Rijtema, P.E.

English

NARCIS 

The calculation of non-parallelism of gamma asccess tubes using soil sampling data

Name not available

Wageningen Yield 

*.*.«,,-,,- » A
 n n riTUUT VOOR CULTUURTECHNIEK EN WATERHUISHOUDING NN31545,0480 
NOTA 480, d. d. 12 september 1968 
The calculation of non-parallel ism of gamma acces s tubes, 
using soil sampling data 
P. E. Rijtema 
BÏBUOTHEEK 
STARINGGEBOUW 
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discussie van onderzoeksresultaten. In de meeste gevallen zullen 
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zoek nog niet i s afgesloten. 
Bepaalde nota's komen niet voor verspreiding buiten het Instituut 
in aanmerking. 
CENTRALE LANDBOUWCATALOGUS Ui 
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1. Introduction 
It has been shown by RYHINER and PANKOW (1968) that the gamma 
transmission method gives good opportunities to measure the variation in soil 
moisture content at various depths in the soil profile with t ime. Absolute 
values of the soil moisture content can be obtained with separate determina-
tions of dry bulk density and organic matter, so some sampling in the direct 
environment of the access tubes will be necessary in that case. 
It is also clear from the discussion presented by Ryhiner and Pankow, 
that the results of the gamma transmission technique are very sensitive to 
small e r ro r s in distance between source and detector. 
An indication of non-parallelism of both access tubes can be obtained 
when comparing the data of soil sampling with the results obtained with the gamma 
transmission method. However, the main difficulty remains the separation 
between the systematical e r ro r , due to non-parallelism and the random 
variation due to sampling e r ro r s and soil heterogeneity. This random variation 
will often be greater than the systematic e r ro r due to non-parallelism. It is the 
purpose of this paper to present a method to calculate the systematic e r ro r s 
due to non-parallelism of the access tubes. 
2. Theory 
When inserting the access tubes in the soil, non-parallelism shall 
mainly appear as a crossing of the tubes in a three-dimensional space, rather 
than as a deviation in a plane. 
Denoting the co-ordinates of the centre-line of the source tube by 
x, y, z and the co-ordinates of the centre-line of the detector tube by 
x' , y1, z1, gives the following equations of the centre-lines of the tubes. 
z-z 
Source tube : — = — = (la) 
a => a , x ' 
x-x y-y 
o _ 3 yo 
a l a2 
o 7 7o z ' -z ' 
Detector tube : p^ = p*- = ;— (lb) 
a l a2 a 3 
where x , y , z and x ' , y1 , z' are the co-ordinates of the respective centre-o ' o o o ' o o 
lines at the soil surface; a . , a? , a , and a ' , a i , a- are the direction coefficients 
of the respective centre-l ines. 
The distance L travelled by the mid-point of the source and by the f 
mid-point of the detector in the corresponding access tubes is presented by 
the following expression 
137/0968/50 
i:r :.:,.; i>:v 
'Li'; • • ' •nao 
ï ••'•,-- 1 "f 
. .uyr i j i ; - ' O J 
:::.-f'-V '.; üis'.o-i 
O"! H l ! • 
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' ! • " 
C -^ i <"• • 
\ ' j C j - h i v - : ;-)i'[J >{d i> •:)![: 
' S r i f l O . ' j . t l " 
L2 = (x-x J 2 + (y-y J 2 + (z-z J 2 = (x'-x' f + (y'-y' f + (z'-z')2 (2) 
Combination of the equations (1) and (2) g ives : 
a 1 L 
x-x 
V a . + a_ + a-1 2 3 
(3a) 
x'-x' = 
a^JL (3b) 
V(a[f + {*tf + (a3) 
Systematic exchange of the coefficients gives the following express ions 
