Calhoun: The NPS Institutional Archive
Theses and Dissertations Thesis Collection
1949
Local change of mean virtual temperature for the
1000-700 mb layer
Somervell, Willis Lewis, Jr.
Monterey, California: U.S. Naval Postgraduate School
http://hdl.handle.net/10945/36329
DUDLEY KNOX LIBRARYNAVAL POSTGRADUATE SCHOOL
_AfONTERfY, CAUfORlUA 83943-5002
LOCAL CHANGE OF MEAN VIRTUAL
TEMPERATURE FOR THE 1000-100 MB LAYER
w. L. Somervell, Jr.
Libro:ryU. S. Naval Postgraduate Sch~Annapolis, Md.
LOCAL CHANGE OF MEAN VIRTUALTEMPERATURE FOR TIlE 1000-700 ME LAYER
byWillis Levds Somervell, Jr.
Lieutenant, junior grade, United states travy
Submitted in partial fulfillmentof the requirementsfor the degree of
MASTER OF SCIENCE IN AEROLOGY
United States Naval Postgraduate SchoolMonterey, California
1~4~
This work is aooepted as fulfillingthe thesis requirements for the degree of
MASTER OF SCIENCE IN AEROLOGY
trom theUnited states Naval Postgraduate School
ChairmanDepartment ot Aerology
Approved:
•Academic Dean
(1)
PREFACE
This paper presents the results of a study of 12 hour local
changes in mean virtual temperature of the 1000 - 700 mb layer.
The objectives werej first, to obtain easily applicable rules for
forecasting local changes in mean virtual temperature under various
synoptic conditions with a forecast mean 5000 ft. wind and the
existing mean virtual temperature gradient, secondly, to determine
quantitatively the effect of such non-advective components as diurnal
heating or cooling and exchange of latent heat of condensation on the
resulting mean virtual temperature change.
Undertaken as the thesis requirement for the degree of Master of
Science in Aerology, this paper was prepared at the U. S. Naval Post
graduate School, Monterey, California during the academic year 1948-1949.
The author is indebted to Associate Professor George J. Raltiner of
the Aerology Department for his valuable assistance and constructive
criticisms during the investigation. He also wishes to acknowledge the
assistance rendered by Associate Professor A. Boyd Mewborn of the
Mathematics Department.
(11)
TABLE OF CONTENTS
CERTIFICATE OF APPROVAL
PREFACE
TABLE OF CONTENTS
LIST OF ILLUSTRATIONS
TABLE OF SYMBOLS
CHAPTER
I. INTRODUCTION
II. TECHNIQUE OF INVESTIGATION
III. RESULTS AND CONCLUSIONS
BIBLIOGRAPHY
(iii)
Page
i
ii
iii
iv
vi
1
10
17
)4
LIST OF ILLUSTRATIONS
Figure 1. Scatter Diagram and Regression Lineof J/oo on "00'
(a) Scatter Diagram and Regression Lineof 110/ on "0/ •
(b) Scatter Diagram and Regression Lineof !o%. on "01'
Figure 2. Scatter Diagram and Regression Lineof 1/06 on X oo '
(a) Scatter Diagram and Regression Lineof Y/o on x,o'
(b) Scatter Diagram and Regression Lineof J/10 on xu'
Figure 3. Scatter Diagram and Regression Lineof !l11I on "/'O •
(a) Scatter Diagram and Regression Lineof y" on Xu •
(b) Scatter Diagram and Regression Lineof 11 '::z. on x,%..
Figure 4. Scatter Diagram and Regression Lineof 1/;1.0 onx~.
(a) Scatter Diagram and Regression Lineof flu on x;u.
(b) Scatter Diagram and Regression Lineof !/:z.:t 0n X ::z.1 ..
Figure 5. Scatter Diagram and Regression Lineof !tOI on x.,.
(a) Scatter Diagram and Regression Lineof .!I// on)C// •
(b) Scatter Diagram and Regression Lineof Y;u on X;1.I'
(iv)
Page
22
24
25
26
Figure 6. Scatter Diagram and Regression Lineof !fO:1. on X 04 '
(a) Scatter Diagram and Regression Lineof !f'4 on X'4 •
(b) Scatter Diagram and Regression Lineof 9:u. on x.u .. 21
Table 1.
Table 2.
Table }.
Table 4.
number of Cases, Arithmetic :Means and StandardDeviations about the 14ean of Local b"VT Changeand of Horizontal Advective Componen1i
Arithmetic Means of the 14agnitude and Extremesof Local MVT Change
Total Correlation Coefficients, ImprovementFactors, Standard Errors of Estimate, RegressionCoefficients, and Regression Equation Constants
Probabilities of No Significant Difference Amongthe Correlation Coefficients
(v)
18
19
20
21
TABLE OF SYMBOLS
EST Eastern Standard Time
f hi Improvement factor in percent of category hi
Fz Sum of the vertical components of ~ngential stresses.frictional forces. and electromagnetio forces
g Acceleration of gravity
h. j Subscripts referring to diurnal segregation
o - no diurnal segregation1 - oases ocourring in the 1100-2,00 period (day to night)2 - cases ocourring in the 2,00-1100 period (night to day)
1, k Subscripts referring to precipitation segregation
o - no segregation as to precipitation1 - cases in which a trace or more of precipitation occurred2 - cases without precipitation
Regression equation constant of category hi in 0C
lmT
+maxhi
mb
,p
Po
hi Poo
Mean virtual temperature
Largest positive local MVT change occurring in category hi in °c
Largest negative local MVT change occurring in oategory hi in °c
Millibar
Number of pairs of observation occurring in category hi
Pressure
Pressure at the bottom of a layer
Probability of no significant difference between correlationcoefficient hi and the "parent" correlation coefficientdetermined from all cases
Probability of no significant difference between correlationcoefficient hi and "parent" correlation coefficient jk
(vi)
II
hiPjk
R
t
y*
u
v
w.
