Vol 19, No. 2 (1968)
SHALLOW-GROUND W A T E R I N T H E D O W N S T R E A M
BASIN O F T H E AYA RIVER,
KAGAWA PREFECTURE
X.
On the Characteristics
of
the Cumulative-Depth-Area-Curve
and Depth-Ar ea-Cur ve
(11)Kiyoshi
FUKUDA
1.
Introduction
From the stand point of water use including irrigation and drainage, it is very important to know the relationship between the water
-
table-depth and the area corresponding to the depth for a given area for a given period.As one of the methods to show the relationship, in the previous paper(4), we drew up the depth-area-curve for the study area for each month during the period from October 1964 to September 1965.
From the character istics shown by these depth-area-curves, we saw the hydraulic charac- teristics of the water-table-depth in the study area, and we detailed them in the previous paper.
We did not, however, analyse mathematically the depth-area-curve, and so we could not obtain the theoretical equation showing the depth-area-curve. If we can obtain the theoretical equation, we will be able to easily estimate the relationship between the water-table-depth and the area by using the theoretical equation.
Therefore, what we have to do is to obtain the theoretical equation showing the depth- area-curve.
The paper presented here reports the derivation by using the statistical method, the characteristics, and the use of the theoretical equation.
2.
Study Procedure
As shown in Fig. 1, the study area was divided into the two regions; 1 ) the right region or AR and 2) the left region or A r . The area of AR was calculated as 492.85ha, and the area of
AL
as 157.94ha. The whole study areaAw,
the sum of the right region and the left region, was found to be 650.79ha.To measure the variation of the water-table-depth from the ground surface of the study area, we used 51 observation wells (the crossed signs) as shown in the same figure.
The data used to calculate the theoretical equation was the observed data from October
1964 to September 1965 obtained in the field investigation and listed on Table 1 in the previous paper(3).
Tech Bull Fac Agr Kagawa Univ
Fig 1. Simplified map of the study area showing the boundaries of the right region and the left region, and the location of 51 observation wells
3.
Theory
Fig. 2 shows the relationship between a,/A, (y-axis) and
D
(x-axis) of the right region in October 1964; where a , is the area having the water-table-depthD
(m), and AR is the whole area of the right region (492.85ha); and a R / A , is a ratio of a area havingD
to the whole area of the right region.T h e dashed curve represents the observed 7
values and the solid curve represents a curve which 60 p h e----Observed o reticai
1
corresponds to the dashed curve. W e called this-
curve the depth-area-curve in our previous paper.
In statistics, a curve resembling the dashed 4 0 / /
![,
$1
curve is called a histogram of a R / A , and the solid~ 1 4
201
curve is called a distribution curve which corre- I I L -- I
sponds to the histogram. ---
W e are able to get this distribution curve by 0
0 0 0 . 5 1 0 1 5 2 0 2 5 3 . 0 statistical calculations if we have the theoretical
equation corresponding to the distribution. T o 1) ( m
obtain this theoretical distribution curve or the Fig 2 Relationship between a / A and theoretical depth-area-curve of the region, we D (the right region in October
applied the statistical method. 1964) The dashed histogram
is the observed values and the As shown in Fig. 3 (A, B and C) in the solid curve is the theoretical previous paper(4), the distribution curves showing depth-area-curve which corre- the relationship between a / A and
D
were all sponds to the histogram skewed curves~ t a t i s t i c s ( ~ ) tells us that the skewed distribution curves will generally be expressed by using the log-normal probability equation. Therefore, we applied the log-normal probability equation to our case.
In a skew distribution curve, the points (D, U X ) in the fractile diagram are located near a theoretical curve and the equation may be written
Vol. 19, No. 2 (1968) 173
where D = t h e stochastic variable.
And the theoretical cumulative distribution function has the equation
and hence
where
p{D) =the distribution function.
