全 文 ：第 49 卷 第 4 期
2 0 1 3 年 4 月
林 业 科 学
SCIENTIA SILVAE SINICAE
Vol. 49，No. 4
Apr．，2 0 1 3
doi:10．11707 / j．1001-7488．20130410
Received date: 2012 － 04 － 01; Reversed date: 2013 － 01 － 22．
Foundation project: Japan Society for the Promotion of Science Program (2007-209) ; The 948 Program (2011-04-67) ．
* Hirafuji Masayuki is corresponding author．
文冠果苗木生长与土壤含水量间关系建模*
王雪峰1 平藤雅之2
(1. 中国林业科学研究院资源信息研究所 北京 100091; 2. 日本农业食品产业技术综合研究机构北海道农业研究中心 札幌 0820071)
摘 要: 对生物燃油原料植物———文冠果苗木的冠幅生长规律进行研究，使用由原野服务器自动获取的土壤水分
及实际测量数据，首先用度量误差模型方法估计模型参数，然后对逻辑斯蒂和理查兹模型进行比较，同时用 3 次
多项式拟合土壤水分、生长日期与单日冠幅生长量的关系。结果表明: 度量误差模型在自、因变量样本测定存在误
差时能够明显提高参数估计精度; 理查兹模型在文冠果苗木生长模拟中表现良好; 苗木早期生长需要更充足的土
壤水分，并且单日最大冠幅生长所需要的最佳土壤含水率随着苗木生长日期的增加而递减。
关键词: 文冠果苗木生长; 度量误差模型; 土壤水分; 建模
中图分类号: S718. 55 文献标识码: A 文章编号: 1001 － 7488(2013)04 － 0070 － 07
Modeling the Relationship between Yellow Horn Seedling Growth
and Soil Moisture Content
Wang Xuefeng1 Hirafuji Masayuki2
(1. Research Institute of Forest Resources Information Techniques，CAF Beijing 100091;
2. National Agricultural Research Center for Hokkaido Region Sapporo 0820071)
Abstract: The objectives of this study were to provide a model reference for estimation of seedling height according to
crown width based on imagery，and to provide technical guidance for the seedling industry． To this end，soil moisture and
seedling growth data were first obtained through the application of automated field sensors． Model parameters were then
estimated using a promising measurement error model method． The next stage of this study was to carry out comparative
research on the fitted results from the Logistic model (Logistic regression) and the Richards model． Finally，relationships
between soil moisture，growth rate，and the daily crown width increment were fitted by a cubic polynomial function．
Results showed that: 1) The measurement error model method provided good estimation accuracy of the parameters when
dependent and independent variables were found within the measurement error． 2) The Richards model was superior to the
Logistic model when simulating yellow horn seedling growth and relationships between crown width and seedling height． 3)
Seedlings require higher soil moisture content during the initial growth stage，and that optimum soil moisture content
demand for the daily maximum crown width increment will decrease with an increase in growth rate．
Key words: yellow horn seedling growth; measurement error model; soil moisture; modeling
For several decades plant data has been obtained
by the application of satellite imagery and aerial
photography ( Johnson et al．，2003; Gong et al．，2002;
Wang et al．，2004) ． Moreover，it has become easy to
automatically obtain high-resolution imagery in recent
years due to field sensor usage． Owing to this， the
acquisition of all manner of information from satellite
imagery and aerial photography imagery has become a
hot research field ( Ke et al．，2010; Somers et al．，
2009) ． For example，Okamoto et al． (2006) carried
out research on the segmentation algorithm of soil and
plants as well as plant classification using hyperspectral
imaging． Similarly， Tsuchiya et al． ( 1999 )
determined tree height and stand density by way of
aerial photography． From the viewpoint of current
conditions， prospects in the development and
application of image data are very broad (Wu et al．，
2004) ． For example，they include the sorting of fish，
第 4 期 王雪峰等: 文冠果苗木生长与土壤含水量间关系建模
the detection of apple surface area，and so on (Yang et
al．，1994; Zion et al．，1999) ．
The vast majority of forestry imagery in existence，
especially with regards to tropical rain forest imagery
(Kuraji et al．，2003 )，are images that show forest
canopy characteristics taken by means of high-altitude
camera devices． If the crown width is known，a greater
amount of information on plant life can be inferred from
an image based on the crown width by applying
mathematical modeling． A large amount of research has
been carried out on mathematical modeling (Mizunaga，
1994; Inoue，2000; Takeuchi et al．，2003) ．
Factors that contribute to the simulation accuracy
of a model can be summarized in two ways． One is the
choice of model type． In recent years a theoretical
model has been widely chosen among researchers． The
other is the selection of model parameters． Little
attention has been previously paid to this latter point．
It is important to understand the time when plants
need to be watered as well as the amount of the water
required at the time seedlings are planted in the
