DEPARTMENT OF COMMERCE
FREDERICK H. MUELLER, Acting Secretary
WEATHER BUREAU
F. W. REICHELDERFER, Chief
MONTHLY WEATHER REVIEW
JAMES E. CASKEY,IR., Editor
Volume 87 Closed July 15, 1959
Number 5 Issued August 1 5 , 1 9 5 9 MAY 1959
SOME STATISTICS ON THE MAGNITUDE OF THE AGEOSTROPHIC WIND
OBTAINED FROM CONSTANT LEVEL BALLOON DATA*
J. K. ANGELL
US. Weather Bureau, Washington, D.C.
[Manuscript received February 11, 1959 revised March 11, 19591
ABSTRACT
On the basis of windspeedsand accelerations derivedfrom U.S. Navysponsored300-mb. constant level
balloon or transosonde flights made during 1953, 1955, and 1956, statistics are presented on the magnitude of the
ageostrophic wind and its variation with latitude andwindspeed. These qtatistics indicate that at 300 mb.
the average angle between wind and geostrophic wind is 11 degrees ,and the mean magnitude of the vector deviation
between wind andgeostrophicwind is 12m.see.-'. The data also show that, through the use of the geostrophic
and gradient wind approximations, half the time errors greater than29percentand11percent, respectively, are
introduced into the derived results.
1. INTRODUCTION
Previous evaluations of the magnitude of the ageo-
strophic wind have in large part been based on comparisons
of wind data with geostrophic wind data obtained from
synoptic weather maps, as illustrated by the works of
Neiburger et al. [l] and Machta [2]. A few evaluations
of the ageostrophic wind have been based on estimations
of the partial wind derivatives in the equu'tions of motion,
as illustrated by the work of Godson [3]. Dusk and Gil-
bert [4] and Neiburger and Angel1 [5] evaluated the
ageostrophic wind by determining the value of the indi-
vidual wind derivatives from constant level balloon flights
and introducing this value into the equation of motion.
Since limited constant level balloon data were available
for analysis in the latter two papers, i t seemed desirable
to carry through further analysis upon the receipt of more
constant level balloon or transosonde data. This paper
'Thiswork was performedunder Office of Naval ResearchContract
Nonr-a33(21) for the Naval Research Laboratory in connection with BuAer
Problem No.TED-NRL-MA505.
51193749-1
presents &d.tist.icson the magnitude of the agemtrophic
wind, the variation of this magnitude with latitude and
speed, and indicates the errors made in assuming the wind
to be geostrophic or gradient, all based on 635 evaluations
of the ageo'strophic wind from 28 transosondei flights made
at 300 mb. Of these 28 transosonde flights, 8 were
launched from Minneapolis, Minn. in 1953, 4 were
launched from Vernalis, Calif. in 1955, and 16were
launched from Oppama, Japan in 1956. The average
duration of these flights was 65 hours and the average
trajectory len@h was 4,700 nautical miles. The flights
were positioned at 1-hourly or 2-hourly intervads by
means of the excellent Federal Communications Commis-
sion (FCC) radio-direction-finding network. Anderson
[6] found, by comparison of FCC positions with transo-
sonde positions determined by phdtographs of tho under-
lying terrain, that within the United States two-thirds of
t.he time the FC'C positions are in error by less than 20
nautical miles. Over the Aitlantic and Pacific Oceans one
would expect this error to be larger owing to the greater
distances between tracking sta'tions and balloons.
