An Investigation Of The Use Of VAD Analysis and UHF Profiler Data To Obtain

A 2-D Wind Field

Dean Reichheld

Deparment of Atmospheric and Oceanic Sciences McGill Universiq, Montréal

July 1997

A thesis submitted to the Faculty of Graduate Studies and Research in partial fulfillment of the requirements of the degree of Masters of Science

in Atmospheric and Oceanic Sciences.

O Dean Reichheld 1997

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Abstract

in this work, we explore the possibility of cornbining information from the VAD

technique, with data from a wind profiler, to obtain the kinematic properties of a linear 2-D

wind field. The idea is f ~ s t tested on an &cial test-bed wind field, and the results show

that the procedure is limited by the degree of non-linearity of the wind field over the wind

profiler. To overcome this problem, some modifications are made to our procedure, which

is then applied to real cases of stratiform precipitation. We find that the non-linearities are

important enough such that the linear approximation is only valid out to about 40 km fiom

the radar. These non-linearïties also affect the retrieval of the vorticity to the extent th& we

had to establish a maximum limit on the uncertainty of the vorticity. These results bring to

question the assurnptions that the non-linearities are relatively unimportant in stratiform

cases.

Résumé

Au cours de ce travail est explorée la possibilité de combiner les informations

déduites de la technique VAD avec les données collectées par un profileur de vent, afin

d'obtenir les propriétés cinématiques d'un champ de vent bidimensionnel linéaire. Cette

approche est testée en premier lieu dans le cas d'un champ de vent simulé. Les résultats

obtenus montrent que la procédure proposée est limitée par le degré de non-linéarité du

champ de vent à la verticale du profileur de vent. Pour pallier cette difficulté, certaines

modifications sont apportées à notre approche. Cette procédure est ensuite appliquée à

différents cas de pluies à caractère stratiforme. 11 est obtenu au cours de ce travail que la

composante non linéaire du vent est suffisamment significative pour que l'hypothèse de

linéarité ne puisse être appliquée que jusqu'à une distance de 40 km du radar. Cette

composante non linéaire affecte également le recouvrement du tourbillon vertical à tel point

qu'il a fallu imposer un seuil de tolérance portant sur l'incertitude sur le tourbillon vertical.

Ces résultats remettent en question l'hypothèse communément acceptée de linéarité du

champ de vent dans le cas des systèmes convectifs à caractère stratiforme.

Acknowledgments

A large thank-you goes to my supervisor, Professor Isztar Zawadzki, for his

support and advice for this project. His enthusiasm for the field, his excitement in new

discovenes, and his constant push for more, have been great inspirations to me over the

past two years.

También quisiera mostrar mi aprecio de todo coraz6n a Ramon de Elia por su ayuda.

Algunos de los resultados de nuestras discusiones me han ayuadado a crear O refinar

algunos de los argumentos que aparecen en esta tesis. Y sus ideas iluminaron algunas

soluciones a mis probiernas m h complejos.

Pour tous ses renseignements et son support, je voudrais remercier Dr. Fred Fabry.

Il m'a donné une perspective fraîche quand j'en avais le plus besoin. Aussi, pour son aide

avec les traductions en Français, je voudrais dire un grand "merci" à Dr. Alain Protat.

1 also owe a large part of my thesis to the efforts of Mrs. Alumulu Kilambi of the

J.S. Marshali radar Observatory at McGill University. She not only provided the data used

in Chapter 4, but was extrernely helpful with the data storage and retrieval systems at the

observatory. The information on the methods used at the observatory to calculate the VAD

c ~ e ~ c i e n t s , dong with some discussions on the results of this work were kindly provided

by Dr. Aldo Bellon. and are dso greatly appreciated.

A lot of valued help with the details of the UHF wind profiler used for this work

came fiom Dr. William Brown. AIso, several good suggestions on the error analysis and

other aspects of the thesis came from Mark Shephard. 1 would like to thank these guys,

and the rest of my peers in the Department of Ahnospheric and Oceanic Sciences at McGill

University, for putting up with me for the past two years.

Finally, I wish to give a special thanks to Dr. Owen Hertzman for persuading me to

go ïnto the Master's program in the f rs t place. AIso 1 wish to express my eternal gratitude

to my parents, Gerald and Lynda Reichheld, for their undying support that eventually saw

me through to this point.

Table of Contents

Abstract . . . . . . . . . . . . . . . . . . Résumé . . . . . . . . . . . . . . . . . . Acknowledgments . . . . . . . . . . . . . . . Table of Contents . . . . . . . . . . . . . . . Statement of Originality . . . . . . . . . . . . .

Chapter 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . 1 . I Introduction

12 Review of VAD Andysis . . . . . . . . . 1.2.1 Use of VAD In DeveIoprnent of VVP Methods . 1.2.2 Development of EVAD Methods . . . . . 1 .2.3 Other Developments of VAD . . . . . . 1.2.4 Characteristics and Cornparisons of VAD Analysis

1.3 Discussion of UIHF Wind Profiler . . . . . . . 1.4 Discussion of Project . . . . . . . . . .

Chapter 2 Linear Wind Retrieval Method . . . . . . . . . . . . . . . . . . . . . . . 2.1 Introduction

. . . . . . . . . . . . . 2.2 General Equations

. . . . . . . . . 2.3 Set Up of The Retrieval Procedure

. . . . . . . . . . . . . . . . 2.4 Surnmary

Chapter 3 Analysis of a Test-Bed Wind Field . . . . . . . 19

3 -1 Wind FieId Description . . . . . . . . 19

3.2 Retrieval of Aaificial Wind Field . . . . . . 20

3.3 Discussion o f Results . . . . . . 21

3.4Summary . . . . . . . . . . . . . . . . 24

. . . . . . . . . . . Chapter 4 Andysis of Real Data 28

4.1 Weather Synopsis . . . . . . . . . . 28

4.2 Wind Retrievai for December 14 1995 . . . . . . . 28

4.3 Error Analysis of Retrieval . . . . . . . 30

LV

4-4 Wind Retrieval for December 9 1995 . . . . 34

4.5 Further Analysis and S u m q - . - . . - . - . . 35

Chapter 5 Conclusion . . . . . . . . . . . . . . 49

5.1 Summary and Conclusions . . . . . 49

5 2 Suggestions and Future Work . . . . . . . . . . 52

Appendix A Equations In Error AnaIysis . . . . . . . . 53

AppendU: B Error Analysis of VAD Coefficients . . . . . . 55

Bibliography . . . . . . . . . . . . . . . . . 58

Statement of Originality

Aspects of this thesis that represent original contributions to knowledge are the foilowing:

A method to separate the kinematic properties of the linear wind from the non-

linear terms, using the range dependence of the VAD coefficients.

A procedure that combines the kinematic properties obtained by the method above,

with information from a UHF wind profiler, to obtain a two-dimensional linear

wind field.

The application of the above procedures to real stratiform snow cases, with the

indication that such cases may be more non-linear than originally thought.

Chapter I

Introduction

1.1 Introduction

Since the advent of Doppler radar, there have been many studies into how to

process, use, and expand upon the data received from it. Some of these studies examined

various techniques to complete the information on the wind field. The most commonly

used is the Velocity Azimuth Display (VAD). The VAD technique, as presently used, is a

diagnostic tool used to obtain kinematic properties of a wind field such as, divergence,

deformation, and the translation wind, ftom single Doppler radar. However, it cannot

determine the voriticity, thus in this work we explore the possibility of cornbining the VAD

technique with data from a wind profder, to obtain a linear approximation to a 2-D wind

field.

In this Chapter, we review the VAD technique and some of its variants. We begin

with a description of the technique as developed by Browning and Wexler (1968). Then

we discuss the papers which rnodify the VAD to obtain two and three dimensional winds

from single Doppler radar. We also review papers that deal with the separation of the

vertical velocity from the divergence, followed by a review of studies analyzing the

characteristics of the VAD analysis. We then briefly discuss papers of the low level UHF

wind profüer, to introduce the reader to the data, and data analysis that is used for this

instrument. Finally we sum up with an outLine for the rest of the thesis.

1.2 Review of VAD Analysis

The use of the VAD technique to determine the kinematic properties of a wind field

was first developed by Browning and Wexler (2968). They used two dimensional Taylor

senes expansions, tnincated at the linear terms, to represent the Cartesian components of

the wind (u, and v), and then transformed them into polar coordinates, to obtain an

equation for the radial velocity (V,). The resultant equation is in the form of a two

harmonic Fourier series, whose coefficients are related to the kinernatic properties of the

wind field (with the notable exception of the vorticity). The procedure was limited by

factors like vertical wind shear, and vertical fail speeds. Since then, studies have expanded

2 upon the VAD, in order to obtain more information about the wind field with single

Doppler radar data.

12.1 Use of VAD In Development Of VVP Methods

One of the earlier modifications to the VAD was what has been since known as the

VARD or VeIocity ARea Display, (Easterbrook, 1975). Easterbrook kept the linearity

assurnption from the VAD, but instead of perfonning the analysis over all azimuth angles

for one range, he fitted two curves within a conical section. Unlike Browning and Wexler,

who set the ongin of their coordinate system at the radar, Easterbrook set the origin at the

center of the conicd section in which he performed his retrievai. In doing this, he found

that he could obtain the divergence and deformations from fitting a curve (similar to the

VAD curve) to the data. However, he also found that the wind components were affected

by the vortÏcity. Thus, he developed a second equation to fit the data in order to determine

the two dimensional wind field within the conical sector.

The VARD was expanded upon by Waidteufel and Corbin (1979), where they

developed the VVP (Volume Velocity Processing). The VVP stiu had the VAD as its base,

but the linearity hypothesis was applied to the whole volume rather than just a range circle

at a single height. This rneans that instead of retrieving a two dirnensionai wkd for a circle,

or a conical sector, they retneved a three dimensional Linear wind within a cylioder. They

found, after testing the procedure on both simulated and real data, that the VVP was

significantly affected by deparhues from linearity, and in most cases the effect was similar

to that of the VAD. However, in the case of the divergence, the VAD obtains the exact

mean divergence within a range circle and so does not depend on the linearity assumption,

whereas, the divergence obtained by the WP is dependent on the linearïty assumption in

b~ are the same manner as the other kinematic properties. The result is that the smaller f e a w

filtered out by the VVP, so that the resultant wind is a rnesosca~e flow. Koscielny et al.

