## 1. Introduction

The long-term plan in Juneau, Alaska, is to develop a terrain-induced wind shear and turbulence hazard warning system for the airport vicinity. Preliminary research indicates that turbulence in this region, as measured by a research aircraft, is well-correlated with the wind speed and vertical shear of the horizontal wind measured with nearby wind profilers and anemometers. Strong vertical shears of the horizontal wind have been measured near the mountain tops surrounding Juneau and are often associated with strong turbulence. Both the shears and turbulence can be a hazard to aircraft operations. The development of a real-time warning system is the motivation for producing rapid update wind and wind shear estimates from Doppler wind profilers. In most profiler applications, a so-called consensus wind is produced based on measurements over a time period of between 30 and 60 min. Because measurements taken over a fairly long time period are used to compute a consensus wind, spurious outliers in the data can be recognized and removed based on this consensus. However, when a rapid update estimate is required, data quality control becomes a more immediate concern. In a real-time warning system false alarms quickly erode user confidence, so the detection and removal of erroneous data is very important. The National Center for Atmospheric Research (NCAR) Winds and Confidence Algorithm (NWCA) has been developed to provide rapid update estimates of wind and vertical shear of the horizontal wind from wind profiler radial velocities. It also provides an indication of confidence in those estimates. In the case when the confidence in the estimates is low, the wind and shear estimates will not be used in the warning system. NWCA can also produce a confidence-weighted average wind, which may be used for similar applications as traditional consensus winds. The NWCA algorithm has been running at three profiler sites in Juneau, Alaska, since 1998.

To produce these products the wind field is assumed to be horizontally uniform and stationary over the measurement volume and duration, while vertical shear of the horizontal wind is allowed. Statistical tests are used to check these assumptions, and when these tests fail a low confidence is given to the wind estimates. Note that failure of these assumptions can be due to atmospheric motions or a result of incorrect moments provided to the algorithm. To test these assumptions, a more general assumption of a stationary, linearly varying wind field is used. This allows for an analysis of biases caused by shears that are not zero, but are set to zero given the horizontally uniform wind field assumption. In Juneau, a four-beam analysis is used, that is, no vertical beam is included in the measurement sequence. In the case of a five-beam system, the assumption of a horizontally uniform wind field can be relaxed to require only that the vertical component of the wind is horizontally uniform. However, it is shown that the additional shear terms (the horizontal shear of the horizontal wind) in this more general wind field are difficult to estimate because they have large variances. In the case of a five-beam system these horizontal shears of the horizontal wind can be estimated by time averaging. With a four-beam system in Juneau, only the vertical shear of the horizontal wind is calculated.

The proposed warning system includes anemometers as well as wind profilers. Because of these additional sensors, the system can continue to function during periods when the assumptions are violated and high-confidence winds are not available at certain heights. This has the effect of reducing false alarms. The warning system is discussed further at the end of section 4.

The assumptions that the wind field is horizontally uniform and stationary are common in wind profiler analysis. For example, estimation of the horizontal wind from radial velocities measured with three-beam wind profilers has been presented by Strauch et al. (1984) and for five-beam wind profilers by Strauch et al. (1987). Koscielny et al. (1984) considered unknown shear terms to be a bias if the wind field is linear rather than horizontally uniform. They applied their work to a three-beam, azimuthal scanning (VAD) and elevation scanning system. There is a long history of the linear wind field assumption for scanning Doppler radars. An early example is the work of Browning and Wexler (1968) where an analysis of the VAD was given in the presence of a linear field, and estimates for mean convergence and shearing deformation were also presented. Waldteufel and Corbin (1979) studied the estimation of various wind field quantities by applying a linear wind field analysis to a volume of radial velocity data (VVP). A more recent example of linear analysis is found in Boccippio (1995), which reviews the VAD and VVP methods using regression diagnostics. All of these analyses assume a linear wind field with additive noise. A regression is performed to estimate some of the linear field parameters by fitting the linear model to measured radial velocity data. In the same spirit, NWCA fits a local linear wind field model to a four- or five-fixed-beam system. Local regressions along fixed beams are used to estimate the horizontal wind and the vertical shear of the horizontal wind field. The appendix further considers these shear terms and effects of noise. A simple mathematical explanation is given for a long observed fact that shears in some directions cannot be individually estimated whereas sums of these shears can be estimated. A confidence index that is used for quality control of the wind estimates has also been developed, based on a variety of tests that determine the suitability of the linearity, spatial homogeneity, and stationarity assumptions.

Although the NWCA algorithm can be used with any moment algorithm that also computes a first moment confidence, it is used in Juneau with the NCAR Improved Moments Algorithm (NIMA; Morse et al. 2002; Cornman et al. 1998). NIMA uses fuzzy logic and global image processing to identify and compute the moments of the atmospheric signal in wind profiler Doppler spectra in the presence of ground clutter, radio frequency interference, and other contamination. A confidence value for the moments is also generated. Using these moments and confidences, the NWCA algorithm then estimates the wind and some components of wind shear from four- or five-beam pointing directions. A study investigating the performance of NIMA and NWCA is described in Cohn et al. (2001), with further results presented in section 6. In this study, human experts identified the atmospheric signal in Doppler spectra, which was then used to compute the first moments, and these first moments were used to compute horizontal wind estimates. These winds were compared to those produced by NWCA. It is shown that for high-confidence winds, there is strong agreement between the human-produced winds and the NWCA-produced winds. This shows the confidence index has skill in identifying high-quality data and in rejecting poor quality data. For a 9-month period, it is estimated that after removing 3% of the data with lowest confidence, the average vector error in the horizontal wind is about 1 m s^{−1}. For the winter period, 18% of the data must be removed to achieve an estimated accuracy of 1 m s^{−1} when compared to a human expert.

## 2. Calculation of horizontal wind and shear terms from a Doppler wind profiler

Assume that the wind field in a local region above the profiler is well approximated by a linear function. Over a relatively small region of space, a linear model should be a good first-order estimate of the wind field. Furthermore, assume that the wind field is stationary during the time that the profiler cycles through the beam directions. These assumptions are made for the region containing the radial data that are used in making a particular estimate of the horizontal wind. For a four-beam system, this region is in the shape of a frustum of a cone, and the size of this frustum increases with height. This frustum is described in more detail later in this section. At the top of our analysis cone (2500 m) the horizontal scale is 1.3 km. Based on these assumptions, a method for estimating the linear wind field parameters, including the effect of shears, is presented for the case of no noise. In this case, an algorithm for computing the horizontal wind and shears should return the exact values of some of the linear wind field parameters. In the appendix an error analysis that accounts for measurement noise is given. Persistent vertical shears of the horizontal wind over hundreds of meters can be a hazard to flight. The assumptions that the wind field is linear and stationary are tested using various statistical techniques as shown in section 4. When these tests indicate that the assumptions might not be correct, a low confidence is assigned to the horizontal wind estimates. Low confidence winds and low confidence vertical shears of the horizontal winds will not be used in a warning system to estimate hazards.