163
164 MONTHLY W E A T H E R R E V I E W MAY 1959
2. DETERMINATIQN OF THE WIND
AND AGEOSTROPHIC WIND
With minor approximations, the vector equat.ion of
motion may be wri&kn :
d_v=.f(V-Vg) d t Xk+F
where V is the horizontal compone'nt of the velocity, V,
is the geostrophic velocity, f is the Coriolis parameter, F
is the frictional force per unit mass, and k is the unit
vertical vecltor. The term on the left hand side of (1)
represents the change of horizontal velocity following
the three-dimensional path of an air parcel and may be
expanded into three terms involving partial derivatives,
where tu is the vertical velocity a,nd V is the horizontal
differential operator. The constant level balloon data
determine the sum-value of the first two terms on the
right hand side of (2). The sum of these two terms (lo-
cal change of velocity plus horizontal advection of veloc-
ity) is hereafter called t.he partial acceleration. The
value of the third term on the right hand side of (2).
t,he veptical advection of velocity, may be approximated
along a constant level balloon trajectory by determining
the vertical motion by the adiabatic technique (vertical
mo't,ion proportional to the ch,ange of temperature follow-
ing the balloon and inversely proportional to the devia-
tion of the lapse rate from the process lapse rate along
the trajectory), and by determining the vertical wind
shear from rawin data. Based on a limited number of
computa.tions from the 1958 flights and r e p o d in reCer-
ence IS], the magnimtude of the vertical advection of veloc-
ity averages about 25 percent of the magnitude of the
partial acceleration, and 'thus is certainly not negligible.
Nevertheless, in this paper it is assumed that the pastial
acFleration data derived from the hransosondes repre-
sents a reasonable approximat.ion to the total accelera-
tion since the bulk of the. transosonde positions are over
mans where it is imprac'tical to estimate the vert,ical
advection of velocity by the techniques describ,ed above.
In order to pass from the value of the acceleration to
the value of the geostrophic deviation it is necessary,
as seen from (1) , to neglect friction. While attempts
have been made to estimate the magnitude of the fric-
tional force from a combination of radiosonde ancl transo-
sonde data, the evaludtions, particularly over the oceans,
are far too crude for inclusion here. Hence, an nddition:rrl
degree of uncertainty (in addi,tion to the neglect of the.
vertical advection of velocity) enters into the values of
t,he geostrophic deviations presented below.
With the neglect of friction and the vertical adveotion
of velocity we find, after taking the clot. product of ( 1)
with the unit vector along and normal to the wind, that,
respect.ively,
d V /d t = fVg sin i (3) and
VdBldt = - f (V- v, cos i) (4)
where d e /d t is the angular velocity of an amir parcel and a'
is the angle of indmft ('the angle between wind and
geostrophic wind). Taking into account the findings of
Neiburger and Angel1 with respect It0 the time intervals
for which the velocity and acceleration should be com-
puted in order to avoid large errors in these parameters
due to radio-direction-finding errors in the positioning of
tlhe transosondes, the speed was determined from a 4-hour
transosonde displacement ; the tangential accelerat,ion
(d V /d t ) was determined from the change in 4-hour aver-
age t,ransosonde speed in 8 hours; and the normal ac-
ce,lerRtion (Vde/dt) was determined a.s the product ofthe
mean transosonde speed during an 8-hour time interval
and the change of 4-hour average transosonde direction in
8 hours.
IJpon eliminating the geostrophic wind speed between
(3 ) and (4), the angle of indraft can be expressed in
terms of t.hree known variables ; namely, the tangential ac-
celeration, t.he normal acceleration, and the Coriolis ac-
c:>ler.ation (to). From knowledge of the angle of in-
draft, t.he geostrophic wind can be evaluated from either
(3 ) or (4). The cross-contour component of the ageo-
strophic wind (V sin i ) can then be determined as well
as the along-contour component of the ageostrophic wind
(V cos i-- V g ). I n the diagrams in this paper V - V , is
substituted for V cos i-Vg since, for the magnitudes of
tshe angles of indraft obtained at 300 mb., the two quanti-
t,ies are practically identical.
<>
3. VELOCITY AND'AGEOSTROPHIC
VELOCITY STATISTICS
I n this section and in sections 4 and 5 we show by
means of histograms and cumulakive frequency curves
the magnitudes of various velocity and ageostorophic ve-
locity parameters obtained from the transosonde flights.