of the co-ord ina tes of the mid-points of source and de tec tor . 
a . l i a 2 L a,]L 
Source : x=x + , •=., y=y 4- , q=-, z=z H- , , , sp (4a) 
a î L a 2 L a 3 L 
Detector : x '=x' + . = , y '=y' + •  , rr , z '=z ' + . . • sa (4b) 
° vj[£F ° \KAtf ° v ^ 7 
As the choice of the origin of the coordinate sys tem and the direct ion 
of one of the axes i s sti l l a r b i t r a r y , it i s useful to take the origin at the point 
of in tersec t ion of the cen t re - l ine of the source tube with soil surface and to 
take the direct ion of the x - ax i s , through the point of in tersec t ion of the c e n t r e -
line of the detector tube with soil surface . In that case the point (x , y , z ) 
has the co-ord ina tes (o, o, o) and the point (x ' , y ' , z ' ) has the co-ord ina tes 
(xo . o, o). 
With th is choice of the co-ordinate sys tem the equations of the mid-
points of source and detector can be wri t ten a s : 
a « j_i a ^ JU a^jXj 
Source : x = ,— ~- ; y = ; y ; z = ,.....- (5a) 
'vjjtf- ' 'vcS?' VIT" 
a^L a 2 L a ' L 
Detector : x ' = x ' + • » ; y ' = , •• ; z ' = -, .. (5b) 
° \/J^f yfïî^F \ / ^ f 
The distance D between the mid-points of source and detec tor i s given 
by the expres s ion : 
D = \ / ( x ' - x ) 2 + (y' - y ) 2 + (z' - z ) 2 (6) 
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Substitution of the equations (5) in (6) g ives : 
_ 
/ a!,L, a , L \ 
(7) 
\ l / a . L a l L | f a ' L a 2 L \ 2 '  ^ L ^ 1 
n° \7^"v™ iv^7"v!J?J Iv^'vsflv 
Rewriting of equation (7) g ives : 
D2 = (xQ)2 + 2 A x ^ L + (A2 + B 2 + C 2 ) L 2 (8) 
w h e r e : A = 
B = 
a l a l 
V ^ ( a ' ) 2 " \ / ( ( a ) Z 
a 2 a 2 
C = 
a 3 a 3 
3. Exper imenta l p rocedure 
Pa r t i cu la r ly in long t e r m exper iments i t i s v e r y useful to obtain the 
n e c e s s a r y information concerning non-pa ra l l e l i sm of the acces s tubes , without 
breaking down the exper imenta l set up. This information can under field 
conditions be obtained, by taking undisturbed soil samples at var ious depths 
of the profile in the d i rec t environment of the acces s tubes . During the sampling 
a lso m e a s u r e m e n t s with the gamma t r ansmi s s ion method a r e per formed at the 
same depths . F r o m the soil samples the dry bulk density, the mois tu re content 
and the organic m a t t e r content a r e de termined. 
Express ing the resu l t s of soil sampling and the data of the gamma 
t r ansmi s s ion method in the equivalent density unit (0. 9 f + p + 1. 01 P ) 
proposed by RYHINER and PANKOW (1968), gives the opportunity to e x p r e s s 
the value of 100[/> /fß - 1. 00) in t e r m s of a deviation in distance.Ax, as was 
shown by the authors mentioned. An example of the soil sampling data, the 
r e su l t s of the gamma t r a n s m i s s i o n s method and the der ived deviation in 
d is tanced x i s given in table 1. 
137 
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Table 1. Values of 0. 9 ƒ* + ƒ> + 1« 01 ß a s obtained by sampling and by the 
gamma t r ansmi s s ion method. The difference between both methods 
is expressed a s a deviation in distance Ax 
Depth 
cm 
10 
20 
30 
40 
. 50 
60 
70 
80 
90 
100 
1. 
1. 
1. 
1. 
1. 
1. 
1. 
1. 
1. 
1. 
0 
a 
658 
651 
624 
573 
656 
702 
650 
638 
654 
592 
»
9
^ s + 
sampl 
b 
1.470 
1.693 
1.634 
1.627 
1.648 
1.690 
1.662 
1.604 
1.600 
1.606 
9 + 1. 
ing 
mean 
1. 
1. 
1. 
1. 
1. 
1. 
1. 
1. 
1. 
1. 
564 
672 
629 
600 
652 
696 
656 
621 
627 
599 
01 P J o 
gamma 
method 
1.638 
1.574 
1.563 
1.521 
1.587 
1.653 
1.648 
1.631 
1.645 
1. 654 
IOO|^/PS -
% 
+ 4. 71 
- 5.87 
- 4 .03 
