(
)
(
)
(
)
(
)
(
)
(
)
+
−
−
+
−
−
+
p
t
t
P
3
3
2
55
,
7
804
,
2
max
6
,
17
029
,
0
exp
1
1
,
0
4
3
,
23
024
,
0
exp
1
5
,
26
(
)
(
)
(
)
(
)
(
)
(
)
−
+
−
−
+
−
−
−
+
+
1
55
,
7
804
,
2
max
max
6
,
17
029
,
0
exp
1
1
,
0
4
3
,
23
024
,
0
exp
1
5
,
26
exp
ln
d
t
t
p
aQ
P
aQ
P
(
)
(
)
(
)
(
)
(
)
(
)
,
6
,
17
029
,
0
exp
1
04
,
0
6
,
17
029
,
0
exp
1
04
,
0
772
,
3
772
,
2
+
−
−
−
+
−
−
−
t
t
(21)
To assess the model parameters, programs have been developed based on the least
squares methods and nonlinear optimization [31, p.5-37]. The values of the parameters
a, p, p
max
, b, c, d, c
1
, c
2
, c
3
, d
1
, d
2
, d
3
, α, γ are chosen, at which the series of obtained
model values best approximates the data, that is, the following problem is solved.
The function is given by formula (19). It is necessary to find such values a, b, c,
d, α, γ, P
max
, p, Q, at which the expression
(
)
max
9
,
0
05
95
,
0
75
,
0
50
10
02
,
0
04
,
0
495
,
0
475
,
0
145
,
0
125
,
0
0025
,
0
0005
,
0
046
,
0
022
,
0
180
100
,
,
,
,
,
,
,
,
,
min
max
30
1
max
2
=
Q
p
P
d
c
b
a
Q
p
P
d
c
b
a
t
G
i
i
reaches a minimum, this minimum should be calculated. Time t
i
takes the
values given in Table 3:
Таble 3 t
i
values
i
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
t
i
3
5
8
11
14
17
20
23
26
29
32
35
38
41
44
i
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
t
i
47
50
53
56
59
62
65
68
71
74
77
80
83
85
88
The results of solving this problem: a = 5,00; b = 0,05; c = 0,0005; d = 0,30; α =
0,30; γ = 0,02; P
max
= 50,00; p = 90,00; Q = 0,40.
The obtained parameters are used to predict the diameter increment of the tree.
Namely, by means of elementary transformations, based on the assumption that the
height of the tree varies in time according to Richards-Chapman's law, and the tree
growth equation is given in the form of the energy conservation law, a mathematical
model for calculating tree diameter increment is obtained:
- 1774 -