Where comparison of two parameters is desirable, histo-
grams and cumulative frequency curves are superimposed
on the same diagram.
The magnitude of the 4-hour average wind speed at
300 mb. for this series of transosonde flights is indicated
in the upper diagram of figure 1. Based upon 755 calcu-
lations, the mean wind speed is 38 m. sec.-l while the mode
is 20-30 m. sec.". Ten percent of the winds exceed 67 m.
see.?. The m.aximum wind speed determined from a 4-
hour transosonde displacement is 150 m. sec.". This value
occurred on Flight 23 approximately 1,000 miles east-
southeast of Tokyo in January 1956, and is undoubtedly
one of the largest, reasonably reliable, values ofwind
speed ever recorded.
MAY 19.59 MONTHLY W E A T H E R R E V I E W 165
FIGURE 1.-Magnitude of thc 300-1nb. wiud sptled, l', (top) ;tnd its FIGURE 2.-Absolute magnitude (top) and algebraic magnitude
zonal, lu/, and meridional, 1 ~~1 , con1poIlc:nts (bottom). (bottom) of angle-of indraft, i, at 300 mb.
The magnitudes of tlle zonal and meridional wind com-
ponents are indicated in the lower diagram of figure 1.
These data demonstrate t,he well known fact that the mean
valueof the zonal wind component (here 30 n1. sec.-l) is
considerably larger than the mean value of t.he meridional
wind component (17 m. sec.?) .
The absolute magnitude of the angle of indraft, indi-
cated in Ithe top diagram of figure 2, is based on 6% calcn-
lations. The mode is indist.inct and lies in the range 0-6
degrees, while the median is 8 degrees and the mean is 11
degrees. I n 15 percent of the cases the angle exceeds 20
degrees,. The considerable difference between median and
me'ari is attributable to a relatively few large values of the
angleof indra$t of doub'tful validity. For comparison,
Godson found by substituting synoptic wind data at 700
mb. in the quahion of motion a mean value of 14 degrees
for the angle of indraft. The larger mean mean d u e
for his data would be expected since the magnitude of
the angle of indraft is inversely proportional to the wind
speed, as shown in section 6 of this paper.
In the lower diagram of figure 2 is shown the algebraic
magnitude of the angle of indraft, considered positive
when the flow is toward low pressure and negative when
the flow is toward high pressure. It is evident from the
cumulative frequency curves in this diagram that in the
rrwm the angle of indraft is of grea,ter magnitude when
the flow is toward high pressure than when the flow is
toward low pressure. It is believed that this tendency is.
at least pnrtially connected n7it.h the occasionalloss' of
geostrophic control in regions of inertial instability in the
deceleration region downstream from sha8rply cuImd.;
ridges.
Tlle nlagnitude of the vect.or geostrophic deviation.
I V-V, 1 is indicated in the upper diagram of figure 3.
Tlle mode is 2 4 m. sec.-l, the n1edia.n is 10 m. sec-l and
tile mean is 12 m. sec.". I n 15 percent of the cases the
tleviation is greater than 20m.sec.-l. For comparison,
Jfachta found an average ~7alue of 13 m. sec.-l f o r the vector
geostrophic deviation at 300 mb. during the winter
nlonths. His chta were obhined by comparing the wind
and pressure gradient (geostrophic wind) on analyzed
synoptic maps.
The absolute magnitudes of the cross-contour (V sin i)
and along-contour (V - T7g) components of the vector
geostrophic deviation are given in the lower diagram of
figure 3. I n the case of the cross-contour component, the-'
.I /,
166 M O N T H L Y W E MAY 1959
I00
90
t eo
p'
:: 50
C
; 70
- 60 a 0
0
b 40
a
330
p 20
a
LL
IO
' 0 2ma" 4 6 . 8 IO 12 14 16 ma' 18 20- 20
FIGURE 3.-Magnitude of the 300-mb. vector gcostrophic deviation,
\V-V,\, (top) and its cross-contour, J V sin il, and along-cont,our,
IV- V,I, components (bottom).
