NCAR-TN-39

Objective Analysison Isentropic Surfaces

RAINER BLECK

PHILIP L. HAAGENSON

December 1968

NATIONAL CENTER FOR ATMOSPHERIC RESEARCHBoulder, Colorado

I

iii

PREFACE

The requirement of highly accurate analyses for isentropic trajec-

tory calculations established the need for the authors to revise the

most recent approaches to generating objective analyses on isentropic

surfaces. Two factors which contributed to generating better analyses

by our computer program were the elimination of time saving approxi-

mations which were used in other methods and, more importantly, the

utilization of a new objective analysis technique developed by Amos

Eddy (University of Oklahoma) which enabled us to make better use of

the actual radiosonde data. Trajectory calculations derived from the

improved analysis justify our efforts.

ACKNOWLEDGMENTS

We are indebted to Dennis Joseph and Roy Jenne, of the NCAR

Computing Facility, for acquiring the input tapes and providing the

routines needed to organize the data for use in our program.

I

v

CONTENTS

Preface . .... .. .. .. ............ iii

Acknowledgments ........................ iii

List of Symbols . . . ... ... . .. .. . .. . . .. .. . vi

INTRODUCTION ......................... 1

COORDINATE TRANSFORMATION OF ISOBARIC DATA .......... 4

OBJECTIVE ANALYSIS TECHNIQUES ................. 6

ANALYSIS OF RELATIVE HUMIDITY FIELDS ............. 13

PROGRAMMING PROCEDURE .................. . 14

Appendix A

SPATIAL INTERPOLATION BETWEEN GRID POINTS .. ..... . 17

FIGURES ... . .................... .21

REFERENCES ......................... 27

I

vii

SYMBOLS

T temperature ("Kelvin)

p pressure (mb)

R gas constant for dry air, 2.8704 x 106 erg g K-1

c specific heat of dry air at constant pressure,

1.005 x 107 erg gl K-l

K R/c , 0.286P

0 potential temperature, T(1000)p

g acceleration of gravity, 980.6 cm sec -2

z geometric height above sea level

TM Montgomery stream function, c T + gz

f Coriolis parameter

v wind velocity vector

y mean tropospheric lapse rate, 0.650/100 m

A gradient operator

I

1

INTRODUCTION

Ever since meteorological upper air observations became available,

three-dimensional weather analysis has been accomplished primarily by

analyzing flow patterns in two dimensions on various isobaric surfaces.

While the choice of pressure offers some distinct advantages over the

choice of geometric height as vertical coordinate, an isobaric repre-

sentation of the air flow still fails to give an adequate picture of

the motion within two-dimensional "material" layers in the atmosphere.

Each-pressure surface cuts through those material layers and hence

picks up information about the motion of fluid elements which are not

closely related to each other.

To a high degree of accuracy, the material surfaces just mentioned1000 K

correspond to surfaces of equal potential temperature = T( ) . Thep

most realistic approach for representing two-dimensional "stratified"

airflow in the three-dimensional continuum is therefore to choose

0 as a vertical coordinate, i.e., to analyze the flow pattern at

constant potential temperature. (Since the entropy is proportional

to the logarithm of potential temperature, the 0 surfaces are commonly

referred to as "isentropic" surfaces.) As on isobaric surfaces,

there exists a geostrophic relation between the wind vector and the

gradient of the so-called Montgomery stream function, YM = gz + cpT.

This means that the flow pattern is most conveniently analyzed by

drawing isolines of M', in much the same way as isolines of gz are

drawn on isobaric maps.

