RU2597645C2 - Method of measuring dynamics of tree leaves growth in clean ecological conditions - Google Patents

Abstract

FIELD: measurement technology.

SUBSTANCE: invention relates to methods of examination of woody plants. Summary: substrate with a white surface is laid from the bottom on the measured leaf, and from above is transparent palette for cartographic measurements. At that, the longitudinal axis of the plant leaf is combined with one of the palette grid lines. Then the sheet is photographed through the transparent palette with the grid, and the image is placed in computer memory. Parameters of leaf are measured by meshes of the palette grid on the magnified image of the sheet. At that, on the side surface of tree top local zone with approximately identical solar illumination is identified, on which group of accounted leaves with marks is selected. Accounted leaves parameters measuring is performed during vegetation period on daily basis from the beginning of leaves bud bursting with photographing each accounted leaf. During vegetation period accounted leaves parameters measuring is performed several times. Then in group by all accounted leaves, taken on one local zone of tree top surface, using statistical modelling average statistical regularities of accounted leaves recorded parameters by separate leaves, growing on one local zone of tree top surface growth dynamics is identified.

EFFECT: broader functional capabilities.

11 cl, 12 dwg, 6 tbl

Description

The invention relates to engineering biology and bioindication of the surrounding part of the crown of trees of the air environment by measuring dynamics, especially during the active life of plants from the beginning of sap flow and budding of the buds to the last stages of ontogenesis, mainly during the life of the growth organs of various types of woody plants, for example, as accounting leaves without taking them from woody plants with a simple and small leaf blade: linden, field or American maple, birch, poplar.

The present invention can also be used in environmental and technological monitoring of the development and growth of young woody plants, in particular for young forest trees, as well as the quality of tree foliage in city parks, alleys, roadside lanes and other tree plantings, natural afforestation of wastelands, former construction and other sites during land reclamation, natural forest restoration on deposits of agricultural land plots.

There is a method of measuring the area of leaves in woody plants (see the same book: Fedorova A.I., Nikolskaya A.N. Workshop on Ecology and Environmental Protection: Textbook Village M: Humanit. Publishing Center VLADOS, 2001. 288 p., P. 123-126), when 20-25 leaves of each tree species are cut from trees growing under different environmental conditions, put into bags, and then dried between sheets of newsprint in the laboratory.

The disadvantage is the low sensitivity (accuracy) of the indication and the high complexity of the practical application of the method due to the mixing of leaves from different trees in one common sample.

There is also a method of measuring the area of leaves in woody plants according to patent No. 2466351, including taking leaves from registration trees growing in different environmental conditions, then each measured sheet is laid on a substrate with a white surface, and a palette for cartographic measurements is applied on top of the sheet, and the axis of the plant leaf is aligned with one of the grid lines of the palette approximately in the middle of the palette, then the leaf is photographed through a transparent palette with the grid, after which the photograph is placed in the computer memory pa, and measurements of the length, width, and area of the leaf are performed on the cells of the palette grid in an enlarged image of the plant leaf.

The disadvantage is the sampling of leaves, which gives only one-time measurements of ontogenesis of leaves, the lack of measurements from the beginning of the growing season by dates, which does not allow to study the dynamics of the behavior of accounting leaves during their development and growth. In this case, the local zone of the crown of the tree is not distinguished, which does not allow one to study the influence of the surrounding part of the crown of trees of the air environment by measuring the dynamics of leaf parameters from the beginning of buds to various stages of leaf ontogenesis.

The technical result is an extension of the functionality of studying the daily dynamics of the average statistical behavior of a particular group of reference leaves, selected without cutting them from the local zone on the surface of the tree’s crown with approximately the same illumination, with a change in time at different stages of vegetation according to four parameters (leaf length and width, perimeter and leaf area), as well as improving the accuracy of measuring the dynamics of these parameters of the reference leaves in time, starting from the beginning of budding.

This technical result is achieved in that a method for measuring the dynamics of growth of tree leaves, including laying a substrate with a white surface from below on the measured sheet, and on top of the sheet lay a transparent palette for cartographic measurements, the longitudinal axis of the plant sheet being combined with one of the palette grid lines, then a sheet is photographed through a transparent palette with a grid, after which the photo is placed in the computer's memory, and measurements of the length and width of the sheet are performed using the cells of the palette grid in an enlarged image of the sheet plants, characterized in that on the side surface of the crown of the tree there is a local zone with approximately the same illumination, on which a group of registration leaves with labels is selected, measurements of the parameters of registration leaves are carried out during the growing season for days from the beginning of the budding of leaf buds with photographing each registration sheet, the first photographing is performed no less than a week and no later than three weeks after the buds of the leaves of the tree open, during the vegetation period, measurements of the parameters taken into account leaves, and then in the group on all user leaves taken on the same local area surface crown of the tree, the average statistical modeling reveal regularities of the dynamics of growth accounting leaves on individual parameters accounting leaves growing on one surface of a local zone of the tree crown.

In the tree’s crown, at a convenient height for measurements, on the side surface of the tree’s crown, a local zone with approximately the same illumination is distinguished, on which a group of at least 10 reference leaves is selected, each of which is marked with a label in the form of a piece of white thread attached to the base of the leaf on the petiole, in this case, the local zone is chosen approximately in the form of a circle with a diameter of not more than 0.5 m, and leaves that are located on the surface of the crown with the side of the crown of the tree illuminated by morning sunshine are taken as accounting.