Thus, the variable
D
itself is not normally distributed, but the function f(D)
is normally distributed.The function f(D) transforms the skew distribution into a normal one; and it is possible to choose a function of the type
If the function f (D) is chosen correctly, the points of a diagram scatter at random about the straight line
In our case, because we used the log-normal distribution, Eq. (6) is usually expressed as log
D
- logD,
u = 1% 6, where Z'(a log D) logD,
= I a[
I
{a(logD
- log D.)'} log 6, = I aI'
(9) and where174 Tech.. Bull Fac. Agr.. Kagawa Univ. log
D,=
the log-geometric mean ofD
log 6',= the log-geometric standard deviation.
Thus, for the log-normal distribution, the cumulative distribution function is defined as log
D
-
logD,
P(D)
= @(u),
u = ,O<D<-
1% 6,Diff'erentiating of Eq. (lo), we get the log-normal distribution function, p ( D ) , d u 1 1 l o g D - - - l ~ ~ D , ) ~ ] d l o g D
p(D)
=p(u)--=dD
log6,d
2;[-T(
log 6, d Dd log
D
For the termdD
,
we used the following transformationsg), d.- d 1 0.4343
d D log
D=-
d D logloD =
loglo e -=---D
D
(12).
,. log10D
logD
.
loglo e =--- = 0.43294L=i= 0.4343log,
D
log,D
Therefore, from Eq. (12), and Eq. ( l l ) , we obtain Eq. (13)
1 1 1 logD-logD,
{D) = 0..4343 -
D
log 6 , d Z[
-
T(
log 6,)4
(13) T o construct a log-normal distribution curve (theoretical depth-area-curve) which has the same area as a given histogram, both sides of Eq. (13) must be multiplied by d(') (the length of each class interval used to draw the histogram) ; and further we put,Then, we obtain Eq. (14).
a
- 0.4343 d 1 log
D-
logD,)2]
A log
6,d2h
D
e x p [ - ~ ( log 6, (14)W e refer to Eq (14) as the theoretical depth-area-curve-equation. The solid curve shown in Fig. 2 was calculated by using Eq. (14)
T o estimate the area a having D, we obtain Eq. (15) from Eq. (14).
Vol 19, No. 2 (1968) 1 75
log-standard deviation, log 8,. The larger the log G,, the more pronounced the skewness. The modal water -table-depth,
Dmoa,,
the median water -table-depth,Dmedi,,
, and the mean water-table-depth,Dm,,,
are obtained by the following equations(').log
Dm,,,
=logD,
-
2.3026(log 8,)' (1 6)log
Dme4xan
= 108Da
(17)log
Dm,,,
=logD,
+
1 .1513(log g,)' (18) From Eqs. (16), (17) and Eq. (18), we see that 1 )Dmode
is less thanDmorjian
which is again less thanDm,,,,
and, 2) the greater the log-standard deviation the greater the difference between these three factors.4. Results and Discussion 4.1. Theoretical depth-area-curve
Using Eqs. (8), (9) and (14), we calculated the data obtained from the field observation^'^),
and we obtained the theoretical depth-area-curves for the right region, the left region, and the whole study area for every month from October 1 9 6 4 to September 1 9 6 5 as shown in Fig. 3(A), Fig. 3(B) and Fig. 3(C). The solid curves show the theoretical depth.area-curve and the dashed histograms show the observed values.
4.2. Variation of theoretical values of
Dmode,
Dmedian
andDm,,,
Using Eq, (17), we computed the theoretical values of the median water-table-depth or
Dmedian,
and using Eq. (16), we computed the theoretical values of the modal water-table. depth,DmOa,,
and also using Eq. (18) we obtained the theoretical values of the mean water- table-depth,Dm,,,,
for the period from October 1 9 6 4 to September 1965.The results calculated by these equations are shown in Fig. 4. The solid curves show
Dm,,,,,,;
the dashed curves showDm,,+,,
and the dot-dashed curves showDm,,,.
For the irrigation period (from June to September), for the non- irrigation period (from October to May), and for the yearly average, the three theoretical water -table-depth,
Dm,di,,,
Dmodc
andDm,,,
are shown in Fig. 5.4.3. Comparison of the theoretical values with the observed values
4.3.1.