IV
y
f
Probability of no significant difference be~Neen correlationcoefficient hi and correlation coefficient jk
Total correlation coefficient for category hi
Gas constant for dry air
Regression coefficient for category hi
Standard error of estimate for category hi in 00
Time
~ean virtual temperature in 00
East-West. X, component of wind velocity
North-South. Y. component of "rind velocity
Vertical, Z, component of wind velocity
Wind velocity
Horizontal wind velocity
Horizontal advective component of M~ change in 00
Arithmetic mean of horizontal advective component in 00
Local MVT change in 00
Arithmetic mean of local :kl1JT change in 00
Arithmetic mean of the magnitude of local 1m change iD. 00
Difference in height, "thickness", between two pressure surfaces
Vector del operator
Density
Angular velocity of the earth
Latitude
(vii)
standard deviation about the arithmetic mean of local M~change of category hi in °c
Standard deviation about the arithmetic mean of the horizontaladvective component of category hi in °c
(vi~i)
CHAPTER I
nrTRODUCTION
The advent of constant pressure charts in the United states,
with their attendant differential analysis, greatly facilitated the
use of mean virtual temperature, or "thickness", in Applied Aerology.
The relationship between "thickness" and mean virtual temperature had
been known for some time, and was used in upper air analysis employing
constant level charts. Its u~e was possible there, however, only after
additional calculations. On constant pressure charts, as shown by
Sverre Petterssen (17), the applioation of "thickness" as an analysis
and forecasting tool is straightforvtard.
In this investigation an attempt has been made to obtain rules
for f'orecasting 12 hour local changes in mean virtual temperature of'
the 1000 - 700 mb layer under various synoptio conditions from the ex
isting distribution and a forecast mean horizontal 5000 ft. wind. l,lVT
change was used rather than "thiokness" ohange because of the greater
probability of its employment as an explicit variate in other oircum
stances. MVT used in this paper is that determined from the "thickness"
of the 1000 - 700 rob layer. It is pertinent to note tha.t the "thickness"
ref'erred to is not the true difference in height between the two surfaces.
Neither, f'or that matter, are the heights of the 1000 and 700 rob surf'aoes
true heights above mean sea level. An explanation is found in a review
of the theory.
(I)
Assuming that the force of gravity may be expressed by a constant
scalar potential, and choosing an orthogonal coordinate system with the Z
axis along the vertical, the X axis directed to the east, the Y axis
directed to the north, and using the notations in the Table of Symbols,
the vertical term of the equations of motion may be vr.ritten
(1) dw= OW + IY. \7w =_.1. 2E -j+ 2/,{(J) COS if + ~dt ot - f a;;?:
Yfuen the tangential stresses, frictional and electromagnetic forces,
and the corioles term are neglected as compared with gravity, (1)
reduces to
If it is assumed that vertical accelerations and vertical velocities
are negligible, (2) becomes
Assuming that local variation of pressure in the vertical may be re-
placed by the individual variation, utilizing the equation of state,
and integrating, the relation between "thickness" and mean virtual
temperature appears as an integrated form of the Hydrostatic equation:
(2)
The vertical distance between two particular constant pressure
surfaces, or Itthicknesstl, i~ thus expressed as a function of a single
variate, mean virtual temperature.
In a discussion of large scale circulations, the assumption that
vertical acceleration is negligible compared with gravity may be warranted,
but in local convective currents it has been found that vertical acceler-
ation and vertical velocity are important factors. The increasing use of
radar together with the radiosonde instrument for obtaining true height
and thickness as well as 1I.'VT require that the artificiality of "thickness lt
as employed be kept in mind.
The local change of MVT, when expanded into its individual and ad-
vective terms, may be written
(5 )--H -jI-
7JT : dT -/Y. r;J =r*at di -
The advective term, when expanded into its horizontal and vertical com-
ponents is
-* -it 07*(6) 1f/.(7 T :; JJ<·17 T + w-
- -H oz:
Thermodynamic considerations indicate that the important components
included in the individual term are: (a) loss or gain of heat. by radiation
diurnal variation; (b) liberation of latent heat by condensation or its
loss by evaporation; (c) turbulent flux of heat to or from the underlying
surface.
Occasions in which anyone of these individual or advective com-
ponents is the sole cause of the local change of MVT are extremely rare,
and certainly not determinable. In middle and high latitudes experience
has shol~ that horizontal advection is ge~erally one of the predominate
components causing local change of MVT. Situations may occur frequently,
however, in which the horizontal advective component is negligible. If
conditions under which the effect of the remaining components are lmcrwn..,
or are negligible) occurj
and the horizontal advective component can be
determined, it should be possible to forecast the local change of MVT.
By expressing the individual change in another way, the concept of
considering the local change as the result of horizontal advection
modified by the vertical advective and individual components is clarified.
(7) aT* ~(fT* 07 "It) -*- :: W - -- -V·f7Tat d%!: O~ -H
Evaluation of horizontal advection occurring during a period depends
upon knowledge of the trajectory, determination of which seemed feasible
by three different methods: (a) Petterssen's (18) system for deter-
mination of trajectories by suooessive approximation from the isohypses
and from the displaoement of the isohypses during the 12 hour period;
(b) determination of trajeotories by assuming oonservation of absolute
vortioity as suggested by Rossby (I?), (11); (0) assuming that isohypses
are streamlines and the flow pattern changes negligibly with time so that
the isohypses can be considered the trajectories along which MVT is dis-
placed. These three methods were those applied by B. Haurwitz and
(4 )
collabora~ors (10) in a study of horizontal advection of density and
of mean density between constant level surfaces.
If MVT a~ a point in an infini~esim.al layer was conservative to
all processes other than horizontal advec~ion. obtaining the trajec~
ories of ~he air particles involved for a 12 hour period would require
no small amount of data. time. and labor. Choosing the proper tra
jectoryto represent the horizontal advection of l~ occurring in the
entire 1000- 700 mb layer introduoes further difficulties. It would
seem reasonable. on the average. ~o use the trajectory obtained from
the 850 mb surface as representative of horizontal advection in the
layer.
The average aerological unit. however. does no~ analyze this ohar~;
and few of those that do have the time and personnel ~o determine tra
jectories of air particles. The 5000 ft. level is normally very close
to the 850 mb surface. and most units which analyze the 700 mb chart
also plot 5000 ft. winds on some auxiliary chart. No extra labor is
introduced if the 5000 ft. wind may be used in place of trajectories.