(1 982)- developed a modified VVP (MVVP), which was a combination of the VVP and the

VARD, such that the VVP was applied to sectors whose widths were only 30' of azimuth.

This method provided some more information on divergence within smaller volumes, but

was very sensitive to strong s m d scde eddies.

Johnston et al. (1990) developed an extension of the MVVP and the RH1 (Range

Height Indicator) methods specifically for rainbands or any 2-dimensional system, called

BVP (Band Velocity Processing). Like the MVVP, the BVP method retrieves the wind

3 field within a selected volume so as to examine the fine scde structures. However, as the

name suggests this procedure is limited only to features that are approximately two

dimensional, or in band structures.

1.2.2 Development of EVAD Methods

Another evolution of the VAD analysis was the EVAD (Extended Velocity Az.imuth

Display) method developed by Srivastava et al, (1986). As opposed to the VVP related

methods mentioned above, the EVAD technique was developed to separate the

contamination of the vertical fail speeds from the divergence determined from the VAD. To

do this, they determined a relationship between the first VAD coefficient and the height and

range. By assurning the vertical velocity was negligible, they could then determine the

divergence and the vertical fall speeds separately, then use the continuity equation to

determine what the vertical air motion should be. However, this technique required that the

wind field be steady over a larger set of ranges since it determines the vertical fall speed

fiom a sample of many VAD over many ranges.

The EVAD was then rnodified by Matejka and Srivastava (1991) so that the Fourier

series coefficients are weighted accordingly to their estimated error. They also developed a

method of removing the outliers frorn the fits, so as to reduce the effect of the srnall scale

perturbations, then they obtained the vertical motion through a variational method that they

developed, simila to that developed by O'Brien (1970). These modifications were made

so as to allow the EVAD to obtain finer structure to the vertical air motion, instead of

requiring the entire field to be unifoxm. However, in order for the EVAD to perform its

best, it requires a scanning procedure within guidelines the authors have set up and even

with these, the features of divergence and vetical motions Iess than the synoptic scale aie

fitered out.

Finally, Matejka (1993) created a fuaher modification on the EVAD, known as the

CEVAD (Concurrent Extended Velocity Azimuth Display). The main daerence in this

method, is the fact that the divergence and vertical motion profiles are solved concurrently.

This means that for each height, the values for divergence and the vertical motion are

required to confonn to the additional constraint of certain boundary conditions. It also has the advantage of imposing these boundary conditions during the calculation of the

divergence and vertical motion, as opposed to the EVAD where these are applied after

values for each height are determined separately.

1.23 Other Developments of VAD

Fuaher modifications to the VAD analysis were proposed by Hanis (1975), and

later used by Testud et al. (1980). Both of these papers used the standard VAD analysis to

obtain a linear wind, however, after they performed the VAD Fourier fit, they subtracted it

from the data that they fitted it to. They observed that the remainder appeared as an

organized wave pattern, and when they performed a perturbation analysis to the radial

velocity, they found that this remainder could be considered as wave perturbations to the

mean horizontal wind (Iinear wind), thus they could represent some of the finer scale

features of the wind field they were studying.

The VAD method has also been rnodified for Dual-Doppler analysis by Scidom and

Testud (I986), to obtain the DVAD (Double Velocity Azimuth Display). In this case a

VAD analysis is used for each of the two radars, then the results are merged to retrieve the

three dimensional wind. The DVAD, however, still employs the Linearity hypothesis for

the analysis from each radar, so Scialom and Lemaître (1994) proposed the QVAD

(Quadratic Velocity Azirnuth Display). The main assurnption in the QVAD is that the wind

is assumed to be quadratic, instead of linear. They then could perform the VAD analysis on

the two separate radars and solve for the quadratic terms when the results were rnerged.

They d s o proposed that the method could be applied with a synthetic Duai-Doppler (single

Doppler scans at two different tirnes).

1.2.4 Characteristics and Cornparisons of VAD Analysis

Other studies have concentrated more on the characteristics of the VAD, and also

comparing it with data obtained by other methods. One such study done by Larsen et al.

(199 1) did a cornparison of vertical velocities inferred by a VAD analysis to those measured

by a VHF profder. Since this study used cases with no precipitation, there was no effect of

the fall speed of the hydrometeors, thus they could use the divergence from the VAD to

infer a mean vertical velocity (in cases of precipitation, refer to the EVAD analyses above).

They found that the VAD results compared well to those fkom the VHF profiler, but only in

situations where the vertical motions were reasonably uniform. Similar results were

obtained by Boccippio and Matejka (1996), where they performed the cornparisons in the

stratiform region of an MCS. However, since their case involved precipitation, they

compared the VHF profiler vertical velocities to those obtained fiom the EVAD, CEVAD, and VVP analyses, rather than the simple VAD. What they found was that the generd

5 structures where sirnilar in all of the single Doppler techniques, with some varïability in

profde heights and velocity magnitudes.

Finally, since this project involves utilizing the charactenstics of the VAD andysis,

it is important to discuss the previous work that examines what is, exactly, obtained from

the VAD analysis. The e s t such study was done by Pasarelli (1983)- where he examined

the mathematics of the VAD technique, to determine the minimum restrictions needed to

obtain the vorticity. What he did, was to not only represent the radial velocity in terms of a

Fourier series as in the VAD analysis, but the tangential velocity, vorticity, and the

divergence as well. He then obtained a relationship between each of the coefficients from

al1 four Fourier senes, finally giving him four equations with six unknowns. He then

discussed various closure schemes to these equations that could be used to obtain the

vorticity. However, he did point out that with this set-up, the linearity hypothesis seemed

to be a unduly restrictive assumption.

Another investigation of the VAD technique was performed by Rabin and Zawadzki

(1984), where they exarnined the behaviour of the only well defined quantity of the VAD

retrieval, the divergence. Some of the things they examined were the effects of beam

smoothing, reflectivity weighting, and the effect due to the geometry a f the conical scan.

To overcome these effects, they developed a deconvolution technique that they then applied

to the divergence profiles that they had obtained. They found that the shape of the profiles

changed very littie after deconvolution, thus they concluded that an average proNe could

provide useful information about the divergence.

Finally, as an expansion on the work of Rabin and Zawadzki (1984), Caya and

Zawadzki (1992) examined the characteristics of the VAD when applied to a non-linear

wind field. They discovered that the coefficients frorn a two harmonic Fourier fit did not

give the linear, or mean, kinematic properties (with the exception of the divergence). In

fact, they found that what is obtained is the linear kinernatic property plus higher order

terms. They noticed that with this, the coefficients varied as even (odd) polynornials for the

odd (even) Fourier coefficients, and that if these polynornids were extrapolated to zero

range (over the radar), one could obtain the Iinear kinernatic properties. Et is from this

study that we will continue, with the addition of the UHF wind profiler data, to obtain an

estimate for the linear vorticity.

13 Discussion of UHF Wind Profiler

Since this project also involves work with a UKF wind profiler, we feel it is

important to introduce some of its basic concepts that have been studied. We begin with the

study by Balsley and Gage (1982), where they review the profiling methods and

instruments of that time and discuss possible future developments. In their study, they

mention some possible developments of the available technology, so as to use it for

operational applications. This lead to the developrnent of a lower Troposphere UHF wind

profiler (Ecklund et al. 1988), on which Our wind profiler is based.

The method we use to calculate the horizontal winds from the wind profiler, was

developed by Wuertz et al. (1988). For cases that we wiLl look at (stratiform precipitation

with relatively uniform horizontal winds), they suggested a 3 beam profiler, where one

beam is pointed to the vertical, and the other two are pointed at small angles from the

vertical and perpendicular to each other. The vertical beam detemines the vertical fall

speed, which can then be subtracted from the other beams, from which the horizontal

velocities are obtained. Some examples of the application of the wind profiler to

precipitation cases can be found in Rogers et al. (2993 and MM) .

1.4 Discussion of Project

As mentioned before, the objective of this project is to obtain a h e a r approximation

to the two dimensional wind using information from the VAD analysis, and a UHF wind

profiler. One motivation for this project is to obtain all of the kinematic properties valid

over the scanning radar, and use them for mode1 evaluation, and initialization. The

procedure could also be beneficial for operational use in terms of aiding nowcasts for some

precipitation events. in order to do this we wiU continue from the work by Caya and

Zawadzki (1992), by accepting that the wind field is non-linear, but still using the

assumption that we can obtain the linear components of the kinematic properties, and that

these are close approximations to the larger scale values.

A mathematical description of the method we are proposing will be developed in

Chapter 2, dong with the set up for the steps in the procedure. In Chapter 3, we wiu apply

the procedure to a non-linear test-bed wind field in order to determine the potential

problems due to the non-linearities themselves. Chapter 4, will concentrate on the

application of the procedure to two precipitation cases and the resulting two dimensional

7 winds- As well we will show our results h m the error analysis performed on the procedure. In Chapter 5, we will discuss the implication of the results fiom Chapter 4, and

describe future work to be done to enhance the procedure, or study its limitations.

Chapter 2

Linear Wind Retrieval Method

2.1 Introduction

In the previous chapter, the VAD analysis technique to retrieve bernatic properties

(Browning and Wexler, 1968), was reviewed briefly. The objective of the present work is

to explore the possibility of using the VAD technique, and data from a UHF wind profiler,

to obtain a 2-dimensional iinear approximation to the measured wind field. Thus the theory

of VAD will be studied in more detail so that the equations can be set up to caiculate the

average vorticity, and then the b e a r wind.

