Environmental Modelling & Software 22 (2007) 1164e1174www.elsevier.com/locate/envsoft

Response of model simulated weather parametersto round-off-errors on different systems

S. Goel b,*, S.K. Dash a

a Centre for Atmospheric Sciences, Indian Institute of Technology, Delhi, Indiab Computer Services Centre, Indian Institute of Technology, Delhi, India

Received 3 August 2004; received in revised form 12 February 2006; accepted 8 June 2006

Available online 30 January 2007

Abstract

In this study, the weather forecasting model of the National Centre for Medium Range Weather Forecasting (NCMRWF) is used for exam-ining the characteristics of round-off-errors on three different computer architectures e PARAM 10K, SUNFIRE 6800 and Dec Alpha for severalmeteorological parameters such as precipitation, temperature at the surface and mid-atmosphere, and upper and lower level winds. It is wellknown that the implementation of floating point arithmetic varies from one computing system to another. As a result, meteorological parameterssimulated by numerical models on two different systems may deviate from each other and the difference field becomes larger as the model isintegrated for longer time, for example, in the scale of several months. This paper focuses on the reduction of such round-off-errors by a simplemethod of modifying the format representation of the initial data supplied to the model. In all the three systems, the model has been integratedfor 4 months starting from 4th May, 1996. It is found that after 5 days of model integration with the modified data, the round-off-errors becomeinsignificant. The rate of reduction of round-off-errors is fast up to a month of model integration and thereafter the rate slows down and stabil-ises. It is further noticed that at the end of four months of integration, the reduction in round-off-errors over the tropical region and oceans ismuch more than over the rest of the globe.� 2007 Elsevier Ltd. All rights reserved.

Keywords: Floating-point arithmetic; Round-off-errors; Model simulation; Iterative process; General Circulation Model (GCM); Spectral method

1. Introduction

It is well known that weather forecasting by numericalmethods is computationally intensive. The code of a NumericalWeather Prediction (NWP) model consists of a very large num-ber of floating point operations, usually of the order of 1012

when the horizontal resolution is about 150 km (Dash andJha, 1996). The number of operations increases with the spatialresolution of the model and the length of time integration. Usu-ally the numbers in the computers are represented by binary bitsgrouped into words. Due to the limitations in the computer ar-chitecture, the exact decimal values of numbers obtained afteran arithmetic operation cannot be stored in the machine with

* Corresponding author.

E-mail address: savita@cc.iitd.ac.in (S. Goel).

1364-8152/$ - see front matter � 2007 Elsevier Ltd. All rights reserved.

doi:10.1016/j.envsoft.2006.06.011

100% precision. As a result, the intermittent values obtainedafter each operation are correct up to a finite number of decimalplaces only. It is accepted that when a code is run on a computingplatform or a compiler, the floating-point environment intro-duces machine-level round-off-errors at the least significantbits of the communication (Rosinski and Williamson, 1997).These uncertainties propagate through the computation, emerg-ing as part of the result. In most cases, the use of double preci-sion arithmetic can reduce the computation error uncertainty.However, uncertainty due to round-off or computer truncationcan become serious for quantities based on a large number ofcalculations, such as iterative processes (Charles et al., 2006),calculations involving matrix products or inverses, calculationinvolving trigonometric or logarithmic functions, etc.

It is well known that the result of a NWP code depends onthe architecture of the computing system, and also on the com-piler being used for the application. It may so happen that