In many cases. the flow pat~ern changes very little over periods as
short as 12 hours and streamlines. isohypses, and trajectories are
nearly the same. FamiliarUy with flow patterns and ~rajec~ories 'V1111
enable the determination of cases where the wind velocity indicates the
advection occurring. This procedure involves foreoasting a mean 5000 ft.
wind for the 12 hour period which is also representative of horizontal
advection; that is, situations wherein the existing flow pattern changes
negligibly during the period. Proceding upwind from the station a dis
tance equal to the wind speed times the 12 hour in~erva1 should give the
(5 )
MVT which '1ill be advected to the station.
Since MVT is not conservative" even if a representative trajectory
for the layer is obtained" it is to be expected that the IJVT at the end
of the trajectory may be considerably modified. Determination of hori-'
zontal advection was accomplished by measurement of lWT at the beginning
of the trajectory and subtracting from this value the 1~ at the station
at which the local change is to be measured. The local change resulting
at the station represents horizontal advection as modified by the indi
vidual and vertical advective components.
The effect of diurnal variation will be to increase MVT during the
day and to decrease it at night. The effect of condensation will be the
liberation of latent heat and an increase of' MVT; that of evaporation"
the reverse. As pointed out by M. H. Halstead (7)" an additional con
tribution of condensation is the precipitation produced falling through
the bottom of the layer and causing a decrease in the height of the
1000 mb surface" thus increasiilg the "thickness" or MVT. Turbulent f'lilx
of heat depends on stability and should be greater when the flux is f'rom
a warmer surface which tends to increase the k~ while decreasing the
stability. vYhen the flux is to a colder surface MVT is decreased" but
owing to increasing stability" this decrease should be less. The vertical
advective component as considered in this paper was restricted to vertical
motions which are of an adiabatic nature (4)" (5)" (15)" (16). Quali
tatively" with this restriction, subsiding motion will result in an in
crease in lWT; upward motion a decrease in }~.
(6 )
The horizontal advective component is usually opposed by one or
more of the other components. Thus. cold advection is, for the most
part, caused by northerly winds which are usually accompanied by hori
zontal divergence and subsidence tending to increase 1~. There may,
however, be convergence rather t~an divergence even with these northerly
winds. Cold advection will also be accompanied by a turbulent flux of
heat from the underlying surface, further tending to reduce the decrease
of h~, but also tending to decrease the stability and MVT by the con
vective activity induced.
Warm advection, being caused principally by southerly winds which
are usually accompanied by horizontal convergence, is opposed by a de
crease of MVT caused by the upward motion. Again, southerly winds may
be accompanied by divergence rather than convergence. Turbulent flux
of heat to the underlying surface will also decrease the effect of war.m
advection.
During the day, insolation will tend to increase MVT at any point,
regardless of the type of advection. Outgoing nocturnal radiation will
tend to decrease MVT, irrespective of the advection occurring.
The effect of condensation and evaporation vdll be to increase and
decrease, respectively, MVT, and will occur with both war.m and cold ad
vection. To qualitatively anticipate what will be the effect of this
component. it would be expected that with war.m advection, due to cooling
of the air particles being advected, oondensation will predominate. With
cold advection warming of the air particles should cause evaporation to
exceed condensation. Condensation and evaporation, then, should assist
horizontal advection in causing local change of MVT.
(7 )
As previously noted, use of occasions on which any single com
ponent is the sole cause to determine the resulting looal ohange, or
even to determine an average value of this component, is impracticable •
. Inasmuch as the average blfVT at various places on the earth remains
appreoiably oonstant over long periods it must be assumed that the average
local change of k~ is zero. Following the same line of reasoning, with
the exception of diurnal variation owing to the earth's radiation un
balance, it must be assumed that the average effect of the advective and
other individual components in produoing a local ohange of MVT at a
particular point over a suffioiently long period is zero.
A procedure, then, to determine the effect of any single com-
ponent or pair of them, would seem to be a statistioal investigation of
cases where all components other than the one, or pair, which it is de
sired to calculate vary in any manner. After compiling a large number or
oases the average or these other components should approach zero, leaving
only the desired component. From the correlation thus obtained a re
gression equation can be formed which may be utilized in future situations
where the same conditions exist to predict the looal change of k~.
In this paper an attempt was made to determine only the effect of
diurnal variation and the effect of condensation with precipitation or
evaporation. These components are those which could be most easily in
vestigated under conditions vmere it might be expected that all other
components would approach zero.
The data used for this investigation were obtained from the surface,
radiosonde, and pibal observations of November 1947. This month was con
sidered to be a period in which all components would be adequately repre
sented, and one which would be indicative of fall, winter, and spring
(8 )
conditions. lNT gradients were strong enough that both war.m and cold
advection could be determined to a reasonable degree o£ accuracy. Out
breaks of polar continental air were well developed. and invasions of
maritime tropical air were still sufficiently intense to be represen
tative. Large high pressure cells in which subsidence should be well
developed were present, as were regions of strong southerly winds with
overcast skies and non-frontal precipitation, suggesting convergence.
Insolation. while less than outgoing nocturnal radiation, should be
suffioiently intense for a determination of diurnal variation for the
season. Areas of wide-spread precipitation were frequent, as were
regions with clear skies. Turbulent flux of heat to and from the under
lying surface with the existing MVT gradients would be perceptible.
1~ny papers have been written by various authors on applications
of mean virtual temperature, or upon other approaches to its prognosti
cation, and, since these features will not be discussed here, the reader
is referred to (1), (4), (6), (8), (9), (12), (15). (17). (20) for ad
ditional information.
CHAPTER II
TECHliIQUE OF IlfVESTlGATION
As Vias mentioned previously, the MVT defined for this paper is
that determined by the "thickness" of the 1000 - 700 mb layer •. The
12 hour periods for which local changes in ~~ were investigated
were fixed by the times of radiosonde observations, 1100 EST and
2300 EST~
Boundaries of the region over which the investigation was con-
ducted were the Rocky MOuntains to the west, the Atlantic ooast to the
east, the Gulf coast to the south, and the Canadian border to the north.
The western boundary was necessary ovnng to the elevation of the terrain
and resulting fictitious nature of the 1000 mb surface and :MVT of the
1000 - 700 mb layer. The eastern and southern boundaries were oaused
by the lack of surface and radiosonde reports over the Atlantic Ocean
and Gulf of Mexico. Imposition of these boundaries, unfortunately,
excluded the possibility of measuring turbulent transfer of heat to the
air from the ocean surfaoe, whioh at times is oonsiderable (2). The
northern boundary was introduced for reasons which are not meteorlogical.