2.2 General equations

In their work, Browning and Wexler (1968) first expressed the radial velocity (V' )

in terms of the standard Cartesian wind components, u, v, and w :

Where, /3 is the azimuth angle, measured clockwise fiom north, a is the elevation angle of

the scanning radar (Figs. 2.1, and 2 2 ) , and thz overbars indicate expandable terms (this

will be explained later).

Fig. 2.1 Definition of the azirnuth angle,

Fig. 2.2 Definition of the elevation angle, a

A similar expression can be written for the tangentid velocity, Vt :

In turn, the u and v components of the horizontal wind can be represented by a Taylor

Series expmsion:

where uo, and vo are the components of the horizontal wind over the scanning radar, taken

to be at the origin of the coordinate system, ux, uy, vx, and vy are the derivatives of the

two wind components with respect to x and y , , and EN are the random non-Iinearities

of the u and v components, and , and ESV are the systematic non-linearities of the u

and v components (the reasons for the distinction will become c1ea.r later on). Strictly

speaking, Browning and Wexler only worked with the linear truncation of the Taylor

Series, but for our purposes it is important that the other terrns are represented.

In (2.3) and (2.4), the Taylor series have been divided into linear and non-linear

parts, where the non-linear parts thernselves, have then been divided into random and

systernatic components. The systematic non-linearities are simply the higher order terrns to

the Taylor Series, and the random non-linearities represent rnainiy small scale eddies not

representable by a finite number of terms in the Taylor expansion. To deal with the random

non-linearities, we appty Iinear perturbation theory, beginning with the separatirig of u and

v into a mean plus a perturbation such as:

- u = u + u r

and -

v = v + v f

where u t , and v' are the random non-linearities, Ü, and ; are the terrns thst can be

represented by a finite number of terms of a Taylor Series. Therefore, to simplify things

we assume that the random non-linearities are fdtered out by the Fourier fitting to the actual

data, and so only substitute 2, and ; into (2.3) and (2.4).

An expression for Vr, as a fùnction of the Taylor Series denvatives, is then

obtained by substituting (2.3a) and (2.4a) into (2.1). For clarity, (2.3a) and (2.4a) are expanded to the third order before the substitution (thus, only two ternis of &su and

are retained) . With some trigonometry , the follo w ing equation is obtained:

This gives a Fourier Series of the form:

Where an and bn are the Fourier Series coefficients for the series. We find from

comparing (2.5) and (2.6) that the Fourier coefficients of a two harmonic Fourier fit to a

non-linear wind, will not give the linear kinernatic properties. This can be seen if we wnte

out the coefficients fkom (2.5) as follows:

r cos a % = 3 ( ~ i v ) + w - sin a

1 - + -unr21 cos a 8 1

where Div is the divergence, Shr is the shearing, and Sr is the stretching of the linear

component of the wind field.

It has been standard practice to assume that the higher order terms in (2.7) - (2.1 1)

were negligible, however, due to the dependance on range in these equations we c m see

that this assumption will become invalid at large ranges. Another interpretation has been to

take the Fourier coefficients as the kinematic properties of the mean wind field, however,

we can show that this is also a misinterpretation for al1 except the divergence, by using the

proof offered by Caya and Zawadzki (1992).

First, we rewrite (2.7) - (2.1 l), neglecting the effects of verticai velocity and the elevation

angle, a .

We can then show that (2.12) gives the mean divergence within a circle of range r, with the

followillg:

which, for a cubic wind gives:

Recall that we represent the divergence of the linear component of the wind, (ux+vy) , in

(2.12) with Div , thus we see that (2.1 8) is exactIy (2.121, which rneans that 2ao/r c m be

interpreted as the mean divergence within the circle of range r. If we repeat this for the

other kinematic propenties, we fhd the following:

When we compare (2.19) - (2.22) to (2.13) - (2.16), we see that the coefficients obtained

h m a 2-harmonic VAD fit do not correspond to the mean kinematic properties. Thus the

most we can Say about the coefficients fiom VAD fi& at any given range is that they contain

the kinematic properties associated with the linear component of the wind, but they also

contain contamination from the non-iinear terms-

To solve the problem of the non-linear contamination, we examine a general form of

(2-12) - (2.16):

~ ( r ) = k + f ( r ) (2.23)

where c(r) is the Fourier coefficient, k is the kinematic property for the linear component

of the wind, and f(r) is an even fünction of range. The implication of (223) is that if we

14 perform VAD fits at many ranges, and plot the coefficients fiom these fits as a fimction of

range, then a polynornial fit of (2.23) to this plot should provide k . If we include the effects of the vertical motion (V+w ) and the elevation angle, a, the relationship will still

hold, however the lefi hand sides of (2.12) - (2.16) are multiplied by cos(a), and w sin(a)

is added to the left hand side of (2.12).

The inclusion of the elevation angle, a, in the above equations means that we

cannot solve for the divergence, (Div), without knowing the fall speed of the particles,

(V+w ) (this was the motivation for the creation of the EVAD, Srivastava et al. 1986). To

determine this we will use the information from the vertical wind profder, which provides

the horizontal wind plus a vertical profde of (V+w ) over a point. For this to be useful, the

wind profiler must be located within the domain of the scanning radar (Fig. 2.3), and we

will assume that in stratiform precipitation, the fall speeds are the same throughout the

domain.

Domain of the scanning

Fig 2 3 Location of the wind profiler (W.P.)

with respect to the scanning radar (R).

Using the profiler to determine the vorticity involves solving for the tangential velocity, VI ,

using a similar method as the one used for the radiai velocity, V , .

We begin by substituting (2.3a) and (2.4a) into (2.2), and applying some

trigonometry, to obtain the following equation:

As with the radial velocity, (2.24) has the f o m of a Fourier series:

Where cn and dn are the Fourier coefficients. As was done before, we equate the

coefficients of (2.25) with the values in (2.24) to obtain the foilowing:

16 Where Vor is the vorticity of the wind field. These equations demonstrate that a linear

approximation of the vorticity is possible, if values of uo, vo, Stretching, Shearing, and a

tangential velocity are known. This is possible at the wind profiler location by using the

kinematic properties obtained fkom the VAD analysis descnbed above, and the tangential

velocity measured by the profiler. This is shown in the following denvations:

Rewriting (2.24):

- ' -~ tr -s in2/3 3 -'- 3 Shr - cos 2B

Since we are interested in determinhg the vorticity for the linear component of the wind, all

t ems in (2.3 1) refer to those associated with the linear wind. However since the profder

measures the true tangential velocity, which may not necessarily be Iinear, we will be

introducing a bias by fncluding this information. This can be seen with the following

relations hip:

Where, VmP is the tangential wind measured by the wind profder, and V, is the tangential

velocity of the lrnear wind.

We then solve (2.3 1) for the vorticity, and subshtute in the values from the wind profiler to

obtain:

Where, is the aziinuth angle of the wind profüer, and rwp is the range of the wind

profder. With the assumption that the vorticity cdculated is valid for the entire domain, the

linear horizontal wind field becomes completely defined.

To obtain an idea of how the precision of this equation will be affected by the enors

in the measured values, an g e n e d error analysis is needed on (233):

Rewriting (2.33):

2 2 - sin p, 2 - cos ,6, Vor = - - Vw - -vo +

Y Y 'Uo

- Sm - sin(2BWp) + Shr cos(2Bwp)

Taking the partial derivative of Vor with respect to; u,, v,, Str , and Shr , we get the

following:

Equation (2.34) shows that the uncertainty in the vorticity is a two-harmonic sine function

of the position of the profiler and the uncertainties in VAD derived kinematic properties,

and proportional to the uncertainty in the tangentid velocity over the wind profder. The

implication of this equation is that the vorticity wili have an uncertainty that is essentially the

sum of al1 the uncertainties from each kinematic property, as well as the degree of non-

Iinearity over the wind profder.

2 3 Set Up Of The Retrieval Procedure

With the mathematical ground work in place, the next step is to combine everything

into a coherent procedure that can be applied to a set of data. First, since Our aim is to

calculate properties of the horizontal wind, only data at constant heights will be used in the

procedure. In other words, to obtain a 2-Dimensional wind at a given height, the only

ranges to be used are those at which the elevation angles intersect the desired height (Fig.

2.4).

Fig. 2.4 Ranges rl , r2, and r3 indicate the ranges used from elevation angles a l ,az , and a3 for a retrieval at height , Z .

The VAD analysis is then performed on the data at each range used. A plot can then

be made to determine the variation of the Fourier coefficients as a fu~ct ion of range, to

which we c m fit ( 2 2 3 ) and therefore find the kinemaùc properties of the linear wind, k . To incorporate the wind profiler data, a fine is fitted to a time series of the u, and v

components of the horizontal wind in order to fiter out the small scale non-linearities. The

values of these linear fits, at the desired thne of the retrieval, are then used in equation

(233) to obtain an estimate for the vorticity. With this full set of kinematic properties, we

can calculate the two dimensional h e a r wind using the Taylor series expansions of u, and

v , truncated at the linear terms.

2.4 Summary

Some difficulties with this retrieval procedure can be anticipated. First, there is the

problem of the non-linearity of the wind field. As mentioned eadier, the coefficients from

the VAD analysis Vary with range for a non-linear wind field, thus, the linear

approximation will have a limited range of applicability . f iom this, it can be concluded

that the procedure should only be applied to wind fields that are reasonably close to linear.

Another concern is the effect of non-linearities over the wind profiler. In other words, how

strongly non-linear does the tangentid velocity have to be, to give an unreasonable error in

the vorticity. To answer this, we will test the method on an artificial two-dimensional

wind, so as to determine the sensitivity of the rnethod. The next chapter will briefly

describe the artScid wind field used, how well the procedure worked, and the conclusions

drawn from the results.

Chapter 3

Analysis of a Test-Bed Wind Field

3.1 Wind field description

In Chapter 2, a procedure was developed to retrieve a linear wind field using the

VAD analysis and wind profiler data, and it was shown that this procedure could be

severely lirnitea by a highly non-hear wind field. To examine the sensitivity of the

procedure to the non linearities, a cubic wind field was created whose non-Linear terms

were relatively important compared to the linear te=, and the procedure was applied to it.