1165S. Goel, S.K. Dash / Environmental Modelling & Software 22 (2007) 1164e1174

a NWP model run on two systems with different architectures,configurations and compilers, yields widely different values ofweather parameters such as wind, temperature, moisture andrainfall due to different rounding-off procedures inherent tothe system. Sometimes, such machine dependent results maylead to unreasonably wrong predictions and hence lead to se-rious consequences. Hence, estimating the error growth hasa practical utility for checking the implementation of a codeon new compiling systems, new libraries and overall on differ-ent computing platforms (Rosinski and Williamson, 1997).Sinha et al. (1996) found significant differences in one-dayforecasts of temperature using several computing environ-ments such as Flosolver, Cray XMP216, IBM-RS6000 andMIPS-R8000. They also found that the intensity of the differ-ence field varies depending on the platform used. In a valida-tion study, Rosinski and Williamson, (1997) introduceda machine level perturbation into the input temperature fieldand studied the error propagation in the climate modelCCM2. In their study, machine rounding level perturbationsare introduced into all the variables. It may be noted thatNWP in the short range of few days is basically an initial valueproblem which depends on observed weather conditions at thezero reference time (UBC weather forecast). When the sameset of initial weather parameters supplied to different comput-ing systems slightly differ due to the difference in the waythose are stored in the machine, the results may be treatedas obtained due to the response of different initial conditions.The accuracy of the prediction using a NWP model dependson the accuracy of the initial condition to a large extent. Itis quite evident that even small round-off-errors in the initialconditions can have a significant negative impact on the modelperformance. Lorenz (1993) in his book ‘‘The Essence ofChaos’’ has postulated that even small undetected errors inthe observations constituting the ‘‘initial conditions’’ fora model can result in big differences in forecasts only a fewdays into the future (Numerical Weather Prediction System).He pointed out that the equations of motion which are non-lin-ear in nature are very sensitive to initial conditions. Such sen-sitivity means that substantially different weather forecasts canresult from slightly different initial conditions. The end resultof this dependence on initial conditions can be a forecast thatdiverges from reality and eventually approaches chaotic state.

As solution in NWP models are based on iterative process,i.e. the result obtained by integrating the model for a day isbeing used as the input to the next run; the inherent round-off-errors becomes more serious in successive integrations ofthe model for a longer time. Recent studies Selvam (1993)show that round-off error doubles on an average for each iter-ation of iterative computations. Round-off-errors propagate tothe mainstream computation and give unrealistic solutions inNWP and climate models that incorporate thousands of itera-tive computations in long-term numerical integration schemes(Selvam and Fadnavis, 1998). Deterministic chaos is a directconsequence of round-off-error growth in finite precision com-puter solutions of error sensitive dynamical systems.

The current study emphasizes that the growth of round-off-errors can not be stopped, yet the rate of growth of

round-off-errors can be controlled by truncating the differentparameters at a suitable decimal places before initiating modelintegration. In short, better the initial quality of data it is morelikely for the model to produce reasonably good predictions.To ensure that the results are meaningful it is necessary toquantify the round-off-errors. In spectral GCMs, there is an in-herent accuracy problem in calculating the Fourier coefficientsdue to the finite word length used in digital computers (Tosh-iba and Bede, 1970). Gentleman and Sande (1966) obtained anabsolute upper bound on the sum of square errors due toround-off accumulation. Two sources play major role in theaccumulation of round-off-errors; the architecture of the ma-chine (as the implementation of the floating point arithmeticon different systems varies), and the compiler implementationof a particular software being used to run the model.

The aim of this paper is to examine the characteristics ofround-off-errors on three compilers and to adopt methods ofreducing their magnitudes. In the current study three comput-ing platforms are used. Amongst these three systems, Dec-Alpha and SUNFIRE 6800 are 64 bit architecture systemsand PARMA 10K is a 32 bit architecture system. However,8 bytes are used for both real numbers and integers while in-tegrating the model. The configuration of these systems usedfor this comparative study is given in Table 1. Section 2 dealswith the T80L18 model description and the specifications. Italso gives a brief description of the complex system of equa-tions involved in a NWP model and the computational powerrequired in solving such system of equations. Section 3 de-scribes the methodology adopted for reducing such errors.The results are analysed in Section 4 followed by the conclu-sions in Section 5.

2. Spectral GCM at resolution T80L18

National Centre for Medium Range Weather Forecasting(NCMWRF), India uses the spectral GCM at resolutionT80L18 for operational forecasting. The accuracy of spectralrepresentation depends upon the type of truncation and the or-der of truncation or the total number of waves. The presentmodel has a triangular truncation with a total wave numberof 80 (hence the name, T80). It has 128 grid points alongthe North-South direction, 256 points along East-West

Table 1

Basic configuration

Computer System: Dec Alpha SUNFIRE 6800 PARAM 10K

Architecture 64 bit 64 bit 32-bit

Operating System Dec Unix Sun Solaris

Ultra III

Sun Solaris

Ultra II

Compiler Dec Unix F77 Sun Workshop

5.0 F77

Sun Workshop

6.0 or Forte

Developer 6

Precision for:

Real 4 bytes 4 bytes 4 bytes

Double 8 bytes 8 bytes 8 bytes

Long int 8 bytes 4 bytes 4 bytes

1166 S. Goel, S.K. Dash / Environmental Modelling & Software 22 (2007) 1164e1174

direction and 18 levels in the vertical direction. This ensuresa horizontal resolution of 1.4� � 1.4�. The radiation calcula-tions however, are done on a coarser Gaussian grid of81 � 162 (Bourke et al.,1977). The spectral model is basedon the equations of conservation of mass, momentum, energyand moisture which are non-linear partial differential equa-tions involving three spatial coordinates and the time as thefourth coordinate. The hydrostatic equation and the thermody-namic equation are combined and similarly the surface pressu-re tendency equation and continuity equation are combined.Hence, finally one gets a set of five coupled equations infive time dependent meteorological parameters such as thetwo horizontal components of wind, temperature, moistureand surface pressure. These five non-linear coupled equationsconstitute a closed system which in principle can be solved atall future times from a given initial condition and with pre-scribed surface boundary conditions. In the numerical solutionof the coupled equations, the differential operators can bediscretized by numerical schemes such as by finite difference,finite element, spectral, Semi-Langrangian etc. In compactform the set of five equations can be written as

vX

vt¼ DðXÞ þPðXÞ ð1Þ

where X is any model variable such as wind, temperaturehumidity and surface pressure, D stands for dynamical pro-cesses such as advection, pressure forces, etc. and P standsfor physical processes such as radiation condensation, surfaceboundary processes etc. In the NCMRWF GCM the spectraltransform method is used for solving partial differential equa-tions on a sphere which is widely used in atmospheric models.The spectral method is popular in NWP because of variousreasons including those of the easy way of handling the poles,less computational time requirements etc. In the spectralGCM, vorticity equation, divergence equation, thermody-namic equation, surface pressure equation and the moistureequation are transferred into spectral form by assuming thatall prognostic variables and tendencies can be expressed inspherical harmonic expansions (Bourke et al.,1977). In thespectral method each of the meteorological parameters X arerepresented by the following double series at any instant oftime.

Xðl; m; s; tÞ ¼XM

m¼�M

"XNðmÞn¼jmj

Xmn ðs; tÞPm

n ðmÞ#

eiml ð2Þ

Here the spherical co-ordinates are l,m (¼ sin f) and s(¼ p/ps). l, f, p, ps represent the longitude, latitude, pressure andsurface pressure, respectively, and t represents the time. Xm

n

are the complex spectral co-efficients and Pmn are the associ-

ated Legendre functions of the first kind of order m and degreen. Also m represents the zonal wave number and n is oftencalled the two dimensional index or the total wave number.M is the highest Fourier wave number included in the east-west representation and N(m) is the highest degree of the asso-ciated Legendre function included in the north-south

representation. As m and n increase, they correspond to in-creasing the model resolution in the horizontal and hencethe number of mathematical operations.

3. Methodology

In this study, T80L18 model has been ported on PARAM 10K and SUN-

FIRE 6800 (Sunfire) installed at the Computer Services Centre of IIT Delhi.

PARAM 10K is a 4-node parallel supercomputing system of the Centre for De-

velopment of Advanced Computing (CDAC). It is a system based on SUN UL-

TRA 250 Enterprise Servers. Each ultra 250 Enterprise is based on Symmetric

Multi-Processor (SMP) architecture with 2 CPUs. Sunfire is a shared memory

supercomputer with a cluster of 24 processors. It is a multi domain system

configured with two domains at present. Number of processors distributed

on these domains is 16 and 8 respectively. T80L18 model has also been inte-

grated on COMPAQ workstation (DEC ALPHA) available at the Centre for

Atmospheric Sciences, IIT Delhi. It has a different architecture with 64-bit

word length. The three compilers involved in this comparative study are

F77 (Fortran 77) of Dec Alpha with Dec-Unix operating system, F77 of Sun

Workshop (Version 6.0) of Sun 6800 and F77 (Version 5.0) of PARAM

10K. The respective compiler options used to get double precision for real

numbers and integers are as follows:

F77 -convert big_endien -integer_size 64 <filename> on Dec Alpha

F77 -dbl -O4 -xarch¼ v8 <filename> on Sunfire

and

F77 -dbl -O4 <filename> on PARAM 10K.