**V**

**x**

**V**

**x**

_{o}

**x**

**x**

_{o}

**x**is a point in space and

**x**

_{o}is an arbitrary reference point, that is,

**V**(

**x**

_{o}) is the velocity at

**x**

_{o}; and 𝗔 is a matrix of the wind shears, that is,

**x**−

**x**

_{o}) is the matrix multiplication of 𝗔 and the column vector (

**x**−

**x**

_{o}). The parameter

*u*

_{x}represents the

*x*direction shear of the wind parallel to the

*x*axis, that is,

*u*

_{x}= ∂

*u*/∂

*x,*

*υ*

_{y}represents the

*y*direction shear of wind parallel to the

*y*axis, etc. Note that 𝗔 is independent of position since a linear-field approximation is based on constant shears over the local domain. Hence the entries in this matrix can be measured at any point and applied to all others. Likewise, the linear assumption allows any convenient point to be chosen as the reference point

**x**

_{o}. That the choice of reference point is arbitrary is shown in Eq. (2) below by adding and subtracting a constant vector

**x**

_{1}:

**V**(

**x**

_{1}) =

**V**(

**x**

_{o}) + 𝗔 · (

**x**

_{1}−

**x**

_{o}).

In the analysis of the profiler measurements, it is convenient to use spherical coordinates as shown in Fig. 1, where the radar is at the origin, *x* = *r* sin*ϕ* cos*θ,* *y* = *r* sin*ϕ* sin*θ,* and *z* = *r* cos*ϕ,* where *ϕ* is the zenith angle, *θ* is the azimuth measured counterclockwise from the *x* axis, and *r* is the length of the vector **x**.

Consider the velocity vector **V**_{V} = **V**(**x**_{V}) = [*u*_{V}, *υ*_{V}, *w*_{V}]^{T} at the point **x**_{V}(*r*_{o}) = [0, 0, *z*_{V}]^{T} = [0, 0, *r*_{o} cos*ϕ*]^{T}, which is directly above the radar at the height *z*_{V} = *r*_{o} cos*ϕ,* where *r*_{o} is the distance from the radar to the center of a pulse volume of a nonvertical beam. The height *z*_{V} = *r*_{o} cos*ϕ* is used since this is the height at which estimates are given in the nonvertical beams. Here the subscript *V* refers to the vertical direction. Capital letter subscripts such as *E,* *W,* *S,* and *N* refer to the directions of east, west, north, and south, respectively. For convenience, these directions are assumed to be aligned with the profiler coordinate system. Thus the direction of *E* corresponds to the positive *x* axis and *N* corresponds to the positive *y* axis, etc. This is done only to facilitate the discussion since it is a simple task to express the horizontal wind vector in terms of any other orthogonal coordinate system, that is, performing a rotation.

The goal of this analysis is to estimate the horizontal component of the wind vector at the point directly above the profiler. As will be shown below, for a four-beam system and the assumption of a linear wind field, this is the only point at which the horizontal velocity component can be estimated. This is due to the fact that shears of the horizontal wind component cannot be estimated. For a five-beam system, these shears can be estimated, however, in general, regardless of the number of beams, the horizontal shears of the vertical component cannot be estimated without additional assumptions. Starting with data from an oblique profiler beam, for example, the east beam, such that *ϕ* ≠ 0 and *θ* = 0, consider the point **x**_{E}(*r*_{o}) = [*r*_{o} sin*ϕ,* 0, *r*_{o} cos*ϕ*]^{T}. The vector velocity at this point is **V**[**x**_{E}(*r*_{o})]. Assuming that the profiler returns an unbiased estimate of the radial velocity at **x**_{E}, the radial velocity at that point is denoted by *V*(**x**_{E}). Thus the radial velocity field is a measured quantity from the profiler. Note that the radial velocity along the east beam is defined as the inner product of the wind vector with the unit vector along the radial, *V*_{E}(*r*) = **V**[**x**_{E}(*r*)] · **e**_{E}, where *r* is a location along the radial and **e**_{E} = **x**_{E}/ ‖**x**_{E}‖ = [sin*ϕ,* 0, cos*ϕ*]^{T}, where ‖**x**‖ is the length of vector **x**.

**x**

_{E}(

*r*

_{o}) [Eq. (1)] and along the radial, that is, where the set of points is restricted to vectors of the form

**x**= [

*r*sin

*ϕ,*0,

*r*cos

*ϕ*]

^{T}, and assuming the vector wind at

**x**

_{E}(

*r*

_{o}) is

**V**[

**x**

_{E}(

*r*

_{o})] = [

*u*

_{E},

*υ*

_{E},

*w*

_{E}]

^{T}yields

*a*

_{E}and

*b*

_{E}may be determined by application of the singular value decomposition (SVD) method to solve a linear least squares fit to the data (see Forsythe et al. 1977, p. 192). These values are estimated quantities. Data from the few nearest range gates

*r*=

*r*

_{o}±

*i*Δ

*r*are used; that is, the linear model in Eq. (3) is fit to the measured radial velocity data at

*V*

_{E}(

*r*

_{o}±

*i*Δ

*r*). In a typical application,

*i*varies from 0 to 2 for a range gate size Δ

*r*of 60 m. The maximum value of

*i*is limited to ensure that the linear model is valid and in order to detect shears on a scale of interest to aircraft response. Confidence in the radial first moments is available from NIMA and the fit is weighted by these confidence values. In general, the confidence-weighted average of the first moments used in the linear fit will be the same as the confidence-weighted average of the linear model. When the confidences are all equal,

*a*

_{E}will be the average of the first moments used in the linear fit. In the absence of noise, the coefficients

*a*

_{E}and

*b*

_{E}are determined exactly in the local linear wind field model. Thus,

Using data from the opposite west beam (i.e., *ϕ* ≠ 0, *θ* = 180°, cos*θ* = −1) and expanding about a point at the same range, *r*_{o}, **x**_{W}(*r*_{o}) = [−*r*_{o} sin*ϕ,* 0, *r*_{o} cos*ϕ*]^{T}, the radial velocity is expressed by *V*_{W}(*r*_{o}) = **V**[**x**_{W}(*r*_{o})] · **e**_{W}, with **e**_{W} = [−sin*ϕ,* 0, cos*ϕ*]^{T}.