The artificial wind was created using (3.1) and ( 3 3 , which are Taylor Series in ci,

and v , that are truncated at the third order tenns.

It can be seen in (3.1) and (3.2) that the cross derivatives were neglected (as was the

vertical velocity , (V+w) ), so as to sirnpl* the calculations. The constant terms of the

velocity field were taken to be: uo = 5.0 m s-1, and vo = 4.0 m s-l , (the other terms are

given in Table 3 -1). Values were then calculated for c i , and v on a 80 km x 80 km grid,

centered on the scanning radar, For the purposes of visual cornparison, this wind field is

plotted over a map of the Montréal region (Fig. 3 -1).

TabIe 3.1

Numerical Values used for the

Taylor Senes' derivatives in cubic wind

Taylor derivatives u component (s - l ) v component (s-1)

3.2 Retrieval of Artificial Wind field

Before applying the procedure fiom Chapter 2, we frrst calculate the radial velocity (Vr ) that would be detected by a radar scanning this wind field. This means that the

horizontal radial velocity is calculated only for the ranges that intersect a given height (Fig.

2.4), that is arbitrarily chosen to be 2.75 km above the Radar. Also, since the radial

velocity seen by the radar is not along a horizontal plane, but along the planes of the elevation angles, av, the horizontal radial velocity is then projected on to these planes.

With these calculations, we obtained a data set that simulates what the Scanning Radar

would "see" under ideal conditions (without random noise or ground clutter).

The first test to the procedure is verifying the range dependence of the VAD

coefficients- As was described in § 2.3, the VAD coefficients of a non-linear wind field

should vary as a function of range that is dependent on the degree of non-linearity of the

wind field. This dependence allowed us to create a general equation (2.23), that could

provide us with the kinematic properties of the Linear wind field. In the case of o u test-bed

cubic wind field, the theory from Chapter 2 tells us that we should expect that the fitted

polynomial, (2.23), to be a second order even polynomial whose fiist coefficient is the

kinematic property for the linear component of the wind field. We see from Figs. 3.2(a-e)

that a second order even polynomial fits the range dependence of the VAD coefficients

exactly, codrrming the mathematics of Our theory. Given this, we can then use the first

coefficients of these polynomials as the VAD derived kinematic properties.

The next step is to determine the vorticity of the linear wind using the data fiom the

wind profiler. As we determined in Chapter 2, it is necessary to do some smoothing to the

profiler data, in order to elimïnate contamination due to non-linearities. What we proposed

was a time senes of u, and v from the profiler (approximately one hour), to which we fit

straight lines, we then use the values from these lines in the retrieval so as to reduce (or

eliminate) the degree of non-lineari~ of the measurement. The underlying assumption in

thïs procedure is that the wind field does not evolve significantly while it is advected over

the wind profùer, so that a time series can be translated to a spatial cross-section (the so-

cded "Frozen Turbulence Assumption"). For our test-bed wind, we imposed an advection

wind of: U = 5.0 m s-1, and V = 4.0 m s-1, to give us a time senes of u, and v in Fig.

(3.3), where tirne t = O is the time of the original wind field, and time t = 60 is the values of

u and v , 60 minutes later under the advection wind (U,V). After taking the tirne senes,

we fit straight lines to u and v (Fig. 3.3), and use the values fiom these fits at time t = O to

calculate VW . We then substitute the VAD denved kinernatic properties and VW into

(2.33) to calculate the vorticity for our linear two-dimensional wind (Fig. 3.4).

3 3 Discussion of ResuIts

To veriQ that the results obtained from our retrieval, we compared them with the

values that correspond to the true linear component of the wind field. We find that the

numencal values we obtained for the VAD derived kinematic properties (Table 3.2) were

exactly the sarne as those we calculated using the linear terms from Table 3.1. However

when we compare the retrieved vorticity for the linear wind to that calculated from the

values in Table 3 -1, we find that they dBer by about 2x10-5 s-1.

Table 3.2

Comparison of Kïnematic Properties

of renieval and true wind fields.

Kinematic Property True Wind field Retrieved Wind field

U O 5.0 rn s-1 5.0 rn s-1

Vo 4.0 m s-1 4.0 rn s-1

Divergence 9.0~10-4s-1 9 -0x 1 0-4s-1

S tretchhg -5 . ox~o-~s - 1 -5.0~10-4s-1

S hearing -7 -7~10-4s- 1 -7 .7~ 10-4s- 1

Vorticity 5.7~10-4s-1 55x10-4s-1

V' of hear wind:

Substituting values into (3.3):

This indicates that if the tangentid velocity from the linear fit to the time series is off by less

than 0.4 m s-l from the tnie 1inea.r tangend& velocS at that point, then the expected error in

the vorticity retneval should be on the order of about 2x 10-5 s- 1. When this is compared to

the case of the artificial wind, we c m see that the retrieved and true vorticity do, in fact,

differ by about 2 x10-5 s-l. As a further test we repeated the retrieval for wind profiler

locations all around the same range circle. The error of the vortkity is then plotted against GV, , so as to venfj the relationship determined by (3 -3). From Fig. 3 5, we c m see that

the errors calculated from each retrieval (stars) match perfectly with the straight line

calculated from (3 -3) (solid line). Figure 3 -5 also gives us an idea of the degree of non-

linearity at the 30 km range, in that there are a significant number of retrievals with an error in the vorticity of 1 x 10-4 s-1 or greater, and that to get errors of this magnitude, 6VI need

only be greater than 1 5 rn s-1 .

Such sensitivity to the non-linearity of the wind requires us to be cautious in using

data fiom the wind profiler without ve-ing the validity of the linear approximation. This

will then require a rnethod of deterrnining the validity of the linear approximation, since we have no way of calculating dV, for a real case without prior knowledge of the complete

linear wind field.

3.4 Summary

In this chapter, we tested the procedure fiom Chapter 2 on an aMcial wind field to

determine its sensitivity to a non-linear wind fieId. In this idealized test, we showed that

the linear values of divergence, translation velocity, stretching, and shearing, are retrieved

exactly, and that the voaicity is obtained within a reasonable rnargin of error. However

this low error in the vorticity seems to be a lucky chance, since there are a significant

number of possible profder locations, dong the sarne range circle, that have unacceptably

large uncertainties .

Thus, we can conclude from the results of the test-bed application, that the

mathematics of the procedure are valid, given the good results when Vwp is within 1 m s-1 of the hear value (Vr ). However, given the sensitiviq of the retrievd to 6Vr, we have to

develop a method of detemiinhg the validity of the linear assurnption without prior

knowledge of the complete wind field. The next step is to test the procedure on real data

(which will have the added effects of srna.ll scale non-linearities), and v e e whether such

departures fiom lïnearity occur in stratiform precipitation, and if they do, how to overcome

them.

Cubic Wind, Projected over Montreal 4 0 t + R 1 ~ + * 6 ~ ; K 1 ~ a t 8

Fig. 3.1 Plot of the artificial cubic wind, projected over a map of the Montreal Region. North is at the top of the page, and East is to the right. X marks the Iocation of the scanning radar (R).

Fig. No te

3.2 that each

20 40 60 80 range (km)

' = VAD coeff.

- = fitted polynomial (2" order)

O 20 40 60 80 range (km)

The plots of the coefficients as a function of range for the artificial follow dong a 2nd order polynomial.

wind.

1 Hour time series over W.P.

W w W w w w W ~ w W W ~ m m m f i n m a m f i , n

0 - - + = u velocity over profiler

1 '= v velocity over profiler

20 30 40 time (minutes)

Fig. 3.3 Time senes of u and v over location of the wind profiler, for the artificial wind field with an advection wind.

Wind retrieval for artificial wind 4 0 r . B 3 r - b s ~ s 1 - ~ 8 G .

Fig. 3.4 Linear Wind retrieval on Artificial wind. WP. marks the location of profiler.

O 1 2 3 4 5 Vt diff. (m s")

Fig. 3.5 Difference of retrieved vorticity from the linear vorticity, as a function of the difference of Vwp from Vl. The stars represent individual realizations, whiie the solid line represents the fitted line.

Chupter 4

Analysis of Real Data

4.1 Weather Synopsis

The procedure described in the previous chapters was f i s t tested on data fiom 1800

to 2200 U.T., (hiversal Tirne) 14 December, 1995. The case was a stratiform snow

event, associated witb a surface warm front of a 1002 W a low pressure system centered

over Lake Huron at 1800 U.T.. What made this a good case to test our procedure was the

fact that the wind field was horizontally uniform in t ems of both radial velocity, and

reflectivity (Fig. 4.1). Also, at any given height over the wind profüer, the winds seem to

vary srnoothly over tirne, although there is a rather significant directional shear with height

at lower levels (Fig. 4.2), and this could affect the measured radial velocities due to

averaging over the width of the radar bearn.

The two instruments used to test these cases are the S-band Doppler scanning radar,

at Ste. Anne de Bellevue, and a UHF wind profder in downtown Montréal. The scanning

radar scans a total of 24 elevation angles, alternating between the even and odd angles to do

two volume scans of 12 elevation angles in a total of 5 minutes. The UHF wind profder

measures profiles every 60 seconds, however, in order to have the same temporal

resolution as the scanning radar, the time steps were set at 5 minute intervals. The

consensus used was 60% of the data within t 2 m s-1 in a 15 minute period, based on Rogers et al. (1994).

4.2 Wind Retrieval for December 14 1995

In order to get an idea of the behaviour of this wind field, we applied the VAD

analysis dong different elevation angles. The results of this analysis can be seen in Figs.