The height of the 700 mb surface was obtained from analyzed 1100 EST
and 2}00 EST synoptic maps, interpolating .vhere accurately possible if
radiosonde observations were not available. The height of the 1000 mb
surface was determined by subtracting the 3 hour pres sure tendency from
the 1330 EST or 0130 EST sea level pressure and then utilizing the~rt1J;,~
pressure-temperature nomogram developed by N.-1!~~on (8). No att~mpt
was made to compensate for the pressure change which occurred in the re-
maining half hour. Neither was the surface temperature corrected for
(10)
changes which occurred during the three and one-half hours between
the time of radiosonde and surface observations. The error intro
duced by neglecting these factors is small, and negligible when com
pared to errors introduced in estimating trajectories.
The thickness obtained by subtraoting the height of the 1000 mb
surface from that of the 700 mb surfaoe was then converted to MVT by
using tables of height evaluation by equivalent isothermal layers (8).
The local change of 1WT occurring at any station in a 12 hour period
was determined by simply subtracting the JiIVT at the beginning of the
period from that at the end.
Evaluation of the horizontal advective oomponent was accomplished
by choosing those stations for investigation of the local change at
which the 5000 ft. wind was a good indication of the horizontal ad
vection. Some experience in determining trajectories is required to
.estimate when this is so. Immediately at and in advance of a irough
line, for eXEpple, even though south or southwes.t winds indicating warm
advection are found, the trajectory would show that the air particles
had just rounded the trough line and the source was northerly. Simi
larly, innnediately at and in advance of a ridge line, with northerly
winds indicating cold advection, a more representative trajectory would
be from the west or even southwest. These two cases were not included
in the data. An inspection of the 1000 mb and 700 mb isohypses to
estimate the 850 mb. isohypses with the aid of the 5000 ft. wind was
always made. The 5000 ft. winds used were the standard 0500, 1100,
1700 and 2300 EST pibal observations. These reports gave a wind ob
servation at the beginning, middle, and end of the 12 hour periods.
(11)
No change in the type of advection was permitted during the period.
Frontal passages and cases in which the wind direction during the
period changed more than 450 were eliminated. The mean wind speed
in knots for the period, multiplied by the 12 hour interval, gave the
distance upwind to the point estimated as the beginning of the tra
jectory ,mere the MVT was determined. The MVT at the beginning of
the period subtracted from that occurring upwind determined the hori
zontal advective component. To improve the objectivity of the data,
this was determined before the resulting local change.
1Vhen the point estimated as the beginning of the trajectory fell
on a station with both surface and 700 mb reports it was used. If
there were no reports at the point, it was not accepted unless the
determination of 1~ by interpolating between several adjacent com
puted points appeared to be accurate. If a 5000 ft. wind was not
available for one or more of the three observations, the station was
discarded. This was too frequently the case in areas of wide-spread
precipitation and low cloudiness. The use of rawins will overcome
this obstacle (14). A frequent reason for discarding cases which met
the trajectory criterion was that the point of beginning of the tra
jectory was outside one of the boundaries and could not be used.
Admittedly, the restrictions applied to acceptability of data
greatly limit conditions under which the method may be used as a fore
cast tool. There are, however, a surprising number of cases where it
can be applied, many more than indicated by the number of cases which
are included. Insistence on ability to measure 1m more accurately and
(12)
lr.lth greater certainty than would normally be practical eliminated
many observations, but was felt essential for the investigation to
be of value. Estimation rather than determination of trajectories
undoubtedly introduced an appreciable error in many cases; but if
these may be considered random, the correlation calculated should
not be greatly affected. The dispersion of the variates about the
regression line, standard error of estimate, would of course be larger.
The data after compilation were segregated into the following
categories:
1. 0° (a) all cases
o\(b) cases with a trace or more of precipitation
./ (c)f)
2. () (a)I.
\.\ (b)
cases with no precipitation
cases from the period 1100 to 2300 EST (day-night)
cases from the period 1100 to 2300 EST with precipitation
\>1 (c) cases from the period 1100 to 2300 EST without precipitation
3. N~' (a) cases from the period 2300 to 1100 EST (night-day)'\\ ,
\,';(b) from the period 2300 to 1100 EST with precipitationcases
"'.~ .. c
" (c) cases from the period 2300 to 1100 EST without precipitation
The segregation of all cases into the 1100-2300 EST period, day to night,
and into the 2300-1100 EST period, night to day, was done in order that the
magnitude of the diurnal variation might be determined, if possible. The
segregation of all cases into those with and without precipitation was done
in order to determine, if possible, the mean virtual temperature change
caused by condensation and evaporation. It is realized that appreciable
condensation may occur without precipitation, but the effect of pre-
cipitation suggested by Halstead (5) may be very important and could not
(13)
be reckoned without this division. The further subdivision was
affected for a n~re critical study of precipitation and diurnal
variation.
For each of these categories, the arithmetic mean and the
standard deviation about the arithmetic mean of the local change
in 1NT and of the horizontal advective component were determined.
The total correlation coefficients between the local change and the
advective component for each of the categories were then determined.
The advantage of combining the horizontal wind and the component of
the gradient of ~WT in the direction of the wind, which form the
horizontal advective term, into a single term becomes apparent.
Total correlation coefficients rather than a number of partial cor-
relation coefficients may be determined.
The scatter diagrams and the correlation coefficients indicated
that the degree of association could be closely approximated by a
linear relation. The linear regression coefficient of the local
change on the advective component, and the regression equation, were
determined and the regression lines plotted for each category.
The distinction between regression and correlation should be
remembered, for the correlation coefficient is frequently an arti-
ficial concept of no real utility. Although there are many situations
where relationship may be studied by means of either correlation or
regression, or both, correlation is the appropriate measure where
neither variate may be looked upon as a consequence of the other.
J-,. .'~
Regression is the proper measure if one variate may be designated as
dependent'on the other, or better, if one may be partly controlled or
caused by the other. The latter is the case in this investigation.