In addition to the length, width and area of the reference leaves, the sheet perimeter is measured taking into account the size of the cells of the transparent palette by the number of cells through which the ridge line of the sheet edge passes, while the sheet area is determined as the sum of the full cells inside the sheet surface and the incomplete cells along the sheet perimeter also given the scale of the cell pallets

The perimeter of the sheet is measured by the number of cells through which the ridge line of the sheet edge passes, taking into account the scale of the cells of the palette 2 × 2 mm, according to the formula:

P = 0.28284I _P ,

where P is the perimeter of the sheet, cm;

I _P - the number on the periphery of the sheet of incomplete cells, pcs.

The leaf area is determined as the sum of the total cells inside the surface of the sheet and incomplete cells along the perimeter of the sheet also taking into account the scale of the cells of the palette 2 × 2 mm, according to the formula:

S = 0.04I _S + 0.02I _P ,

where S is the leaf area, measured by the number of cells with dimensions 2 × 2 mm inside the sheet and the number of cells of the perimeter of the sheet, cm ² ;

I _S is the number in the image of the sheet of full cells, pcs .;

I _P - the number on the periphery of the sheet of incomplete cells, pcs.

Measurement of parameters of registration leaves is carried out during the growing season for days from the beginning of the blooming of leaf buds with photographing each registration sheet through a transparent palette with a 2 mm grid, and the first photographing is performed no less than a week and no later than three weeks after the buds of the tree leaves open according to the date of the beginning of the growing season, the current time in days is taken as the influencing variable, and the value of the influencing variable at the beginning of the growing season is taken to be zero, and visually the parameters are taken into account leaf length and width, perimeter and area records leaves in the early growing season also take a zero value, the photographing is performed until the end of the growing season at least 10 times.

In the group for all at least 10 reference leaves taken on one local area of the tree crown surface, statistical modeling reveals the average statistical patterns of growth dynamics of reference leaves growing on one local area of the crown of the tree, according to four parameters of reference leaves - the length and width of the leaf, perimeter and leaf area, and, if necessary, identify biotechnological patterns and the estimated relative parameters of the reference leaves.

Measurements and statistical modeling are carried out from the beginning of budding of buds to any stage of ontogenesis of reference leaves for trees - also taking into account the time of active life of plants from the beginning of budding of buds, while the time is calculated in days, that is, in the Earth’s circulation cycles around itself, and measurements with photographing the registration leaves through a transparent palette with 2 mm divisions is carried out at approximately the same time of the day, for example, about 15 hours, to increase the accuracy of identifying patterns of dynamics of the parameter tally sheet statistical modeling measurements on the median strip of the European part of the Russian Federation is carried out at least until the end of August.

Each sheet is measured without cutting from the plant and is photographed through a transparent palette for cartographic measurements, for example, using a cell phone with a memory for storing photos as one single object, and the parameters of each sheet are measured on a computer using a photograph enlarged to A4 size of the sheet through the cells after 2 mm on the grid near the transparent palette, and before photographing, the longitudinal axis of the leaf of the plant is aligned with one of the grid lines of the transparent palette approximately in the middle the width of the palette.

To identify biotechnological patterns of dynamics of the length and width of the sheet during the growing season, a generalized two-term formula is used:

where a _t , b _t - time-varying average statistical length or width of a population of at least 10 reference leaves, mm,

- знак логических отношений «или»,

- a sign of logical relations "or",

t is the vegetation time from the beginning of leaf blooming, day,

A is the amplitude (half) of the vibrational disturbance of the length or width of the leaves in the population growth dynamics of at least 10 leaves, mm,

p is the half-cycle of the fluctuation of length or width of at least 10 reference leaves, day,

a ₁ ... a ₁₂ - model parameters that receive numerical values after statistical modeling by identifying the model according to measurements of length or width for at least 10 reference leaves, with the alternation of signs:

+ a ₁₀ , -a ₁₂ - take for regularity the dynamics of the average statistical length a _t in a population of at least 10 reference leaves;

-a ₁₀ , + a ₁₂ - take for regularity the dynamics of the average statistical width b _t in a population of at least 10 reference leaves.

To identify biotechnological patterns of perimeter dynamics and leaf area over the growing season, a generalized formula is recommended:

where P _t , S _t - time-varying average statistical perimeter (cm) or area (cm ² ) in a population of at least 10 reference leaves,

- знак логических отношений «или»,

- a sign of logical relations "or",

t is the vegetation time from the beginning of leaf blooming, day,

A is the amplitude (half) of the vibrational perturbation of the perimeter (cm) or area (cm ² ) in the dynamics of the average population growth of at least 10 reference leaves,

p is the half-period of vibrational disturbance when adapting to external influences of the perimeter or area of the reference leaves, day,

a ₁ ... a ₁₂ - model parameters that receive numerical values after statistical modeling by identifying the model according to perimeter or area measurements for at least 10 reference leaves.

The essence of the invention lies in the fact that in the crown of the tree at a convenient height for measurements on the side surface of the crown of the tree, a local zone with a uniform external exposure to sunlight is allocated, on which at least 10 reference leaves are selected, each of which is marked as a piece of white thread, tied to the base of the leaf on the petiole. In this case, the local zone is chosen approximately in the form of a circle with a diameter of not more than 0.5 m, and leaves that are located on the surface of the crown with the side of the crown of the tree illuminated by morning sunshine are taken as accounting.

The invention also consists in the fact that during the growing season, for days from the beginning of leaf bud blooming (according to the beginning of the growing season, time t is taken to be zero) with photographing each sheet through a transparent palette with a 2 mm grid. At the same time, photographing before the end of the growing season is performed at least 10 times, starting at least a week after the buds of the leaves of the tree open.