4.3 (a-e), where we see that in some of the coefficients there is a darnped oscillatory

dependence on range and distance. This seems to ve r i e the findings of Rabin and

Zawadzki (1984) as weU as Caya and Zawadzki (1992). Looking at constant heights,

(represented by symbols) it c m be seen that some part of this non-linearity is i fûnction of

29 range only, so at f i s t glance we can Say that even in a stratiform precipitation event, the

linear approximation will only have a limited region of applicability. With this in mùid we

begin the test of ouï retneval procedure on the data.

As was the case in Chapter 3, our f i s t test is the validation of the variation of the

VAD coeff~cients with range. In this case we want to determine if we can fit a polynomial

of the form of (223) to the data, as was done in Chapter 3. However, unlike the test-bed

wind field (which we knew was cubic), we do not know the actual degree of non-linearity

of the wind field of the real case, thus we use (4.1) as a e s t guess, assuming the constant

term (which is equivalent to the kinematic property of the linear wind) will not change

significantly with the addition of higher orders:

The results of the fit of (4.1) to the coefficients are in Fig. (4.41, and they show that

although the frts are not perfect, (4.1) does offer a reasonable approximation to the large

scale non-Linearity of the wind field. However, we also note that there are a significant

amount of small scale non-linearities visible in each of the kinematic properties. The

problem with these is that, although they are smoothed out by the fitting of (4.1), their

presence will increase the standard deviation about the coefficients of (4.L), and thus

increase the uncertainties in Our kinematic properties for the linear wind (kg ). Another

consequence of both the large, and small scale non-linearities, is that they can increase the

degree of non-linear contamination of the profder information, and therefore we need to

find a method of determining the limit of our linear approximation.

There is no way of determining the degree of non-linearity of the wind field,

without knowing ail components, however if we make the assumption that the degree of

non-fine- is the same in the tangential and radial components, then we can use the radial

velocity as an indicator. We begin by rewriting (2.6), using the constant terms of (2.7) to

(2.1 1) for the coefficients (which, if we recall from Chapter 2, are the kinematic properties

of the linear wind), to give us an equation for the radial velocity for the linear wind.

+ (Shr) r - cos a r - c o s a

sin 2P + (S t r ) - cos 2P 2 2

We then substitute the retrieved vaIues of the kinematic properties into (42) to calculate the

radial velocity of the linear wind, and compare this with the true radial velocity . If this

cornparison is done for several ranges, then we may have a reasonable idea of the

maximum range of validity for the linear approximation. Some examples of this

cornparison for the Decernber 14 1800 U.T. data can be seen in Figs. 4.5 (a-d)- From

these plots we can see that the retrieved linear radial velocity ( V A ) , fits quite well on the

measured radial velocity up to at least 36 km from the radar as can be seen in Fig. 4 5 (d).

From such cornparisons we can then conclude that for the radial component, the linear wind

seems to be a reasonable approximation of the true wind, up to about a range of 40 km for

this pariicular case. Thus, using Our assumption that the behaviour of V, is representative

of Vr, we can then Say that the Linear approximation should be representative of the full

wind field up to 40 km. As a final check, we examine the tirne series of the winds over the

profiler for the previous hour (1700 - 1800 U.T.), and we find that a straight line fits the

data reasonably well, (Fig. 4.6) indicating that the profiler information is not contaminated

significantly by any s m d scale perturbation.

With the maximum range for the retrieval established, we apply the retrieval to the

data and find that the retrieved wind (Fig. 4.7), has a strong anticyclonic vorticity (about

-2.0 x IO-^ s-1). Considering the fact that there is an approaching synoptic scale low

pressure systern, we would expect at !east cyclonic (positive) vorticity. On the other hand,

since we are only retrieving a section of the wind field which is 80 km in diameter, perhaps

this is an accurate representation of the mesoscale around the radar. However, in order to

have ony confidence in our results, we need to perform an error analysis on the procedure

at severai times during the precipitation event.

4.3 Error analysis of Retrieval

To obtain a reasonable estimate for the uncertainty in the retrieved vorticity, we have

to analyze every step of the procedure to see what are the sources of errors for each. We

can then combine the errors of each step, and determine the overall uncertainty in the

retrieved vorticity .

Our first source of uncertainty in the retrieval cornes from the errors due to the VAD

analysis performed at each range. As we demonstrated in Chapter 2, one source of error

will be due to the non-linear terrns that are within the VAD coeffcients. However, while

31 these non-linearïties will lead to misinterpretations of the coefficients, we have developed a

method in Chapter 2 that will separate the non-linear terms fiom the linear terms, therefore

we will ignore these errors for the moment. Other sources of unceaainty in the VAD

analyses include random noise of the measurement, beam smoothing, and reflectivity

weighting . B eam smoothing , and reflec tivity weighting s hould have minimal effects since

we have picked a case where the reflectivity is reIatively unifonn, and the velocity has a

reasonably smooth gradient. On the other hand, random noise could have a significant

effect on the VAD analyses, and thus should be included in the enor estimate.

One of the methods we use to estimate the errors due to random noise is the

standard deviation about the coefficients on the two-harmonic fits, the definition of which is

given in Appendix A. In essence, it gives a quantitative value of the scatter about the

coefficient which we take to be the uncertainty. This uncertainty would be due to the

combination of the random noise and s m d scale eddies. Since this is a combination of

effects, one of which we do not want to include (that of the s m d scale non-linearïties), the

results of this error analysis would be an overestimation of the error in the VAD analysis.

However, it should still provide a reasonable estimate of the uncertainty of the coefficients

at a given range.

Another method of estimating the error in the VAD coefficients estimates the root-

mean-square error of the VAD coefficients. In this method we caiculate the difference

between the coefficients of fits from adjacent ranges, square hem, then take the mean of the

squares to determine the contribution to the error due to random noise. This procedure is

described in more detail in Appendix B. The result of this procedure is that it estirnates the

contribution of the random variations to the variation of the coefficients in range.

With the results of both of these error estimates, we can determine the overali efiect

of the random error on the VAD coefficients. The f i s t method gives us the variation of the

coefficients at a given range, due to the random error. This gives us an idea of the

uncertainty of the points to which we fit our polynomial in the next step. As we can see

from the error bars in Fig. 4.4, the errors are relatively srnall cornpared to the measured values, and thus they do not have a significanî contribution to the uncertainty in the

polynomial fit. The second method gives the contribution of the random error to the

coefficients variation with range. Again we can see from the errors obtained from this

method (Table BA), that they are an order of magnitude less than the coefficients

themselves. We can conclude fiom this that the random error does not have a signifkant

effect on the variation with range itself.

The next step of the procedure is the fitting of (4.1) to the coefficients as a function

of range. To estimate the uncertainties in the kinematic properties obtained fiom these fits

for a given time, we will use the variances about the polynomial fits. This wiil give us the

uncertainty due to the fact that (4.1) is not an exact representation of the non-linearity of the

wiod field. We fmd fÏom such an analysis on the data frorn 1800 U J. (Table U), that the

resultant errors are relatively small compared to the retrieved values.

Table 4.1

VAD denved linear bernatic properties

with uncertainties for 1800 UT.

December 14 1995

Kinematic Property Retrieved Value Uncertainty

Divergence -1.12 x 10-4 s-1 + 2.0 x 10-5 s-1

v o 11.6 m s-1 k 0.5 m s-1 S tretc hing 4.55 x 10-5 s-1 + 1.3 x 1 0 - 5 ~ - ~

It should be noted that the errors in Table 4.1 are also overestirnates, since they include the

effects of random noise, and higher order non-linearities. While overestimating the error in

the retneved kinematic properties helps to illustrate the point that it is relatively unimportant,

it will be of no value to do so when the error is used in the estimation of the uncertainty in

the vorticity. Therefore, we need to estirnate the true variance of the retrieved kinematic

properties, which we can then use to find the uncertainty in the retrieved volticity.

In order to find the variation of the retrieved kinernatic properties, we do an

autocovariance on a time senes of two hours. While the statistical sample is not very large,

we believe that with the smoothing of the VAD fitting, and the polynomial fitting, the

autocovariance should be stable enough to provide a reasonable approximation of the

random variations. To do this, we assume that the random error is completely uncorrelated

by the first time lag, so that the autocovariance at the "zeroth" lag is a combination of

random and non-random errors, while the other lags contain only the non-random error.

To find the contribution of the non-random error at the "zeroth" lag, we determine the trend

33 of the autocovariance for the other lags (1 to x ), and extrapolate this trend to the "zeroth"

lag. This is then subtracted from the total value of the autocovariance at that lag, giving us

the random error. Continuing with the same assumptions, if the random error was large,

then we would see a sharp change in the values of the autocovariance from the "zeroth" to

the firs t lag .

The autocovariance curves are shown in Fig. 4.8, and from these we can see that

the random variations are very small, since we do not see the sharp peak that is associated

with the randorn errors (as described above). To show how these errors influence the

retrieved vorticity, we modified equation (2.34), such that we ignore for the moment the

error due to the proNer then we take the absolute value of each of the remaining terms.

When equation (4.3) is caiculated for each five minute time step of the two hours of

data, we obtain the error bars on the vorticity in Fig. 4.9. What this clearly shows is that

the random errors from the VAD derived coefficients do not significantly affect the

precision of the retrieved vorticity. The next step is to investigate the uncertainty that arises

from the degree of non-linearity of the tangential veIocity measured by the UHF wind

profiler (VW ). This is important, since we assume in calculating the vorticity in (2.33),

that Ilfwp and al l of the kinematic properties (u,, v,, Stretching, and Shearing) we

retrieved, are those of the h e a r component of the wind, meaning that the vorticity that we

obtain is also the vorticity of the linear wind.

In the previous section, we made the assumption that due to the stratiform nature of

the precipitation, the statistical behaviour of Vf should be the same as V,, and we used this

in a qualitative sense to determine a maximum range for the retrieval. Since we now need a

quantitative value of the degree of non-linearity of V,, we will use the same assurnption,

and determine the mean deviation of the measured V, about the linear V, . In other words,

if we find that the measured V, has a mean deviation from the linear Vr of +x m s-1, then

we WU assume that any given measurement of Vf should not deviate from hnearity by

more than +x m s-1. Thus, we estimate the deviation of Vr at the closest range to the

profiler for which we have a VAD (recall we only have VAD at certain ranges, as shown in Fig. 2.4), and use this as an approximation the value of 6Vf for (2.32). When we apply

34 this to the retrieval at 1800 U.T., we see that this gives us an uncertainty about Vwp of

20.34 m s-1, and when this error is included, we obtain an overall uncertainty in our

retrieved vorticity of k4.2 x 10-5 s-1 .