Modal water-table-depth,Dmode
Fig. 6 shows the variation of the theoretical values of
Dmodc
(the solied curves), the observed values ofDmode
(the dashed curves) and the values ofDemode
(the modal water- table-depth.class) (the shaded region) during the period from October 1964 to September 1965.There are slight differences between the theoretical values and the observed values of
DmOa,
i~ every month of the period. However, the curves showing the values of theoreticalDmode
run within the shaded region throughout the period.T o see the deviation of the observed values from the theoretical values,
a
(%), we usedTech Bull Pac Agr Kagawa Univ,
D ( m )
D ( m )
D
( m ) D ( m ) D ( m )October November December
D
(4
January
1964 1965 February March
Time (month)
Fig 3(A) Theoretical depth-area-curves (the solid curves) computed by Eq (14) and the observed values (the dashed histograms) for the right region (the top, A,), the left region (the middle, A,) and the whole study area (the bottom, A,,) during the period from October 1964 to March 1965
Vol.. 19, No.. 2 (1968)
D ( 4 D (m) D (m) D (m) D (m)
D
( 4April May June July August September
Tzme (month)
Fig 3(B) Theoretical depth-area-curves (the solid curves) computed by Eq (14) and
the observed values (the dashed histograms) for the right region (AR), the left region ( A , ) and the whole study area (Aw) during the period from April
Tech. Bull. Fac. Agr Kagawa Univ.
D
( m )D
( m )D
( m )Irrigation Non-ir rigation Yearly average period period
Time
Fig 3(C) Theoretical depth-area-curves (the solid curves) computed by Eq (14) and
the observed values (the dashed histograms) for the right region (A,), the
left region ( A , ) and the whole study area (A,) for the irrigation period (June to September), non-irrigation period (October to May) and the yearly average
Vol 19, No. 2 (1968)
Fig 4 Variation of theoretical values, DmedLsn (the solid curves), Dmode (the dashed curves) and D,,,, (the dot-dashed curves) for the right region (A,), the left region (A,) and the whole study area (A,) during the period from October
1964 to September 1965
Fig 5 Variation of theoretical values, DmadIsn (the solid curves), Dmode (the dashed curves) and Dm,,, (the dot-dashed curves) for the right region (AR), the left region (A,) and the whole study area (A,) for the irrigation (I), non-irriga- tion (N) periods and the yearly average (Y)
Fig 6 Comparison of the theoretical modal-depth, (Dmode) t, (the solid curve) with the observed values, (D,,,,) o, (the dashd curve) for the right region (A,), the left region (AL) and the whole study area (A,v) during the period from October 1964 to September 1965 The shaded region shows the observed values of the modal-depth-class, DC,,,,
Tech Bull. Fac. Agr. Kagawa Univ..
where
(Dmodc)o =the observed values of Dmodc, m.
(Dmodc)t=the theoretical values of
Dmode
calculated by Eq. (16), m.The results computed by Eq. (19) were as follows : 1) the values of
a
varied from 3.8 per cent (December and February) to 23.5 per cent (May) and the mean was 13.1 per cent in the right region; 2) the values ofa
varied from 1.6 per cent (December and 50 February) to 41.5 per cent (September)
-
403
and the mean was 14.6 per cent in the left " 30 region ; and 3) the values of
a:
varied from 20 3.1 per cent (December and February) to 10 23.0 per cent (September) and the meanD
f9$M
AM
J J A Swas 12.8 per cent in the whole study area. Tzme (month)
These are shown in Fig. 7.
Fiq 7 Variation of n (%), the deviation of the theoretical D,,,, from the 4.3.2. Relatzonship between Dmode,
observed Dmodo, for the right region
D m e d i a n and D m e a n (the solid circles), the left region
According to statistics, the Dmode may be replaced by its approximate values,
( D m e a n
-
3(Dmean-D m e
lian)) (1) when the(the open circles) and the whole study area (the crosses) during the
period from October 1964 to Sep- tember 1965
depth-area- curve is a moderately asymmet-
rical or skewed one. Therefore we obtained Eq. (20)
Using Eq. (20) we estimated the calculated values of
Dmode
after substituting the Dmedinndata cmputed by Eq. (17) and the
Dm,,,
data computed by Eq. (18) for the first and the second terms of Eq. (20), respectively. W e referred to the values calculated by Eq. (20) as6'
calculated Dmode", or (Dmodc)c.