**x**(

*r*) = [−

*r*sin

*ϕ,*0,

*r*cos

*ϕ*]

^{T}again in the direction of the radial, and assuming the vector wind at

**x**

_{W}(

*r*

_{o}) is

**V**[

**x**

_{W}(

*r*

_{o})] = [

*u*

_{W},

*υ*

_{W},

*w*

_{W}]

^{T}yields

*a*

_{W}and

*b*

_{W}are again determined by application of the SVD method. Again in the absence of noise,

**x**

_{W}as they are at

**x**

_{E}and so the terms from opposite beams may be combined. The difference of the

*a*terms in Eqs. (4) and (6) yields

*a*

_{E}

*a*

_{W}

*u*

_{E}

*u*

_{W}

*ϕ*

*w*

_{E}

*w*

_{W}

*ϕ.*

*u*

_{E},

*u*

_{W},

*w*

_{E}, and

*w*

_{W}, Eq. (1) is applied to determine the wind directly over the profiler, at the point,

**x**

_{V}(

*r*

_{o}) = [0, 0,

*r*

_{o}cos

*ϕ*]

^{T}. First from the east beam:

*w*

_{x}= 0 so that

*w*

_{E}=

*w*

_{W}. Subsequently, when considering the N–S beams, a similar condition for

*w*

_{y}will be imposed. The assumption that

*w*

_{x}=

*w*

_{y}= 0 implies that the vertical component of the wind field is locally homogenous. This is a common assumption in profiler wind measurements, although it may be violated, for example, in a terrain-induced (stationary) gravity wave. If this assumption is violated there will be an error in the wind calculation. In this case, the wind estimates will be biased as shown in Eq. (A7) in the appendix. This is similar to a result found in Koscielny et al. (1984). In section 3 it is shown that in the case of a linear wind field, the terms

*w*

_{x}and

*w*

_{y}cannot be determined without additional assumptions. An indirect test for the assumption that the vertical component of the wind is horizontally uniform is discussed in section 4. In most cases, the system should assign low confidence to such estimates when this assumption is violated.

*w*

_{x}= 0, Eqs. (7) and (10) can be combined to find the component of the horizontal velocity along the E–W axis at the point (0, 0,

*z*

_{V}):

*w*

_{x}= 0. Note that since the linear field is expanded around (0, 0,

*z*

_{V}), the effect of the

*u*

_{x}shear is zero. By similar reasoning (and assuming

*w*

_{y}= 0), an estimate for the component of the horizontal velocity along the N–S axis at the point (0, 0,

*z*

_{V}) is given by

*a*

_{E}and

*a*

_{W}from Eqs. (4) and (6) gives the following result:

*a*

_{E}

*a*

_{W}

*w*

_{E}

*w*

_{W}

*ϕ*

*u*

_{E}

*u*

_{W}

*ϕ.*

*a*

_{E}

*a*

_{W}

*w*

_{V}

*ϕ*

*u*

_{x}

*r*

_{o}

^{2}

*ϕ.*

*z*

_{V}) is thus

*u*

_{x}

*r*

_{o}sin

^{2}

*ϕ*)/(cos

*ϕ*). If

*u*

_{x}= 0.01 s

^{−1}, which is a fairly large horizontal shear,

*ϕ*= 15° (typical for wind profilers), and

*r*

_{o}= 2500 m, then (

*u*

_{x}

*r*

_{o}sin

^{2}

*ϕ*)/(cos

*ϕ*) = 1.73 m s

^{−1}. This is a significant error. If a horizontally uniform wind field is assumed, instead of a general linear field, the shear terms would vanish and the two vertical estimates would be

*u*

_{x}if a vertical beam is available, that is, in a five-beam system. However, for a wind profiler operating without a vertical beam, as is the case in the current Juneau operations, Eqs. (18) and (19) must be used. This requires the assumption of a horizontally uniform wind field. The variances for these estimates in the presence of noise are discussed in the appendix. Equations (18) and (19) are used in section 4 to test the assumption of a horizontally uniform wind field.

## 3. Closer examination of the shear terms

*w*

_{z}can be directly estimated and it is possible to give an estimate of

*u*

_{x}. If an additional azimuth such as

*θ*= 45° were available, the quantity

*υ*

_{x}+

*u*

_{y}could be estimated as well.

**e**is given by

*V*

_{r}=

**V**(

**x**) ·

**e**=

**V**(

**x**

_{o}) ·

**e**+ 𝗔 · (

**x**−

**x**

_{o}) ·

**e**, where

**e**=

**x**/‖

**x**‖ = [sin

*ϕ*cos

*θ,*sin

*ϕ*sin

*θ,*cos

*ϕ*]

^{T}. In considering only the radial component of the wind seen by a single beam,

**x**and

**x**

_{o}are restricted to be along the radial,

**x**=

*r*

**e**and

**x**

_{o}=

*r*

_{o}

**e**so that

**x**−

**x**

_{o}= (

*r*−

*r*

_{o})

**e**, where

*r*and

*r*

_{o}are the distances of the points

**x**and

**x**

_{o}, respectively, from the profiler. Then

^{T}is antisymmetric; that is, (𝗔 − 𝗔

^{T})

^{T}= 𝗔

^{T}− 𝗔 = −(𝗔 − 𝗔

^{T}), where 𝗔

^{T}is the transpose of 𝗔. This can be shown by considering any antisymmetric matrix 𝗠, that is, a matrix for which 𝗠

^{T}= −𝗠. The identity (𝗠

**y**) ·

**y**=

**y**· (𝗠

^{T}

**y**) =

**y**· (−𝗠

**y**) = −(𝗠

**y**·

**y**) can only be true if 𝗠

**y**·

**y**= 0. Thus the only measurable terms in Eq. (21) are the shears in (𝗔 + 𝗔

^{T})/2 (the symmetric part of 𝗔):