The essence of the invention also lies in the fact that for all at least 10 registration leaves taken on one local area of the tree crown, statistical modeling reveals the average statistical pattern of growth dynamics of registration leaves growing on one local area of the tree crown, according to four parameters of registration leaves: length and sheet width, perimeter and sheet area. If necessary, patterns can also be identified in the calculated relative parameters of the reference leaves.

The essence of the invention also lies in the fact that measurements and statistical modeling are carried out from the beginning of the growing season to any stage of ontogenesis of reference leaves for trees - also taking into account the time of active vital activity of plants from the beginning of budding. Moreover, time is calculated in days, that is, in cycles of the Earth's revolution around itself. And measurements with photographing of registration leaves through a transparent palette with 2 mm divisions are carried out at approximately the same time of the day, for example, about 15 hours. To improve the accuracy of identifying patterns in the dynamics of the parameters of accounting leaves by statistical modeling, measurements in the middle band of the European part of the Russian Federation are carried out at least until the end of August.

The novelty of the technical solution lies in the fact that for the first time the average dynamics of the behavior of at least 10 reference leaves in one local zone on the surface of the crown of a tree is considered. At the same time, the average statistical pattern of 10 registration leaves is identified by measurements at least 10 times from the beginning of budding, and at least until the end of August.

At the same time, each registration sheet is measured without cutting from the plant and is photographed through a transparent palette for cartographic measurements, for example, using a cell phone with a memory for storing photos, as one solid object. And the measurements themselves on a computer on an enlarged to A4 format photographs of the registration sheet are performed on the cells after 2 mm in the palette. Moreover, before photographing, the longitudinal axis of the leaf of the plant is aligned with one of the lines approximately in the middle of the width of the palette.

The positive effect is that on the surface of the tree’s crown it is possible to distinguish different local zones, taking into account the illumination of the crown and other factors influencing the development and growth of accounting leaves. Acceptance of at least 10 reference leaves from one local zone is sufficient to identify the trend with the wave component of the biotechnical regularity by four primary parameters (length, width, perimeter, area) of reference leaves. For a deeper scientific analysis, it is possible to calculate additional secondary parameters.

The positive effect also lies in the fact that for the first time it is precisely possible to identify patterns of dynamics of behavior according to the measured 10 registration sheets in a particular one local zone of the tree crown. Even in such a local zone, each of the 10 linden leaves turns out to be very different in growth dynamics, which indicates the high resolution of the proposed method, taking into account various factors of the local zone on the surface of the tree crown on the growth dynamics of each accounting sheet. In this case, it becomes possible, in addition to the absolute values of the measured four parameters of the leaf, to additionally take into account the calculated parameters in the form of the speed and acceleration of growth of the leaves of the tree from the beginning of bud blooming to any stage of ontogenesis of the counted leaves until the end of the growing season.

Thus, the proposed scientific and technical solution has significant features, novelty and a positive effect. In the scientific, technical and patent literature, information materials discrediting the novelty of the invention have not been found.

In FIG. 1 shows a satellite image of part of the city of Yoshkar-Ola, where the arrow shows the linden tree, and the top of the arrow is oriented to the local zone of the tree’s crown, where 10 reference leaves are highlighted; in FIG. Figure 2 shows an enlarged satellite image of the linden tree, and the local area of the linden crown is located on the side of the botanical garden and it is in a clean air environment (the high fence and the entire linden crown protect against the pollution from the street and the relatively low intensity of car traffic); in FIG. Figure 3 shows a photograph of three lindens, of which on the middle linden the circle zone of the linden crown is highlighted in a circle to isolate 10 leaves of leaves from the foliage; in FIG. 4 graphs of two laws of distribution of all 100 measurements are given (10 reference leaves of 10 measurements for each sheet for the growing season); in FIG. 5 shows spatial graphs of changes in the four parameters of the counted leaves depending on the time of the growing season and leaf number; in FIG. 6 as an example, photographs of sheet No. 1 for the growing season are shown (the difference in the photographs is explained by the rotation to the camera with the front or back of the sheet); in FIG. 7 shows a graph of the average pattern of leaf length dynamics; in FIG. 8 is the same in FIG. 7 leaf widths; in FIG. 9 is the same in FIG. 7 perimeters of leaves; in FIG. 10 is the same in FIG. 7 leaf areas; in FIG. 11 shows the dynamics of the perimeter of each sheet from a group of 10 leaves on one local area of the surface of the crown of linden; in FIG. 12 is the same in FIG. 11 area dynamics of each sheet.

A method for measuring the growth dynamics of tree leaves, for example, from the beginning of budding of buds to any stage of the growing season, at least until the end of August, includes such actions.

In contrast to the prototype, the selected registration sheet is not cut off, but is marked with a tag in the form of a tag with the number of the counted sheet on a white thread tied to the base of the counted sheet by the petiole. Each sheet is measured. Therefore, the measurement method is individual, and the identification of patterns is group based on at least 10 accounting leaves isolated on the surface of the tree’s crown with approximately one light in the form of a local zone.

A transparent palette for cartographic measurements with a grid with 2 mm divisions is fed to each measured sheet without damage, and manually take with the fingers of the left hand the petiole of the counted sheet, and the palette is laid on top of the registration sheet. Then the pallet is pressed to the sheet and held with the left hand so as not to tear off the registration sheet. After that, a cell phone is taken in the right hand, and the sheet is photographed through a transparent palette. In this case, the longitudinal axis of the tree leaf is aligned with one of the grid lines of the palette approximately in the middle of the palette.