- 4 .96 
- 3. 95 
- 2 .55 
- 0.50 
+ 0. 60 
+ 1.11 
+ 3.45 
•'} à* 
en 
+ 1, 
- 1. 
- 1. 
- 1. 
- 1. 
- 0. 
- 0. 
+ 0. 
+ 0. 
+ o, 
r 
i 
.20 
60 
15 
37 
12 
76 
18 
,13 
.27 
,84 
The depth given in th is table r e p r e s e n t s in fact the distance t rave l led 
by the mid-poin ts of the source and of the detector in the i r respec t ive a c c e s s 
tubes . 
When the repl ica t ions of the soil sampling data in the top soil show-
l a r g e differences, which can be very often the case in a rab le land, it will be 
wise to m e a s u r e di rect ly the distance between the two cen t r e - l i nes at the 
soil surface , which reduces the p rob lem to the solution of an equation with 
two unknown constants* When th i s , due to any reason , h a s not been per formed 
it might be useful to set into the s e r i e s of data a lso the value L = o, combined 
with the theore t ica l dis tance D of 40 cm and to solve the equation a s one with 
th ree unknown constants using the re la t ion 
D 2 = a L 2 + bL + c (9) 
It appea r s from equation (8) in section 2' that the constant a mus t be 
2 -1 g rea te r than jf b c - When a i s smal le r than this value the random var ia t ion 
due to sampling e r r o r s and soil heterogenei ty i s such that the th ree dimensional 
c ross ing of the tubes cannot be shown. The only remaining possibi l i ty in that 
case i s to show a deviation in a plane with in tersec t ing cen t r e - l i ne s . The 
137 
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problem reduces then to a linear relationship between D and L.. 
In table 2 the data of L and D are given. As a result of the calculations 
it appeared that the relation between both factors can be given best by the 
expression: 
D 2 = 0. 05742 L 2 - 5.15427 L + 1630. 0005 
i , 2 -1 
(10) 
It follows from the constants that ; b c equals 0. 00407, so a is greater than 2 - 1 jç b c and the given condition of crossing of the access tubes is fulfilled. 
The values of D calculated for each level with equation (10) are also given in 
table 2. From the data presented by Ryhiner and Pankow the corresponding 
correction factors for each level are calculated by which the original data 
from the gamma transmission method must be divided. The corrected gamma 
data are also given in table 2, as well as the mean values of the soil sampling 
data. The remaining deviation between the corrected data of the gamma t rans -
mission data and the soil sampling data is due to random variation. 
Table 2. Values of L and D, as derived from table 1 and the values of D calcu-
lated with equation (10). The correction factors due to non-parallelism 
are also given, as well as the density derived from the original gamma 
data, the corrected values and the mean values of soil sampling 
L, 
(cm) 
0 
10 
20 
30 
40 
50 
60 
70 
80 
90 
100 
D 
(cm) 
40.00 
41.20 
38.40 
38.85 
38.63 
38.82 
39.24 
39-82 
40.13 
40.27 
40. 84 
D(eq. 10) 
(cm) 
40.38 
39-28 
39-37 
39.07 
38. 92 
38. 93 
39.08 
39-44 
39.81 
40 .39 
41.10 
Correc t ion 
factor 
1.0169 
0. 9943 
0. 9797 
0. 9681 
0. 9613 
0. 9615 
0. 9685 
0. 9816 
0. 9945 
1.0171 
1. 0435 
0 ,9 /> s 
original 
gamma data 
-
1.638 
1.574 
1.563 
1.521 
1.587 
1.653 
1.648 
1.631 
1.645 
1.654 
+ p + 1 . 0 1 
* w 
co r rec t ed 
gamma data 
-
1.647 
1.607 
1.615 
1.582 
1.651 
1.707 
1.679 
1.640 
1.617 
1.585 
ß J
 o 
mean 
sampling 
-
1.564 
1.672 
1.629 
1.600 
1. 652 
1.696 
1.656 
1.621 
1.627 
1.599 
137 
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4. Discussion 