W e estimated the deviation of the values of
(DmOae),
from the values of (Dmode)t,@
(%), by using Eq. (21), and the results are shown in Fig. 8.T h e maximum values of' @ were 1) 5.6 per cent (October) in the right region; 2) 9.4 per cent (September) in the left region; and 3) 6.0 per cent (August) in the whole study area. However, the mean values of' @ were 2.12 per cent in the right region, 1.9 per cent in the left region, and 2.4 per cent in the whole study area. These are shown in Fig. 8.
Vol 19, No 2 (1968) 181 perimental errors, the values calculated by Eq. (20)
are close to the values calculated by Eq. (16).
From this, Eq. (20) is enough to use to obtain the values of
Dm,
,,.
It is important that the theoretical values of
a / A at
Dm,,l,
agree closely with the observed ones, because a / A at Dm,dc occupies the largest area in a given region for a given period. W e compared the theoretical values of a / A at Dm,,,, with the observed ones and the results are shown in Fig. 9.In this figure, we see that the theoretical values (the solid curves) agree closely in practice with the observed values (the dashed curves).
" K h D J F M A M J
J A S
me
(month)Fig 8 Variation of ,B ( % ) for the right region (A,), the left region (A,) and the whole study area (A,) during the period from October 1964 to September 1965
60
40 40
T i m e (month)
Fig 9 Variation of a/A at Dm,,, for the right region (A,), the left region (A,) and the whole study area (Aw) during the period from October 1964 to Septem- ber 1965 The theoretical values are shown by the solid curves and the observed values are shown by the dashed curves
4.3.4. Median water -table-depth,
Dmeai,,
orD50
According to the book named ~ u n t a i ( ~ ) that we cited, the values of Dmedian that were read
from the distribution curve (which was shown in Fig. 10 in that book) were larger than the values of D,,,1,,, that were read from the cumulative distribution curve (shown in Fig. 12).
These two values, however, have to be the same values.
T o make these two values coincide or to obtain the correct values, we have to introduce an adjusting number and that number was calculated as 0.25 m.
T o compare the theoretical values of
Dm,lian
computed by Eq. (17) with the observed ones, we adjusted the observed values of Dm,~i,, by using Eq. (22).where
(Dmolion)o,=the observed values of Dmedian or
D 5 ~
that were measured from thecumulative depth-area-curves shown in Fig. 4 (A to 0 ) in the previous paper
c6)
182 Tech. Bull Fac. Agr Kagawa Univ. Time (month)
Fig 10 Variation of D,,,,,, or D,, for the right region (An), the left region (A,) and t h e whole study area (A,) The solid curves are the theoretical values and the dashed curves are the observed values
Fig. 1 0 shows the comparison of the theoretical values of
Dm,
li,n (the solidcurves) with the observed values that were calculated by Eq. (22) (the dashed curves). On the graph, the theoretical values are larger than the observed values in the five months (December to April) in the non-ir- r igation per iod,and July in the irrigation period.
T o see the deviation of (Dme,i,,)t from (DmediaJo, we used Eq. (23) and the results calculated by the equation are shown in Fig. 11.
Fig 11 Variation of 7 (%), the deviation of the theoretical Dmedi,, from the observed Dm,,,,,, for the right region (the solid curve), the left region (the dashed curve) and the whole study area (the dot-dashed-curve) during the period from October 1964 to September 1965
The graph shown in Fig. 11 tells us that the theoretical values agree closely in practice with the observed values.
4.3.5.
Mean water-table-depte,Dm,,,
As shown in the previous paper(4), the arithmetic mean of the water-table-depth of the region D,,, was calculated by Eq. (24),
where
a = a area having D, ha
Vol. 19, No. 2 (1968) 183 Here, we used this
D,,
for the observed values ofDm,,,.
On the other hand, the theoretical values of
Dm,,,
can be calculated byEq.