After carrying out all the operations with all the accounting leaves, the photos are placed in the computer’s memory, and measurements of the length, width, perimeter and area of the sheet are performed on the cells of the palette grid on the image of the plant leaf enlarged to A4 format. After taking measurements on the required group of leaves, statistical modeling reveals the biotechnical patterns of changes in leaf parameters depending on the time in days.

Before taking measurements on the required leaf group, the edges of the photographic image of each leaf are first cut off so that the leaf appears in the described rectangle with centimeter cells, after which the leaf pattern is enlarged to A4 format, for example, by positioning the pattern of the measured leaf of the plant along the long side of A4 format, Moreover, the clarity of the image here does not matter - if only small cells were clearly visible at the grid of the palette.

On the side surface of the crown of the tree, a local zone with approximately the same illumination is distinguished, on which a group of registration leaves with labels is selected. Measurement of parameters of registration leaves is carried out during the growing season for days from the beginning of leaf bud blooming with photographing of each registration sheet. Moreover, the first photographing is performed no less than a week and no later than three weeks after the budding of the leaves of the tree. During the growing season, the parameters of the counted leaves are measured, and then in the group for all the reference leaves taken on one local area of the crown surface of the tree, statistical modeling reveals the average statistical patterns of growth dynamics of the reference leaves for individual parameters of the reference leaves growing on one local area of the crown surface tree.

In the tree’s crown, at a convenient height for measurements, on the side surface of the tree’s crown, a local zone with approximately the same illumination is distinguished, where a group of at least 10 reference leaves is selected, each of which is marked with a piece of white thread attached to the base of the leaf on the petiole . In this case, the local zone is chosen approximately in the form of a circle with a diameter of not more than 0.5 m, and leaves that are located on the surface of the crown with the side of the crown of the tree illuminated by morning sunshine are taken as accounting.

In addition to the length, width and area of the reference leaves, the perimeter of the sheet is measured taking into account the scale of the cells of the transparent palette by the number of cells through which the ribbed line of the sheet edge passes. In this case, the leaf area is defined as the sum of the total cells inside the surface of the sheet and incomplete cells along the perimeter of the sheet also taking into account the scale of the cells of the palette.

The perimeter of the sheet is measured by the number of cells through which the ridge line of the sheet edge passes, taking into account the scale of the cells of the palette 2 × 2 mm, according to the formula:

P = 0.28284I _P ,

where P is the perimeter of the sheet, cm;

I _P - the number on the periphery of the sheet of incomplete cells, pcs.

The leaf area is determined as the sum of the total cells inside the surface of the sheet and incomplete cells along the perimeter of the sheet also taking into account the scale of the cells of the palette 2 × 2 mm, according to the formula:

S = 0.04I _S + 0.02I _P ,

where S is the leaf area, measured by the number of cells with dimensions 2 × 2 mm inside the sheet and the number of cells of the perimeter of the sheet, cm ² ;

I _S is the number in the image of the sheet of full cells, pcs .;

I _P - the number on the periphery of the sheet of incomplete cells, pcs.

The parameters of the registration leaves are measured during the growing season for 24 hours from the beginning of leaf bud blooming with photographing each registration sheet through a transparent palette with a 2 mm grid. Moreover, the first photographing is performed no less than a week and no later than three weeks after the budding of the leaves of the tree. At the same time, according to the start date of the growing season, the current time in days is taken as an influencing variable. Moreover, the value of the influencing variable at the beginning of the leaf vegetation is taken to be zero, and visually the parameters of the reference leaves along the length and width, perimeter and area of the reference leaves at the beginning of the growing season are also taken with zero values, while photographing before the end of the growing season is performed at least 10 times.

In the group for all at least 10 reference leaves taken on one local area of the tree crown surface, statistical modeling reveals the average statistical patterns of growth dynamics of reference leaves growing on one local area of the crown of the tree, according to four parameters of reference leaves - the length and width of the leaf, perimeter and sheet area. And if necessary, biotechnological patterns are also revealed by the calculated relative parameters of the reference leaves.

Measurements and statistical modeling are carried out from the beginning of bud blooming to any stage of ontogenesis of reference leaves for trees - also taking into account the time of active vital activity of plants from the beginning of bud blooming. In this case, the time is calculated in days, that is, in cycles of the Earth's revolution around itself, and measurements with photographing of registration leaves through a transparent palette with 2 mm divisions are carried out at about the same time of the day, for example, about 15 hours. To improve the accuracy of identifying patterns in the dynamics of the parameters of accounting leaves by statistical modeling, measurements in the middle band of the European part of the Russian Federation are carried out at least until the end of August.

Each sheet is measured without cutting from the plant and is photographed through a transparent palette for cartographic measurements, for example, using a cell phone with a memory for storing photos, as one solid object. And the measurements of the parameters of each record sheet are carried out on a computer on a photograph enlarged to A4 format photographs of the record sheet by cells through 2 mm on a grid near a transparent palette. Moreover, before photographing, the longitudinal axis of the leaf of the plant is aligned with one of the grid lines of the transparent palette approximately in the middle of the width of the palette.