nursery，but no one can tell the exact optimum soil
moisture content required． To determine this，yellow
horn was used for the study due to its excellent biofuel
reputation． Model parameters were aligned to the
measurement error model method． The relationship
between soil moisture content and seedling growth was
then determined to ascertain the optimal amount of soil
moisture content needed to achieve maximum seedling
growth per day． Since the significant correlation
between soil moisture content and species genetic，the
results of the research will provide methodological
guidance for yellow horn planting in the nursery．
1 Material and method
1. 1 Material source
The experiment was carried out indoors． The total
dimension of the experimental nursery was
1 800 mm × 1 500 mm × 400 mm ( Fig． 1 ) ． It was
divided into 30 small square plots， each being
300 mm × 300 mm，and numbered from 0 to 29． In
the centre of each plot was planted with one yellow
horn seedling． A special water vat that has good water
permeability was positioned on the left corner of the
experimental area so that water could penetrate through
the vat and into the soil at a slow rate of absorption．
This allowed for variability since soil closer to the vat
will experience greater soil moisture content．
Moreover， two distinctive groupings of light sources
were positioned above the test area ( Fig． 1) that was
automated to turn on and off in accordance to sunset
and sunrise times． Two on the left was positioned near
to the test area and therefore offered higher light
intensity while six on the rightwas positioned far from
the test area and therefore offered weaker light
intensity． Two Ricon cameras were positioned over the
test area，and two Canon cameras were positioned on
the upper right hand side of the test area．
Fig． 1 The layout of test area
The No． 21 yellow horn seedling area was chosen
as the object for which to measure soil moisture owing
to its particular light intensity and moisture rate． An
ECH2O EC-5 Soil Moisture Sensor was positioned at
the roots of the seedlings 10 cm below the soil surface．
The probe was placed perpendicular to the main root
( Fig． 2 ) ． Measurement accuracy of the sensor was
approximately 1% to 2% ． Both moisture sensors and
cameras were connected to a field sensor that
automatically recorded the soil volumetric water content
and took photographs every 10 minutes． The average
soil volumetric water content in a 24 hour period was
then recorded，and the result was taken as the soil
moisture content for that particular day．
The height and crown width of each seedling was
measured in an east-west and south-north direction
every morning at 8: 30 a． m． after seedlings sprouted．
17
林 业 科 学 49 卷
Fig． 2 The soil moisture sensors setting positon and direction
The average of the two measurements was taken as the
seedling crown width． The test period lasted for 107
days in total．
1. 2 Model selection and equation parameter
solutions
Suppose the growth of the seedling crown width is
greater than 0． If c0 represents the largest growth of the
crown width starting from t = 0， and w ( t ) is the
cumulative growth of crown width at time t［y = w( t) /
c0］，then y satisfies the following growth rate equation:
dlgy
dt
= f( y，t) ．
where f( y，t) is a continuous function of 0 ＜ y ＜ 1，
t ＞ 0． The most common form of f ( y， t ) is the
equation developed by Turner:
f( y，t，c) = λ(1 － ym) 1 － p( y －m － 1) p ．
where λ，m ＞ 0 ( France et al．，1996 ) ． For p = 0，
m = 1，dlgy /dt = λ(1 － y) ． This translates into:
w( t) ≈
c0
1 + ec1 － c2 t
as is represented by the Logistic model．
For p = 1，m ＞ 0，dlgy /dt = λ( y －m － 1)，as
represented by the Richards model， its solution is:
w( t) = c0(1 － e
－c1 t) c2 ．
The two models represented above are referred to
as “ theoretical models ” in biological statistic
modeling． They are often used to simulate the S curve．
This is especially true for the Richards model that has
an explicit solution for t = 0 and w( t) = 0． It is flexible
on the whole． Moreover，the subsection fits well and is
predictive． It has been widely used owing to these
benefits．
Generally，observed values(Xi，Yi) are not equal
to actuarial values ( xi，yi)，that is:
Xi = xi + ui，Yi = yi + ei ．
where ( ui， ei ) is the measurement error． The
regression equations of model Yi = f ( xi， ei， c1 )
(where ui = 0 and ei≠0) and model Xi = f ( yi，ui，
c2) (where ei = 0 and ui≠0) differs in relation to the
different error structures． At the same time，ui and ei