(18).Therefore, we compared the theoretical values of
Dm,,,
with the observed values(D,,)
and the results are shown in Fig. 12. T h e solid curve shows the theoretical values ofDm,,,
Fig 12 Variation of Dm,,, (the solid curves) Fig 13 Variation of 6 (%), the deviation and D,, (the dashed curves) for the Dm,,, from D,,, for the right region right region (A,), the left region (the sloid circles), the left regiou
(the open circles) and the whole
( A , ) and the whole study area ( A , )
study area (the crosses) during the during the period from October period from October 1964 to Sep-
1964 to September 1965 tember 1965
and the dashed curve shows the observed ones.
T h e deviation of
Dm,,,
from the observed one,6
(%), was calculated byEq.
(25) and it was 0.0 to 2.2 per cent through both regions, (the right and left regions) and the whole study area, and the mean was
below 0.8 per cent. These are shown in Fig. 13. T h e solid circles show the values of
6
for the right region, the open circles are for the left region and the crosses are for the whole study area.
Because these values are generally within the permissive range fbr experiment errors, the theoretical values agree well with the observed values.
XI
. -
-
3 - OA,' ' .Aw '3
24.4. Statistical characteristics of theoretical depth-area-curve-equation 4.4.1. Standard devzatzon, 6 ,
According to statistics, the skewness of a depth-area-curve depends upon the standard deviation, and the larger the standard deviation, the more pronounced the skewness.
By using
Eq.
(9), we calculated the standard deviation, C,, and the results of the com- putation are shown in Fig. 14.O ~ N ~ . ! ? M A A ? ~ A S
1964 1 9 6 5 ~ 1 m e (month)
d
1 ,
co
period from October 1964 to Sep- tember 1965
-
1
Ti
!
1
1
,r
0.2 . -'A; I Fig 14 Variation of the standard deviation,
-
,,
'
co.
1I
7
--- -. , a,, for the right region (the solid
curve), the left region (the dashed
O'OO N D J
b~
curve) and the whole study atea184 Tech. Bull. Fac.. Agr. Kagawa Univ.. The solid curve shows the values of 6, for the right region; the dashed curve is for the left region; and the dot-dashed curve is for the whole study area.
From the graph shown in Fig. 14, it is evident that the values of 6, varied from 1.20 (December) to 1.68 (August) in the right region, from 1.29 (April) to 1.78 (September) in the left region, and from 1.25 (December) to 1.63 (August) in the whole study area.
The mean values of 6, are 1.39 (the right region), 1.41 (the ieft region), and 1.40 (the whole study zrea).
As the periodical averages of 6,, they are 1.38 (the right region), 1.39 (the left region) and 1.38 (the whole study area) for the irrigation period; and 1.27 (the right region), 1.32 (the left region) and 1.28 (the whole study area) for the non-irrigation period; and also 1.36
(the right region), 1.43 (the left region) and 1,38 (the whole study area) for the yearly average from October 1964 to September 1965.
4.4.2. Skewness, @
In statistics, the departure of a distribution from a normal probability curve is called the asymmetry or skewness('). In our case, the asymmetry or skewness,
o,
is defined by Eq. (26)*where
6,= the standard deviation.
When
Dm,,,
is equal toDm,ae,
W is equal to zero, and then the curves are symmetrical curves.The results calculated by Eq. (26) are 0.2
!---A---
---
.':--y--/
shown in Fig. 15. The solid curve shows
the values of
o
fbr the right region, the - 1dashed curve is for the left region and the O " 8 ) N D J F M A h l 1964 1965 Time (month) J J A S
curve is the study Fig 15 Variation of the skewness, w, for the
area. right region (the solid curve), the
The values of' @ fluctuate within the range of 0.05 (the right region, December) to 0.19 (the left region, September) and
left region (the dashed curve) and the whole study area (the dot-dashed curve) during the period from Oc- tober 1964 to September 1965 the mean values are 0.11 (the right region),
0.13 (the left region) and 0.12 (the whole study area).