Isentropic charts were introduced into meteorology in the early

1940s. At that time, however, analysis difficulties which arose from

large errors in the reported values of the Montgomery stream function

prevented these charts from becoming very popular. Much later it was

discovered (Danielsen, 1959; p. 299) that smooth patterns of TM can

be obtained when it is recognized that at any isentropic level z and

T are not independ ent of each other but are related through both the

hydrostatic equation and Poisson's equation. This relationship implies

that an error in the term c T produces an error in pressure, and hence,

2

an error in gz of the opposite sense. Correspondingly, an error in

the first term gz produces a compensating error in c T. This fact

greatly reduces the sensitivity of TM to miscalculations of either

its first or its second term, and makes it possible to calculate

fields of M values which are just as easy to analyze as the geopo-

tential fields on isobaric surfaces.

Gustafson (1964), in his computer program for generating objective

analyses on isentropic surfaces, avoided those earlier problems and

opened the way for another attempt to introduce isentropic maps into

weather analysis. His program converts isobaric grid analyses into

isentropic analyses of T and TM by interpolating between the two

mandatory pressure levels adjacent to a particular isentropic surface.

Such an interpolated field is not satisfactory due to the large

vertical distance between the pressure surfaces; Gustafson, therefore,

uses it only as a "first guess" for subsequent objective analysis

scans based on actual radiosonde observations.

The need of highly accurate f analyses for isentropic trajectoryM

calculations led the authors to revise Gustafson's approach and to

eliminate most of his simplifying assumptions regarding the vertical

interpolation between pressure levels and the integration of the

hydrostatic equation in 0 (potential temperature) coordinates.

Because we were not concerned with the execution time of the program,

we were able to avoid time saving approximations wherever "exact"

answers could be obtained. Another reason for writing this program

was the recent development of a new objective analysis technique by

Eddy (1967) which, for data-sparse regions, excels the technique

applied in Gustafson's program. The quality of the resulting maps

seems to justify our efforts.

In view of the intended application of the grid point data in moist-

and dry-adiabatic trajectory calculations, the program is set up to

generate analyses of Montgomery stream function, pressure, and relative

humidity on 0 surfaces. The procedure to generate these analyses, as

in Gustafson's program, consists of two steps:

3

1. Objectively analyzed data fields on several pressure surfaces

(made available by the National Meteorological Center, NMC) are con-

verted by means of a "coordinate transformation," into data fields on

isentropic surfaces. Due to the large vertical distance between the

standard pressure surfaces, this conversion process produces only

approximate values of the desired parameters on the isentropic surfaces.

2. Radiosonde data, processed to make them applicable in an isen-

tropic coordinate system, are used to improve the analysis from step (1).

Since there is very little chance to improve the analysis over ocean

areas (because only regular radiosonde observations are used), it is

important to perform the transformation from isobaric to isentropic

data (step 1) as accurately as possible. (The data sources available

to the NMC enable them to arrive at fairly accurate isobaric analyses

even over the ocean.)

4

COORDINATE TRANSFORMATION OF ISOBARIC DATA

The NMC delivers analyzed temperature and geopotential fields at

seven pressure levels: 850, 700, 500, 300, 200, 150, and 100 mb.

In order to perform the transformation to isentropic coordinates, a

fictitious sounding has to be constructed from the 2 x 7 = 14 sets of

data at each point of the data grid. The analyzed temperature fields

immediately define seven p,T points for each of these "soundings."

Six additional points -- one between any two of the mandatory points --

can be constructed by requiring that the temperature and geopotential

data at the mandatory levels are consistent in the vertical. Considering

two mandatory points i and i+l, this requirement can be expressed as

i

g z(i+) -- z(i) + T d In p980 980 980

i+l

With z(i), z(i+l), T(i), T(i+l) given, and under the assumption that T

varies piecewise linearly in p , this equation generally can be fulfilled

only after introducing a level i+;, that is, a pair of variables p(i+2)

and T(i+½), with T(i+½) deviating from the linear lapse rate between

T(i) and T(i+l).