FIGURE 4.-Algebraic magnitude of the 300-mb. cross-contour flow,
V sin i, (top) and along-contour component of the vector geo-
strophic deviation, V - V,, (bottom).
mode, the median, and the mean are respectively 0-4 m.
set.", 4 m. sec.-l, and 6 m. sec.-l whereas for the along-
contour component they are, respectively, 0 4 , m. se~.-~,
7 m. sec.-l, and 9 m. see?. In the case of the cross-con-
tour component, 10 percent of the values exceed 13 m. sec."
whereas in the case of the along-contomur component, 10
percent of the values exceed 19 m. sec.-l. For comparison,
Neiburger et al. found at 7'00 mb. a mean value of 3 m.
sec.-l for the along-contour component of the vector
geostrophic deviation. This smaller value would be ex-
pected since the magnitude of this component is a func-
tion of the wind speed, as will be shown below.
The algebraic magnitude of 'the cross-contour flow is
shown in $he upper diagram of figure 4. I n agreement
with the findings for the angle of indraft, the cumulative
frequency curves indicate that the magnitude of the cross-
contour flow is greater when the flow is toward high pres-
sure than when the flow is toward low pressure.
The algebmic magnimtudeof the along-contour com-
ponent of the vector geostrophic deviation is given in the
lower diagram of figure 4. I n this diagram the difTer-
ence between the cumulative frequency curves is striking
indeed, with the difference between the wind speed and
geostrophic wind speed being much larger, in &he mean,
when the flow is cyclonic than when the flow is arlti-
cyclonic. The respective median values are -8 m.sec.-l
and 6 m. see-l. It may be that this asymmetry is due to
the existence of, a type of upper bound on the degree to
which the wind may be supergeostrophic on ridges (twice
the geostrophic value), whereas there is no such bound in
troughs.
4. ERRORS RESULTING FROM THE USE OF
GEOSTROPHIC AND GRADIENT WIND EQUATIONS
The m a g n i t u d e of the partial acceleration
(W /a t +V -V V ) in comparison with the Coriolis ac-
celeration (fv) is of interest since the geostrophic as-
sumption states t,lmt the paptial acceleration is negligible
in comparison wikh the Coriolis accelera,tion. The mag-
nitude of the ra'tio of partial acceleration and Coriolis
accelera.tion is given in the upper diagram of figure 5.
The mode of the ratio is roughly 0.10-0.20 while the
median is 0.29 and in 10 percen,t of the cases the ratio
exceeds 0.83. I n ot,her words these data suggest thlt,
MAY 1959 M O N T H L Y W E A T H E R 167
90 635 CASES
- ; 80
:70
I
: 60
9 50
40
30
F
a
a
; 20
b
IO
' 0 .IO . .20 .30 .40 .x) .60 .70
~ ~
.SO -00 C o o w 100
FIGURE 5.-Ratio of p a r t i a l a n d Coriolis acceleration,
~~~.v v [, a t 300 mb. (top) and the ratio of the zonal,
100
90
c eo - L
%70
s 50
-
0
t 60
- - * 40
30 a
U. : 20
IO
0 0 .x) -40 ,60 .EO 100 1.20 1.40 180 WO 200'290
FICVRE Ci.--Ratio of tangcnt,ial and normal components of partial
]au'a'$yvuI] and meridional, l a u ' a t ~v ' v u l , -~ components (bot- FIGURE 7.-Algebraic ratio of the angular velocity of the wind and
the angular velocity of the earth about the local vertical (dO/dt)/st,
tom). a t 300 mb.
through the introduction of the geostrophic approxima-
tion, half the time a.n error exceeding 29 percent is intro-
duced into the derived results.