are not equal to 0 for actuarial reasons． Moreover，
greater errors are bound to occur if conventional
methods are used to estimate parameters． Fuller
( 1987 ) discussed such an issue in detail from a
statistical point of view with a model called a
measurement error model．
For this study，both the Richards model and the
Logistic model were selected to simulate seedling
growth and to compare their fitting results． The
measurement error model method was used to select the
parameters applied to the models． The general formula
of the measurement error model was:
f( yi，xi，c) = 0，
Yi = yi + ei，
E( ei) = 0，Var( ei) = Σ， i = 1，…，
{
n．
(1)
where f = ( f1 f2 … fm)
is a known m
dimensional vector-valued function; 1 × p-dimensional
vector Yi is the observed value of the true value yi，ei is
its error; 1 × q- dimensional vector Xi is the observed
value that possesses no error; Σ is a known or
unknown p × p positive matrix; the k × l dimensional
vector c are the general parameters; and p≥m． If f is
a bilinear function of ( yi，xi) and c，equation (1) is
considered a linear measurement error model or a
nonlinear model otherwise． If y·， x·． are fixed
variables，they are considered to be function relation
models． If y·，x·． are random variables，they are
considered to be structural relation models． If y· is a
random variable and x·． is a fixed variable，they are
considered to be superstructural relation models．
A key problem of model development is the
estimation of parameters． If the model is a function
relation model where ei has normal distribution， the
logarithm likelihood function of ei or Yi is:
－ 1
2
lg( | 2πσ2Ψ | ) －
1
σ2Σ
n
i = 1
(Yi － yi)Ψ
－1(Yi － yi) ． (2)
The estimation of model parameters turns into a
27
第 4 期 王雪峰等: 文冠果苗木生长与土壤含水量间关系建模
nonlinear programming problem with constraints． The
maximized objective function is:
l( y，c) = Σ
n
i = 1
(Yi － yi)Ψ
－1(Yi － yi) ． (3)
The constraint is:
f( yi，xi，c) = 0， i = 1，…，n． (4)
For constraint (4) where the solution of equation
(3) is denoted as ( y^ i，c )， the estimation of σ
2
will be:
σ^2 = 1npΣ
n
i = 1
(Yi － y^ i) ． (5)
Fuller (1987) employed the algorithm of (3) and
(4) when f was the function． Tang et al． ( 1996 )
employed the algorithm of (3) and (4) when f was a
vector function．
It can be determined from equation (1) that for
ei = 0，Yi = yi，the measurement error model is，in
fact， the parameter estimation model often used in
biological statistics． That is to say that the method of
solving regression parameters is one that is typical of
the measurement error model method．
2 Results and conclusion
Individual records were made for each seedling
each day after the buds first sprouted． A total of 2 858
crown width and seedling height data points were
collected． The following crown width simulation
equation parameters were taken from the 2 858 data
points．
2. 1 Comparison between the measurement error
model and the least squares method
Some errors were detected in the measured
seedling height ( Hi ) and crown width (Wi ) ( Hi =
hi + ui and Wi = wi + ei) ． Such objective conditions of
existence and measurement error were found to be in
line with each other when seedling height and crown
width satisfied the following two polynomial
relationships:
h = c0 + c1w + c2w
2 ． (6)
where h is the seedling height and wis the crown width．
Equation (6) was fitted to the measurement error
model method and the least squares method using the
seedling height and crown width data obtained
throughout the 107 day experimental period． Model
parameter calculations are provided for in Tab． 1．
Tab． 1 Parameters of seedling height-crown width
model obtained by the measurement error model
and the regression method
Estimation method Measurement error model Least squares method
c0 6. 619 3. 635
Parameters c1 － 0. 351 0. 582
c2 0. 058 0. 028
Definite index 0. 951 0. 919
Tab． 1 reveals that estimation accuracy and results
were inconsistent when different parameter estimation
methods were applied to the same model． Compared to
the least squares method，the measurement error model
contained a larger definite index． This indicates that
the measurement error model performed better than the
least squares method． The measured data， since it
contained error，will undergo greater error in parameter
estimation without the necessary condition for the least
squares method．
2. 2 Crown width growth equation
Let Ti = ti and Wi = wi + ei ． This means that the
growth rate Ti retains no error，but the measured crown