To identify biotechnological patterns of dynamics of the length and width of the sheet during the growing season, a generalized two-term formula is used:

where a _t , b _t - time-varying average statistical length or width of a population of at least 10 reference leaves, mm,

- знак логических отношений «или»,

- a sign of logical relations "or",

t is the vegetation time from the beginning of leaf blooming, day,

A is the amplitude (half) of the vibrational disturbance of the length or width of the leaves in the population growth dynamics of at least 10 leaves, mm,

p is the half-cycle of the fluctuation of length or width of at least 10 reference leaves, day,

a ₁ ... a ₁₂ - model parameters that receive numerical values after statistical modeling by identifying the model according to measurements of length or width for at least 10 reference leaves, with the alternation of signs:

+ a ₁₀ , -a ₁₂ take for regularity the dynamics of the average statistical length a _t in a population of at least 10 reference leaves;

-a ₁₀ , + a ₁₂ take for regularities the dynamics of the average statistical width b _t in a population of at least 10 reference leaves.

To identify biotechnological patterns of perimeter dynamics and leaf area over the growing season, a generalized formula is recommended:

where P _t , S _t - time-varying average statistical perimeter (cm) or area (cm ² ) in a population of at least 10 reference leaves,

- знак логических отношений «или»,

- a sign of logical relations "or",

t is the vegetation time from the beginning of leaf blooming, day,

A is the amplitude (half) of the vibrational perturbation of the perimeter (cm) or area (cm ² ) in the dynamics of the average population growth of at least 10 reference leaves,

p is the half-period of vibrational disturbance when adapting to external influences of the perimeter or area of the reference leaves, day,

a ₁ ... a ₁₂ - model parameters that receive numerical values after statistical modeling by identifying the model according to perimeter or area measurements for at least 10 reference leaves.

Example. The experiments were carried out in the city of Yoshkar-Ola (Fig. 1) during the growing season of 2014 on the same 10 reference leaves of linden (Fig. 2) located in the botanical garden on Krasnoarmeyskaya Street. Accounting leaves were selected in one local zone (Fig. 3) at a height of about 1.6 m from the eastern side of the linden lighting.

Each sheet was marked with a tag in the form of a piece of white thread with a tag with the sheet number tied to the petiole at the base of the sheet.

The method for measuring the area of linden leaves includes the following actions: a transparent palette with a grid, for example, with small cells 2 × 2 mm in size, is placed on top of the sheet so that the middle line along the palette coincides with the axis of the longitudinal vein of the sheet. Then, the sheet with the pallet is photographed, for example, with a digital camera with the functions of photographing and storing many photographs in memory.

Take a photo from each sheet and cut off the excess edges on the computer, and then increase for the convenience of cell counting to A4 format.

The initial data on measurements are given in table 1, at first without indicating the date of measurement. Therefore, first we consider all 100 measurements as one statistical sample for classical analysis.

=

Table 1 gives the following conventions for the physical parameters of the sheet:

a is the length of the leaf along the main vein, measured from the junction of the petiole with the leaf plate of the plant to the end of the leaf tip, mm;

b is the width of the sheet at the extreme points across the sheet plate or the total width of the sheet in the largest cross section of the sheet, mm;

P is the perimeter of the sheet, cm;

S - leaf area, measured by the number of cells with dimensions of 2 × 2 mm and the number of cells of the perimeter of the sheet, cm ² .

The perimeter and area of the registration sheet is calculated by the formulas:

For a set of 100 measurements, the sheet length varies from 29.4 mm to 92.2 mm. The width of the sheet varies between 32.6-80.2 mm. The perimeter varies from 10.18 to 36.77 cm. The sheet area varies from 7.8 to 62.68 cm ² .

To determine the discharge width Δx, an approximate formula proposed by G.A. By strajes:

where x _max - x _min - the range of variation; N is the number of measurements.

In our case, N = 100. Therefore, the denominator will be the number 1 + 3.322 × 2 = 7.644. The value x _man - x _min along the sheet length varies in the range of 62.8 mm, at a sheet width of 47.60 mm, a perimeter of 26.59 cm, and a sheet area of 54.88 cm ² .

The Δx value will be equal to: Sheet length 8.21 or approximately 8 mm: 96, 88, 80, 72, 64, 56, 48, 40, 32, 24.

Sheet width 6.22 or about 6 mm: 84, 78, 72, 66, 60, 54, 48, 42, 36, 30. Perimeter 3.47 or about 4 cm: 40, 36, 32, 28, 24, 20 , 16, 12, 8. The sheet area is 7.17 or about 7 cm ² : 63, 56, 49, 42, 35, 28, 21, 14, 7.

The normal distribution law plays an extremely important role in the existing theory of mathematical statistics and therefore now occupies a special position among other distribution laws. This is the most common law of distribution in practice.

In our case, the distribution curve according to the normal law has a symmetrical hill-shaped form, which corresponds to the standard normal distribution.

A biotechnical law is more accurate than normal because of the asymmetric arrangement of the two parts of the graph

Figure 4 shows graphs of the laws of distribution of the length of the reference leaves (for the rest of the parameters of the reference leaves, similar distribution laws were obtained) of linden (table. 2):

- normal

- biotechnical

The correlation coefficient of the normal law is 0.9566, while that of the biotechnical law is 0.9731.

Therefore, ceteris paribus, the biotechnological law more accurately reflects the distribution of actual values for all four parameters of linden leaves.

From table 2 and figure 4, the residues from formulas (4) and (5) are visible, which can be further identified by wave functions.

Table 3 shows the source data for constructing spatial dynamics graphs of the four main parameters of the reference leaves (Fig. 5) and subsequent statistical modeling of the dynamics without taking into account the number of reference leaves.