The model has been integrated on all the three machines for four months

starting from 4th May 1996. The duration of each time step of integration is

15 min. Hence, in one day, there are 96 time steps. Each time step involves

computation of spectral transforms twice, first for the dynamics and second

for the physics computations. This is done in two different modules. The

model simulated meteorological fields obtained from both PARAM 10K, Sun-

fire and Dec Alpha have been compared using Grid Analysis and Display Sys-

tem (GrADS). To show the small values clearly different scales and contour

intervals are chosen in the figures.

The meteorological parameters obtained after model integration are post

processed in order to investigate the errors due to round-off in all the three

compilers. To start with the error growth has been analysed for a short period

of 5 days only in case of surface temperature. Considerable differences have

been noticed between the results obtained from PARAM 10K, Sunfire and

Dec Alpha. At the initial stage itself, the difference in surface temperature

at the second decimal point is observed at eight grid points of the model.

This difference in temperature is really large.

Thus to minimize the round-off-errors, emphasis has been given on the im-

provement of the initial data presentation. It may be noted that the input data

set for the model integration primarily contains three files: (i) the initial data

for surface parameters such as surface temperature, surface pressure, snow

amount, soil temperature, roughness length and plant resistance; (ii) atmo-

spheric parameters such as wind, divergence, vorticity, and specific humidity

at 18 levels of the atmosphere; and (iii) sea surface temperature (SST). Out

of these three files, the first two are unformatted, i.e. in binary form and the

SST file is an index sequential file. Since binary system of data writing/reading

entirely depends on the architecture of the computer, the same data file may be

read up to different accuracy by different systems. To make initial data iden-

tical up to a certain decimal point, all the meteorological parameters are read

in to an ASCII file with a suitable format by fixing the number of significant

decimal places. Since the fluid dynamic equations used in NWP models deal

with very small values, as small as of the order of 10�23 and also simulta-

neously very large values of the order of 1017, it is necessary to check the pre-

cision of data on all three platforms by reading those to a binary file and then

back to ASCII file. Such checking procedure has been done outside of the

model code. Since SST data is read from an index sequential file, in which re-

cords are not read sequentially, simply truncating the last digits and fixing the

SST data up to 6 decimal places makes the files identical on both the systems.

S. Goel, S.K. Dash / Environmental Mo

4. Results and discussion

As mentioned earlier, the results of model integrations con-sist of several atmospheric fields such as wind, temperatureand humidity at 18 vertical levels and also surface pressureand rainfall at different stages of integration. In this section,the above meteorological parameters are examined but mainlyrainfall and temperature at 500 hPa are considered for discus-sion after 5, 15, 30 and 122 days of model integration. In orderto facilitate the comparison of meteorological fields obtainedfrom all the three computing systems, the results are consid-ered at fixed intervals.

Precipitation is the most important output of any weathermodel. Fig. 1 shows the difference in precipitation fields after5 days of model integration on PARAM 10K and Dec Alpha.The difference field ranges from �0.07 cm to 0.06 cm asshown in Fig. 1. The difference field obtained after the modi-fication in the initial data files is shown in Fig. 2 and it rangesbetween �0.02 cm and 0.03 cm. It may be noted that over therest of the globe not shown in Figs. 1e4, the differences aremuch less in comparison to the displayed region. The purposeis to show the extreme values of round-off-errors. ComparingFigs. 1 and 2 it is seen that the modifications in the input datafiles have enabled to reduce the round-off errors considerably.From Figs. 1 and 2 it is clear that not only the magnitude ofround-off-errors is reduced but also there is no round-off-errorat most of the places over the globe after the modifications inthe initial data files as input for the model integration havebeen introduced.

Rainfall differences are also examined after 15 days ofmodel integration, and the range of round-off-error is foundbetween �0.04 cm and 0.07 cm. After modification in the in-put data, the range of difference has been reduced to between

�0.03 cm to 0.05 cm. Differences in the round-off-errors after15 days of model integration are not reduced as much as incase of 5 days integration. However, the reduction is signifi-cant at a number of places around the globe.