When the error anaiysis was performed over the 2 hour period, we find that the

uncertainty in the vorticity increases dramatically after 2920 UT. (Fig. 4.10). The reason

for this is that the rnean deviation of V, from its linear component also increases

dramatically, which means that Our profiler rneasurement would most likely be significantly

dBzrent frorn V, . For example, at 2000 U.T. the deviation is approximately 4 mis from

linear, which contributes to an overall uncertainty in the vorticity of about k2.9 x 1 0 - ~ s-1.

To determine the cause of this deviation, we examined the cornparisons of the linear V, with

the measured V, at 2000 U.T. in Figure 4.11. We see fiom this cornparison at the four

ranges used widiin the retrieval domain, that there is a d e f i t e systematic deparnire from

the Iinearity hypothesis at this time. To examine this further we take a time series of the

divergences obtained from the VAD at each of these ranges, for the full two hours, and

compare their evolutions (Fig. 4.1 2). We find from these evolutions that up to about 1920

UT., one value of divergence could reasonably approximate the whole dornain. However

after 1920 UT., the rnean divergence within the 21 km range smoothly increases by 2 x

10" s-1, whereas the mean divergences at the other ranges do not increase so dramatically.

The smoothness of the change, and the fact it reaches a peak and levels off, strongly

indicates that this is a naturd variability. Also, the fact that it is more noticeable at the

shorter range (where smaU perturbations have a stronger effect on the mean divergence)

seems to support this idea. In any case, this non-linearity is rather strong and quite

obviously makes the retrieved vorticity very unreliable. Thus, in order to maintain the

uncertainty in the retrieved vorticity below a certain value, a threshold of 1 rn s-1 was

imposed on the deviation from linearity, so that for any deviation beyond 1 m s-L no

retrieval will be performed (Fig. 4.13). The choice of a 1 m s-1 threshold is rather

arbitrary, and was only chosen for dernonstration purposes.

As another check to the retrieval procedure, we tried it on another stratiform snow

case, that occurred December 9 1995.

4.4 Wind Retrieval for December 9 1995

The second case chosen to test the retrieval procedure, was also a stranform snow

event that occurred over Montréal on December 9 1995, starting at around 2000 U.T.. As

35 with the previous case, it was chosen for its lack of noticeably strong small scale feanires in

both reflectivity and velocity, and because the velocity profiles appear to be relatively

uniform over tirne. Examples of the instrument output for this case are given in Figs. 4.14

and 4-15

Repeating the process described in section $2.3 for the data at 22 15 U.T. 9

December 1995, we determine the VAD coefficients at each range at 2.0 km height (Fig.

4.1 6) , find the VAD derived linear coefficients (Table U), retrieve the 2 dimensional wind

field (Fig. 4.17), and calculate the h e m vorticity (6.7 x 10-5 si). In this case, as opposed

to the 14 December case, we obtain a positive (cyclonic) vorticity. To determine how

certain we are in this value, we repeated the retrieval and error analysis from section $4.3

on a 1 hour time series (2200 - 2300 UT.). We can see from the time series of this

retrieval (Fig. 4-18}, that the bernatic properties in this case are Iess variable than those in

the 14 December case, thus reducing the randorn error contribution to the vorticity. We cm

also see fiom the small error bars on the retrieved vorticity, that the wind field is very close

to linear, since small error bars indicate small variations about the linear radial velocity.

Table 4.2

VAD derived linear kinematic properties

with uncertainties for 22 15 U.T.

December 9,1995

Kinematic Property Retrieved Value Uncertainty

uo 11.6 m s-1 k0.77 rn s-1

VO 12.7 m s-1 *O -48 m s-l

Divergence 3.4 x 103 s-r 12.7 x s-l

S tretching -1 .O x 10-5 s-1 5~2.2 x 10-5 s-1

Shearing 3.0 x 10-5 s-1 el -7 x 10-5 s-l

4 5 Further Analysis and Summary

In this chapter we have tested the procedure on two winter cases, to see how it

would perform in an actual operational situation. We have verified that the random errors

have minimal contributions to the overall uncertainty in the system, as shown by the

smooth variations in tirne of al1 the kinematic properties. However, as we discovered in

Chapter 3 with the test-bed wind field, the main contribution to the uncertainty in the

retrieved vorticity, is due to the deviation from linearîty of the wind field at the position of

36 the wind profiler. In December 14 case, this was revealed dramatically with the sudden

increase in non-linearity after 1920 U .T..

As was mentioned in section $42, we used ody second order even polynomialç to

obtain the VA.D derived Linear kinematic properties. The question is, how would Our

results improve if we used higher order polynomials, (especially in the case of December

14). To determine this we repeated the retrieval for the December 14 case, with the o d y

clifference being that we used 4th order even polynomials for the determination of the VAD derived linear kinematic properties (Fig. 4.19). The results of this retrieval are in Table

4.3, and if we compare them with those fiom Table 4.1, we find that for the most part, the

retrieved linear kinematic properties from each method are within each other's value of

uncertainty. Also, if we examine the two hour bme senes (Fig. 4-20), using the 4th order

polynomials, we find that, although there are differences from the curves in Fig. 4.13,

there are no significant irnprovernents fiom the values derived f?om the second order fits.

TabIe 4.3

VAD derived hear kinematic properties

from 4th order polynomials for 1800 U .T. Decernber 14,1995

Kinematic Property Retrieved Value Uncer tainty

"O 4.5 m s-1 a . 3 7 m s-1

VO 11.5 m s-1 4.57 m s-1 Divergence -1 -4 x 10-4 s-1 22.5 x 10-5 s-1

Stretc hing 5.8 x 10-5 s-l -t 1.3 x 10-5 s-1

Shearing 4.8 x 10-5 s-i -cl -9 x s-l

As a counter-example, we also calculated the mean values of the coefficients, and used these as the kinematic properties for the linear wind. The resulting errors from the

scatter about the means were twice those obtained from the 2nd order even polynomial fits.

In tems of the constant wind (uo,v0) this means errors that are still within 1 rn s-l, which is

still a reasonable estirnate. However, in tems of the other coefficients, this means that we

now have errors of the order of the retrieved values. This tells us that for this case, the first

and second order derivatives are of equal importance.

37 Finally, we noticed that the errors due to the VAD derived linear kinematic

properties were srnall relative to the retrieved vorticity, and therefore not the main source of

ccncem in using the procedure. However, as was discovered in Chapter 3, the uncertainty

in the degree of non-linearity of die tangential velocity used (VW ) is a big concern in

using the procedure and should be kept as smali as possible, given the data avadable.

These, and other final conclusions about the procedure WU be presented in Chapter 5.

Radial VeIocity at height = 2.75 km

Reflectivity at height = 2-75 km Ref (d Bz)

Fig. 4.1 Constant Altitude Plan Position Indicator (CAPPI) plots of Radial Velocity

and Reflectivity, for 1800 U.T. Decernber 14 1995. The CAPPI's were made according to

the description in section $2.3 (Fig 2.4).

Fig. 4.2 Time series of the horizontal wind over the profiler, from the surface to 4.2

km, for 1705 - 2000 U.T. Dec. 14 1995.

Wind Profiler data from 14/Dec/95

t2o

I I L I 1 8 1 1 I l I I 3.5-----LA-d,-J-/-/,,L///,,J,,,,,, b t I I I l i l

/ / c c / z / c r / M / / - / / / / / / ~ / f , / / / . / / / / / -

- / / / / / / / / / / / / / / / / f / / / / / / / / P P / / f - - , / / / r / / r / / / / r r / " / " f / / f / / / / / P / / / / - - / r r r / / / r r / / ~ / r r / / / / / r r / / / f / / / / -

r / / f f f / r P / / / / / / / r / / / / / f / / / / f f f - 3 * O ~ , / / . , / / / / / , , , / , , , , , , / / , / I I f f f , -

/ / / / / / / / / / . / / / / / / / / / / / f f f f f / f /

- E

I m .- al -t

/ / / / / / f / / / / / / / / / / / f f f / f f / f / f f f - / / / / / / / / / / / / / / / / / / / f f f f f f f / / f f

- f f / / / / / / / / f f f f f / / r / / f f / / / / / / f / -

2 . 5 - t r r t ~ ~ r ~ ~ ~ r r t t r t t r r / 1 / / ~ ~ / / r t r - - i i r r t . t t t t t t t t t t t / / f f / / / / / f f f f r r t t t t r r t f t t t t t t t t t t t t t t f f f / f f / - - t t t t T t r t t t t t t t t t 1 t t t t t r t r r t t f f - - t t t t t t t t t t t t t t t f t f t t t r t t t r t r r , -

s 2 . 0 - t t t t t t t t t t t t t t t t f t t t t f t t t r t t t I - - t t t t t t t t t t t t t t t t t t t t f r 1 t t t t r t t -

' t \ t t \ \ \ t t t t t t t \ \ \ t t t t t r f f f f r r f - - t t t \ t \ \ \ \ \ \ \ \ \ ~ \ ~ \ t f r r ~ ~ ~ f f ~ ~ f - - \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ t \ \ t t ~ f r t f f f f f r -

1 . 5 + \ \ ~ ~ ~ \ \ \ t \ \ t t t t t t I r t t t t I I r f -

- \ \ \ \ \ \ \ \ \ t t \ \ t t t t t t t r t r t I r , I r , -

- \ \ \ \ \ \ \ \ \ \ \ \ \ t t t t t T t t t t t t t t t r t t t - - \ \ \ \ \ t t t t t t t t t t t t t t t t t t r i t - - \ \ \ f \ I \ t t z t t f t t t t t t f t t t T t t t t t t -

1 * 0 - \ \ \ \ \ \ \ \ \ z ~ f l f f l t l l l 7 t t t t t t t t t - - \ \ \ \ \ \ \ \ \ \ \ t \ t \ \ \ \ \ \ \ \ \ t \ \ \ \ f f - . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . - \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ - - \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ -

0 . 5 - . . ~ ~ - - - - - - ~ - - ~ - - - - - - - ~ ~ ~ ~ - ~ ~ . . - _ - - - - - - - - - - C C - - Z Z C & _ _ - C - - - - - - - - - - - - - - -

- / . * . 0 # / @ / / / 0 / / / M / 0 / / / 0 / - / / / / / / -

- , a . . , / / y / - - > I I I I I I , I I I , 1 I l I I l I I - - - ---

18:OO 18:30 19:OO 19:30 20:OO time (U.T.)

