130 Tech. Bull. Fac. Agr. Kagawa Univ.
VARIATION O F
WATER
TABLE FLUCTUATION IN
PADDY FIELDS
SHOWN BY COEFFICIENT OF VARIANCE
Kiyoshi FUKUDA,
Katsuhiko I z u ~ s u , Tadao MAEKAWA
1.
IntroductionIn order to show the local variation of tbe water table fluctuation in a given region during an annual cycle, the whole quantities of the fluctuation of the water table for a well for the cycle were calculated in the previous paper(3). And as a mathematical method, SHIMPSON'S 1/3 rule method was used to calculate the quantities.
This time, however, a stochastic method was used to estimate more generally the variation of the fluctuation in the region during an annual cycle. This is the 16th report of "Shallow Ground Water in the Downstream Basin of the Aya River".
2.
Mathematical TechniqueThe ratio of the rootmean square deviation, o, to the arithmetic mean, M , is called
the coefficient of variance, C,cl). This dimensionless coefficient gives an idea of the variability of certain phenomena. The coefficient of tariance is written as
where xi =magnitude of variable, M =ar ithmetic mean,
n=number of terms in the series.
For a wel1,during an annual cycle, the values of C can be calculated by using Eq.(l) when n=12.
And for the series of C during m years, the average values of C or can be obtained by Eq
.
(2)where Ci=the magnitude of C for a well for an annual cycle, m=number of annual cycles.
3..
DataData used in this paper was obtained from observations made during the period from March 1965 to February 1969 for all the observation wells in the study area in the down
stream basin of the Aya River in Kagawa Prefecture(Fig. 1).
Fig. 1 Simplified map of the study area in Kagawa Prefecture The shaded regions show the right region (AR) and the left region ' (AL). The crosses show the observation wells The solid circle on AR5 is the base point a t the fanhead of the study area. From this point on the map, the values of R ( k m ) were measured
The observation wells and the base point (AR5) are shown in Fig. 1. From this base point, the distance of wells from the fanhead (AR5) was measured on the 1/50000 topographic map.
The values the height above sealevel of the well sites used here were the ones :hat were listed in Table 1 in the previous paper(2).
Table 1 Major values of C for the four annual cycles from March 1965 to Februar y 1969.
Annual cycle Values of C
Max
..
Min. 
Mar. 1965Feb 1966 0.603 (AR4) 0 039 (FL1) Mar .I966 Feb 1967 0.647 (AR4) 0 025 (FL1) Mar 1967Feb. 1968 0.505 (BR1) 0.048 (ER1) Mar. 1968Feb. 1969 0 519 (AR2) 0.031 (FL1)
Average 0.569 0.036
4.
Results and Discussions4.1
Average Values of CBy using Eq. (I), the values of C for each well were calculated for the four annual cycles (from March 1965 to February 1969).
To save space, only the major values of C (maximum, minimum, and mean) are listed in Table 1. Values of C vary from 0.647 (AR4, March 1966

February 1967) to 0.025 (FL1 ,March 1966February 1967) .The average value of maximum of C is 0.569., and the minimum is 0.036 for the four annual cycles.By using Eq.(2), the average values of C or C were calculated for each well for the period from March 1965 to February 1969, and the results are listed in Table 2.
Tech Bull Fac. Agr Kagawa Univ. Table 2 Values of for each well for the four annual cycles from March
1965 to February 1969
Observation Observation Observation
 
C

well well C well C
AR1 0.174 DR3 0.327 CL 1 0.294 2 0.443 4 0.228 3 0 189 4 0 507 5 0.212 4 0.205 6 0.203 6 0 095 DL1 0.312 BR 1 0 466 ER1 0 075 2 0.187 2 0 324 2 0.155 3 0.289 3 0.113 EL 1 0.234 4 0.220 4 0.181 2 0,254 5 0.288 3 0 152 7 0 204 FR 1 0.086 4 0.148 2 0.116 CR1 0 169 3 0 107 F L1 0.040 2 0 295 5 0 071 3 0.293 6 0.092 DEL1 0.116 4 0 373 5 0 318 DER1 0.107 6 0 093 Max 0.507 BL1 0.223 Min. 0.040 DR1 0.168 2 0.231 Mean 0 215 2 0.303
4 . 2
Relationship between Cand HIn order to see the relationship between the values of and the height above sealevel of the well sites or
H,
(m), the values of H, were classed into ten classes having 1.00minterval from O.Om to 10.0m a s listed in Table 3.
Table 3 H,(m) classed into ten classes and their average values or & ( m )
Class Range Wells
2 2 002 99 ER4 2 72 2.57 FR6 2.63 FR2 2.13 FR3 2.30 FR5 2.02 CL4 2.93 ELI 2.57 EL3 2 19 FL1 2.77
There is no data in class0 and class8 as shown in Table 3.
For a class, the average values of
C
or can be obtained by using Eq. (3). _{1 }
_{" }
c,=
_{p }
z
G
134 Tech Bull Fac Agr Kagawa Univ. where cT=magnitude of in a class,
p = t h e number of terms in the series in a class.
By using Eq.(3), the values of were calculated from the data of H,(m) listed in Table 3, and the results axe shown in Fig. 2. The open circles are for the right region, and the solid circles are fox the left region. In order to obtain a linear equation for the whole region as listed Eq. (4). This data was used and calculation was made by the least square method.
r=0.879
where r = the correlation coefficient of and H.
Because both sides of Eq.(4) have to be a dimensionless variable, the first term in this equation is divided by HI ( = lm)
.
This same division is also done in Eqs. (5), and (6).o 6
_{ }