5. Summary
Applying the theory of log-normal distribution, we obtained the theoretical equation showing the relationship between
a / A
andD
as listed on Eq. (14), and we referred to this equation as the depth-area-curve-equation. Also we used Eq. (16), Eq. (17) and Eq. (18) to calculate the values ofDmode,
Dmedian
andDm,,,,
respectively.Vol. 19, No. 2 (1968) 185 values that were observed and the results were as follows:
1) The deviation of the theoretical values of
Dmoae,
(Dmodo)t, &om the observed values calculated by Eq. (19) was 41.5 per cent at the maximum. Its mean value was, however, 14.6 per cent.2) The theoretical values of a / A at
Dmode
agree closely practice with the values ob- served.3) The values of Dmedianr the theoretical values agreed well with the values observed.
4) The theoretical values of
Dm,,,
agreed with the values observed. For the statistical characteristics of the theoretical equation, we obtained the following.5) The values of the standard deviation, B,, varied from 1.20 (the right region, De- cember) to 1.78 (the left region, September), and the mean was 1.40 (the whole study area). 6) The skewness of the depth-area-curves, w, varied from 0.05 (the right region, December) to 0.19 (the left region, September), and the mean was 0.12 (the whole study area).
From these results, we conclude that we will be able to use the theoretical equations obtained here to estimate the depth-area-curve, DmOae,
Dmeai,,
andDm,,,
for a given region for a given period.6.
Acknowledgements
W e wish to thank to Emeritus Professor Dr. Hitoshi FUKUDA and Associate Professor Dr. Hiroyuki OGATA, both of Tokyo University for their encouragement. A part of the expenditure was defrayed by the research fund of the Ministry of' Education granted to Pro- fessor Dr. Eiichi ODA of' Tokushima University and Kiyoshi FUKUDA.
References
1) DALLAVALLE, J M : Micromeritics, 54 and 58,New York, Pitman Publishing (1948).
2) DIXSON, W., MASSEY, F : Introduction to Sta- tistical Analysis, 48-49, New York, McGraw- Hill (1957)
3) FUKUDA, K
,
NAGANO, T, MAEKAWA,
T. :Shallow-ground water in the downstream basin of the Aya river IV. T e ~ h Bull Fa6 Agr Kaga- ma Unzv ,l8,186-207 (1967)
4) FUKUDA, K. : Ditto, V Ibid.., 18 208-229 (1967) 5) HALD, H.: Statistical Theory with Engineer- ing Applications, 1.59-I73, New York, John Wiley & Sons (1952)..
6) KUBO, K., MIZUWATARI, E., NAKAGAWA, Y., HAYAKAWA, S. : Funtai, '79 Tokyo, Maruzen (1 964)..
7) NIPPON KIKAIGAKKAI : Kikaikogaku-Binran, 2..6, Nippon Kikaigakkai (19.57)..
Tech Bull Fac Agr Kagawa Univ.
%T7k@iD@S D(m) jS~'ffEjl$.8@i@%, a(ha)
t:
L,
%%rC@-DEi%%, A(ha) L-f 8. a / A ( % ) .%. y BiZ, Da.
.xBiz:
2 5 2 , &T7jxEDi%S 2 %D?CAt!E@%~%T&%Sr (%J%~cBL~-c, Depth-Area-Curve 2g.M.vf:)
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~%iEB5+j.;mt&%$S%~&TC2&5T, % 9 k , + ; h i 2 S,i
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fx
<,
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%Ozm@i@sM.1.%%~sf-~.3D@th~T&8. +$LC&,H(a1og D) log D, =Ta-
8 {a(log D- log D,)2]
logog=[ I a
I"
(9)log D,=D ~$$j&gfl$.iS~.f$i, loga,=$$&@l;jS@@Eg, d=histogram D interval(m).
L f:&, 1964LglON -+1965$. 9
a
D 1 /+f&z, 51 @J,$;jS\ l;fq.f: dataiz
& 9 Zbg~gE ?LL~, %lOliiTkE%Lf:. $$%I'd, Fig 3 (A, B BkZF C)izZf
k5
iz,
d:<
k.5 2jSiihjS17f:" %8llOli&t:@%iij$iD8%,%&Z LT$&%f: (Fig 4-Fig 15) jSf,