The special case, p(i+i) = p(i) p(i+l) , is the simplest from

the computational point of view and is tried first. If the resulting

temperature gradients (between level i and i+~-, and between i+- and i+l)

both turn out to be subadiabatic, the problem is solved. If a super-

adiabatic lapse rate is generated, an attempt is made to remove it by

changing the pressure value p(i+½) within a reasonable limit. If con-

sistency among the four values z(i), z(i+l), T(i), and T(i+l) cannot

be reached at all, or only by introducing an unreasonable temperature

inversion somewhere between p(i) and p(i+l), the mandatory geopotential

and temperature values at this particular grid point must be corrected

slightly. But before the algorithm can make another attempt to generate

a meaningful point p(i+i), T(i+i), the changes in z(i) and T(i) require

it to recalculate the significant level i-½, which in turn might require

5

changes in z(i-l), Z(i), T(i-l), and T(i). (In an almost dry-adiabatic

atmosphere the generation of 13 consistent significant points can

become a quite lengthy procedure!)

Each fictitious grid point sounding is now processed by a

routine (Duquet, 1964) which calculates the pressure and the Mont-

gomery stream function, YM = c T + gz, for every 5° of potential tem-

perature e = T ( 1 ) . Combined over the whole area covered withp

grid points, these results define the desired first guess analysis

of YM and p on isentropic surfaces.

6

OBJECTIVE ANALYSIS TECHNIQUES TO IMPROVE ISENTROPIC ANALYSES

Whereas 13 significant points, if they were actually measured by

radiosonde, would be sufficient to define the vertical temperature dis-

tribution as well as 1M(0) and p(9), the 7 points taken from isobaric

analyses, together with the 6 points inferred from the hydrostatic

relation, do not provide the accuracy needed in isentropic trajectory

computations. Wherever possible, the resulting fields must be compared

with actual radiosonde measurements and any differences eliminated.

As it is set up now, the program reads the radiosonde data from

a tape prepared by the Air Weather Service at Offut Air Force Base,

Nebraska. Because the data are compiled from the teletype communica-

tions network, significant levels in the individual soundings are

somewhat sparse. No estimate can be given at this time as to how this

affects the final results.

The individual soundings are processed by the same program that

generated the fictitious grid point soundings. In order to compare

these values with the first guess fields, an interpolation scheme must

be selected which can infer values of Montgomery stream function or

pressure at any point from surrounding grid point values. (Appendix A

describes the scheme presently used.)

Considering an individual 6 surface, each sounding location can

now be assigned a "difference" between the actual, say, M value (as

inferred from the radiosonde observation) and the YM value obtained from

the first guess field by interpolation. If the resulting field of

differences is sufficiently smooth (i.e., if it contains no irregularities

with a "wavelength" shorter than two or three times the mean distance

between adjacent radiosonde stations) then the first guess field can be

improved by objective analysis techniques.

In order to remove any possible roughness from the first guess field

which would disturb the difference field, a proper smoothing operator

is applied before the differences are evaluated. This is justified if

one assumes that the roughness is introduced by the method of

7

constructing the fictitious grid point soundings, which does not neces-

sarily assure horizontal continuity.

Before one can improve the first guess fields it is necessary to

convert the difference fields, defined above with their irregularly

spaced data points, into fields of grid point values. In the program

under consideration, this is accomplished by a multiple linear regression

technique which was introduced into meteorology by Eddy (1967). Its

basic principles are outlined here for convenience.

Given nom observations of a variable Dkt at n locations

(k = l,...,n) and at m different times (t = l,...,m), the issue is to

find the best linear prediction of this variable at location (n+l):

nD Z caDn+l,t kl k Dkt (t=l,...,m) (1)

If "best" is interpreted in the Gaussian sense, then the following

expression must be minimized:

m n

tl (Dn+lt k=l ak Dkt) = mi

By setting the derivatives of this expression with respect to the ak

equal to zero, the following n equations are generated:

I ^(Dit Dkt k Dn+l t Dit (i=l,...,n)k=l, .. . ,n t =l,. ... ,m nk=l,...,n t=l, . . . ,m

If it can be assumed that

m m m1 t2 2

DDt = D =...= Dnltt=l1 t=l t=l n

8

the above equations can be rewritten as

n

Z r ik k = rin+l (2)k^l ik k i,n+l (i=l,...,n) (2)k=l

where the rik are the spatial auto-correlation coefficients defined by

t=l

m

Z D Dt=1 it kt

ik/m 2 m 2

t=l t=l

Hence, the weights ak' by which the "predictors" Dkt must be multiplied,

are given by the solution of a system of linear algebraic equations

whose coefficients are the auto-correlation coefficients between any

two of the predictors.

