Some problems of determining the water equivalent
of the snow cover
Ì . H egedus and Ê . Szesztay
Research Institute for Water Resources D evelopment
Budapest, H ungary
sv MMARv : D uring recent years 52 stations and 2 experimental areas have been installed in the
framework of preliminary investigati ons for elaborating à national programme on âï î ÷
measurements. On evaluating the fi rst results of these measurements t he following problems are
discussed:
— T he conditions of equal accuracy in snow depth and water equivalent measurements;
— T he real accuracy in determining âï î ÷ density from depth and water equivalent data;
— Factors aff ecting variability of simultaneous snow depth and density values along snow courses
and within small areas;
— Theoretical bases of pl anning à combined application of snow depth and snow density
measurements.
itasv M : D urant les ñ1åãø åãåâ àï ï ååâ 52 st ations et 2 terrains expbrimentaux ont ece btablis dans
les limites des recherches prbliminaires pour Ãå1àÜî ãàñ1î ï d'un programme national de mesures
de la couche de nei ge. Cvaluant les premiers rdsult ats de ces mesures, on discute les problbmes
suivants :
L es conditions de ò åò å ðãåñ1â1î ï pour les mesures de hauteurs et ñÃåñ1ø ÷à!åï ñ li quide de la
couche de nei ge;
Üà ðãåñ1â1î ï ãåå11å de l a dáterminati on de l a densitb de neige ñÃàðãåâ les donn6es sur hauteur
et Ãåñ1ø ÷à1åï ñ liqui de de l a couche de neige;
les facteurs exercant une infl uence sur la variation des donn6es ÿ ò ï !ñàï ååâ obtenues pour
l a hauteur et l a ñ1åï ÿ ñå le l ong des lignes ø ÷î ò åñã1ñ1èåâ et dans les petits terrains;
L es bases thboriques pour btablir un plan d'application ñî ò Û ï åå des mesures de hauteurs et
de densitbs de la couche de neige.
I n H ungary (93,000 sq. km) the depth of snow cover has been systematically recorded
for about 80 years, through à national rain gauging system consi sting now of 1000 stations.
For registering the water equivalent of the snow cover , à national network with 52 stations
(fi g. 1) has been established and gradually developed by the Research Institute for
Water Resources D evelopment since 1960. Some results of investigations carr ied out in
order to devel op measurement techniques and network operation in H ungary, are
discussed in this paper . M ost of the results rely on measurements made at snow density
stations and in two hydrological experimental areas.
A s regards the cal culation and forecast of fl ood r unoff , application of i ndices character izing areal values in an indirect way may often be suffi cient. For determining the
attainable accuracy and for working out computation specifications, however, à detailed
knowledge of the snow regime is indispensable even in this ñàâå.
M ET H OD O F M EA SU R E M EN T S
The value of the water equivalent of the snow cover at à point may be determined directly
by sampling and by measuring the weight of the sample. Sampling is rather troublesome,
especially if the âï î ÷ cover is thick, while snow depth is easy and simple to measure
6 16
Ì . JI egedus and Ê . Szesztay
U sually à combined approach is used: çï î ú depth is measured at à large number Î Ãpoints
and both sets of data (çï î ÷ depth and water equivalent) are obtained at some of the
points. Average water equivalent of the snow cover (along snow courses or within an
area) is then calculated as à product of the average values obtained from snow depth
and âï î ðó density data.
One of the most important problems of this sol ution is assessing the r atio of depth
measurements and sampling points. In deciding on thi s ratio the conditions of equal
accuracy of the two kinds of measurements on the one hand, and the variability of snow
depth and density (along à course or within an area), on the other, must be taken into
consideration.
CON D I TI ON S OF EQUA L A CCU RA CY OF SN OW D EPTH A N D
WA TER EQUI VA L EN T M EA SUREM EN TS
D ue to conditions of measurements and instrumentation, the depth of snow cover is
measured with an accuracy of 1 cm and its water equivalent with an accuracy of 1 mm.