After 30 days of model integration, the range of rainfall dif-ference becomes�0.09 cm to 0.18 cm as shown in Fig. 3. Aftermodifying the format of the initial data, the range is reducedfrom (�0.09 cm, 0.18 cm) to (�0.06 cm, 0.04 cm) as shownin Fig. 4. To improve the visualization, the region where differ-ences occur is chosen for display in Figs. 3 and 4 is and also thevalues marked in these figures are in 10�3 cm. Comparing Figs.3 and 4, the reduction in round-off-errors for the tropical regionis evident. After 30 days of model integration, the round-off-er-rors have reduced not only in terms of the range of the errorsbut also in terms of absolute magnitude. After 2 months ofmodel integration, the interval is reduced from (�0.8 cm,1 cm) to (�0.8 cm, 0.6 cm). Similarly, after 4 months of modelintegration, the range of round-off-errors is reduced from(�0.7 cm, 0.5 cm) to (�0.7 cm, 0.4 cm). It is observed thatthe rate of reduction of errors in the beginning of model inte-gration is very high with considerable change up to 30 daysand afterwards the rate of reduction becomes too slow. Ulti-mately, after 4 months of model integration, the error becomesstable. Thus, the round-off-errors do not reduce further after4 months of integration.

In the atmosphere, the vertical level at 500 hPa is very sig-nificant. This level represents the level nearest to the severeweather events. Most of the clouds form close to this mid-level. It is also the level of non-divergence. So we examinethe temperature field at 500 hPa level as simulated by themodel with and without modifying the format of the initialdata. The difference in temperature at 500 hPa betweenPARAM 10K and Dec Alpha is quite apparent since the

1167delling & Software 22 (2007) 1164e1174

Fig. 1. Differences (PARAM 10K e Dec Alpha) in rainfall in 10�2 cm after 5 days of model integration. Contour interval is 1.

1168 S. Goel, S.K. Dash / Environmental Modelling & Software 22 (2007) 1164e1174

Fig. 2. Same as in Fig. 1 except with modified data.

beginning of model integration or after one day integration.The difference grows fast after each day of integration there-after. After 5 days of model integration (Fig. 5), temperaturedifference increases to extreme with �14 �C at one pointover the globe 175�E 40�S. Similarly the extreme positive dif-ference field is 10 �C at one of the point 175�E 75�N. Aftermodifying the format of initial data representation, for the firstthree days of model integration, the difference is almost niland after 5 days there is a drastic reduction in the errors. Asshown in Fig. 6 the round-off-errors reduce to (�1.2 �C,1 �C) after modification of initial data format. The occurrence

of these errors is confined to only three small areas in thesouthern belt of 30�e60�S in Fig. 6. Comparison of Figs. 5and 6 clearly shows the minimization of error to a great extent.It is found that after 15 days, not only the range is reduced butalso error has vanished at a number of places around the globe.

In a seasonal run on PARAM 10K and Dec Alpha, the er-ror growth averaged over four months is depicted in Figs. 7and 8, after using the initial data without modifications andwith modifications respectively around the globe. The aver-age temperature error before modification of data and afteris in the range (�6 �C, 4 �C) and (�3 �C, 2 �C) as seen in

Fig. 3. Differences (PARAM 10K e Dec Alpha) in rainfall in 10�3 cm after 30 days of model integration. Contour interval is 10.

1169S. Goel, S.K. Dash / Environmental Modelling & Software 22 (2007) 1164e1174

Fig. 4. Same as in Fig. 3 except with modified data.

the Figs. 7 and 8 respectively. Similarly, after the seasonalrun for 4 months on Sunfire and Dec Alpha, the averagetemperature error before modification of data and after isin the range (�3.5 �C, 2.5 �C) and (�2 �C, 2 �C) as seenin the Figs. 9 and 10 respectively. Even without modification,the average growth of round-off-errors is in the range(�3.5 �C, 2.5 �C) which is quite low in comparison towhat has been seen in the case of PARAM 10K and Dec Al-pha. This implies that the average growth of round-off-errorsremains low in the higher bit architecture systems. Neverthe-less, it is important to note that the modifications in the

initial data still have considerable effect in reducing the av-erage growth of round-off-errors, as it has reduced from(�3.5 �C, 2.5 �C) to (�2 �C, 2 �C) in the case of Sunfireand Dec Alpha (Figs. 9 and 10).