The magnitudes of the ratios of zonal and meridional
components of partial acceleration and Coriolis accelera-
tion are given in the lower diagram of figure 5. These
data suggest that the neglect of the acceleration in the
zonal component of the equation of motion results in an
untenable approximation, since for this component the
mode of the ratio is 0.30-0.40 and the median is 0.41. I n
the case of the meridional component of the equation of
motion, however, the ratio of partial acceleration and
Coriolis acceleratio8n has a modeof only 0-0.10 and a
median value of only 0.25. I n this diagram the relatively
large percentage of cases where the ratio exceeds 1.00 is
due mainly to the existence of "cusp" points in the tra-
jectories. At such points the velocity component may go
toward zero at a time when the component of the partial
acceleration is far from zero.
The magnitude of the ratio of tangential and normal
components of partial acceleration is of interest since the
gradient wind approximation assumes the tangential com-
ponent to be negligible in comparison with the normal
component. It is seen from figure 6 that the mode of this
ratio is 0-0.20 while the median is 0.67. I n 38 percent
of the cases the tangential component of the partial accel-
eration is larger than the normal component. From con-
sideration of the median ratio of the tangential and nor-
mal components of partial acceleration and the median
ratio of partial acceleration and Coriolis acceleration we
estimate that through the use of the gradient wind ap-
proximation half the time an error exceeding 11 percent is
introduced into the derived results.
5. FREQUENCY O F OCCURRENCE OF "ABNORMAL"
FLOW
I n figure 7 is indicated the ratio of the algebraic angular
velocity of the air parcel or transosonde and the angular
velocity of the earth about the local vertical. It is seen that
in 27 percent of the cases the angular velocity of the air
parcel exceeds that of the earth about the local vertical
168 MONTHLY WEATHER REVIEW MAY 1959
1
\
\-"
\
\
\ &
\
\
\
-\\
\
\
10 IS 20
1
I
-4
20
.90
I I n a"
- " I
I
70 - I " I
I
I
I 30 - - I
I
I
30 - 7'
I
f
I IO - I r=.25 .
I 0 mr-1 5 10 15 20
FIGURE 8.-Variation with latitude of angle of indraft, l i l , (lower
left), and of the vector geostrophic deviation, /V-V,J, (upper
left) and its along-contour, 1 V- V81, (upper right) and cross-
contour, itr sin il, (lower right) components at 300 mb.
FIGURE 9.-Variation wilh speed of angle of indraft, l i l , (lower left),
and of the vector geostrophic deviation, IV-V,I, (upper left) and
its along-contour, I Ti- V l (, (upper right) and cross-contour,
11' sin il, (lower right) components at 300 mb.
when the angular velocity is positive (cyclonic flow), but
that this criterion is satisfied in only 9 percent of the 398
cases when the angular velocity of the air parcel is nega-
tive (anticyclonic flow). Thus in 5 percent of the total
635 ca.ses the flow allegedly possesse's anticyclonic rotation
in space. Flow which satisfies this criterion is called "ab-
normal" or "anomalous" flow by Holmboe et al. [?] and
Gustafson [8].
6. VARIATION OF AGEOSTROPHIC PARAMETERS
WITH LATITUDE AND SPEED
In this section and in section 7 the data are presented in
the form of group means, regression lines, and correlation
coefficients (figs. 8-11). On either side of the group means,
lines have been drawn extending a distance equal t,o two
standard errors from the mean. If the data are randomly
drawn from a normal population there is only a 5 percent
chance that the true group mean lies outside the extent of
these lines. An estimate of significance of variability at
the 5 percent level can then be obtained by noting whether
a straight vertical line can be drawn so as to intersect all
these lines extending either side of the group mean. The
regression lines and correlation c,oefficients serve to yield
an indication of the overall trend but it is not meant to
imply thereby that these trends are actually linear in
nature. It is also possible to determine the correlation
coefficient which would indicate significance at the 5 per-
cent level. Based on the z transformation presented in
Hoe1 [ 93, it is found that for the number of cases available
here a correlation greater than "0.08 is significant at the
5 percent level.