It is directly conceivable that conditions of equal accuracy are fulfi lled only in the case
of 0.10 g/cu. cm snow density. In this ñàçå à one-centimetre variation in snow depth
represents à one-millimetre variation in water equivalent. Should the density be higher
than 0.10 g/cu. cm, it is the water equivalent, and should an opposite case exist, it is
the snow depth, that measured with higher accuracy than the other.
When examining the conditions of equal accuracy, the thought may be raised that for
the factor measured ÷÷1Û î ú åã accuracy the diff erence might be compensated by increasing
the number of measurements (repetitions). I f the statement of the theory of random errors
0 ,6
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Con s i der i ng
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an d 1 mm i n me asu r i ng sn ow dep t h
an d wa t er e qu i v al en t r e sp e c t i v e l y
an d su pp os i ng à Gau s s i an
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Number of r epet i t i on s r equ i r ed f or equal
5
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20
50
100
ac c ur acy of t he measur ement s
F1GURR 2. Condi ti ons î / ' equal accuracy i n snow dep th and water equi valent measurements
6 18
Some p r oblems of éeter mò ò g the water equi valent of the sno w cover
accor din g t o which r esult ant accuracy is var ying w it h t he squar e r oot of r epet it ionsi s accepted as an appr oxim at i on , the con dit i ons of equal accur acy ar e expressed by t he
curve of fi g. 2.
I n evaluat ing the data of fi g. 2 t he f act sh oul d be consi dered, th at samp li ng cannot
actually be rep eated because of t he prev ious m easur emen t . A n ap pr ai sal of measurement
accuracy cannot be separated t heref ore f r om t he analy si s of v ar iabili t y of t he dept h and
d ensity al o n g sn ow cour ses and wit hin areas.
A C CU R A CY OF SN OW D EN SI T Y D ET ER M I N A T I ON S
A t calcul at in g aver age densit y val ues f or à cour se line or f or an ar ea, it i s t he densi ty :
h
10 v
com puted f or the indi vidual sam pl in g point s t hat m ust be t ak en as à st ar t in g p oin t
(h repr esents t he w at er equ ivalent in m ill im et res, v t he sn ow dep th in cm , w hil e t the
den sity in g/cu . cm) . I n obtainin g k now ledge of the m easurement accur acy of h an d î
( 1 m m an d 1 cm , respectivel y) t he quest ion ar i ses, h ow m any decim al s sh oul d be accepted
in com p ut in g t he density accor ding t o equat ion ( 1).
I n or der t o deter m ine t he r eal accur acy of t val ues, the densit y changes correspond in g
t o un it var iati on s i n sn ow depth and w ater equivalent under var i ous con dit i on s ar e t o
be exam ined . T hese are expr essed by the part ial diff erenti al s.
É
t)h
1
10 v
(2 à)
ot
Ää
h
Ääã
( 2 Ü)
an d
A s indicated by t he rel at i onship in fi g . 3, un der usual con di ti on s of àï î â m easurem ent s
an accur acy t = 0.00 1 cannot be achieved, even i f only the h igher of t he tw o er r or s i s
taken int o accoun t and t he possi bil it ies of superp osit ion s ar e neglect ed . I f i t is necessar y
t o mak e f ul l use of the i nf orm at i on obt ained f r om t he measurement s, i t m ay be recom mended t p use the th ir d decim al of t he den si t y v al ues in t he cases above l ine À Â Ñ in
fi g. 3, wh iIe t he sam e i s per mi ssibl e in t he r ange above l ine àÜñ. T he t hir d decim al ,
however , sh oul d not be empl oyed in t he case of v an d h v al ues in dicat ing p oint s bel ow
line abc.
F r om à pr act ical p oint of view , t he r esul t s of t he above analysi s m ay be sum med up
by t he r ule that — if the accuracy of depth measur em ent s is 1 cm , wh il e t hat of w ater
equivalent m easur em ent s 1 m m , in the case of âï î ÷ dept hs sm al ler t han 20 cm t he
density i s t o be cal cul at ed usin g tw o decimal s, w her eas at à depth above t his li mi t w it h
three decim al places m ay be used .