The standard error of estimate about the regression line of the
local change obtained through knowledge of the advective component
was computed for each category. If it is assumed that the two variates
come from a normally distributed bivariate universe, the following in
terpretationmay be given to the standard error of estimate and the
regression equation. The probability is about 0.68 for a deviation of
the standard error of estimate on either side of the predicted value.
The chances are even for a deviation of 0.6745 the standard error of
estimate on either side of the predicted value.
The improvement factor, an indication of the improvement of pre
diction of local change of mean virtual temperature with knowledge of
the advective component over that of a mere guess based on the mean,
was also determined. Methods used in calculating the foregoing
statistics are those of J. F. Kenney (1).
Criteria developed by R. A. Fisher () for testing the signi
ficance of correlation coefficients based on the number of degrees
of freedom were then applied to those obtained.
The first test applied was whether each correlation coefficient
itself was significant; that is, the probability that such a cor
relation coefficient will oocur by random sampling from an uncor
related population. If the probability was less than 0.01, the
oorrelation coefficient was regarded assignifioant.
(15)
For test of signifioant differenoes among the oorrelation
ooeffioients, the null hypothesis that there was no signifioant
differenoe was assumed. The probability that suoh a hypothesis
was tenable was determined, and if less than 0.01, the null hy
pothesis was oonsidered denied. If the probability was greater
than 0.01, it was oonsidered that the null hypothesis was tenable;
that is, the oorrelation ooeffioients may have oome from the same
. population.
The oorrelation ooeffioient determined from all oases was taken
as a·theoretioal oorrelation ooeffioient for the relationship be~Neen
looal ohange of MVT and horizontal adveotion. The oorrelation coef
fioient determined for eaoh other oategory was then tested with the
"parent" for signifioant difference.
Similarly, eaoh major subdivision was further broken down for
tests of signifioant differenoe under the "parent" oriterion; e.g.,
cases with preoipitation into day to night with preoipitation and
night to day with preoipitation.
Categories of a dissimilar nature were tested for signifioant
differenoe between them to determine whether they oame from equally
oorre1ated "parentI' populations; e.g., day to night without pre
oipitation and night to day without precipitation.
(16)
CHAPTER III
RESULTS AND CONCLUSIONS
The determination of local MVT change and of horizontal ad-.. v-vection described in Chapter II was made for one lmndJ;"ed ninety-
eight cases. The distribution of these pairs of observations into
the categories in which they occurred is shown in Table 1 with the
arithmetic mean and· standard deviation of the local MVT change and
of the horizontal adveotive oomponent. Asa measure of the average
local change of WIT which migh'b be expec'bed. the arithme'bic mean ot
the magnitude of the local MVTchangewas computed for each category
and is 'babula'bed in Table 2 with 'bhe extreme local changes which oc-
curred , Listed in Table ~ are 'bhe 'bo'bal correla'bioncoei'ficien'b, 1m-
provemnt factor. s'bandard error of estimate. regression coefficient. and
regression equa'bion constant for each category. Table 4 contains proba-
bilities of Whether the correlation coefficients determined were trom 'bhe
sam~ population. Scatter diagrams with regression lines plotted thereon
for each of the categories are presented on Figures 1 to 6. Different
categories are presented on the same figure to facilitate visual com-
parison of the results obtained. .
In general. the degree of association determined between horizontal
advection.and the resulting local MVr change supported that which was
anticipated from experience ';'-the importance of horizontal advection in
controlling the resulting 100al MVT change. All correla'bion coefficients
were significant for representing this dependenoe. bu'b i'b may be seen in
Table 4 that there was no significant difference between several categories.
(17)
TABLE I
All Cases Cases With Precipitation Cases Without Precipitation- •.J r( =0.09497 - =-0.58405 0.50020!Joo !fIJI r~iI. :::
XoJ··~·'l').l·~-o. 00161 ~, ::: 0.25684 x,,2. - 0.12766-Oi.. ::: 3.25935 OJ., ::: 3.77508 ~~a. ::: 2.83097
<J: =5.68934 cr: - 6.41801~2. = 5.20078
"06 "II -JY.o. :r 198 JY., - 74 1Y.;;t - 124- -
Day to Night Day to NightDay to Night With Precipitation Without Precipitation- =0.44902 - = -0.64144 - 1.37489y", ~II $2 --
X'D =0.57221 - = -0.56416 .x;J 1.53706XII --<t:': ::: 3.40019 0311 ::: 3.71409 OJ/S. - 2.79256~,. -
<Ji',. - 5.52918 Gilt - 6.52491~.t - 4.51851- - -
tr;. = 98 /'1" = 45 ~.z = 63
Night to Day Night to DayNight to Day With Precipitation Without Precipitation
y,.. =-0.25199 f,., • -0.49500 ~1. • -0.15273
)(~O =-0.59266 ~ - 0.22003 x..%,% ::: -0.92460-C1j7..0 =3.07586 OJ,v - 3.86863 ~ - 2.67922- :r~4 -e1i'"'.t.P - 5.68892 0;- - 6.21852
<1i'~ - 5.42292- - -N
~o = 100 IYv - 29 ~~ - 71- -
NUMBER OF CASES, ARITHMETIC MEANS, AND STANDARD DEVIATIONS
ABOUT THE MEAN OF LOCAL MVT CHAlTGE AND OF HORI:ZONTAL .ADVECTIVE
COMPONENT.
(18)
All Cases
Day to Night
l.!fl,. :: 2.63606
-I'n~,. = -13.15
Night to Day
-'!I'~ = 2.37867
+mc.x~ 5.56
~~ - -11.536""-.010-
TABLE 2
Cases With Precipitation
tyl"l = 2.90827
+mc.I.,:: 5.51
-17I('X.,::· -13.15
Day to NightWi th Precipitation
iii" = 3.01744
+I71Q,= 5.51
-~. - -13.15",.. I' -
Night to DayWith Precipitation
'!I'~ = 2.73886
4.67
-1'1IcJc = -11.536iJI
Cases Without Precipitation
'''0.2 = 2.26604
-thl4U.~= 5.66
-"'4",,~= -5.838
Day to NightWlthout Precipitation
r;I/% II 2.31224
+m4A~ 5.66
-l11c'\-4= -4.965
Night to DayWithout Precipitation
ARITmmTIC ME..l\NS OF THE MAGNITUDE AND
EXTREMES OF LOCAL MVT CHANGE.