These are the shears shown in Eq. (20) and the additional shear term *υ*_{x} + *u*_{y}, which may be found with an additional beam azimuth such as *θ* = 45°. Thus a profiler cannot measure *u*_{z}, *υ*_{z}, *w*_{x}, or *w*_{y} directly but it can measure *u*_{z} + *w*_{x} and *υ*_{z} + *w*_{y}. If *u*_{z} is replaced by *u*_{z} + *c* and *w*_{x} is replaced by *w*_{x} − *c,* where *c* is an arbitrary constant, then the radial velocity field given in Eq. (21) will not change. This shows that for a linear wind field, it is impossible to measure *w*_{x} and *w*_{y} directly without further assumptions. So in order to estimate *u*_{z} and *υ*_{z}, it must be assumed that *w*_{x} and *w*_{y} (the horizontal shears of the vertical component of the wind) are zero or at least small. This is equivalent to assuming that the vertical component of the wind is constant in a horizontal plane, an assumption that could fail with a complicated wind field, such as that near significant terrain. Section 4 discusses an indirect test for this assumption. Shear estimates are typically quite noisy and hence these estimates must be averaged over time and/or space. The space averaging is accomplished by the linear fit, Eq. (A6). The shear estimates are noisy because sin*ϕ* is quite small for small angles *ϕ* and sin*ϕ* appears in the denominator in Eq. (20), magnifying the error in the case of noise. Estimates for the vertical shears of the horizontal component of the wind, *u*_{z} and *υ*_{z}, are often found by direct computation from averaged measurements of *u* and *υ* over range gates along a beam. In the case when the wind field is linear these average wind shears are approximations to *u*_{z} + *w*_{x} and *υ*_{z} + *w*_{y}. The shear of the horizontal wind field is defined to be the length of the vector difference of the horizontal winds at two different heights above the profiler divided by the change in height. In the case of a linear wind field, this equals (*u*^{2}_{z}*υ*^{2}_{z}^{1/2}. In order to estimate this shear in the vertical wind, the assumption that the vertical wind is horizontally uniform is required, that is, *w*_{x} = *w*_{y} = 0. See appendix for an analysis of the shear terms *u*_{x} and *u*_{z}.

**V**

**x**

**V**

**x**

_{o}

**x**

_{o}

**x**

*u*

_{z}in 𝗔 with

*u*

_{z}+

*c*in 𝗔′, where

*c*is some constant. Also, replace

*w*

_{x}in 𝗔 by

*w*

_{x}−

*c*in 𝗔′. The new linear wind field is then given by

**V**

**x**

**V**

**x**

_{o}

**x**

_{o}

**x**

It is easy to see that these two wind fields will have the same radial wind values at all points and all beams. It is also easy to see that horizontal wind fields do not agree at *x*_{0} where *x*_{0} is on the *z* axis. See the discussion before Eq. (22). It is not likely that this exact set of circumstances could occur in real data, but it does point out the possibility that two radial wind fields could look very much alike, but the horizontal winds would not agree. This occurs in the case of large horizontal shears in the vertical wind or in the case of large temporal changes of the vertical wind. These horizontal shears of the vertical wind cause a bias in the estimates for the horizontal winds [Eq. (A7)], and are a fundamental limitation of wind profiler geometry. Thus it is important to detect such situations, and give such winds low confidence. This is discussed is section 4, where benefits of incorporating a vertical beam are also discussed.

## 4. The horizontal wind confidence estimate

From an operational point of view, it is important to give a confidence estimate for each horizontal wind estimate. For example, if the winds are used to detect an apparent operational hazard, data may not be used if the wind confidence is low. Cohn et al. (2001) demonstrate that the current wind confidence algorithm does have skill in predicting the similarity of NWCA winds to those considered to be truth by an expert. Based on this study some performance results are given in section 6. Cohn et al. (2001) also give a comparison to aircraft wind measurements.

The confidence estimates are produced using fuzzy logic methods. See Cornman et al. (1998) and Morse et al. (2002) for a more complete description of fuzzy algorithms applied to profilers. A human expert might determine the wind confidence based on assessing the validity of the assumptions that were used to calculate the winds. These assumptions are that

the radial velocity moments used in the SVD determination of

*a*and*b*were correct;the local linear wind model is a good model of the atmosphere;

the vertical wind has no horizontal shear, that is,

*w*_{N}=*w*_{S}=*w*_{E}=*w*_{W},*w*_{x}=*w*_{y}= 0, and*u*_{x}=*u*_{y}= 0 in the case of a four-beam analysis; andthe atmosphere is relatively stationary in time and space over the period when the various radar beams were collected.

*C*is the final confidence,

*c*

_{i}are the individual confidence factors chosen such that 0 ≤

*c*

_{i}≤ 1 for each

*i,*and

*d*

_{i}are the weights of the individual factors. The combined confidence

*C*has the property that if each confidence value is the same then the combined confidence will equal this common value. In addition, a zero or small confidence in one value makes the combined confidence zero or small. This technique requires each confidence to be close to 1 in order that the combined confidence be close to 1. The weights

*d*

_{i}could be chosen from analyzing data to determine which confidence indices best predict overall confidence. In the present algorithm, each

*d*

_{i}= 1.

Confidence estimates are calculated separately for each of the horizontal wind components, that is, *C*_{u} and *C*_{υ} for the *u* and *υ,* respectively. These are based on four individual confidence factors, described below. First consider the confidence estimate, *C*_{u}, for *u,* the *x* component of velocity. The confidence *c*_{1} is determined by the average of the confidences associated with each Doppler first moment used to produce the *a* values at a given height [see Eqs. (4) and (6), and the remarks before Eq. (4)]. These first moment confidences are determined based on such factors as the Gaussian fit to the spectral signal, the signal-to-noise ratio, and the continuity of the first moments as a function of range, as described by Morse et al. (2002).

*c*

_{2}is based on how well the data fits the linear model, for example,

*V*

_{i}=

*a*+

*b*(

*r*

_{i}−

*r*

_{o}), and is based on a chi-square test of this assumption:

*σ*

^{2}

_{i}

*r*

_{o}is given as

*i*Δ

*r.*For a discussion of the chi-square test see Freund (1992, pp. 309 and 512). In order to compare the fuzzy algorithm for computing

*c*

_{2}to a statistical analysis, it is assumed that the variances are all nearly equal and this allows for estimating

*σ*

^{2}

_{i}

*σ*

^{2}

_{i}

*σ*

^{2}

_{i}

The theory for the chi-square test makes several statistical assumptions that are often violated in practice. The standard chi-square test assumes the data are given by a linear model plus Gaussian noise and *σ*^{2} is the variance of the Gaussian process. The coefficients *a* and *b* are estimated via the SVD method, which minimizes a residual [given by Eq. (A4)]. In order to connect these two methods, the *σ*^{2}_{i}*c*_{i} in Eq. (A4) are assumed to be nearly constant. Then the estimates for *a* and *b* from each of these methods are nearly the same. Also the chi-square test assumes the data samples are independent with a normal distribution, while the first moment data could be dependent and the errors do not always have a normal distribution. However, it can be seen that the larger *χ*^{2} is, the less likely it is that the computed linear model is a good fit to the data. This chi-square test compares well to a statistical analysis when the errors in the Doppler first moments are Gaussian and the confidences are all nearly equal, thus matching the equal-weight assumption implied by using the average *σ*^{2}_{i}