The results presented in the previous section show that it is possible 
to determine factors to correct for non-parallelism of the access tubes. The 
practical meaning of this procedure with respect to the determination of the 
change in moisture content between two successive measuring dates, as well 
as the consequences in the determination of the absolute moisture contents 
must be further considered. For this reason the original data obtained under 
dry and wet conditions are given in table 3. Moreover, the corrected values 
are given, as well as the uncorrected and corrected values of the change in 
moisture content. This table also presents the original and the corrected values 
of the absolute moisture content. 
Table 3 . Values of the or ig ina l and corrected da ta , obtained by the gamma transmission method 
under dry and wet condi t ions, as well as the change in moisture content and the 
values of the absolute moisture content 
Depth 
10 
20 
30 
40 
50 
60 
70 
80 
90 
100 
0.9 Ps 
original 
dry wet 
I.492 
1.489 
1.403 
I.374 
1.^ 77 
1.546 
1.591 
1.59^ 
1.628 
1.634 
1.638 
1.57* 
1.563 
1.521 
I.587 
I.653 
1.648 
I.631 
1.645 
I.654 
+ P + 1.01 P 
J
 V * 0 
corrected 
dry wet 
1.501 
I.520 
I.449 
1.429 
I.536 
I.596 
I.621 
I.603 
I.601 
1.566 
1.647 
I.607 
I.615 
I.582 
I.651 
I.707 
I.679 
1.640 
I.617 
I.585 
Change in 
moisture 
content 
original 
14.6 
8.5 
16.O 
14.7 
11.0 
10.7 
5.7 
3.7 
1.7 
2.0 
corrected 
14.6 
8.7 
16.6 
15.3 
11.5 
11.1 
5.8 
3.7 
1.6 
1.9 
dry 
original c 
25-9 
25.5 
23.2 
27.9 
34.O 
29.3 
35-7 
43-5 
55.9 
64.4 
Moisture 
lorrected 
26.8 
28.6 
27.8 
33.4 
39.9 
34.3 
38.7 
44.4 
53.2 
57.6 
content 
wet 
original 
40.5 
34.O 
39-2 
42.6 
45.O 
40.0 
41.4 
47.2 
57.6 
66.4 
corrected 
41.4 
37-3 
44.4 
48.7 
51.4 
45.4 
44.5 
48.1 
54.8 
59-5 
The change in moisture of the whole profile according to the original 
data is 88. 6 mm, whereas it is after correction 90. 8 mm. The e r ror due to 
non-parallelism is 2. 5%. The er ror in the separate layers varies from o to 
4. 5%. As the systematic e r ror operates each measuring date in the same 
direction the effect of non-parallelism on the determination of changes in 
moisture content is not so very critical. 
137 
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•;r.srio {•• -.o 
'^ :1: 
The e r ro r s in the absolute moisture contents are under the presented 
conditions of non-parallelism in the order of 10 to 20% of the original moisture 
content. Particularly when these moisture content data are used in a drying cycle 
to derive the suction profiles in the field, the correction of non-parallelism 
is very important. In this type of investigations the determination of non-para-
llelism is always necessary. 
5. Summary 
An equation has been derived to determine the e r ro r s due to non-
parallelism of gamma access tubes with the aid of soil sampling data, taken 
from the direct environment of the measuring spot. 
It is shown that the determination of the change in moisture content is 
not so very sensitive to e r ro r s due to non-parallelism. When the absolute 
values of moisture content are needed, it is always necessary to determine the 
e r ro r s due to non-parallelism of the access tubes. 
6. References 
RYHINER, A. H. and J. PANKOW. 1968. Soil moisture measurement by the 
gamma transmission method. I. C W. -nota 475: 9 p« 
137 
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The calculation of non-parallelism of gamma asccess tubes using soil sampling data

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