I N T ER PR ET A T I ON O F W A T ER E QU I V A L EN T V A L U ES BY ST A T I ON S
I n deter mining air temper at ur e an d hum idity , pr ecipit at i on , w ind vel oci ties an d m any
other m eteor ol ogical and hydr ol ogical f act or s, observat ions ar e based on t he assum p ti on
that , in the case of appr opr iately selected st at i ons, obser vat i on s at à sin gl e point yiel d
6 19
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values characteristic of à l arger environment. It is known that this assumption is not
entirely valid even in the case of the above "conventional " factors.
For observations based on à single measurement (sample) this view becomes untenable
in the case of factors which show considerable changes even within very small distances.
The data of fi g. 4 give à numerical characterization on the well-known fact, that the water
equivalent of snow pertains to thi s second group of factors.
The upper part of fi g. 4 represents, on the basis of average values of 20 samples taken
along an observation course of 100 ò length, daily variations in depth and density of
snow cover . I n l ower part of the fi gure, the distribution of individual values obtained
from the 20 sampl ings are indicated, based on data of measurements carried out on
17ãÛ Ýåñåï é åã.
D istribution of the 20 samples was analysed in à similar way for the 22 other occasions
indicated in upper part of the fi gure. This provided the possibility of following the
time-variations of snow depth and density variability along à course. For this purpose
the amplitudes of the whole r ange of fl uctuations (Dv and Dt), their relative values
(100Dv : v~ and 100Dt : te), as well as the coeffi cients of variability (Ñî and Ct ) have been
determined for every each instant examined.
D ata il lustrated in the middle part of fi g. 4 show that the variability of snow depth
samples is considerably greater than that of çàï î ÿ density. Essential changes of the
variability parameter s occur usually in the period of considerable variations in depth or
density. À summary of this process is given in fi g. 5 where the variabil ity parameters have
been pl otted in ñî -ordinates î and t, with indication of the chronological order.
I n the fi rst years of network development, 20 snow depth and 20 snow density samples
were taken at all stations along an observation line of 100 ò length. A verage values
computed on the basis of these measurements have been accepted as values of the water
equivalent of snow cover at stations. It is on the basis of this experimental period,
yielding certainly ò î ãå samples than reasonable, that we intend to investigate the observational methods developed abroad and to work out à solution adaptable to conditions
in H ungary.
ÒÍ Å R A T I O O F ÒÍ Å D EPT H A N D W A T ER EQU I V A L EN T O F SA M PL I N G S
Should the time required to fi nd access to the individual observation points be shorter
than that required to execute the measurements, then à reduction in time of measurement
will become important. À way of attaining this reduction ò àó consist in measuring only
snow depth and dispensing with sampling at certain points. The necessity of à combmed
evaluation of data referring to depth and density ò àó arise also in other cases(snow
depth observations carried out by aerial surveys, j oint evaluation of data obtained at
stations installed for depth measurement and sampling, etc.).
The ratio of the two measurements is fi rst examined from the aspect of snow courses,
and later from that of areal averages. The composition of the observational data has
been expressed by à percentage value:
ë„ = 100 %„ : Ì „
giving the relative number of the sampl ing data. The accuracy of the averages is indicated
by the deviations related to the most reliable value h~, determined for è„ = 100 per cent :
À Ü = / '(ëä) ,
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(4)
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o f d a t a o b t a i n ed a t o n e o f t h e h i g h est st at i o n s i n t h e c o u n t r y . D at a o b t a i n ed f r o m
30 m easu r em en t s w er e a n al y sed . E ach m easu r em en t c o n si st s o f N Ä = 20 sam p l i n g s a n d
N Ä = 20 üï î ÷ d ep t h m easu r em en t s, a l l car r i ed o u t al o n g à b a se l i n e o f 100 m l en gt h .
A v er a ge w a t er eq u i v al en t h o, accep t ed a s à b asi s f o r co m p a r i so n , i s t h e ar i t h m et i c m ean
o f t h e 20 m ea su r em en t s. T h e n u m b er o f sam p l i n g p o i n t s w as t h en g r ad u al l y r ed u ced
t r y i n g t o m ai n t a i n à u n i f o r m d i st r i b u t i o n al o n g t h e co u r se — à ï ñ1 f o r av er age v al u es.