With regard to the variation of the parameters with
respect. to latitude it must be emphasized t,hat the resu1,ts
are biased since the transosondes sample chiefly cyclonic
flow patterns at southerly lati,tudes and anticyclonic flow
patterns St northerly latitudes. If transosondes werere-
leased from stations along a given meridian but at differ-
ent latitudes, it is probabsle that different mean values
would bs found from those presented below.
Figure 8 gives the variation with latitude of the ab-
solute magnitude of the angle of indraft and the vector
geostrophic devi'ation and iits along-contour and cross-con-
tour components. All four parameters decrease in mag-
nitude with increasing latitude, with the smallest decrease
being found for V sin i (r= -0.12). The group means
show that ithe variability is ragged. The magnitude of
MAY 1%9 MONTHLY WEATHER REVIEW 169
the vector geostrophic deviation varies from a mean of
10 m. sec." at northerly latitudes to 14 m. sec.-l at south- 60-
-
L
art latitude 35O. The absolute magnitude of the angle of g50-
I \ erly latitudes with most of the change occurring abruptly
I I 60- -
\ 1
1 - 5 0 - I
.indraft varies from a mean value of about 8 degrees at 0: \
\ I
Q I
northerly latitudes to a mean value of 13 degrees at south- I \ I
erly latitudes, while the absolute magnitude of 'v sin i -\
varies from a mean value of 4-5 m. sec.-l in northerly \
latitudes to a mean value of 7-8 m.sec.-l in southerly 530- \A \ - -30 - 2 I -
latitudes. \ I \ I -I
magnitude of the angle of indraft, and the vector geo- r s -.32
strophic deviation and its along-contour and cross-contour
components. While the angle of indraft shows a signifi- FIGURE lO.-Variation with latitude of ratio of partial and Coriolis
cant decrease in magnitude with increase in speed acceleration, ~f t v ' v v l , (left) and ratio of tangential and
(T=- 0.27) the ageostrophic parameters show significant
increases in magnitude with increase in speed. It is of normal components of partial acceleration, 1-1, (right) at 300
I
\
-
- 40-
\ " I I
Figure 9 gives the variation with speed of the absolute 20- "20- - \I -1 I r
- -.03
0 ' 2 0 .40 .60 -80 20 -40 .60 -80 lo0
d V/dt
interest to note that the ma,gnitude of the cross-contour mb.
flow increases with increase in speed despite the counter-
acting tendency of the angle of indraft. The vector
geost.rophic deviation varies in average magnitude from
8 m. sec." at low speeds to 18 m.sec.-* at high speeds. The 90-
mr' \ \ I absolu'te magnitude of the angle of indraft varies from 17 mc-'
" 90-
'