If the data Dkt are homogeneous, the auto-correlation coefficient

becomes a function of only the distance dik between two predictors

i, k. Without questioning whether it is justified or not, we introduced

this simplifying assumption into our computations and determined the

auto-correlation curves shown in Fig. 1. Each pair of radiosonde

stations was put into one of eight classes according to the distance

between the two stations. The class width was chosen to be 200 km,

so that all station pairs farther apart than 1600 km had to be discarded.

Limiting the maximum separation distance to 1600 km was justified later

by the results of the computations, which indicate that there is no

correlation between stations 1000 km or more apart.

Only two pairs of North American radiosonde stations were found

in the class 0-200 km, and due to some systematic differences in their

reported results they turned out to be negatively correlated. They

were considered nonrepresentative and are not included in Fig. 1.

9

By taking data from ten consecutive sounding times and several

0 levels it was possible to fill each of the seven remaining classes

with at least 4000 pairs of difference values. Figure 1 shows that

both r( M)(dik) and r(P)(d) are linear functions of dik forboth r .1k (d ik ik

dik > 200 km. The curves were simply extrapolated to dik = 0.

Curves computed from the first five sounding times turned out to

be very similar to those computed from the last five sounding times,

which means that the spatial auto-correlation of the difference fields

does not depend critically on the synoptic situation. The influence

of seasonal changes has not yet been investigated.

To determine the grid point values of a difference field from data

given at the radiosonde locations, each grid point, one after another,

is now chosen as the location (n+l) in Eq. (1), whereas the n predictors

are defined as the radiosonde data surrounding the particular grid

point. Given a spatial auto-correlation function rik = r(dik), the

weights ak which are needed in Eq. (1) to compute the grid point value

are found by solving the linear system, Eq. (2). Because the correla-

tion drops to zero at 1000 km in the YM difference field, and at 800 km

in the p difference field, all radiosonde stations which are farther

away from the particular grid point than 1000 or 800 km respectively

are excluded from Eqs. (1) and (2).

After the grid point corrections of Montgomery stream function and

pressure are computed in this way, the "second guess" fields can be

generated by adding the corrections to the first guess fields. If the

new fields are treated as the first guess fields were before (i.e., if

an attempt is made to improve them further by again comparing them to

the actual radiosonde data), it is found that the remaining spatial

auto-correlation between the new difference values is negligible.

Figure 1 shows the resulting correlation curves, which indicate that

it is superfluous to iterate the process described above.

It is possible, however, to further improve the analysis of the

Montgomery stream function YM (Danielsen, 1968, private communica-

tion) by utilizing the information contained in the wind measurements.

10

This additional improvement can be made because the velocity vector v

and the horizontal gradient of YM are approximately related through

f v = k x V M (3)

where k is the unit vector pointing vertically upward. The subscript

0 indicates that the gradient must be evaluated at constant 0.

The predictors Dk entering into Eq. (1) have previously been

defined as the differences between the actual observed values and the

interpolated values obtained at the radiosonde locations from the

first guess field.