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as w el l as d ev i at i o n s h ; w er e c o m p u t ed f r o m i = 18, 16, 14 .. . 2 sam p l in gs.
F r o m à su r v ey o f r el at i o n sh ip s t l h = f (n Ä) b y i n d i v i d u a l ser i es o f m easu r em en t s,
ev i d en ce h a s Üååï o b t a in ed t h a t i n t h i s r el at i o n sh i p , t h e d ep t h o f sn o w co v er p l a y s an
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e q u i v a l en t
s amp l i n g s ,
,
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a long à snow cour se
6 2 4
Á î ò å p r o b l e m s of
d e t e r m i n i n g t h e w a t e r a q u i v a l e n t of
t h e s n o w c o v er
important r ole and it is the expression
ÀÉ = / '(è„ , î )
(7 )
that it seems most useful to start from.
D ata in fi g. 6 show that in the ñàâå of the station considered— disregarding snow depth
values lower than 5 cm— accuracy is generally decreased by only 2 to 3 per cent, if sampling
is executed at only 20 to 40 per cent of all the points. For two curves (15 cm and 50 cm
depths) the points serving as basis of the analysis have been also given. Scattering of the
points representing averages of 5 to 10 cases directs attention to the fact that the relationship examined is continually varying in time according to rules discussed relating fi g. 4.
Under unfavourable conditions deviations three times greater than those read from
fig. 6 may occur .
A n example of the relationship according to equation (5), in the ñàâå of areal averages,
is presented in fi g. 7.
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À = 32 sq. km,
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Ðþ òë è 7 . E+ ec t of
r el a ti ve ï è ò Üåò of
sa mp li ng p oi n t s o n a cc ur a cy of a ver ag e w a t e r eq ui va l en t
wi t hi n a n a r ea
I n à part of the 32 sq. km extension of the M irhogyolcs experimental area, characteristic of flatland areas in the country, à detailed survey was eff ected during the period
given in the fi gure, with measurement points at an average spacing of 1.2 km. The
average water equivalent of the snow cover within the area (ho), accepted as reference
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Some p robl ems of deter mi ni ng the water equi va lent of the snow ñî âåò
basis, has been determined by à map î é âî ø å1ï ñ lines, constructed from the data collected
at the 20 measurement points. Then, on the basis of areal averages h; computed by
equation (6) and of relative deviations according to equation (4), the correlation
ÀÜ = / '(è„) characteristic of the individual measurement series has been plotted for
cases i = 17, 14, 11 ... 2, 1.
In fi g. 7 à comparison is made between two cases, where values h~ are approximately
identical (32 and 29 mm) and only variations accompanying the compaction pr ocess
cause à considerable divergency in formation of the relationships. A s shown by the data
represented ø fi g. 7, the number of àï î â density points may be reduced under the
conditions considered to about à half, without appreciably decreasing the accuracy
of the survey.
A CC U R A C Y O F A V ER A G E V A L U E S W I T H I N A N A R EA
In solving problems connected with calculation and forecast of fl ood runoff , an accurate
knowledge of the water equivalent of the snow cover in à drainage area is in gener al of
ï î great importance. Since in most cases derivation or application of empirical relationâÛðû â in question, it is suf5cient to determine such indices which vary in close correlation
with the actual averages. In order to be able to prepare recommendations presenting à
basis for the selection of these indices, however , the relationship between fully detailed
observational data and accuracy of territor ial averages must be known.
For small areas first of all the results of detailed area surveys carried out in hydrological
experimental areas may be relied upon. I n the M irhogyolcs experimental area covering
61 sq. km and including also the earlier-mentioned part, water equivalent and snow
depth measurements were performed on some characteristic days in winter 1963-1964
at 43 observation points with an average spacing of ' 1.3 km. Such measurement series
are intended to be taken as à basis to analyse statistical laws and rules, characterizing
the variability of depth, density and water equivalent of the snow cover within à region.