TABLE 3
All Cases Cases With Precipitation Cases Without Precipitation
t;~ - 0.891625 10, - 0.955610 1;.1 .. 0.855184- .. ..~. =54.72 ~, =70.86 ~2 :: 48.17
.s:... - 1.47575 -5." .. 1.10024 S"z. = 1.46736.. ..R.., - 0.51080 R o, .. 0.56283 R ....1 .. 0.4655.. - ..K... .. 0.096 If., =-0.729 x;.% :: 0.440-
Day to Night Day to NightDay to Night ~th Precipitation Without Precipitation
1,4. - 0.882296 Ii, = 0.955245 .r,; = 0.762797-£~ =52.93 f{, =70.42 f{z. = 35.34
.s:. - 1.60045 .5;, - 1.09868 ~.t = 1.80577.. -~~ - 0.53293 R" = 0.54374 R,z :: 0.47143-/(,0 .. 0.189 /(, =-0.334 /(,z. .. 0.650- -
Night to Day Night to DayNi~ht to Day With Precipitation Without Precipitation
/i~ • 0.902143 r:zr - 0.931140~ :: 0.908365-
~ - 56.86 ~, • 63.53 ti:z =58.18-S.c~ = 1.31928 ~, = 1.410745 ~1 = 1012039
Rz~ .. 0.48777 .Rz, = 0.57927 ~u :: 0.44878-~~ - 0.037 k; = -0.622 ~2 - 0.262- -
TOTAL CORRELATION COEFFI CI ENTS. IMPROVEMENT FACTORS.
STANDARD ERRORS OF ESTIMATE. REGRESSION COEFFICIENTS.
AND REGRESSION EQUATION CONSTANTS.
(20)
TABLE 4
Cases Wi. th Preoipitation Cases Without Precipitation
D,F:. :: 0.00006 • o;zF:.. :: 0.02719
"D.l P;;." P,,;(. < 0.0001 '" , 0.0001 •
Day to Night Day to NightDay 110 Night With Precipitation Without Preoipitation
,,, f=:.. .0.6672 I/r::. • 0.00295 * 'a.1;. • 0.00252 '"/(
" .' , ,I.~" < 0.0001 0,/ \ ,,1::, :: 0.9182 /Z.~2, =0.05612
, ,"RtI -0.00114 * ,,ZR. =0.00676 '"-
p "" Pf,;.. ( 0.0001 * "'~ /:z F:, < 0.0001 l4r
" "/I~/ • 0.06652 /:Z~.l. <0.0001 *
Night to Da.y Night to Da.yHight to Day Wi th Precil'i tation ~thout Precipitation
Z.OF:D .0.59962 v e. • 0.22805 21.f:" • 0.46638p /
%.Z ~::uP:.. < 0.0001 • 2' F:., :: 0.22502 :: 0.0431I I
Z/~O :: 0.35052 %2~O :: 0.2262
" "2/P;2, :: 0.3366 :Z2~' :: 0.3366
"-lIFt, :: 0.06652 zz~~ <0.0001 •
PROBABILITIES OF NO SIGNIFICANT DIFFERENCE
AMONG THE CORRELA.TIOH COEFFICIENTS.
* Probability is significant.
(21)
+. • ,-.p. / ..++
-,......._.
•
++
....... - .
++ . -
1-
.....-
+++
+-
+ / + ,,,'.1-+
+
+.+
..
+ .... ;;
+++
/'
..... /
+ . +
++.+', ;-
'.+,- ••+~+
++
. ..04-
+
+
+
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++ +
M+
+
-(>.- +
+
.~
/+ +
-- . + ++ + t
.+ +)
.-
+
-- .---.--__ -----l
,++
++
+ + Cases It/I"!IJ a. Trace cr more at
.1.. L . l-__ . ...J ;
I
?recipi tcr...1/ol7
CQS CS 11/ ;1h no Prec;pitav"on
u on X o : & == '0.5/08)( + 0.096.::too ° 0'00 00
j See =;: ~7S8 °c ; 100 =:0 0.g'1/6
-10,-
+
.--- '_.- ._ ...... ,-- .--_._..-
•
Re.gress/on aT'J/oo =O. O?50 °c j 0;;;,0 =...1'. 2S7~ "e
---. ------- Re.:;re ss icc ol
LI -:: - G58¥-O ce : cY;-- =3.776/ °CdOl / <;IN
.j!OI.,/
Hi X OI : -!lOI ::= 0.56-28xo, ~ 0.729Sc,=I.I002 cC )/0, -=0.9566
...r:j!I.
-I¥-I- - •-IS
+
x 1-
-/61
-1+,
. ~--_.,~ -;2-
. (22)
....J .. - ... 1 • . _. __._
-Ie -8' -6
- _. - - - RCJlrcs5icn
Jo~ =0.5002 'oC ,) CFioz.:: 2. 83/0
2
of' ..Yc;z. on XC;l. =d/C7- ;:: O. '1-655 X 0..:. .+ O. 7"f'-ODC ./ S,,~ = ;. '1-67 'f °C )~:I. -= O.8.5S2
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,
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,I
III
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il']/;;;;'/ Ie da..7' I
O.5/08x . + 0.096~l' .
/;c '" {). 8 CJ/6
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.,!fco en xCC' : -<-4f.e e -:::
-j 5cu "" I. ¥ 7S8°C.
.. i.- ....1--
Ferloa' /1 CO - 23CJ(J EST
Rr/cd 2300-//:-0 EST
;':e.yreSSlon oj"
) <J;"r::: 3. 25 9If C C
- I
.: . '.- ~ .., ._ .......
C a.. 5es r rom the
+ C Q$es from the
+ + -+:; .
+ +".+ -t-
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(/~/..... ,~"':.
r
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II
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I
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+
• -- ._- ---.- ... - - .. f· - -.- ..
+ '+IY
+ +
+ ++
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.--''" 1. ,
i
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1.2.x10
T O.18C;·
0.8'823
= O.JI-71tj-x~:<.- + 0.650
.7 r;.=. = O. 7628
6
cf' }ju() C • c
) -'I.J..
2.