From the relationship shown in Eq. (3) it is clear that whenever

the wind vector is reported at a radiosonde location, the scalar YM

difference at that point can be supplemented by a gradient vector of

the PM difference field: namely, by subtracting the gradient vector

of the raw YM analysis at that point from the TM gradient implied by

the observed wind. Instead of using the difference value itself as

a predictor, as we did previously, we now are able to use the value

which the difference field, according to its gradient vector at the

radiosonde location, is likely to assume at the grid point. Due to

the apparent nonlinearities in any TM pattern this value will not be

the correct one, and the error in the mean will increase with the

distance between the grid point and the radiosonde location. However,

it can be assumed that an analysis scheme defining the predictors

in this way will result in fields of TM grid point data that are more

reliable than those obtained from the purely scalar technique discussed

previously. In particular, the wind field derived from such a YM

analysis agrees much better with the observed winds than the wind

field derived from the scalar analysis.

In order to arrive at a meaningful auto-correlation curve for the

gradient method, the PM field had to be smoothed by a standard 9 point

smoother before the difference values and the gradient vectors of the

P difference field were calculated. Then the data at each (wind reporting)M

11

station were used to estimate difference values for all surrounding

stations, and finally those values were correlated with the actual

EM differences at respective locations. The result of this calculation

is shown in Fig. 2. For computational purposes the curve(Tr d -0.0017 d

r(d) = 0.43 cos ( - 10 0 ) e (d in km) is used to fit the

correlation points.

Special consideration must be given to the zone where a 0 surface

intersects the ground. A grid point close to this intersection would

not be surrounded by radiosonde data as in the regular case, because

difference values can only be defined where the 0 surface exists.

The lack of data in the southern (or, more precisely, the warmer)

semicircle as seen from the grid point will make the predictions of

Eq. (1) less accurate; this is especially precarious in the case of

the pressure analysis, because the e surface generally becomes very

steep when it approaches the ground, resulting in large isentropic

pressure gradients which are hard to analyze correctly.

One way to approach this problem is by defining an underground

continuation of the 0 surface. If a temperature sounding is extrapo-

lated downward from the surface point by assuming a lapse rate of, say,

0.65°C per 100 m, a first guess analysis of both M and p can be defined

by continuing the fictitious grid point soundings down from the 850 mb

level, and thus difference values can be defined by comparing those

extrapolated first guess fields with the extrapolated radiosonde data

-- downward as far as desired.

These manipulations at least provide estimates of YM and p values

in a 1000 km strip parallel to the intersecting curve of the 0 surface

with the ground. Figure 3 shows an example of how the pressure analysis

on a 0 surface close to a cold front can be improved by utilizing the

extrapolated underground data.

The specific formulae for extrapolating downward from a point

p(i), T(i) are as follows:

1

p(i-l) = p(i) ( i - ) R/g - 0.286

12

0.286T(i-L) = (i-l) [ P() ]

1000

z(i-l) = z(i) - - [T(i-l) - T(i)]

where y is the chosen lapse rate, 0.65°C per 100 m, and e(i-1) is the 0

level at which T and p have to be computed.

13

ANALYSIS OF RELATIVE HUMIDITY FIELDS

Provisions have also been made to generate grid data of relative

humidity on e surfaces. (These data are needed for moist-isentropic

trajectory calculations.) Because the standard grid used by the NMC

is not adequate to resolve moisture patterns developing in connection

with cyclonic disturbances, the resulting fields are not as reliable

as the fields of Montgomery stream function and pressure.

The NMC does not provide isobaric analyses of moisture. Hence,

a first guess field must be generated exclusively on the basis of

radiosonde data. Following Eddy, a field of moisture values varying

linearly with geographic latitude is chosen for this purpose. The

field is generated by a one-dimensional regression analysis of all

moisture data available on the particular 0 surface. Because the 0

surface, in the mean, slopes down toward the south, the regression

analysis normally produces a relative humidity field with values

increasing in the same direction.