\\-!-
/I degrees at low speeds to 7 degrees at high speeds, while ,o-
\ I
!-
the absolute magnitude of the cross-contour flow varies \
from 4 m. w .-l a t low speeds to 11 m. sec." at high speeds. \ I
I
\
- 7 0 - I
I
I
\
50 5 0 - - 4 - - 0
\ w \ \
7. VARIATION WITH LATITUDE AND SPEED k!
GEOSTROPHIC AND GRADIENT WIND EQUATIOFS
OF ERRORS RESULTING FROM THE USE O F
\
The two diagrams in figure 10 give the variations with
I
ao- -\
I \ - 30-
\
\ \
IO 10- - - \. -
r.7.38 \I 1 -T r * -.02
latitude of the ra.tio of partial acceleration and Coriolis -60 -60
acceleration and the ratio of tangential and normal com-
ponents of partial acceleration. As would be expected, the
ratioof partial accelerat.ion and Coriolis accelera,tion
varies greatly with la,titude ( Y=- 0.32) , ranging from a
mean value of 0.30 at northerly latitudes to a mean value
of 0.70 at southerly latitudes. The group means indicate
that the ra'tio is almost constant north oflatistude 40",
with a rapid, nearly linear increase in the value of the
ratio as one progresses southward from latitude 40". The
ratioof the tangential and normal components of the
partial acceleration shows little variation with latitude
("0.03). The group means a,lso indicate no significant
variations except at latitude 60° where the large values of
the normal components of partial acceleration on the crests
of traj&ries overbalance the tangent,ial components,
The two diagrams in figure 11 give the variations with
speedof the ratio of partial acceleration and Coriolis ac-
celeration and the ratio of tangential and normal com-
ponent~ of partial accelerat'ion. As would be expected,
the ratio of partial acceleration and Coriolis acceleration
r
shows a significant decrease in magnitude with increase
in speed ("0.38) with the mean value of the ratio
10
FIGURE 11.-Variation with speed of ratio of partial and Coriolis
acceleration, ~2 f i y v ' v v ~, (left) and ratio of tangential
and normal components of partial acceleration, Iyg'l, - (right) at
300 mb.
varying from 0.66 at low speeds to 0.20 at high speeds.
The ratio of tangential and normal components of partial
acceleration shows no significant variation with sped
(r = -0.02) , the group means at all speeds resting close to
0.70.
8. CONCLUSION
Statistics based on 635 evaluations from 300-mb. con-
stant level blalloon data indicate that through the use of
the geostrophic wind approximation half the time an er-
ror greaiter than 29 percent will be introduced into the
derived results, whereas through the use of the gradient
wind approximation half the time an error greater than 11
percent will be introduced into the derived results. On
it percentage basis both khe geostrophic and gradient wind
170 MONTHLY WEATHER REVIEW MAY 1959
approximations are about twice as bad at latitude 20” as
&t latitude 60” and about twice as bad at speeds of 10 m.
sec.-l as at speeds of 50 m. set.?. The magnitudes of
angles of indra-€ t and sgeos;trophic parameters are in fair
agreement with results obtained by other techniques, with
the transosonde data yielding a mean angle of indraft of
11 degrees, a mean cross-contour flow of 6 m. sec-I, a mean
deviation between wind and geostrophic wind speed of
9 m. sec.-l, and a mean vector geostrophic deviation of
12 m. sec.-l.
REFERENCES
1. M. Seiburger et al., “On the Computation of Wind from Pres-
sure Data,” Journal of Meteorology, vol. 5, So. 3, June 1948,
2. L. Machta, A Study of the Observed Deviations from the
Geostrophic Wind, ScD thesis, Dept. of Meteorology, Mas-
sachusetts Institute of Technology, 1948. (Unpublished).
pp. 87-92.
3. W. 1,. Godson, “A Study of the Deviations of Wind Speeds and
Directions from Geostrophic Values,” Quarterly Journal of
the Royal Meteorological Society, 001. 76, No. 327, Jan. 1950,
pp. 1-15.
4. C. L. Durst and G. H. Gilbert, “Constant-Height Balloons”
Calculation of Geostrophic Departures,” Quarterly Journal
of the Royal Meteorological Society, vol. 76, So. 327, Jan.
5. X. Neiburger and J. K. Angell, “Meteorological Applications of
Constant-Pressure Balloon Trajectories,” Journal of Veteor-
ology, vol. 13, So. 2, Apr. 1956, pp. 166194.
0 . A . D. Anderson, “A Study of the Accuracy of Winds Derived
from Transnsonde Data,” (Memo. Rep. 498), Naval Research
Laboratory, Washington, D.C., 1955,lO pp.
7. 5 . Holmboe, G. E. Porsythe, and W. Gustin, Dynamic Meteor-
ology, New York, John Wiley and Sons, 1945,378 pp.
8. A. F. Gustafson, “On Anomalous Winds in the Free Atmos-
phere,” Bulletin of the American Meternological Society, vol.
34, So. 5, May 1953, pp. 19&201.
9. P. G. Hoel, Introdnction to Mathenzatical Statistics, New York,
John Wiley and Sons, 1947,258 pp.
1950, Pp. 75-86.