In comparison with the data of table 1 in table 3, the first three columns are supplemented: date, time t, day, and sheet number. Time t becomes an explanatory variable, and the parameters of the record sheet become dependent parameters (factors) or indicators. Therefore, statistical modeling identifies patterns in the form of structures a = f (t), b = f (t), P = f (t) and S = f (t).

The sheet number is only an auxiliary variable that does not have physical (in our case, physiological) meaning. We used it only for constructing spatial graphs showing the difference in the behavior of each record sheet. Thus, it becomes clear that each registration sheet experiences a different external impact. This allows us to take into account in future experiments those influencing factors of the environment, illumination, location in the tree crown, and others that distinguish the behavior in the life process of each record sheet from another.

In table 3, the date of 02.05.2014 is taken as the beginning of the growing season, that is, the start date of budding of the buds or the process of visible growth and development of reference leaves. For this date, zero values of all four studied basic parameters of accounting leaves are accepted.

Apparently, the adoption at the beginning of the growing season of the beginning of the sap flow process (approximately 04.21.2014) in the future will allow us to study the physiological processes occurring in the foliage of the tree.

From the graphs in figure 5, it is noticeable that before the period of growth of reference leaves at 30 days, all four indicators are smooth in dynamics. Here, rapid growth occurs without wave disturbances.

It is obvious that then there is an optimal value of the second (the first measurement is visual by date 02/05/2014 with zero parameter values) measurements ranging from 7 to 21 days from the beginning of the growing season for budding. Less than a week after the buds open, the leaves are still very small, and therefore the measurement error of the parameters of the registration leaves in a 2 mm grid of a transparent palette increases significantly. If the second measurement is carried out after three weeks, then there is a risk of insufficient fixation of the onset of vibrational disturbances in the growth of recorded leaves. From the data of table 3 it can be seen that in our experiments the threefold measurement was carried out after 19 days, which is within the acceptable interval of 7-21 days.

Further comparison of the graphs in figure 5 shows that after 30 days the dynamics of the behavior of each record sheet varies greatly. This is due to the fact that in the population dynamics of registration leaves vibrational disturbances are observed that are different for the habitat of each registration sheet. About a month after the buds open, the leaves begin an independent life, right up to the end of their existence. These processes can be studied through additional parameters in the form of speed and acceleration of leaf growth.

A comparison of the graphs in figure 5 also shows that linear parameters (length, width and perimeter) are more sensitive to the dynamics of growth of registration leaves, and areal parameters are less sensitive and the dynamics of the area of registration leaves is average statistical justification of individual effects on each recorded sheet are aligned. Therefore, it should be expected that when measuring the volumetric parameters of the reference leaves, an even greater leveling of vibrational perturbations of the individual reference leaves is formed.

But one thing is clear, which is shown by the time series in figure 6 of changes in sheet No. 1, that the linear parameters of the sheet have the greatest perspective on sensitivity. In this case, the shape of the sheet also changes over time: some parts of the recorded sheet develop faster than other parts. This is the future for finding new technical solutions for measuring the dynamics of leaves as a whole plant, for example, the dynamics of veins and angles between them.

Next, we show the regularities of the average dynamics of the four basic parameters in 10 reference linden leaves.

Table 4 shows a fragment of the data in table 3 on sheet No. 1.

Table 4 removes the “Sheet Number” column. You can simulate data for each record. However, we do not know the influencing factors among a population of 10 leaves selected as accounting in the local zone on the surface of the linden crown. Therefore, it will search for all 10 registration leaves for all 10 registration leaves from the data in Table 3.

The length of the sheet varies (Fig. 7) according to the formula

A = 4.75240 · 10 ^-33 t ^20.44582 exp (-0.063164t ^1.19166 ),

p = 2.99774 + 0.10895t ^0.94969 ,

where a _t - time-varying average length of leaves, mm,

t is the vegetation time from the beginning of leaf blooming, day,

A is the amplitude (half) of the vibrational perturbation of the leaf length in the population growth dynamics of 10 leaves, mm,

p is the half-period of fluctuation in the length of the reference leaves, day.

The first term of formula (1) is a biotechnical law, and half of the amplitude of the second term in the form of a wave function is also a biotechnical law. In this case, the second term has a positive sign in front of it, which indicates a positive effect of wave disturbance on the growth of leaf length. Thus, it becomes clear that the wave adaptation of plants to the environment is the main mechanism of their growth and development. At the same time, at the beginning of the growing season, when the buds opened on 02.05.2014, the period of oscillation was 2 × 2.99774≈6 days, and the beginning of the oscillation occurred a little later than the date of 02.05.2014 (about 3-5 days). Then, as the leaf organism grows older, this period increases, that is, the vibrational adaptation calms down.

The adequacy of formula (1) for the correlation coefficient is 0.9765, and this level of tightness of communication is an extremely strong (allowable correlation coefficient of more than 0.95) factorial relationships.

The width of the sheet varies (Fig. 8) by a similar two-term formula in construction

A = 1.00341 · 10 ^-22 t ^13.54328 exp (-0.0025789t ^1.72624 ),

p = 192.24495-29.76665t ^0.30703 .

Although the construction of the formula is the same, the behavior of the width of the population of 10 reference leaves is different.

The correlation coefficient is 0.9400 less than that of the formula for the dynamics of sheet length, and relates formula (2) to the strongest (more than 0.9) bonds. In this case, a fundamentally different half-period of oscillation. At the same time, the beginning of the variation in the leaf width is much earlier than the term for the buds to bloom (apparently, the increase in the leaf width was laid genetically much earlier than even the beginning of sap flow). As of May 2, 2014, the oscillation period is 2 × 192.24495≈384 days. This is more than a year period of revolution of the Earth around itself in 365 days.

