ) 10−4 S 2. Andreas and Treviño (1997) also use instrument accuracy as a criterion for deciding how to proceed with a turbulence analysis.
SPECTRAL ANALYSIS OF NONSTATIONARY TURBULENCE
ˆ where R is a β-dependent approximation to the normalized autocorrelation of χ(γ ), t − T ≤ γ ≤ t.
As β varies from zero to T, L equals in those cases when T and the autocorrelation of χ(γ ) is greater than or equal to zero for all values of τ. For ¯ our data, the condition T is satisﬁed when (σ /S) 0.03. In those cases when T and the autocorrelation becomes negative for some values of τ, L ¯ is greater than. In the cases when (σ /S) 0.03, L can still be determined, but the measured signal is highly concentrated (small deviations) about its mean.
In all cases, though, (6) is a measure of the separation in time beyond which variables cease to be correlated. It is thus a measure of the degree of randomness.
Small L corresponds to a highly random (short memory) ABL; and large L, to not-so-random. Since classical analysis methods do not incorporate effects of such variability, they can be designated as time-invariant memory methods (TIMMs).
The method formulated here, conversely, is designated the time-dependent memory method (TDMM). We show later, though, that TDMM reduces to TIMM when the ABL is stationary. No TIMM, though, can produce the same information as does TDMM when the ABL is nonstationary. VITA, on the other hand, also incorporates effects of averaging window variability, but its averaging window is deﬁned by the user.
238 GEORGE TREVIÑO AND EDGAR L. ANDREASIn view of this interpretation, then, the magnitude of AW is taken to be 10L.
Since L is always greater than or equal to, it follows that AW ≥ 10. This ensures that, (i) a representative sample of the random part is analyzed, and (ii) all correlations are lost within AW. As indicated in (7) below, deﬁning AW this way assigns uniform weight to values of χ(γ ) that are backward from t by an amount less than or equal to 10L and assigns no weight to values of χ(γ ) backward from t by more than 10L. It also assigns no weight to elements of the ABL forward in time from t. This is compatible with the fact that, in physical phenomena, the future cannot inﬂuence the present.