A method for converting this *χ*^{2} value into a confidence index between 0 and 1 is required. A reasonable choice for this method is the probability, *Q*〈*χ*^{2} | *ν*〉, that *χ*^{2} is greater than or equal to *χ*^{2} if the model is correct. The number of degrees of freedom *ν* is one less than the number of points used in the fit minus the number of independent parameters replaced by estimates. In the current application, the number of points is 2*K* + 1, the number of parameters estimated is 2, *a* and *b* for the linear field, for a resulting *ν* = 2*K* − 2. The probability *Q* is calculated using the incomplete Gamma function using a routine in *Numerical Recipes in C* (Press et al. 1988, see formula 6.2.18 p. 177). The confidence based on how well the data fit a linear model is just this probability *Q,* but the fuzzy methodology allows for scaling this value if need be. Thus this “chi-square” test should be viewed strictly as an input to a fuzzy algorithm, which is motivated by statistics, and not as a rigorous statistical test. As in all fuzzy algorithms, the final test is performance (see section 6).

The assumptions that *u*_{x} = *υ*_{y} = 0 (in the four-beam analysis) require testing. There is no direct test for the assumption that *w*_{x} = *w*_{y} = 0 [see the discussion directly following Eq. (20)]. However, the assumption *u*_{x} = *υ*_{y} = 0 implies that the two estimates for the vertical component of the wind given by Eqs. (18) and (19) are equal when there is no noise. This implies that, assuming local stationarity in the wind, a time series of estimated vertical components of the wind using alternate north–south and east–west beams should be nearly constant. If *w*_{x} = *w*_{y} = 0, the estimates of the vertical wind component should also be constant as the wind field advects past the profiler over time. Thus to indirectly test the assumptions that *u*_{x} = *υ*_{y} = *w*_{x} = *w*_{y} = 0, a confidence index *c*_{3} should be defined to test the condition that the time series of the estimates of the vertical components of the wind is nearly constant.

In fact, estimation of the confidence *c*_{3} is itself a fuzzy module with three factors and is calculated analogously to the confidence in Eq. (23). The factors include a test of the temporal stationarity and spatial homogeneity of the vertical component of the wind (estimated from all beam directions), whether the current estimate of the vertical component is consistent with a prediction based on the time series, and finally the variance of the estimates in the time series.

*w*

_{obs}is a good fit within the LSAP model

*w*

_{pred}, where

*w*

_{pred}is the predicted value from the time series fit. An outlier value for the vertical component should result in a low confidence in the associated horizontal wind component. A test that performs this function utilizes the

*Z*statistic,

*σ*

^{2}

_{pred}

*Z*value gives a high confidence and a large

*Z*value gives a low confidence. However, a small

*Z*value could also result if

*σ*

^{2}

_{pred}

The final criterion *c*_{4} for the confidence in the *u* wind component is based on applying an LSAP model to the time series of *u* wind components and assessing the *Z* statistic calculated analogously to Eq. (25).

The confidences *c*_{1}, *c*_{2}, *c*_{3}, and *c*_{4} are combined by Eq. 23 to give *C*_{u}. Note that if any input to the confidence is zero, the final confidence will also be zero. A similar analysis is performed to calculate *C*_{υ}, the confidence in the *υ* wind component. The overall confidence in the horizontal wind speed is calculated using the confidence in the *u* and *υ* wind components; that is, *C* = *C*^{2}_{u} + *C*^{2}_{υ})/2

In a five-beam case (includes a vertical beam) refinements to the system could be made. A rapid temporal change in the measured vertical wind would be an indicator that the time stationarity assumption has failed. This would lower the confidence in the horizontal wind estimates, but this rapid variation of the vertical wind could be used as a regressor for turbulence. A large vertical variation in the vertical wind would (by mass continuity) indicate a large two-dimensional divergence or convergence in the horizontal wind near the profiler (horizontal shear). The horizontal shears of the horizontal winds could also be estimated. These estimates have a large variance, but they could be time averaged. Such persistent shears could be used as a regressor to estimate turbulence. These matters are under study.

In addition to testing assumptions of the wind field, a practical application of the NWCA confidence algorithm is removal of outlier radial velocities. NIMA removes many outliers from the first moment estimates, but it does not remove all outliers (see Cohn et al. 2001). Winds produced from first moment data containing outliers usually do not satisfy all of the wind tests and usually (but not always) have a low NWCA confidence. If the measurement errors in *a*_{E}, *a*_{W}, *a*_{N}, and *a*_{S} are, respectively, *e*_{E}, *e*_{W}, *e*_{N}, and *e*_{S}, then from Eqs. (18) and (19) the two estimates of the vertical wind will be the same only if *e*_{E} + *e*_{W} = *e*_{S} + *e*_{N}.

It follows that if one beam has a large measurement error and the other beams have small measurements errors, then the vertical wind estimates will not agree. This should result in a low confidence because the time series of the vertical wind will not be nearly constant. This will be true even if a single beam error persists over time, since the times series of the vertical winds alternate between north–south and east–west estimates. A large error in one beam will show up as a choppy vertical component as these estimates alternate. This is a case where the horizontal component of the wind field would be nearly constant over time. This component would be in error, but the NWCA wind confidence would be low. A persistent error such as this can occur if the moment finding algorithm selects a persistent signal such as radio frequency interference (RFI) instead of the atmospheric signal. This can occur in a peak finding algorithm. NIMA rarely selects RFI, but in weak atmospheric signal cases NIMA may select a nonatmospheric signal, and in rare cases this error can persist in time. If the errors satisfy *e*_{E} + *e*_{W} = *e*_{S} + *e*_{N}, and this holds over time, then the NWCA confidence would be high. This is not likely in the case of outliers, but it can occur. Isolated errors in a beam will be detected when the linear fit test along the beam fails. NIMA usually selects continuous moments in range, but a peak finding algorithm may select point targets that show up as isolated outliers in the first moments. In order for an outlier wind to be assigned a high confidence, the outlier error in the first moment must persist over space and time, and occur in more than one beam and the errors must satisfy *e*_{E} + *e*_{W} = *e*_{S} + *e*_{N} in time as well, an unlikely occurrence. For these reasons, the wind confidence algorithm correctly assigns a low confidence to most wind outliers caused by first moment outlier errors.