Fig 2 Relationship between C and %w
_{( m ) }_{. }
_{R }_{shows the right region, }
1%L is the left region and W, the whole region
0 2
00
0 1 2 3 t 5 6 7 8 9 1 0
H ( m )
Eq. (4) is shown by the solid line in Fig. 2.
Using the same method, for the right region, this data (the open circles in Fig. 2) was used and a linear equation, Eq. (5), was obtained.
0G Fig. 3 Relationship between
c
andH,. R is the data obtained
0 4 from the right region, and L,
the left region.
0 2
0 1
0 0
0 1 2 3 4 5 6 7 8 9 1 0
The dashed line in Fig. 2 shows this equation.
On the other hand, from the data of c a n d H,, f o ~ the whole region, a scattered map was made as shown in Fig. 3. Calculating by the least square method, the data in Fig. 3 was used and the linear equation, Eq. (6), showing the relationship between @ and H ,
for the whole region was obtained.
Y =0.532
Eq. (6) is shown by the solid line in Fig. 3.
These equations are drawn in order to show the relationship between
C
and H, orHg,
i.e. the higher the well sites are situated above sealevel, the larger the values ofc.
In other words, the higher. the elevation of the well sites, the larger the fluctuation of the water table of shallow groundwater in the paddy fields tends to be.
For the whole region, two equations (Eqs. (4) and (6)) were obtained. However, much the same results are obtained from both equations using the same data.
4.3
Relationship between C and R (km)In order to see the relationship between the distance of the wells (varying from 0.93 to 4.94 km) from fanhead (AR 5) or R (km) and C the values of R were classed into five classes having 1 kminterval each as listed in Table 4. For a class, the values of CC
were calculated by using Eq. (3)
Table 4 R (km) classed into five classes and their average or ~ ( k m )
Class Range Wells
136 Tech. Bull. Fac. Agr Kagawa Univ. ER 1 ER2 ER3 ER4 o 6
Fig 4 Relationship between C and 2 ( k m ) .
0 5
0 2
0 1
Fig. 4 shows the relationship between the values of R (km) and
c.
By using the least square method, the data shown in Fig.4 was calculated, and the linear equation, Eq. (7), was obtained, for the whole region,


C,
=o.
393 0.0558(R.)R1
(7)Y=0.948 and for the right region,
Fig. 5 Relationship between
3
and R ( k r n ) . 1 0 0 3 0 2 0 1 0 0 IAlso, from the scattered map of C and R shown in Fig. 5 , Eq. (9) was obtained.
 _{R }
C=O. 4010.058()
R1 (9)
r =  0.628
Because the values of
Cc
or are dimensionless variabie, the values of the right hand side of Eqs. ( 7 ) , (8), and (9) have to also be dimensionless variable. Therefore the second term in those equations was divided be RL( 1 km), respectively.From these equations, it is seen that the larger the distance of a well from the fan head (AR5) or R (or
B),
the smaller the values of C, (orC).
In other words, the greater the distance of a well from the fanhead, the smaller the fluctuation of the water table of shallow groundwater in paddy fields tends to be.
For the same region (the whole zegion), the results obtained from both equations, (Eqs. (7), and (9)) will be much the same.
5.
Summary1) In order to show more generally the fluctuation of the water table in a paddy field for an annual cycle, the coefficient of variance or Eq. (1) was used.
2) The coefficient of variance of the monthly average of water table depth (measured
from the ground surface) in an annual cycle or C was calculated for all the wells. During the annual cycles(March 1956 to February 1969), the values of C varied from 0.647 to 0.025, and the average of C for the same period varied from 0.569 to 0.036.
3) The relationship between the height above seelevel of well sites and C (the degree of fluctuation of the water table of shallow groundwater) was obtained as listed in Eqs.
(4), (5) and (6). As shown by these equations, the higher the height of the well sites, the larger the values of C (shown in Figs. 2 and 3).
4) The relationship between the distance of wells from the fanhead and C was obtained a s listed in Eqs. (7), (8) and (9).
These equations show that the greater the distance of the wells from the fanhead, the smaller the values of C as shown in Figs. 4 and 5.
6.
AcknowledgementsThe writers wish to thank M r . Yoshikazu KUSANAGI, a graduate student a t Kagawa University, for his help in carrying out this investigation.
138 Tech. Bull. Fac. Agr, Kagawa Univ. A part of the expenditure was defrayed by a research grant f ~ o m the Ministry of Education.
References
(1) CHEBOTAREV, N P : Theory of Stream Runoff (translated from Russian), Israel Program
for Sczenti f zc Translatzons, Jerusalem 1966,1824.
(2) FUKUDA, K., et a1 : On the Fluctuation of the Water Table through both the Irrigation and NonIrrigation Period (October 1964 to September 1965), Tech. Bull Fac Agr Kagawa Univ , 18, 5469, (1966).
13) FUKUDA, K : On the Whole Quantity of the Belt of Fluctuation of the WaterTable and its Characteristics, Tech. Bull. Fac Agr
.
ICagawa Unzu , 19, 159170. (1968).32 tj
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