The deviations from this humidity field are correlated as shown in

Fig. 4. These points, like the points in Fig. 1, have been obtained

from ten consecutive sounding times. The curve chosen to fit the

points in Fig. 4 is

d -0. 0015 dr(d) = cos ( -1 00-) (d in km)

The boundary problem, similar to the one discussed at the end of

the previous section, is taken care of by extrapolating the reported

surface humidities down into the ground. In a few situations this

might hinder rather than help the analysis but, overall, the extrapo-

lation seems to be beneficial.

14

PROGRAMMING PROCEDURE

The complete program is quite lengthy and contains many subroutines;

a flow chart (Fig. 5) has been provided as an aid to better understanding

of the programming aspects involved.

It was mentioned earlier that two input tapes are needed: one

provides the grid point data on several pressure surfaces, the other

supplies the original radiosonde data. The first tape is prepared by

the NMC. It can either be used directly, or be improved by eliminating

excessive horizontal smoothness in the analyzed fields, which in many

instances impedes the construction of the fictitious grid point soundings.

This "improving" is done by a separate routine that computes, for each

pressure surface and for each sounding location, the differences between

the observed geopotential height and temperature values and the respec-

tive "analyzed" values. By means of Eddy's analysis technique, these

differences are then converted into corrections to the grid point values.

It must be mentioned that the auto-correlation functions obtained

from the isobaric difference fields were quite erratic. According to

the theory, this would indicate that no improvement should be expected

from applying Eddy's analysis technique to the NMC data. However, an

arbitrarily chosen auto-correlation curve with a "zero lag" of 0.5 and

a linear slope reaching zero at 1000 km (identical to the TM curve shown

in Fig. 1) leads to gz and T fields which are more consistent in the

vertical than the original fields. As a result, these more vertically

consistent data require considerably less adjustment when constructing

the fictitious grid point soundings.

The routine which produces the improved isobaric fields outputs

them in the same format as the original NMC data; thus either data

source can be used by the main program with no changes required.

Aside from data cards containing the Eddy weights, ak (Eq. 2),

only two other data cards are needed. The first card, which is read

in at the beginning of the program, contains six values. The first two,

which give the grid point coordinates of the North Pole on the desired

15

grid, enable the program to pick any grid array location from the

standard NMC grid. The array size used in our program is 21 x 21, but

the program is not confined to this particular grid size. The third

read in value determines how many consecutive sounding times will be

processed. Up to 14 have been processed with one submittal. The next

two card values tell the program which sounding time to begin with, and

whether more date times are needed than are on the tape. If the extra

radiosonde tape is needed, the fifth value initiates the read statement

that inputs the tape. The last card value is the total number of pres-

sure levels desired.

The second data card, read in from subroutine Objan, lists the

pressure levels needed for generating the 0 surfaces. If any desired

level is missing from the NMC grid tape, the program automatically

adjusts. If two levels are missing a new card listing the remaining

millibar levels can be read in if the lack of data is not considered

too serious.

After storing the Air Force data on a drum (which provides for

increased flexibility and speed), the Eddy weights are read in either

by cards or by tape. The task of computing the weights is treated by

another secondary program written to supplement this one. The smaller

secondary program punches one data card for every grid point. Each

card gives the weights for all radiosonde locations affecting that

grid point. Since most of the sounding locations are permanent land

stations, the smaller program need be submitted only once for a given

grid location, regardless of how many date times are used. If the

number or location of observations affecting a grid point does not

agree with the data card, the weights are recomputed by subroutine Eddy

which is called from subroutine Eddy2 (see Fig. 5).

The flow chart is self-explanatory with respect to the mechanics of

the program. Function Bint, referred to within the Bjenne block, per-

forms the interpolation described in Appendix A. There are a few small

subroutines that do not appear on the flow chart. Two of them (Filt3

and Extrap) should be briefly mentioned because of their important

16

contribution to the final output: Filt3 handles the smoothing of the

grid point values and Extrap performs the extrapolation of the soundings

(fictitious and real) below the ground.

