文章信息
- 王雪峰, 平藤雅之
- Wang Xuefeng, Hirafuji Masayuki
- 文冠果苗木生长与土壤含水量间关系建模
- Modeling the Relationship between Yellow Horn Seedling Growth and Soil Moisture Content
- 林业科学, 2013, 49(4): 70-76
- Scientia Silvae Sinicae, 2013, 49(4): 70-76.
- DOI: 10.11707/j.1001-7488.20130410
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文章历史
- 收稿日期:2012-04-01
- 修回日期:2013-01-22
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作者相关文章
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
2. National Agricultural Research Center for Hokkaido Region Sapporo 0820071
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 st and 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, 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 underst and 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 sourceThe 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 h and side of the 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. 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 solutionsSuppose 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:
$\frac{{{\rm{d}}{\mathop{\rm l}\nolimits} gy}}{{{\rm{d}}t}} = f\left({y, t} \right).$
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:
$f\left({y, t, c} \right)= \lambda {\left({1 - {y^m}} \right)^{1 - p}}{\left({{y^{ - m}} - 1} \right)^p}.$
where λ, m > 0(France et al., 1996 ). For p = 0, m = 1, dlgy / dt = λ(1 - y). This translates into:
$w\left(t \right)\approx \frac{{{c_0}}}{{1 + {{\rm{e}}^{{c_1} - {c_2}t}}}}$
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 - c1t)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:
${X_i} = {x_i} + {u_i}, {Y_i} = {y_i} + {e_i}.$
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:
$\left\{ \begin{array}{l} f\left({{y_i}, {x_i}, c} \right)= 0, \\ {Y_i} = {y_i} + {e_i}\\ E\left({{e_i}} \right)= 0, {\rm{Var}}\left({{e_i}} \right)= \sum {}, i = 1, \cdots, n. \end{array} \right.$ | (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; $\sum {} $ 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 r and om variables, they are considered to be structural relation models. If y● is a r and om 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:
$\begin{array}{l} - \frac{1}{2}\lg \left({|2\pi {\sigma ^2}\psi |} \right)- \\ \frac{1}{{{\sigma ^2}}}\sum\limits_{i = 1}^n {\left({{Y_i} - {y_i}} \right)} {\psi ^{ - 1}}\left({{Y_i} - {y_i}} \right). \end{array}$ | (2) |
The estimation of model parameters turns into a nonlinear programming problem with constraints. The maximized objective function is:
$l\left({y, c} \right)= \sum\limits_{i = 1}^n {\left({{Y_i} - {y_i}} \right){\psi ^{ - 1}}{{\left({{Y_i} - {y_i}} \right)}^\prime }.} $ | (3) |
The constraint is:
$f\left({{y_i}, {x_i}, c} \right)= 0, i = 1, \cdots, n.$ | (4) |
For constraint (4) where the solution of equation(3)is denoted as $\left( {{{\hat y}_i},c} \right)$, the estimation of σ2 will be:
${\hat \sigma ^2} = \frac{1}{{np}}\sum\limits_{i = 1}^n {{{\left( {{Y_i} - {{\mathop y\limits^ \wedge }_i}} \right)}^\prime }} .$ | (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 conclusionIndividual 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 methodSome 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:
$h = {c_0} + {c_1}w + {c_2}{w^2}.$ | (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 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 equationLet 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 = \frac{{{c_0}}}{{1 + {e^{{c_1} - {c_2}t}}}}, $ | (7) |
$w = {c_0}{\left({1 - {e^{ - {c_1}t}}} \right)^{{c_2}}}.$ | (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.
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.
2.3 Seedling height-crown width curveMeasuring 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:
${c_0} + {c_1}w + {c_2}h = 0.$ | (9) |
Tab. 3 is the parameter estimation of the three model types.
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 moistureSoil 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 underst and 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:
$dw = f\left({t, m, c} \right).$
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:
$dw = m'Ct.$ | (10) |
where dw is the daily increment of the crown width at t days, and m, C, t are:
$m = \left(\begin{array}{l} 1\\ m\\ {m^2}\\ {m^3} \end{array} \right), C = \left(\begin{array}{l} {c_{11}}\;{c_{12}}\;{c_{13}}\;{c_{14}}\\ {c_{21}}\;{c_{22}}\;{c_{23}}\;{c_{24}}\\ {c_{31}}\;{c_{32}}\;{c_{33}}\;{c_{34}}\\ {c_{41}}\;{c_{42}}\;{c_{43}}\;{c_{44}} \end{array} \right), t = \left(\begin{array}{l} 1\\ t\\ {t^2}\\ {t^3} \end{array} \right).$
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:
$C = \left\{ \begin{array}{l} \;\;5.504\;\;\;\;\;\;\;\;0.578\;\;\;\;\;\;\;1.492\;\;\;\;\; - 0.167\\ 321.448\;\;\; - 116.341\;\;\;18.528\;\;\;\;\; - 1.198\\ 308.516\;\;\; - 193.262\;\;\;31.782\;\;\;\; - 1.020\\ 212.523\;\;\; - 236.103\;\;\;66.127\;\;\;\; - 6.239 \end{array} \right\}.$
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 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 me. According to equation(10), let αi = $\sum\nolimits_{j - 1}^4 {{c_{ij}}{t^{\left({j - 1} \right)}}} $, i = 2, 3, 4, then
${m_{\rm{e}}} = \frac{{ - {\alpha _3} \pm \sqrt {\alpha _3^2 - 3{\alpha _2}{\alpha _4}} }}{{3{\alpha _4}}}.$ | (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.
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 discussionThe 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 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. That's 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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