width Wi retains error． Tab． 2 is the result when
Logistic model and Richards model are fitted to the
measured crown width data．
w =
c0
1 + ec1 － c2 t
， (7)
w = c0 (1 － e
－c1 t) c2 ． (8)
Fig． 3 provides a comparison draft of the discrete
points of all measured values as well as the fitted
seedling crown width growth curve．
Tab． 2 Fitted parameters of the seedling crown width
growth model［equation(7) and (8)］
Parameters Logistic model Richards model
c0 31. 961 34. 165
c1 2. 089 0. 036
c2 － 0. 076 1. 447
Definite index 0. 954 0. 955
Fig． 3 reveals that during the initial stage of
seedling growth the fitted curve of the Richards model
was more significant than the fitted curve of theLogistic
model． Taking the overall curve into account， the
definite index of the Richards model slightly larger than
that of the Logistic model． That is the Richards model
performed better with the simulation of the growth
process in relation to yellow horn seedling crown
width．
37
林 业 科 学 49 卷
Fig． 3 The fitted curves and the discrete points
of Logistic model and Richards model
2. 3 Seedling height-crown width curve
Measuring seedling height is easier than measuring
crown width． It is better to estimate crown width with
seedling height and then examine competitiveness
between seedlings．
According to the data，the scattered distribution of
seedling height and crown width are S shaped．
Therefore，both the Logistic model and the Richards
model equation can be used to simulate relationships
between them． Some researchers believe that a linear
relationship exists between them (Chen et al．，2003) ．
Therefore， for the purpose of comparison， a linear
parameter equation was also provided for in this study．
The linear equation is:
c0 + c1w + c2h = 0 ． (9)
Tab． 3 is the parameter estimation of the three
model types．
Tab． 3 Fitted parameters of the seedling
height-crown width curve
Parameters Beeline Logistic model Richards model
c0 － 5. 117 37. 043 40. 355
c1 0. 574 1. 934 0. 037
c2 － 0. 226 － 0. 083 1. 298
W definite index 0. 916 0. 949 0. 957
The above parameter estimation was carried out
under the hypothesis that seedling height h and crown
width w exist within the measured errors． Tab． 3
provides the estimated parameters， and the definite
index of the crown width for the linear equation，the
Logistic model，and the Richards model equations． It
reveals that the Logistic model and the Richards model
equations performed better than the linear equation． It
was the Richards model equation that provided the best
overall results．
2. 4 The relationship between daily increment of
crown width，time，and soil moisture
Soil moisture is a key factor for the survival of
plants，especially those located within arid zones． It is
correlated to soil structure， ground water， and
precipitation． Kashiwabara et al． (1999) carried out
research on soil moisture movement． It has also been
found that plants affect soil moisture (Kobayashi et al．，
2000) ． It，however，is more important to understand
how soil moisture impacts seedling growth and how
much soil moisture is best for seedling growth within a
nursery environment．
Genetic factors and soil moisture (m) became the
deciding factors of seedling growth when nutritional and
light conditions were reproduced for this experiment．
Genetic factors are directly related to the number of
growth days ( t) ． Therefore，the daily increment of the
crown width (dw) can be expressed as:
dw = f( t，m，c) ．
If figures on genetics and soil moisture content are
revealed by way of the seedling growth data，then the
above equation can be written as a cubic polynomial:
dw = mCt． (10)
where dw is the daily increment of the crown width at t
days，and m，C，t are:
m =
1
m
m2
m

3
，C =
c11 c12 c13 c14
c21 c22 c23 c24
c31 c32 c33 c34
c41 c42 c43 c

44
，t =
1
t
t2
t

3
．
where m is the soil moisture content at t days; cij
( i，j = 1，…，4 ) is the unknown parameter matrix;
and t is the growth days．
When daily volumetric soil moisture content and
daily increment crown width data are obtained from the
water sensor，equation (10 ) can be fitted，and the
following parameter matrix is revealed:
C =
5. 504 0. 578 1. 492 － 0. 167
321. 448 － 116. 341 18. 528 － 1. 198
308. 516 － 193. 262 31. 782 － 1. 020
212. 523 － 236. 103 66. 127 －










6. 239
．
Its determination exponent R = 0. 999 and the
residual sum of squares SSD = 7. 377． This indicates
that equation (10) can provide a good description on
47
第 4 期 王雪峰等: 文冠果苗木生长与土壤含水量间关系建模