All the soundings for a given sounding time are processed and

stored for later use by subroutine Bjenne. The values stored are

block number, latitude, longitude, e level, YM' p, relative humidity,

wind speed and wind direction. The e level values are calculated and

returned to Bjenne from subroutine Duquet. The stored wind data are

later returned to subroutine Haag for use in calculating the gradient

difference fields. (For a given date and time the average number of

soundings that affected our North American grid and had to be processed

was about 150.) The fictitious grid point soundings are processed and

the needed values stored by subroutine Objan. Again, the e level

values are supplied by subroutine Duquet. Subroutine Haag, after

receiving the scalar difference fields from Bjenne, computes the M

difference gradient field and then completes the calculations required

to arrive at the "final guess" fields.

The output given by the program comprises the Montgomery stream

function, pressure, and relative humidity grid point values on any

number of isentropic surfaces, ranging from 270 to 400°K in 5°

intervals. The "Call Haag" statement in the main program asks for

five levels; therefore, if additional levels are desired or more

sounding times needed, recycling is provided for as shown by the

program flow chart (Fig. 5).

Finally, the execution time on the Control Data 6600 for one 9

level and one sounding time is approximately 30 sec.

17

Appendix A

SPATIAL INTERPOLATION BETWEEN GRID POINTS*

*Taken from Rainer Bleck, 1968: A Numerical Technique for Calculating

Dry- and Moist-Adiabatic Trajectories in the Atmosphere, doctoral disser-

tation, Pennsylvania State University.

I

19

Appendix A

SPATIAL INTERPOLATION BETWEEN GRID POINTS

The task of interpolating in a two-dimensional grid is split up

into a series of one-dimensional interpolations in the x and y

directions. These interpolations in one dimension are performed

throughout by using four successive data points (a, b, c, d in

Fig. 6), two on either side of the point E under consideration.

Instead of using the standard procedure of fitting a third order

polynomial to these four points, two parabolas are fitted to the

two three-point sets (a, b, c) and (b, c, d). These parabolas

(dotted and dashed, respectively, in Fig. 6) are combined into one

curve between the two inner points, b and c, by taking a weighted average

of them (solid curve). The weighting is done linearly with respect

to the distance from the two points b and c, assuming that each of

the two parabolas gives best results at its center point. The linear

combination of the two parabolas results in a third order polynomial

which, of course, does not generally go through all four data points.

However, its inner section (between b and c) connects more smoothly

to the inner sections of the polynomials to the right and left (if

they are constructed in the same manner) than do the inner sections

of "common" third order polynomials. This is a desirable result,

because the interpolated field should have a transition as smooth

as possible across grid lines.*

If the two parabolas were weighted on the basis of the square ofthe distance from b and c, rather than linearly, the resultingsection of the curve would connect to the adjacent sections withno break even in the first derivative. The linear weighting waschosen, however, because a "quadratic" transition between the twoparabolas in the interval (b, c) seems to be too rapid aroundthe center of the interval.

20

The procedure of interpolating in two dimensions is illustrated in

Fig. 7. The point i, I, where the function value has to be calculated,

must be surrounded by 16 data points. In a first step, the one-dimensional

interpolation scheme is used to compute the function values at four

points A, B, C, D. These interpolations are performed at y = const.

Thereafter, the value at i, n is obtained by interpolating along the

line n = const, using the values at A, B, C, D.

The result is not altered if the points E, F, G, H (Fig. 7) are

chosen instead of A, B, C, D; i.e., if four interpolations at x = const

are performed first, and if the final answer is obtained by interpolating

along the line i = const.

21

0.6 I 1

0.5

0.4

0.3:

0.2

AFTER ONE.1 -ITERATION

-0.1 i 0 2 4 6 8 10 12 14

SEPARATION DISTANCE x 100 Km

Fig. 1 Auto-correlation curves for 'M and p difference fields.

22

0.6 1 11 1 I I

0.5

0.4

0.3

0.2

0.1

0.0

-0.I i II I I 0 2 4 6 8 10 12 14

SEPARATION DISTANCE x 100 km

Fig. 2 Auto-correlation curve for ~M difference field (gradientmethod).