Two hypotheses can be put forward here:

1) in antiquity more than 100 million years ago, linden was with year-round foliage, and then a change in the global climate on the planet forced this woody plant to switch to annual cycles of bud bloom, leaf growth and decay;

2) at the genetic level, linden laid the period of the vegetation cycle of 384 days in order to reliably exceed any periodicity of the Earth's rotation around itself (thereby linden will survive further inhibition of the Earth's rotation around the Sun for many millions of years ahead for the future).

The negative sign in the half-period formula shows that linden leaves quickly reduce the value of the oscillation period by phenotypic wave adaptation. This a posteriori fact shows that linden leaves on the width as it were, quickly “land” and, apparently, the discharge of foliage is associated precisely with the dynamics of the width of the linden leaves.

For the length and width of the sheet, a generalized formula is recommended

where a _t , b _t - time-varying average statistical length or width of a population of at least 10 reference leaves, mm,

- знак логических отношений «или»,

- a sign of logical relations "or",

t is the vegetation time from the beginning of leaf blooming, day,

A is the amplitude (half) of the vibrational disturbance of the length or width of the leaves in the population growth dynamics of at least 10 leaves, mm,

p is the half-cycle of the fluctuation of length or width of at least 10 reference leaves, day,

a ₁ ... a ₁₂ - model parameters (3), receiving numerical values after statistical Modeling by identifying the model (3) according to measurements of length or width for at least 10 reference leaves, with the alternation of signs:

+ a ₁₀ , -a ₁₂ take for regularity the dynamics of the average statistical length a _t in a population of at least 10 reference leaves;

-a ₁₀ , + a ₁₂ take for regularities the dynamics of the average statistical width b _t in a population of at least 10 reference leaves.

The perimeter of the sheet P _t varies (Fig. 9) according to the formula

A = 6.82928exp (0.00028005t ^1.76364 ), p = 8.57053 + 0.49522t ^0.81824 .

The adequacy of formula (4) for the correlation coefficient is 0.9585, and this level of tightness of communication is an extremely strong (allowable correlation coefficient of more than 0.95) factorial relationships.

The beginning of the perimeter unrest occurs before 05/02/2014 (minus sign before the wave shift). In this case, the oscillation amplitude varies with time according to the law of exponential growth, starting from the date of bud dissolution at 2 × 6.82928≈13.7 cm. Then, the oscillation amplitude increases and such a perturbation apparently leads to exceeding the permissible amplitude of the perimeter vibrational perturbation and then to tear off the leaf from the twig.

At the time of budding, the oscillation period is 2 × 8.57053≈17.2 days. Then the wobble calms down. Thus, the perimeter is characterized by an increase in amplitude and an increase in the period of oscillation.

The sheet area S _t varies (Fig. 10) according to the formula

A = 4.46628exp (0.00011616t ^1.88117 ), p = -9.37122 + 32.12674t ^0.10813 .

The adequacy of formula (5) with respect to the correlation coefficient is 0.9838, and this level of tightness of communication is the largest of the four parameters of scientific leaves, and at the same time it is super-strong (allowable correlation coefficient of more than 0.95) factor relationships.

The oscillation amplitude as of May 2, 2014 is 2 × 4.46628≈8.93 cm ² , and this is much larger than the cross section of the kidney itself. With further growth, the counted leaves sharply increase the oscillation amplitude. Apparently, there comes a moment when the amplitude exceeds the permissible limit. In this case, the perimeter and area act in the same way.

The construction of formula (5) is completely analogous to regularity (4). However, the initial value of the half-cycle is negative and we still cannot explain this feature of the biotechnical regularity. But already at t = 1 day, the half-period becomes positive and equal to 22.8 days. Thus, at the time of budding, the registration sheet receives some abnormal phenomena and processes.

For the perimeter and area of the sheet, a generalized formula is recommended

where P _t , S _t - time-varying average statistical perimeter (cm) or area (cm ² ) in a population of at least 10 reference leaves,

- знак логических отношений «или»,

- a sign of logical relations "or",

t is the vegetation time from the beginning of leaf blooming, day,

A is the amplitude (half) of the vibrational perturbation of the perimeter (cm) or area (cm ² ) in the dynamics of the average population growth of at least 10 reference leaves,

p is the half-period of vibrational disturbance when adapting to external influences of the perimeter or area of the reference leaves, day,

a ₁ ... a ₁₂ - model parameters (6), receiving numerical values after statistical modeling by identifying model (6) according to perimeter or area measurements for at least 10 reference leaves.

In relation to the dynamics of the parameters of each of the 10 linden leaves, the general formula (3) turned out to be unstable, and the generalized formula (6) for describing the dynamics of the perimeter (Table 5, Fig. 11) and area (Table 6, Fig. 12) turned out to be high sustainable.

We write both terms in each model (4) and (5) in the form of a wavelet signal coming from each registration sheet about its behavior.

The wave equation is an asymmetric wavelet of the form

where A _i is the amplitude (half) of the wavelet, p _i is the half-period of the oscillation.

According to formula (7) with two fundamental physical constants e (Napier number or time number) and π (Archimedes number or space number), a quantized wavelet signal is formed from within the studied phenomenon and / or process. The concept of a wavelet signal allows us to abstract from the physical meaning of the series themselves (in the general case, not only dynamic ones) and consider their additive decomposition.

For the perimeter of 10 reference leaves, the wavelet signal parameters are given in Table 5, and the corresponding graphs are shown in Figure 11. In all models, the correlation coefficient, due to the excess of the number of model 13 parameters over the number of observations 11, is 1.0000.