Variability in the areal distribution of these factors is expressed by the relationship
between spacing of observation points (L) and é éåãåï ñå of measured data, i .e.
À î = / '(Ü)
(8 à)
(8 Ü)
as shown in fi g. 8.
For de<j gmining the relationships accor ding to (8) all data of an observation point
were ñî ø ðàãåé to data of every other point, that is, according to the formula of the
sum of arithmetical progressions, on the basis of 43 observation points all in all 903 pairs
of values L ;, Àî ; and L ,, tl t,, respectively, have Úååï defined. By grouping the diff erence
in values according to adequately chosen distance classes and by forming their averages,
the full-line relation curves in fig. 8 have been obtained. For characterizing the scattering
of diff erence values pertaining to identical distances L , the l ower and upper quartiles have
also been determined (see dotted lines ø fi g. 8).
Based on the results of preliminary methodi cal examinations computer programmes
are being completed, in order to ensure the most comprehensive data processing.
REFER EN CE S
1. Î ëêçòêë, W .U . (1964): Snow and snow survey, Section 10. I ï : " H andbook of applied
hydrology " , edited by × åï Òå Ghow, M cGraw-H ill, N ew-Y ork , 1964.
627
Ì
. R a c h n er
2. N astavlenie gidometeorologitsheskim stanciam i postam. (1946): v ipusk 3, Tshast 1, Gidrometeoisdat , L eningrad.
3. Pacifi c Southwest Inter-A gency Committee. (1966): L imitations in hydrologic data as applied
to studies of W ater Control and W ater M anagement , 129 ðð.
Some results of investigation and representation
of snow supply and the melting of snow
Ì . Rachner , M eteorological Î ãï ñå,
H a11e, GD R
SUMMARY: Results of experiments to investigate and represent the snow-supply and the ablation
of snow cover for the Selke catchment area up t o the gauging station at M eisdorf are presented.
M ethods of surveying snow cover are outlined. The abl ation of àï î â cover is calculated by the
aid of radiation measurements and the enregy budget . Special measurements (gradients of
climatic elements) showed si gnifi cant l ocal diff erences of snow-cover ablation. Finally, the
importance of the experiments with regard to water management and hydrology is discussed.
êéàî ì é : On pr6sente les r6sultats des expáriences dont le but est 4 '6tudier et de pr6senter les
donn6es concernant le st ock nei geux et ÃàÛ àã|î ï de la couche de nei ge dans le bassin versant
Selke j usqu' à l a st ation de j augeage M eisdorf . On d6crit bribvement les ãï áãéî äåü appliquáes pour
les reIev6s de l a couche de nei ge. Ü'ablation 4å la couche de neige est calculde par l a máthode du
bilan de radiati on et d'6nårgtå. L es mesures sp6ciales (les mesures de gradient des 6I6ments
climatiques) ont d6montr6 les é 1ãåãåï ñåâ l ocales considárables dans ÃàÛ àã|î ï de l a couche de
ï ñ å . Enfi n on indique Ãí ï ðî ããàï ñå de ces exp6riences pour Ãèã|1ü àã|î ï pratique des eau x et
pour ÃÜó4ãî 1î à|å.
1. Ï ~1ÒÊ 0 13Ñ ÑÒ10 1~1
In the Selke river catchment area (à hydrometeorological experimental area in the
H arz-M ountains) extensive investigations of water balance dependent on meteorological
conditions are performed.
Áî ò å results of snow ablation measurements and computations at the end of the winter
Season 1965-1966 (February 1966) are discussed.
M eteorological measurements are performed at the base station of the observations
Siptenfe1de-H anichen (ãð = 51' 38'.8 N ; Ë = 11' 05'.3 Å ; Í = 395 m above sea level ;
see also fi gs 1 and 2).
2. M EA SU R I N G M ET H OD S
Fig. 3 gives an impression of the measuring program. In addition to the measurements
and records at the station Siptenfe1de-H anichen à network of secondary stations with
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