the relationship between growth days，soil moisture，
and the daily increment of crown width．
According to equation ( 10 )， the crown width
increment can be easily calculated under certain soil
moisture content on a given day． Given a certain
growth day ( t )，the daily increment of crown width
will be the maximum value specified at the position
where the first order derivative value of the soil
moisture content (m) in equation (10) equals zero．
This is referred to as the optimum water content which
we called m e ． According to equation (10)，let α i =
Σ 4j = 1 cij t( j －1)，i = 2，3，4，then
m e =
－ α3 ± α
2
3 － 3α2α槡 4
3α4
． (11)
Equation (11 ) provides two solutions and it is
easy to judge which one is the efficient solution
according to the definite issue． The crown width growth
curve of t = 35 and t = 65 are provided for in Fig． 4．
Fig． 4 The relationship between yellow horn seedlings crown growth
and the soil moisture content every growing day
a． The predictive value curve of the daily increment of crown width
under different soil moisture contents; b． Trend of optimum soil
moisture content required for maximum crown single-day growth．
According to Fig． 4a，the optimum water content
reached on day 35 of the experiment was 55% while on
day 65 it was 38% ． This shows that a declining trend
occurred as growth days increased． Fig． 4b shows the
optimum soil moisture content required at different
growing days，which was calculated through equation
( 11 ) ． And the curve clearly performed this
relationship． Furthermore， the daily crown width
increment increased along with an increase in soil
moisture content prior to reaching optimum soil
moisture content on day 35． After this point， a
decreasing trend was detected． Moreover， this
decreasing rate in water content was faster than the
increasing rate prior to when the optimum soil moisture
content was reached ( day 35) ． This means that the
capacity of the seedling to resist drought was better
than its capacity to resist excessive soil moisture
content． Its drought resistance capacity is therefore
stronger than its resistance to waterlogged conditions．
Another finding of the study is evident through a
comparison of the two curves prior to the optimum soil
moisture content being reached (day 35) ． The slope of
day 35 was greater than it was on day 65． The yellow
horn seedling therefore has a stronger dependence on
water content during its early growth stage，but this
dependence diminishes with an increase in growth
period． Its capacity to resist drought increases over
time． The curves also show that the declining rate seen
after day 65 becomes more pronounced． This means
that excessive water content has a greater negative
effect during the latter stages of seedling growth．
Therefore，an adequate supply of water during the early
stages of development and a subsequent reduction
during the latter stages will benefit overall seedling
growth and development．
3 Conclusion and discussion
The growth rate of yellow horn seedlings were
examined in detail for this study using data obtained
through a controlled experiment carried out in a nursery
environment． Results show that the measurement error
model method works well for parameter estimation，
especially when dependent and independent variables
are found within the measurement error． When this is
the case，more significant results will be obtained． The
Richards model proved superior at simulating data
compared to the Logistic model with regards to crown
width growth and the relationship between crown width
growth and seedling height for different growth periods．
Yellow horn seedling needs more soil moisture in the
57
林 业 科 学 49 卷
early growth，and the optimum soil moisture content
required for optimum growth decreased along with an
increase in growing days． In addition，it clearly shows
that yellow horn has stronger drought tolerate compared
to water-logging tolerate． Thats why we should avoid
soil moisture logging in actual operation．
Furthermore， this study provided a quantitative
description of seedling growth with respect to the
relationship between soil moisture and seedling growth．
Only a limited amount of outdoor testing was reserved
to the time of day and to constraints related to
environmental conditions． The experiment revealed that
light intensity and soil temperature have a great
influence on seedling growth． It should be noted that
further research in this respect must be carried out．
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