It is found that tropical regions in both the hemispheres arethe main regions of error growth. Areal average growth rate ofround-off-errors for southern and northern tropics from day0 to 30 is investigated on PARAM 10K & Dec Alpha aswell as on Sunfire & Dec Alpha and the results are shownin Figs. 11e12 and 13e14 respectively. For convenience allthe values in Figs. 11e14 show absolute values only. In

Fig. 5. Differences (PARAM 10K e Dec Alpha) in temperature in �C at 500 hPa after 5 days of model integration. Contour interval is 2.

1170 S. Goel, S.K. Dash / Environmental Modelling & Software 22 (2007) 1164e1174

Fig. 6. Same as in Fig. 5 except with modified data.

general, the positive round-off-errors are dominant in northernand negative round-off-errors are dominant in the southerntropical regions. The possible reason for such differences isthat the northern tropical region is full of land areas whereassouthern tropic is covered by oceans. In the ocean area, modelis prescribed with observed SST whereas over land area thesurface temperature is computed by the model based on ther-mal balance equation.

For both the pair of systems PARAM 10K and Dec Alpha(pair I) & Sunfire and DecAlpha (pair II), southern tropical av-erage growth rate for round-off-errors has reached maximum

up to 2.35 �C (Figs. 11 and 13), and in northern region the av-erage growth of round-off-errors has maximum value 4.53 �C(Figs. 12 and 14). After modifications, in southern region ofpair I, the averaged growth rate of round-off-errors diminishedsignificantly up to 8 days and up to 12 days for pair II. Also,after modifications in the initial data, for southern tropic thisgrowth remains at maximum value 0.86 �C which is less than1 �C occurred for pair I. For pair II, where both the systemsare with 64 bit architecture, the round-off-errors growth rate re-mains strictly less than 0.30 �C. In northern tropical region theaverage growth of round-off-errors after format modification

Fig. 7. Differences (PARAM 10K e Dec Alpha) in temperature in �C at 500 hPa after 4 months (seasonal) of model integration. Contour interval is 1.

1171S. Goel, S.K. Dash / Environmental Modelling & Software 22 (2007) 1164e1174

Fig. 8. Same as in Fig. 7 except with modified data.

has diminished significantly up to 8 days for pair I and up to19 days for pair II. It is interesting to note that the averagegrowth rate for round-off-errors is always less than 0.74 �Cfor pair I and less than 0.9 �C for pair II. In 90% of the casesfor pair II in the northern tropic it has reduced to even lessthan 0.5 �C after modifications. Figs. 13 and 14 show that inthe northern tropical region the average round-off-error growthremains much lower after modifications. This clearly showsthat the round-off-errors reduced further in case of Sunfireand Dec Alpha both being 64 bit architecture systems. After

4 months of integration, the averaged round-off-errors liewithin the maximum limit of 1.5 �C in any of the tropics andirrespective of any of the pair of systems.

Such saturations can be easily explained by stating that themodifications in initial data format help in reducing round-off-errors to a large extent. However, in the model there aremillions of floating point arithmetic, which is the source ofgeneration of round-off-errors at the least significant decimalposition. As implementation of the floating-point arithmeticis a system architecture dependent feature, the round-off-errors

Fig. 9. Differences (Sunfire e Dec Alpha) in temperature in �C at 500 hPa after 4 months (seasonal) of model integration. Contour interval is 1.

1172 S. Goel, S.K. Dash / Environmental Modelling & Software 22 (2007) 1164e1174

Fig. 10. Same as in Fig. 7 except with modified data.

due to limitation of the number of bytes used for the variablescan not be controlled easily at each arithmetic operation duringthe course of model integration. Once the error has been prop-agated at the least significant place, it goes on accumulatingwith successive integrations of the model for more numberof days.

The analysis on three different systems strengthen the viewthat the rate of growth has reduced up to a month considerablybut after that round-off-errors can not be reduced significantlyfurther with this methodology. In each case, the initial differ-ence is quite high for all the parameters considered. Also glob-ally, after a month of model integration, the error growth ismore or less stable up to 4 months.

0.00

0.25

0.50

0.75

1.00

1.25

1.50

1.75

2.00

2.25

2.50

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29

PreviousModified

Tem

pera

ture

in ºC

Number of Days

Fig. 11. Growth of round-off-errors (PARAM e Dec Alpha) in temperature at

500 hPa averaged over southern tropical region.