The warning system in Juneau will include a system of anemometers and three profilers. An anemometer is collocated with each profiler and pairs of anemometers are located on several nearby peaks. The profilers are located near arrival and departure flight tracks where the aircraft are changing altitude. See Cohn et al. (2001) for a map of the area showing the location of these sensors and the flight tracks. On several of these flight tracks the aircraft is making a sharp turn. A large vertical shear of the horizontal wind over hundreds of meters could be a significant hazard to flight. High-confidence large vertical shears of the horizontal wind have been detected in Juneau. Smaller-scale shears are classified as turbulence. The warning system uses various combinations of regressors to estimate turbulence intensity. Among these are wind speed estimates from the three profilers and vertical shear of the horizontal wind at various heights. In preliminary studies, these quantities add skill to the system when high-confidence data is used. Other regressors are second moment (spectrum width) data from the profilers and anemometers and wind speed estimates from the anemometers. Vertical shear of the horizontal wind may also be estimated from the anemometers at the peak top and the anemometer at the runway. These estimates give no indication of the height of the strongest part of the shear. Each of these regressors have data quality control problems. Because so many combinations of regressors are available, it is possible to use only high-confidence regressors. Using a large number of regressors also allows for the possibility of detecting or mitigating the occasional outlier incorrectly assigned high confidence. In the rare event that there were only a small number of high confidence regressors available, a warning would be sent that the system was impaired. Using a subset of a large number of regressors has the effect of removing false alarms while still allowing for detection using high confidence data.

## 5. Confidence-weighted average winds

*u*

_{A}is the confidence-weighted average value for the

*u*wind component

*C*

_{uA}

*u*

_{A}, the

*u*

_{Vi}

*n*wind components calculated [Eq. (12)] from consecutive sets of east and west beams collected during the averaging interval, and the

*C*

_{ui}

*u*

_{Vi}

*υ*wind component. When reporting winds as wind speed and direction, the confidence is reported as

*C*

^{2}

_{uA}+

*C*

^{2}

_{υA})/2

Currently the operational system in Juneau reports 10-min confidence-weighted average wind values calculated in this manner. Winds with confidence values less than 0.5 are reported as “not available.”

## 6. Some wind confidence performance results

A confidence algorithm may be used to reduce false alarms in a warning system. In order to make use of a confidence algorithm in this way, it is necessary to quantify the relationship between confidence and measurement errors. Wind estimates should be of high quality when the wind confidence is high. To test this assumption, a dataset was used from an intensive data collection effort in Juneau, Alaska, during February–April in 1998. Three 915-MHz boundary layer profilers were employed in this experiment and are still operational in Juneau. Operational parameters included a pulse repetition frequency of 34 500 s^{−1}, a pulse width of 400 ns, and a dwell time of 30 s. The profilers were located near precipitous terrain and aircraft flight tracks to be able to measure the disturbed winds and turbulence generated by the terrain. The weather conditions in Juneau during this time period ranged from strong frontal passages with significant embedded precipitation (rain, snow, and rain–snow mixture), to strong winds associated with clear-air downslope winds off the Taku Glacier (so-called Taku winds). This combination of weather extremes and nearby terrain made for very challenging conditions for profiler operation. The dataset included a limited number of cases when a research aircraft was flying near the profilers and the aircraft was providing wind and turbulence estimates. Around 174 profiles (of 36 range gates) were selected for human verification. Most of these profiles in the database were selected to coincide with times when the aircraft was flying near one of the three profilers. The database had Radio Frequency Interference, ground clutter, point targets, and low signal-to-noise cases (SNR). The RFI problem in Juneau occurs frequently since two of the profilers are in a direct line of sight with each other. NIMA has the capability of identifying and tracking RFI in time (see Morse et al. 2002). NIMA rarely selects RFI instead of an atmospheric signal unless the atmospheric signal is near to the RFI in the given beam. NIMA rejects point targets and rarely selects ground clutter unless the ground clutter is located near the atmospheric signal. Some additional data from a larger 9-month database were also analyzed. These were cases where the NWCA winds had large temporal discontinuities, and contained low SNR cases where NIMA generated incorrect first moments. In addition, the verification database contained data from the South Douglas Island profiler where a beam-switching problem made the wind estimates unreliable. These data and the additional data from the 9-month database were included in the study as a test of the NWCA wind confidence algorithm. This database was chosen to include very challenging data scenarios, and hence some of these data are not typical of profiler data collected in Juneau.

Human verification was done on the Doppler moments in the database. A graphical interface was available to the human experts performing the verification (see Cohn 2001 for a description of the methodology used in this verification effort.) Using this interface, the human experts chose the location of the atmospheric signal, which was then used to compute the moments. The experts also estimated a confidence for each of these moments. The error in a first moment is defined as the difference between the human first moment estimate and the NIMA first moment estimate. Here we are assuming the human estimate to be truth. The absolute error is defined as the absolute value of the error. There were wind profiles containing over 6000 first moment estimates in the database provided by human verification. The average absolute error was 0.49 m s^{−1} with a root-mean-Square (rms) error of 1.32 m s^{−1}. This rather large rms indicates the presence of outliers. Most of these outliers are caused by low SNR cases as well as some cases where NIMA selected ground clutter as part of the atmospheric signal. When the first moment estimates with the lowest 20% of NIMA-calculated confidence are removed, the average absolute error was approximately 0.3 m s^{−1}, which is the spectral resolution of the profilers used in Juneau. The rms was reduced to slightly less than 1 m s^{−1}. This indicates that NIMA first moment confidence has skill in reducing first moment outliers and errors.

NIMA first moment confidence is a factor in computing NWCA wind confidence (see section 4). Although NIMA first moment confidence has skill in identifying outliers, it can, in rare circumstances, give outliers high confidence (Cohn et al. 2001). In fact, outliers in the first moment measurements will often violate many of the assumptions made in computing the horizontal wind (section 2) and should result in a lower wind confidence (see the discussion in section 4). The NWCA wind confidence should have skill in removing additional outliers that were given a high NIMA first moment confidence.

As part of the verification exercise, the profiles were verified in groups of four (north, south, east, and west beams). A human-verified wind was produced using Eqs. (12) and (13) where the *a*_{E}, *a*_{W}, *a*_{N}, and *a*_{S} values are the human-verified first moments. The NWCA error is defined as the difference between the human-verified horizontal wind components and the NWCA horizontal wind components. The NWCA absolute wind error is defined as the square root of the sum of the squares of these errors. The NWCA wind speed error is the difference between the human-verified wind speed and the NWCA-produced wind speed. The absolute error in the wind speed is the absolute value of wind speed error. The verification dataset contains over 2400 human-verified horizontal wind components.