F/GUJ?E-'3 !- .- -I - ..----- .--- .. - _ .•--" - _•. --•. ._.. ---l- . • 0 •• -----.- '-" -'-TII
,
perioo' //00 - :2300 E 5 T li/iii7 Prr.:;ciF i 7a ii:J n
o
•Re..yrt55 ie-/')• rr- = r;.; 'IfiLl-,V;;jll -..t. f.
frcm 11K..
-;2..~-~---~--_.,-- --_._.. -_.. ~~-_ ..- ----._-_.._-_.__._._--,~,_ •._--- --_..•_---- -_. -" ---'-'-.-_._----_.,._---._- ---- -_ .. --,-
+ Ca.ses
P.-:::;ressiol7
Yo. ~ /. 3777 °c -j ~'l. = 2.7926
(
-----.------------ ,R2.//'t::::;"::'/:J/; or
ylC =O.'fi-i/() 't .7 0]10 == 3./1002 DC
--I-6
+
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Freeipi (q j;O/7
Prec I p/ t£z 1/0 1",
+
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w//;;cut
++
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r ..
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230U -//Ou ES T
Per/cxI
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+
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+
+
•
,
/.
• Ca.ses
+ CCLses rrom the
- .._._._.__ ..._._.._.~_.__._ Re'l~s:.;i(">17 or LI en X ~ LI ~ 1.~.:';-27Px + C. 03'/d -4,.0';"0 4 ~ ?-c
}j;.:u = -0.2520 JC ---= CS-"c = 3.0'/5'1 C( .3 S~o = /.3/ t;.3' C L· j r;;o : O. YO;?/
+
..
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+
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+of I./~ en x~ : J/l. ::; C. 57'/3 x.</. • . I ,..../
.!j:<"1 = -(1'/-'/5C "'c ) OJ;.. = 3. 8686 °c ; S~I = I. '1-107 °C .3 0.1 := 0.93//
x/c6"-- -----------------
R~/"t:~s /cn
; U:;.;:2.L'1<;2_ -4 A4
-2-----_ __ _--------_._---
,2
i/n= -0./527 °C
-- ..- - - - - -.
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fI
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-/8 X -/6 -/'f- -f2
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11 I, I, Ii
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t,
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+ +
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.!IN cV7 X rl :..!IC'I ::: O.562Sxcl -0.729) 5~ I -= /. /002 '-~ :;-; I :-= O. 7'566
------.------- ----- RellressfC'/i ri
)jc/ = -0.581}-0 '(" ..; 0;" c= 3.7'76/ C'c
-8'- . ....
-I;) ~:...._,--.-.-.------'----------
+
--_..._--- - _._- ••• _ ~_.__._•• ._. •• _. _ .. _._.,._ - ••• •.• ~ __ 'O
Ca::es /r(//7
+ Case..c, Trom
/);e re.r/cd //00 -;230(' EST Ir//".f
ilJe. Pcrlcd 230C/--I/OC' EST tv///1Preci,Plk !/C/l
Pre c ipi7a ITo /7
II
I_-----l.
iII
IIIi
+-1,2~ ._-----_.._--_._-- -_.... - _._--_.- --_._.-
u/ :: -0 6'1-1'1, <"C0::1 /
- R e:Jre 5SIC/? C.'f
: u:" '" :':1. ] / 'Y'I C C~ <I'
X It : ..,,'//.1 .= o. fi '1-3'7 X II
/. r 'lb' '7 'c .,: 0:; ::: l;
-o.33~
9S52~t:
.!/~/ on X,v: Jj;:.; ::: 0 ..5'/'13 X;?'I - 0.622
./ 5;./ =I. -r/O /' cc _,' ~ :: C.93//
-/If'--f? x
..-/6 -12
(26)'/0
.. -. -- _.-
'-8 -6 -'!-
Rc.j' re~:';CJrl or) o;~ :: 3.8686 cc
o-._- ---- ---/-_ .._------- -.::-~---_.._- -
l;"' ,'> /0 x
- - - - - - - Re.Jresslol7 or .j;!;l. 0/1 xAZ : }f;u." O. 'l-rff8 )<;1.2. +0.262
!f2;2.=-O.l52.7 °C .j0i42.~ 2.~67~2 °c j s~ -::; /./201f- "C ) 02 =: o~ 90S'f
/2
I
1I
x10
on Xo.: .!j';J. :: 0/1-7/ 'f-x,2 + 0.6505,.2 = 1.8058 '" ; t;~:: 0.7628
-2
-.-~- R,e,ilress 1ol7 01 )/,;'!l1~= /.37'19 DC , ) eJ"i,z £ 2.7'126 c>c ;
--'------~--------~----"---"----'----. -6 -'7-
i: .
6
Where no significant differenoe was found it cannot be assumed.that
either is better for prediction of local MVT change, or that there
is an inequality be"tween them ",hich is a measure of any component in
produoing local MVT change. The statistios are included and the re
gression lines plotted for all categories, but no inference other than
no differenoe may be made where no signi.ficant difference was obtained.
Determination of the "diurnal variation" proved to be important
for it is concluded that there was no appreciable MVT ohemgee.ttribute.ble
to loss or gain of heat by radiation. Antioipation of this could have
been made from the times of the observations, which serve to mask that
which actually occurs. For e. determination o1'diurnalvariation of MVT,
allowing for lag in the upper air, observations should be taken about
sunrise and three to four hours after local· noon. At thf!se times the
efreots of insolation and nooturnal radiation would be most pronounced.
Radiosonde ascents in the United states over the region investigated are
made approxtmately halt'waybetween· the optimum· times· for determination
of diurnal variation. Halving of the diurnal variation 'thus serves to
mask that whioh aotually occurs. The probabilities in Table 4 do show
a significant difference for diurnal varla'tion. For reasons 'Which will
appear later, however, it is felt that correlations determined 'With pre
cipitation are the most representative, while those oocurring without
preoipitation are the least representative. Preoipitation correlation
ooeffioients· between day to night and night to dayaro not significantly
different, and the means and standard· deviations· are very similar. Fur
thermore, the regression ooefficient for cases with preoipitation from
(28)
day to nigh~ is less than ~hat for preoipi~a~ion from night to day.
while ~he reverse is true when no preoipitation oocurs. Corralation
ooefficients between day ~o night and night to day when no preoipi-
~ation occurs do show a significant dif'ference. but ~he difference
in size of ~hese correlation ooeffioients is so great that it cannot
be acoepted. This is in view of absenoeof signifiean-c difference
be1:iween diurnal preoipita1:tion segregation. the times of observations.
and the non-representativeness of non-oocurrence of preoipitation
oorrelation. The significant difference between day to night ca$es
and night to day cases is simply a reflection of non-occurrence of
precipi1:tation cases in the periods.