Horizontal Wind (U-VI

(b) i 1- /

/ f /

I 1

1 I

! / /

20 40 60 80 range (km)

- = elv. angle 1.8 - = elv. angle 2.7 - = elv. angle 4.1 -. = elv. angle 5.9 t = 2.5 km height * = 3.0 km height

O 20 40 60 80 range (km)

Fig. 4 3 Plots of the VAD coefficients dong eievation angles, for the case of Dec. 14 1995, at 1800U.T..

O 20 40 60 80 range (km)

G 20 40 60 80 range (km)

l = VAD coeff. - = fitted polynomial (2" order)

Fig . 4.4 Kinematic properties as a function of range for 1800 U.T. Dec 14 1995. Error bars i~dicate the mean variation of adjacent measurements over half an hour. Soiid lines are 2nd order even polynornial fi&.

O 100 200 300 O 100 200 300 Azimuth angle (degrees) Azimuth angle (degrees)

Fig 4.5 Cornparisons of retrieved radial velocity (solid lines), the rneasured radiai velocity (dots), and the two-harmonic Fourier fits (dashed lines), for Dec. 14 1995 1800 UT.. At; a) range of 21 km from the radar, b) range = 26 km, c) range = 3 1 km, and d) range = 36 km.

-2 - + = u velocity over profiler -

4- ' = v velocity over profiler

- 6 - 1 ~ . ~ . ~ ~ 1 ~ i ~ t ~ . . . ~ . t ~ . . . ~ . . . . . ~ ~ ~ t . t t . r ~ i ~ ~ , , . , . . , ~ . , , , , , , . , - O 10 20 30 40 50 60

time (minutes)

Fig. 4.6 Time series of u and v over profder for 1 hour, for Dec. 14 1995. From 1700 - 1800 U.T..

Wind retrieval at h=2.75 km, 14lDec195 1800 UT.

t f f i r r r

f t i : t t t

t f t 1 t : t

t t t t t t t

f t t t i i r t i t t

l ; ? l \ ! t \ : t ! r l f l t l t \ t t \

-40 -20 O 20 40 x(km)

Fig. 4.7 Plot of retneved 2-D wind field over Montréal, Dec. 14 1995 1800 UT..

Autocovariance of Kinematic Properties -- 3.0

5 20 x - 5 1.0 O

., O

m

Fig 4.8 Autocovariances of the VAD derived kinematic properties £rom a two hour time series of; a) Divergence, b) Uo, C) Vo, d) S hearing , and e) Stretching .

Time Series of Retrieved Kinematic Properties 20 k (a) - . - . = i / w p i

- - - = Vorticity

18:OO 18:30 19:OO 19:30 20:OO Time (U.T.)

Fig. 4.9 Tirne series of al1 retrieved kinematic properties, error bars on vorticity are calculated oniy fiom the random perturbations due to the VAD denved kinematic properties. The error bars on the rest of the plotted values are due to the variation about the polynomial fits at each time.

Time Series Of Divergence and Vorticitv

18:OO 18:30 19:OO 19:30 20:OO Time (U.T.)

Fig. 4.10 Two hour tirne series of Divergence and retrieved vorticity at a height of 2-75 km for Dec. 14 1995 from 18:OO to 20:OO U.T.. Error bars on vorticity are calculated Çom the random errors fiom the VAD derived kinematic properties, and the mean deviation of the measured radiai velocity fiorn the linear radiai velocity.

Time series of Divergence at different ranges

- . - . 36 km range - - - - - - - - - - -

= fitted divergence

18:OO 18:30 19:OO 19:30 20:OO Time (U.T.)

Fig. 4.11 Two hour time series of mean divergence from four ranges within the retrieval domain, and the retrieved h e a r divergence.

O 100 200 300 O 100 200 300 Azimuth angle (degrees) Azimuth angle (degrees)

Fig. 4.12 Cornparisons of retrieved radial veIocity (solid lines) to the measured radial velocity (dots), and two-hamonic fits (dashed), for Dec. 14 1995 2000 2. At; a) range of 21 km fiom the radar, b) range = 26 km, c) range = 3 1 km, and d) range = 36 km.

Time Series Of Divergence and Vorticity

18:OO 18:30 19:OO 1 9:30 20:OO Time (U.T.)

Fig. 4.13 Two hour time series of Divergence and retrieved vorticity at a height of 2.75 km for Dec. 14 1995 from 18:OO to 20:00 U.T.. No values of vorticity are retrieved at times where the degree of non-linearity of V, exceeds i m/s.

Radia1 Velocity at 2.00 km Vr (m/s)

Reflectivity at height = 2.00 km Ref (dBz)

Fig 4 -14 Constant Altitude Plan Position Indicator (CAPPI) plots of Radial Velocity and Reflectivity , for 2215 U.T. December 9 1995. The CAPPI's were made according to the description in section $2.3 (Fig 2.4).

Wind Profiler data from 9/Dec/95

-1 -2 1.1 O 20 40 60 80

range (km)

O 20 40 60 80 range (km)

I = VAD coeff.

- = fitted polynomial (2" order)

Fig 4.16 Plot of VAD denved kinernatic properties as a function of range for 2215 U.T. December 9 1995, with polynomiai fits for a) divergence, b) uo, c) vo, d) shearing, and e) stretching.

Wind retrieval at h=2.00 km, 9/Dec/95 2215 UT.

4 0 -20 O 20 40 x ( W

Fig 4.17 Vector plot of retrieved 2-dimensional wind field for 22 15 U.T. December 9 1995, projected over the island of Montréal.

Tirne Senes of Retrieved Kinernatic Properties - = vo -.-.=vwp

22XlO 2212 2224 222% a48 Time (U.T.)

Fig 4.18 One hour t h e senes (2200 - 2300 U.T. Dec. 9/95) of retrieved kinematic properties and profiler Vt ; a) uo, vo, V-. b) divergence, and vorticity (with error bars), c) saetching, and shrearing.

O 20 40 60 80 range (km)

1 = VAD coeff. - = fitted polynomial (4" order)

0.2 - O 20 40 60 80

range (km)

Fig 4.19 Plot of VAD derived kinematic properties as a function of range for 1800 U.T. December 14 1995, with fourth order polynomial fits for a) divergence, b) uo, c ) vo, d) s hearing , and e) stretching .

18:OO 18:30 19:OO 19:30 20:OO Time (U.T.)

Fig 4.20 Two hour tirne series (1800 - 2000 U.T. Dec 14/95) of retrieved kinematic properties (from 4th order polynomial fits) and profiler Vt for; a) uo, vo, Vtwp, b) divergence, and vorticity (with error bars), c) stretching , and shrearing.

Time Series of Retrieved Kinematic Properties 20 (a) -.-.=vwp i

---=uwp -- 15p -- - = vo 4

Chapter 5

Conclusion

5.1 Summary and Conclusions

Since the developrnent of the VAD analysis two assurnptions have been made about

the meaning of the coefficients. The first was that for large scale precipitation, the non-

linearities could be neglected, and therefore the coefficients fiom a two-harrnonic Fourier

series fit would correspond to the kinernatic properties of a Iinear wind. The second

assumption, is that the coefficients were the means of the kinematic properties within the

scanned circles, as is the case with the divergence. m e sorne papers descnbed concerns

about the limitations of the linear hypothesis (e-g. Pasarelli, 1983), Caya and Zawadzki

(1992) demonstrated how both interpretations of the VAD coefficients could have serious

implications.

As a continuation of the work by Caya and Zawadzki (1992), we have added a

further modification to the VAD andysis. In Chapter 2, we began with the proof fiorn

Caya and Zawadzki (1992) that the VAD coefficients could be interpreted as a kinematic

value of the linear wind plus the non-linear terms. We then use this property of the wind

field to separate the linear frorn the non-linear terms, to obtain the kinematic values of the

iinear wind. At this point, we have two things of importance: first, we have the kinematic

properties (except the vorticity) of the wind field directly over the radar itself, and second,

we have an indication of the degree of non-iinearity of the wind field. As an addition to this

information, we developed a procedure that incorporates the tangential velocity over a wind

profiler and the retneved kinematic properties, to calculate a two dimensional linear wind

field.

In Chapter 3 we tested our procedure on a test-bed cubic wind field. When the

procedure was applied, we retneved the exact values of the VAD derived iinear kinematic

properties (divergence, u,, v,, stretching, and shearing), and obtained a vorticity within 2

x 10-5 s-1 of the true linear voaicity. However, this apparently good result was due o d y to

the face that we happened to use a tangential velocity (VmP ) at a location where the wuid

50 field was very close to linear. When we chose other points around the same range circle

that were not so linear, we found that we had tangentid velocities that deviated from

Linearity by as much as 4 m s-1. When an error analysis was performed on our equations,

we found that the error in vorticity is proportional to the deviations fiom linearity by 2/r

(where r is the range of the profiler). This rneant that at the range we "measured" our

tangential velocity (30 km) we had a 4 m s-1 deviation fiorn linearity , leading to an error in

our vorticity that was as large as 2.67 x 10-4 s i . To deal with such deviations from

hearity, originating from srnall scale or random perturbations, we decided to use a one

hour time series of the wind profiler data and apply a linear fit, as a way of approximating

the linear tangential velocity. As will be discussed later, this rnay not be a helpful or even

necessary modification.