5. Summary and conclusions

Weather forecasting models are among the most computa-tionally sensitive applications. Such applications involvea large number of floating point operations leading to round-off-errors. Whenever such applications are run on differentcomputing environments with different compilers and archi-tectures, deviations in simulations occur. The characteristicsof such round-off-errors in temperatures at the surface and at500 hPa, and rainfall are examined using NCMRWF modelat resolution T80L18 in case of three different systemsnamely, PARAM 10K, SUNFIRE 6800 and Dec Alpha. Thesesystems have different architectures and compilers. After

0.00

0.50

1.00

1.50

2.00

2.50

3.00

3.50

4.00

4.50

5.00

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29

PreviousModified

Tem

pera

ture

in ºC

Number of Days

Fig. 12. Growth of round-off-errors (PARAM e Dec Alpha) in temperature at

500 hPa for northern tropical region.

1173S. Goel, S.K. Dash / Environmental Modelling & Software 22 (2007) 1164e1174

5 days of model integration on both the systems separately, therate of growth of the round-off-errors in surface temperatureremains quite high which gives the indication to provide theinitial data to the model as accurately and identically as pos-sible. It is noticed that in the free format, the representationof the initial weather parameters varies with the type of thecomputer architecture and hence the initial data on two differ-ent systems differ to start with. In order to reduce the growthof round-off-errors, a suitable format is provided to each of theparameters in the initial data file so as to keep adequate deci-mal points fixed. Thus initial data identical up to a particulardecimal representation is supplied to both the systems.

In general, after a model run on PARAM 10K and Dec Al-pha in the time scale of 4 months, the average difference intemperature at 500 hPa is in the range (�6 �C, 4 �C). Thisrange reduced to (�3 �C, 2 �C) after modifications aroundthe globe. Thus, by making the initial data exactly the same

0.00

0.25

0.50

0.75

1.00

1.25

1.50

1.75

2.00

2.25

2.50

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29

PreviousModified

Tem

pera

ture

in ºC

Number of Days

Fig. 13. Growth of round-off-errors (Sunfire e Dec Alpha) in temperature at

500 hPa for southern tropical region.

0.00

0.50

1.00

1.50

2.00

2.50

3.00

3.50

4.00

4.50

5.00

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29

PreviousModified

Tem

pera

ture

in ºC

Number of Days

Fig. 14. Growth of round-off-errors (Sunfire e Dec Alpha) in temperature at

500 hPa for northern tropical region.

up to a fixed floating point on both the systems, the round-off-errors can be reduced up to 50%. When the model runson Sunfire and Dec Alpha for 4 months, the average differencein temperature at 500 hPa is in the range (�3.5 �C, 2.5 �C).This range of round-off-errors reduced to (�2 �C, 2 �C) afterinitial data modifications. Thus the round-off-errors reducedapproximately 30%. It is noticeable that without initial datamodifications, the range of round-off-errors between Sunfireand Dec Alpha is much lower in comparison to range ofround-off-errors between PARAM 10K and Dec Alpha foran average run of 4 months.

It is observed that growth of round-off-errors for tempera-ture at 500 hPa is more in the tropical regions. In the southern(northern) tropic, the average growth of round-off-errorshas the maximum absolute value of 2.35 �C (4.53 �C). Aftermodification in the format of initial data, on PARAM 10Kand Dec Alpha the averaged growth rate of round-off-errorshas reduced to 0.86 �C and 0.74 �C for southern and northerntropical regions respectively. On Sunfire and Dec Alpha thecorresponding values are 0.3 �C and 0.9 �C respectively.

It is well known that the round-off-errors cannot be elimi-nated completely since during the course of model integrationa large number of intermittent floating point operations arecarried out. Such operations propagate the error. Nevertheless,the simple method of data formatting in the input data for themodel integration has helped in reducing the difference in theround-off-errors between three different computing platformsas shown in this study.

Acknowledgements

The authors would like to acknowledge NCMRWF for pro-viding the code of weather forecasting model at resolutionT80L18 and the necessary initial data. Thanks are due toMr. Manik Bali of C-DAC and Mr. Subrat K. Panda of CASfor their initial help in porting the model on PARAM 10Kand Dec Alpha respectively. One of the authors, Ms. SavitaGoel is very much thankful to Prof. B.P. Pal, Head, CSC forhis support and encouragement.

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