Inspection of Table 1 reveals that the arithme~ic mean of local
MVTohange for precipitation,categories is appreciably less than the
mean of categories without preoipitation. It should not be construed.
however. that the effect of precipitation as compared to non-occurrence
is a decrease of lm~l~)Itima.Ybe seen from Figure 1 and Table 2 that the;
oause of this difference is greater MVT decrease when cold advection is
aooompanied bypreolpitation. A possible explanation is that the pre
cipitation is an indioation of ~nstability in place of the subsidenoe
whioh is frequently assooiatedwith northerly flow.
In this investigll.tionit'was a1:ttempted to eliminate frontal cases
in order that the looal MVTohange and the horizontal adveotive com-
ponent would be a measure of the individual and advootive components
and not a refleotion of the front itself. Exoept at fronts. MVT
gradients will normally be larger with cold adveotion than with warm
adveotion. The oontribution of turbulent flux of heat from the
underlying surfaoe is to increase instability and cause a rapid
modifioation of the air mass during oold adveotion. With warm
adveotion turbulent flux to the underlying surfaoe tends to in
crease stability and result in a slow modification and smaller MVT
gradients. In figure 1 it may be seen that the greatest horizontal
adveotive oomponents occurred with oold advection. Not with oold
adveotion alone, however, but cold adveotion accompanied by pre
cipitation, a further indication of instability. Preoipitation is
infrequent when subsidence is marked.
The standard deviation about the mean and the average local MVT
change are greater when preoipitation oocurs. These are an indication
only of the greater dispersion of looal MVT changes which are found with
procipitation. The correlation ooefficients disolose a higher degree of
assooiationbetween horizontal advection and the resulting looal MVT
change for precipitation oategories. The ocourrenoe of precipitation
with cold advection has just been disoussed. Reoalling that the effeot
of preoipitation and oondensation should be to assist warm advection in
produoing local ohange of MVT, the reason for the high correlation ob
tained is evident. The correlation ooefficient determined from pre
cipitation cases is signifioantly different from both all cases and
oases with no preoipitation.
It should be remembered that appreoiable condensation is possible
without preoipitation. The segregation of precipitation non-ooourrence
does not take this into consideration, and for this reason is not
<:'0)
considered representative. Explanation of the smaller correlation
coefficients for non-occurrence of precipitation may thus be found
in the choice of segregation. If a further segregation was made to
imply condensation and evaporation by the amount of cloudiness, it is
believed that the correlation would improve. The correlation coef-
fioient determined; while significantly different from precipitation
cases, shows no significant difference from the correlation obtained
for all cases and further Bupports this belief.
Regardless of the non-representative nature of correlation deter-
clned from eases in which no precipitation occurred, an inspection of
the regression ooeffioients of precipitation and non-precipitation
cases yields useful qualitative information. In all precipitation
categories the regression coeffioients indicate that the resulting
looal change will ordinarily be greater than one-half of the horizontal
advective oomponent. With warm. advection the effect of precipitation
and condensation together with the horizontal advective component is
usually greater than twioe the cooling due to turbulent flux of heat
to the underlying surface and to upward motion,of the air particles.
Similarly, with oold adveotion the horizontal and vertical adveotive
components are normally more -than double the warming due to turbulent
flux of heat trom the' underlying surtaoe and to latent heat of oon-
densation released.
In oases without preoipitation the regression coefficients in- ~~./\
dioate that the looal MVT ohange will ordinarily be less than one-halt
the horizontal advection. With cold adveotion the horizontal adveotive
oomponent and the prooesses tending to deorease MVT, evaporation and
en)
asoending motion are usually less than twice the inorease of MVT due
to turbulent flux of heat trom the underlying surface, subsidence,
and oondensation. During warm advection the horizontal advective
component and those prooesses tending to inorease MVT, condensation
and subsidence, are normally less than double the deorease of JfVT
due to turbulent flux of heat to the underlying surfaoe, loss of
latent heat by evaporation, and ascending motions.
In f'uture studies of local Jm ohange following this preliminary
investigation~ further segregation of non-ooourrence of preoipitation
would be desirable. Corresponding to the forecast and verification
cloud coverage, mostly clear - less than three-tenths, partly cloudy
three to seven-tenths, and clOUdy - greater than seven-tenths, would
be adequate and preferable segregation. Additional separation into
well defined subsiding and asoending motion categories for the deter
mination of the degree of association 'WOuld enable inoluding the effect
ot pronounoed vertical motions when expected. Evaluation of turbulent
flux ot heat does not seem praotioable awing to the great dissimilarity
in type and effect of underlying surfaoe encountered.
Practical applioation to forecasting with these reaul1;s is limi1;ed
to the ocourrenoe or non-ooourrenoe of precipitation. The standard
error of estimate about the regression line for oases with preoipitation
is an improvement of 70.86% over a prediction of looal MVT ohange using
only 1;he ari1;hme1;io mean and "standard deviation of 1;he looal ohange.
Utilizing 1;he regression equa1;ion of looal MVT ohange for preoipitation
oases and oonverting "thickness'may be predioted to ±37.4 feet with a
probability of 0.68. The ohanoes are even for a "thiokness" prediotion
of ±25.2 feet.
02)
When no precipitation is expected the results f'or all cases
should be used rather than those of' cases without precipitation.
there being no signif'icant dif'f'erence. The improvement factor f'or
'1/'"I'!
all cases is 54.72%. utilizing the regression equation for all cases.
"thickness" may be predicted to ±50.1 feet with a probabi Iity of' 0.68.
and to ±33.8 feet vdth a "fifty-fifty" chanoe.
These results do indicate tha.t a continuation of' ~e investigation
is warranted. More accurate determination of the ef'fects of condensation
and evaporation and vertical motions will increase the accuracy of locel
MVT change prediction.
//
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