In order to make a further comparison, POP [Profiler On-line Program, Carter et al. (1995)] were also compared to human-verified wind speeds. This algorithm is based on locating the peak intensity of the signal at each range gate and following that signal to the noise floor. The signal thus collected is used to calculate the POP first moment. This algorithm produces high-quality first moments for uncontaminated spectra. However, when contaminants have spectral intensities greater than that of the atmospheric signal, erroneous moments are often generated (Morse et al. 2002). In traditional profiler applications such erroneous moments are not particularly problematic as they are eliminated through consensus averaging over time periods of 30 min or more. Rapid update winds are required in the airport hazard application described in the introduction. The POP algorithm returns first moment estimates approximately every 30 s, and these may be used to compute rapid update winds similar to the way human-verified first moments are used to produce horizontal winds. It is these rapid update winds from POP moments that are compared to the human-verified horizontal winds. Figure 2 shows a scatterplot comparing the rapid update POP wind speeds to the human-verified wind speeds in meters per second. Notice the large number of outliers that give an rms of 9.9 m s^{−1} and an average absolute error of 4.7 m s^{−1}. These outliers are caused by several factors including RFI, ground clutter, point targets, and low SNR. In addition, the beam-switching problem at South Douglas Island produced unreliable winds. The South Douglas Island profiler is located in the Gasteneau Channel where the winds are often dry and have a low SNR. If we remove the South Douglas Island data (about 35% of the data), Fig. 3 shows a definite improvement. The rms is now 6.7 m s^{−1} and the average absolute error in the wind speed has been reduced to 3.4 m s^{−1}. The data in Fig. 3 does not contain the beam-switching problem, but the two remaining profilers have a significant RFI problem since they are in direct line of sight of each other.

Figure 4 shows a scatterplot of NWCA wind speeds in comparison to human-verified wind speeds. Notice there are significant outliers, but many fewer outliers than in the POP analysis. The average absolute error is 1.8 m s^{−1} and an rms of 4.4 m s^{−1}. These statistics are an improvement over the POP analysis even when the South Douglas Island data were removed from the POP analysis in Fig. 3.

Figure 5 shows the same data as Fig. 4 except the South Douglas Island data have been removed. Now the average absolute error in the wind speed is 0.9 m s^{−1} with an rms of 1.6 m s^{−1}. This shows that many of the outliers have been removed, but there remains a cluster of outliers at (20,10) in Fig. 5. Comparing Figs. 2 and 3 to Figs. 4 and 5, it is clear that rapid update NWCA wind speeds compare better to human-verified wind speeds than those produced by the peak-finding algorithm. This is because NIMA produces first moments that are less prone to RFI, point targets, and ground clutter contamination. The peak-finding algorithm produces better results when SNR is used as a confidence value. However, to obtain an average absolute error in first moment estimates of 0.3 m s^{−1}, 60% of the data with the lowest SNR must be removed. In contrast, this level of performance is obtained for NIMA first moments when the lowest 20% of first moment data is removed as measured by the first moment confidence algorithm.

With these results as a baseline, a study of the wind confidence algorithms is provided. By removing the South Douglas Island data profiler from the analysis, the human-estimated wind speeds were in better agreement with the NWCA wind speeds. Since there were problems in the South Douglas Island wind data due to the beam-switching problem, much of the data should be classified as poor quality data by the wind confidence algorithm. To test this proposition, about 30% of the data as classified by low wind confidence were removed from the dataset (this corresponds to removing all data below the threshold of 0.56 wind confidence). Figure 6 shows a scatterplot of the resulting data compared to the human-estimated wind speeds. Notice that the average absolute error is now 0.74 m s^{−1} and has an rms of 1.26 m s^{−1}. The cluster of outliers at (10, 20) remains, but the number of points has been reduced from 12 to 3 points. Thus many of the poor quality South Douglas Island data were removed by the wind confidence algorithm, and additional outliers in the remaining data were also removed. This is because some of the South Douglas Island data did not satisfy the wind assumptions (discussed in section 2) because of the switching error, and subsequently had low confidences based on the application of the tests described in section 4. In fact, this switching problem was discovered because the South Douglas profiler reported persistent low confidence winds. Additional first moment outliers are detected because the resulting winds do not satisfy the tests described in section 4, and hence the resulting winds have low wind confidence. The NWCA wind speed error indicators such as absolute error and rms error continue to decrease as the wind confidence threshold is increased. To see this in a different way, the average absolute error in the horizontal components (viewed as a two-dimensional vector) as a function of the amount of data removed as ranked by NWCA wind confidence is examined.

Figure 7a shows the average absolute vector error as a function of the data removed. The vertical axis in the absolute vector error, and the lower horizontal axis, is the percent of data removed. Thus when the data with the lowest 40% of NWCA confidence have been removed, the average absolute vector error is about 1 m s^{−1}. The upper horizontal axis is the equivalent NWCA confidence threshold. Thus when all data with a wind confidence less than 0.6 are removed, 40% of the data have been removed. Figure 7b shows the rms as a function of percent of data removed. Notice both curves in Fig. 7 continue to decrease as the quality of the data improves, where the quality of the data is measured by the wind confidence. The absolute vector error as measured by the length of the error vector is a strict test. By the triangle inequality, the absolute vector error dominates the absolute error in wind speed. Figure 7 shows that the wind confidence algorithm may be used to ensure high-quality wind estimates.

As mentioned above, the evaluation dataset is not representative of the data in Juneau. The average wind confidence for the 9-month period of 18 November 1998–31 July 1999 was 0.78 compared to the average value of 0.66 for the verification dataset. This 9-month period did not contain the beam-switching problem at the South Douglas Island profiler. However, the winter months are of interest since this is the period of high turbulence in Juneau. Data for the period of 7 December 1999–14 February 2000 were analyzed. Figure 8a shows an estimate of the probability distribution function for wind confidences. The mean is 0.68, which is close to the value of 0.66 for the verification dataset. An estimate for the cumulative distribution function for the wind confidences is shown in Fig. 8b. This dataset did not contain the South Douglas Island beam-switching problem. Notice that to remove all data with a threshold less that 0.557 requires the removal of only about 18% of the data. Assuming performance is similar for this dataset as for the verification data, this implies that the horizontal wind vector estimate has an average vector error of about 1 m s^{−1} and an average absolute error in the wind speed of 0.74 m s^{−1} after 18% of the data is removed. During other times of the year, even fewer data need to be removed to obtain this level of performance. In the 9-month database, the confidence of the confidence-weighted average wind was less than 0.56 only 3% of the time. These fractions are representative of this Juneau data only and should not be extrapolated to other sites or profiler operating parameters. For example, if measurements were routinely made at heights much greater than 2.5 km where the signal strength is less, a larger fraction of measurements would be removed by the confidence threshold of 0.557 seemingly needed to achieve an average error of 1 m s^{−1}.