644X641 494

801\ X491 448 7801 450793 617 40497

X 617 497. 617 518 X

8350 ig 773 X510 541 53878 V _594 _ · 54/ * 609

1n670 60 551 of 592 w 611.670 6300 , 5949 64

\X 765 )599 643 ,843 X 7 7 6 642 63 69

699 674 0*67, 7IX , 720 71 723

~/ X "^ ^^755 ~ ,,-Z03 _ 705

/ % ?-82 7

8 804 749

\^ N. 766 '768" ,839 809

%, x( 829\ x ^ X

Fig. 3 Pressure analysis on 0 surface, 3100 K, 7 March 1967, 00 GMT, showing cold front (heavy

line) and intersection of surface with ground (dashed line). Lower grid point values

are with the underground continuation of the e surface, and upper values are without.Values at black dots are actual radiosonde values.

24

1.2 I

0.8 —

0.6 0-

0.4

0.2

0 00.0

-0.20 2 4 6 8 10 12 14

SEPARATION DISTANCE x 100 km

Fig. 4 Auto-correlation curve for relative humidity difference field,

25

MAIN PROGRAM

Input desiredgrid location& sounding time. SUBROUTINE BJENNE

Put A.F. tapeon drum.

on .~. ~Decode sounding .SUBROUTINE OBJAN SUBROUTINE ADDPT..... Decode sounding

_ Call Haag date and time.

Recycle for Call Objan Input Isobaric Fit significantmore levels. ..... grid point data. points between

..... Process sound- ..... standard levelsRecycle for ings. Call Addpt to achieve verti-new sounding ..... ..... cal consistencytime. Call Duquet — Call Duquet of geop. height

and temp.End. Compute differ- Smooth fields of .....

ence fields (PM first guess Returnand p), use func- values.tion Bint to in-terpolate. Return with first

SUBROUTINE..... guess fields of~~SUBROUTI NE HAAG ~ Calculate first Tmand pCalculate first P I I \I/Mand P' SUBROUTINE DUQUETguess latitudi-DUQUETnal R.H. values.

Input Eddy .....weights (precom- Solve for R.H. _ Compute TM, p and

puted for a difference field R.H. at variouscomplete set of using Bint. e levels.radiosonde ..... .....stations). Return with dif- Extrapolate values

ference fields of below the ground.Call Bjenne M, p and R.H.

Return

Remove meandifferences.

Call Eddy2 --Call Edd2 SUBROUTINE EDDY2

Add TMscalercorrections togrid. Calculate grid

~~~..... ~point correct-Recycle for p & ions using EddyR.H. corrections. weights. (If sta-

~~~..... ~tions are missingCompute TMdif- compute newference gradient weights).field. .....

~~..... ~ Return

Recycle for TMgradientcorrections.

Return with fi-nal M, p andR.H. fields.

Fig. 5 Program Flow Chart

a b c d

Fig. 6 Bi-parabolic fit to four points.

o- -o l o---—0

0-- --- 0- -T0- -- --

I I I

II

°- " F -- o ....--

Fig. 7 Two-dimensional interpolation in a field of 16 grid points.

27

REFERENCES

Danielsen, E., 1959: The laminar structure of the atmosphere and itsrelation to the concept of a tropopause. Arch. Meteorol. Geoph.BioklimatoZ. U2(3), 293-332.

Duquet, R. T., 1964: Data Processing for Isentropic Analysis.TechnicaZ Report No. Z, Contract No. (30-1)-3317, Air ForceCambridge Research Laboratories.

Eddy, Amos, 1967: Statistical objective analysis of scalar datafields. AppZ. Meteorol. 6(4), 597-609.

Gustafson, A. F., 1964: Objective Isentropic Analysis. TechnicaZMemorandum No. 30, National Meteorological Center.

I