We then applied the procedure to a stratiform snow event fiom December 14 1995.

We first applied the procedure to only one specific time (1800 U.T.), and retrieved a

vorticity that was negative (anticyclonic). This was such an unexpected result, that we

performed an error analysis on each step of the procedure, to verîQ whether this was an

actual feature of the wind field, or an artifact of the collection of uncertainties in the

procedure.

We deterrnined fkom the error analysis on the retrieval of the VAD derived linear

kinematic properties, that the uncertainty in each of the properties at a given time were an

order of magnitude less (or smaller) than the retrieved value. Also, an autocovariance on

the full two hour time senes showed that the random variations were actudy even smaller,

thus convincing us of the reiiability of the retrieved values. Thus, the negative vorticity that

we observed was not due to errors in the VAD measurements, as their contribution to the

total uncertainty was negligibly small. We then came back to the data from the wind

profiler, which we already knew from the results in Chapter 3, could make a sizable

contribution to the total error in the vorticity.

The problem with detennining the degree of non-linearity of the profiler tangentid

velocity is that, unlike the test-bed wind field, we have no linear tangential velocity to

compare it to. Our solution from Chapter 3, was to simply take an hour long time series of

the profiler winds, perform a Linear fit, and use the value from the linear fit in the retrieval.

The difficulty with this is that such a procedure requires the assumption that the wind field

does not evolve as it advects over the profiler (Frozen Turbulence assumption). While this

may be a very good assurnption over short time periods, it becomes Iess realistic over

51 longer time periods. Also, it turns out that small non-linearities would most likely be

srnoothed out by the 15 minute consensus tirne used by the profiler, and thus the linearizing

would be useless, whether the Frozen Turbulence assumption was valid or not. To see this

another way, if we assume that the system progresses at a speed of about 1 km min-'. then

a non-linearity of the order of 40 km will advect over the profüer in 10 minutes, and thus be

smoothed out by the 15 minute consensus t h e .

Since we cannot eliminate the effects of the large scale non-linearities, we found a

way to determine their importance. We did this by assuming that the wind field behaved in

such a way that the magnitude of the departures fiom Linearity of the tangential velocity

would be the same as that of the radial velocity. So that when we determined the mem

deviation frorn Linearity at a given range for the radial velocity, we had an estimate of the

rnost likely deviation from linearity we would have on a given rneasurernent of tangential

velocity. In essence, we established a way of evaluating the validity of the iinear wind

hypothesis at the profiler site, that is used to calculate the vorticity. In the case of 1800

U.T. December 14 1995, we find that the mean deviation of radial velocity at the range of

the profder is only about 0.3 m s-1, and thus we conclude that the value we use for the

tangential velocity is likely very close to the Linear tangential velocity, and that the retrieved

vorticity should be reasonably accurate (within F 4.2 x s-1). On the other hand, we

discovered that by 2000 U.T. on the same day, the deviation from linearity of the radial

velocity was about 4 rn/s, indicating that our measurement of tangential velocity would add

a significant degree of uncertainty to the retrieved vorticity. Overall, we found a way of

d e t e d n i n g the degree of non-linearîty of the wind field, and if we impose a certain

threshold on the deviation fkom lineariiy (e.g. 1 m s-l) then we cm keep the accuracy of the

retneved vorticity within a certain limit, and that the anticyclonic vorticity is indeed a real

feature of the wind field that we studied.

In conclusion, we found that the procedure can give a meaningfui result, based on

the information we have to test it with, as long as the errors are kept at a minimum. To do

this, we impose a threshold on the mean deviation from linearity of the radiai velocity . This

was confmed from the results of the application to another case (2200 - 2300 U.T.

December 9 1995). However, we also discovered that non-linear terms can be very

important, even in stratiform precipitation, even though the wind field appears to be

relatively uniform. This is significant, since it easily dernonstrates the problerns that a i se

by interpreting the VAD coefficients as the kinematic properties of the linear wind.

52 Our analysis aiso exposes a problem in the method used by Harris (1975) and

Testud et al. (1980), where they use the VAD coefficients as the kinematic properties of a mean flow, and interpret the difference between the data and the two harmonic fit as a wave

perturbation. This problern lies mainiy in the fact that the VAD coefficients are not

equivalent to the mean kinematic properties, and they may not even be close

approximations, since the non-linear terms can be of equal importance as the linear terms.

This could possibly be overcome by initially assuming that a wave function, instead of a

Taylor senes, best represents a wind field, and determining the phases and amplitudes from

a VAD analysis.

5.2 Suggestions and Future Work

If this procedure is to be implernented either operationaiiy, or as a research tool, we

suggest some further work be done in evaluating its results. An ideal evaluation would be

to do a direct cornparison with output from another source of wind measurements, such as

dual Doppler, or bistatic systems. The goals of this s o a of evaluation would be to

deterrnine whether the linear wind provides a reasonable approximation to the wind field,

and if oot, would any Taylor series be able to represent it accurately with a reasonable

number of tenns.

If the previous evaluations tum out positive results, the procedure should then be

tested on stratiform rain events. In this thesis we concentrated mainly on the snow events,

due to the overall uniforrnity of both the reflectivity and velocity fields. However, the rain

cases would provide a challenge, in that there are more fluctuations within both fields, due

to things like embedded convection. Perhaps a combination of some of the modified VAD

techniques (EVAD, CEVAD , etc.) with the profiler would be able to work around this.

Appendix A

Equations In Error Analysis

A.1 Standard Deviation

In Chapter 4, we estimated the error at each range for the VAD coefficients (Two

hamonic Fourier fits), by using the standard deviation of the data about the coefficienl. In

tems of a Gaussian fùnction, the standard deviation is the region where there is a 68% chance the measured value will be. The general equation to cdculate a standard deviation of

a given set of data is:

Where, S is the standard deviation, x is the mean value of the measured data, q, and N is

the number of data. ln Our case, since we are using a least-squares method to fit a cuve to

the data, Y, is the value of the fitted curve, rather than the mean value. This was also used

to determine the uncertainty of the linear components of the VAD derived kinematic

properties .

A.2 Autocovariance

To ob6iaïn an approximation of the random error, we perfonned an autocovariance

on the twc hour time series on 14 December 1995. The equation for the autocovariance of

a data set, x, is:

n-k

A k = i=l n - k

Where, n is the number of data, k is the time lag, and x ' , is the perturbation of the data

point fiom the mean value of the complete set,

The assumption in using the autocovariance is that the random errors would be

completely decorrelated by the fmt t h e lag, (tf+r ). This means that the value of Ak at

zero lag would be due to both the random noise, and perturbations correlated over several

time lags . In this case, we determined an upper limit to the noise by taking the ciifference

of Ag, and AI, and taking the square root to the result.

In order to determine the degree of non-linearity of the radial component of the wind

at a certain range, we calculated the mean deviation of the measured values from the iïnear

values. This measurernent will give us the effect of the larger scale non-linearities while

fdtering out the smaller scale. The equation for the mean deviation of data set x is:

Where: d is the mean deviation, xi is the measured value, xfi is the linear value, and N is

the number of data points.

Error Analysis of VAD Coefficients

8.1 Description

One of the methods described in Chapter 4 to determine the uncertainty in the VAD

coeffkients, is to perform VAD fits on adjacent ranges, take the difference between the

coefficients, square the differences, and take the mean of the squares. The idea behind this

method is that it can give an estimate of how much the rândom error affects the variation in

range of the coefficients.

We begin the description with the Fourier series equation used in the VAD fit at a

single range:

Where the terms a,, and bi , are the VAD coefficients, and /3 is the azimuth angle. We then

fit (B. 1) to adjacent ranges at different heights and elevation angles, as in Fig . (B. 1).

r

Fig, B. l Ranges and heights where VAD were performed. Numbered regions indicate

adjacent ranges where differences between coefficients were taken. Elevation angles are indicated by ar,.

We then have two sets of VAD coefficients for each of the regions (1-8) in Fig. B .1. We

then take the difference between the coefficients in each region to obtain a set of differences

56

for each coefficient. We then represent the set of differences for a given coefficient by Ai,

which can be written as:

where E~ is the contribution due to the random error, and 6, is the contribution due to the

non-linearity of the coefficient in range. The obvious dficulty is the separation of the

random error from the dependence in range. To do this, we square the differences of each

of the eight regions, and take the mean of the squares, the square root of the result gives us

the rms (root-mean-square) error of the VAD coefficients (B .3).

where N is the number of differences used (in this case eight), and ErmS is the rms error.

To show how this could separate the two sources of error, we square (B.2) to obtain the

following:

Then, if we take the mean of (B.4), and assume that the two sources of error are

uncorrelated with each other, then the product of the two wiil go to zero so that we obtain:

However, since the differences used in this calculation are d l in different locations, rhen

there is no reason to believe that they are correlated with each other and thus, the mean of 6,2 should go to zero. Then the only remaining term is the square of the random enor,

which we find by taking the square root of (B S).

We can be more certain of our assumptions if we increase the number of

independent realizations. One way of doing this is to use the same eight pairs as in Fig.

57 B -1, for several tirne steps. For example, if we did this in five minute time steps for a total

of 30 niinutes, we would have a set of 56 differences for each coefficient, instead of just

eight. When this was done for the time period of 1800 - 1830 UT. 14 December 1995, we

obtained the errors given in Table B. 1 .

Table B.1 Random Error Approximations for

Kinematic Properties

14 December 1995

Kinernatic Property Uncertainty

Divergence 7.38 x 10-6 s-i

u o 0.28 m s-1

vo 0.42 rn s-l Shearlng 1.68 x 10-5 s-1

S tre tc hing; 1.89 x 10-5 s-1

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