## 7. Conclusions

An analysis describing the estimation of linear wind field parameters in a four- or five-fixed-beam profiler system has been presented. It has been shown that it is possible to estimate the horizontal components of the wind in a reliable way with a typical profiler geometry under the assumptions that the wind is modeled by a linear wind field, that the vertical wind component is constant in the horizontal directions, and that the wind is stationary over the sampling region. It has also been established that it is possible to give reasonable estimates for the vertical shear of the horizontal components of the wind. When the assumptions upon which the analysis is based break down, confidence indices are available that can be used both to identify the wind estimates as unreliable and to remove outliers in the wind estimates. A description of these wind confidence indices and how they are derived based on testing the validity of those assumptions has been given. These confidence indices are also used to generate a confidence-weighted average wind value. The performance of the NWCA algorithm was discussed and it was shown that the confidence indices show skill in reducing first moment and wind estimate errors, based on “truth” values available from human expert analysis.

The NWCA algorithm has been running in near real time in Juneau, Alaska, since February 1998. Modifications continue and it is expected that this algorithm will eventually be an integral part of a turbulence and shear warning system for the Juneau airport.

## Acknowledgments

This research is in response to requirements and funding by the Federal Aviation Administration (FAA). The views expressed are those of the authors and do not necessarily represent the official policy or position of the FAA.

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*Computer Methods for Mathematical Computations*. Prentice Hall, 249 pp.Freund, J. E., 1992:

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## APPENDIX

### Measurement Error Effects on Wind and Shear Estimates

_{x}, ɛ

_{y}, ɛ

_{z}are assumed to be independent with expected values of zero and each with the same variance

*σ*

^{2}. Furthermore, it is assumed that these errors are homogeneous in space and stationary in time. In the case of atmospheric turbulence these error terms along a radial will not be independent. To study this, a model of the autocovariance function is required. This depends on the particular model of turbulence studied. Equation (5.3.5) in Priestley (1981) shows the variance as a function of the autocovariance function. The more correlated the data, the more averaging that is required to reduce the variance. It is beyond the scope of the present paper to include this type of analysis. For the case of small measurement errors, it is reasonable to assume independence. In order to compare our results to a statistical analysis, the first moment confidences are all assumed to be equal as well.

**e**= [sin

*ϕ*cos

*θ,*sin

*ϕ*sin

*θ,*cos

*ϕ*]

^{T}, the radial velocity becomes

*V*

_{r}=

**V**(

**x**) ·

**e**=

**V**(

**x**

_{o}) ·

**e**+

*A*· (

**x**−

**x**

_{o}) ·

**e**+

*·*

**ɛ****e**. Analogously to Eq. (3), the radial velocity can be modeled as

*V*

_{r}

*a*

*b*

*r*

*r*

_{0}

_{r}

_{r}is given by

*E*(ɛ

_{r}) = 0, and the variance Var(ɛ

_{r}) =

*σ*

^{2}, and holds for any given radial. This follows from the fact that Var(Σ

_{i}

*a*

_{i}

*ɛ*

_{i}) = Σ

_{i}

*a*

^{2}

_{i}

_{i}) if the ɛ

_{i}are independent, and

*E*(Σ

_{i}ɛ

_{i}) = Σ

_{i}

*E*(ɛ

_{i}).

*V*

_{ri}

*r*

_{i}=

*r*

_{0}+

*i*Δ

*r*along the radial, where

*i*= −

*N,*… , −1, 0, 1, … ,

*N,*and

*a*and

*b*are estimated by

*â*and

*b̂,*where

*â*and

*b̂*are chosen to minimize

*R*is the residual between the data and the model.In Eq. (A4), the confidences (here weights in the minimization process)

*c*

_{i}are assumed to be equal and nonzero. Taking the partial derivatives of

*R*

^{2}with respect to

*â*and

*b̂,*and setting these equal to zero,

*â*and

*b̂*are given by

*â*is an unbiased estimator of

*a*with variance

*σ*

^{2}/(2

*N*+ 1). Also

*b̂*is an unbiased estimator of

*b*with variance

*u*

_{V}is

*û*

_{V}for

*u*

_{V}is a biased estimate with bias

*w*

_{x}

*r*

_{0}cos

*ϕ*and variance

*σ*

^{2}/[2(2

*N*+ 1) sin

^{2}

*ϕ*]. In the case where

*w*

_{x}= 0,

*û*

_{V}is an unbiased estimator for

*u*

_{V}. In a typical application,

*N*= 2 and

*ϕ*= 15°, so that the variance is about 1.5

*σ*

^{2}. A similar result holds for

*υ*

_{V}. In Cohn et al. (2001) it is established that after the data with lowest 25% NIMA first moment confidence is removed, there is a measurement error of

*σ*= 0.6 m s

^{−1}when comparing NIMA first moments to human estimates. If this value for

*σ*is used, a variance of 0.54 m

^{2}s

^{−2}results.

*u*

_{x}yield

^{′}

_{V}

*w*

_{z}. Thus

*û*

_{x}is an unbiased estimator for

*u*

_{x}with variance

*N*= 2,

*ϕ*= 15°, Δ

*r*= 60 m, the resulting variance is about 0.0085

*σ*

^{2}. If again the value of

*σ*= 0.6 m s

^{−1}is used, a variance of 0.003 s

^{−2}results. This represents a large shear, [Var(

*û*

_{x})]

^{1/2}= 0.05 s

^{−1}. This shows why it is difficult to estimate

*u*

_{x}without averaging over a large amount of time. Thus only persistent shears in

*u*

_{x}can be found. At present in the Juneau system, there is no attempt to estimate

*u*

_{x}. To estimate

*u*

_{x}a vertical beam would be required and some time averaging would be required to further reduce this variance.

*u*

_{z}, again by applying Eq. (20) to yield

*b̂*

_{E}−

*b̂*

_{W})/(2 sin

*ϕ*cos

*ϕ*) is a biased estimate for

*u*

_{z}with bias

*w*

_{x}and variance

*û*

_{x}). This will be an unbiased estimate assuming

*w*

_{x}= 0. With

*σ*= 0.6 m s

^{−1}, this becomes Var(

*û*

_{z}) = 0.000008 s

^{−2}and [Var(

*û*

_{z})]

^{1/2}= 0.0089 s

^{−1}. This is more than an order of magnitude smaller than the variance for finding

*u*

_{x}. With some time or spatial averaging it should then be possible to measure

*u*

_{z}and

*υ*

_{z}for persistent shears. Such shears have been detected in Juneau and they are often associated with strong turbulence.

^{*}

The National Center for Atmospheric Research is sponsored by the National Science Foundation.