All eight parameters, except for the wave shift a ₈ , have a significant interval of variation of their values. This fact indicates the high sensitivity of the proposed method and the practical feasibility of studying the influence of various factors on the dynamics of physical parameters in one sheet of linden and other trees.

For the area of a population of 10 reference leaves, the wavelet signal parameters for each reference sheet are given in Table 6, and the corresponding graphs are shown in Figure 12. In all models, the correlation coefficient, due to the excess of the number of model 13 parameters over the number of observations 11, is 1 , 0000.

Out of 10 registration leaves, only sheet No. 2 has a negative half-period at the beginning of budding. The interval of change of the half-period a ₅ is in the range from -3.57412 to 21.47660 days.

By the area of each record sheet, it is also possible to identify patterns of influence of various external and even internal (for example, physiological) factors. Therefore, the proposed method has a significant prospect of methodological improvement.

The advantage of the proposed method is the technical simplicity of execution, since it is only necessary to make a transparent palette from the equipment, for example from sheet plexiglass, and a cell phone with a photographing function is now very common and almost every student or student has it. In addition, the advantage is the identification of time series of measurements of the average statistical growth dynamics of at least 10 accounting leaves by two generalized functions containing at least two components. The first component is a trend in the form of a biotechnical law, and the second is a wave equation that is positively directed at leaf growth. At the same time, the second term, showing the average vibrational perturbation of the recorded leaves, has a half amplitude for the length and width of the recorded leaves, is a biotechnical law, and for the perimeter and area of the recorded leaves, the half amplitude changes according to the law of exponential growth.

The invention can be widely implemented in school environmental circles, school forestry, and even in kindergartens, as well as in any geographical and other expeditions. To use the proposed method, it is necessary to make a transparent palette with a grid of cells 2 × 2 mm in size, have a cell phone with the functions of photographing, storing photos in memory and transferring them to a personal computer. The computer must have a software environment such as CurveExpert to identify persistent patterns.

Claims (11)

1. A method of measuring the dynamics of growth of tree leaves, including laying a substrate with a white surface on the bottom of the measured sheet and on top of a transparent palette for cartographic measurements, the longitudinal axis of the plant sheet being combined with one of the grid lines of the palette, then the sheet is photographed through a transparent palette with a grid , after the photo is placed in the computer’s memory, and measurements of the length and width of the sheet are performed on the cells of the palette grid on an enlarged image of the plant’s leaf, characterized in that on the side surface of the tree’s crown allocate a local zone with approximately the same sunlight, on which a group of registration leaves with labels is selected, the parameters of the registration leaves are measured during the growing season for days from the beginning of leaf bud opening with photographing each registration sheet, the parameters of the registration leaves are measured several times during the growing season and then in the group for all accounting leaves taken on one local zone of the crown surface of the tree, statistical modeling reveals the average statistical patterns spine dynamics of growth accounting leaves on individual parameters accounting leaves growing on a local surface area of the tree crown. 2. The method according to claim 1, characterized in that at a convenient height for measurements on the side surface of the tree crown, a local zone with approximately the same illumination is allocated, on which a group of at least 10 reference leaves is selected, each of which is marked with a white piece mark threads tied to the base of the leaf on the petiole, while the local zone is chosen approximately in the form of a circle with a diameter of not more than 0.5 m, and leaves that are located on the surface of the crown with the side of the tree illuminated by morning sunshine are taken as accounting. 3. The method according to claim 1, characterized in that in addition to the length, width and area of the reference leaves, the perimeter of the sheet is measured taking into account the size of the cells of the transparent palette by the number of cells through which the ribbed line of the sheet edge passes, and the sheet area is determined as the sum of the total cells inside the surface of the sheet and incomplete cells along the perimeter of the sheet also taking into account the scale of the cells of the palette. 4. The method according to claim 3, characterized in that the perimeter of the sheet is measured by the number of cells through which the ribbed line of the sheet edge passes, taking into account the scale of the cells of the palette 2

2 mm according to the formula

,
Where

- perimeter of the sheet, cm;

- the number on the periphery of the sheet of incomplete cells, pcs. 5. The method according to claim 3, characterized in that the area of the sheet is defined as the sum of the total cells inside the surface of the sheet and incomplete cells along the perimeter of the sheet also taking into account the scale of the cells of the palette 2

2 mm according to the formula:

,
Where

- leaf area, measured by the number of cells of size 2

2 mm inside the sheet and the number of cells of the perimeter of the sheet, cm ² ;

- the number on the image of the sheet of full cells, pcs .;

- the number on the periphery of the sheet of incomplete cells, pcs. 6. The method according to claim 1, characterized in that the measurements of the parameters of the registration leaves are carried out during the growing season for days from the beginning of the blooming of leaf buds with photographing each registration sheet through a transparent palette with a 2 mm grid, and the first photographing is performed at least a week later and no later than three weeks after the buds of the leaves of the tree open, and according to the start date of the growing season, the current time in days is taken as the influencing variable, and the value of the influencing variable at the beginning of the growing season is equal to th zero, and visually the parameters of the reference leaves along the length and width, perimeter and area at the beginning of the growing season are also taken with zero values, while photographing before the end of the growing season is performed at least 10 times. 7. The method according to claim 1, characterized in that in the group for all at least 10 reference leaves taken on one local area of the tree crown surface, statistical modeling reveals the average statistical patterns of growth dynamics for four parameters of the reference leaves - length and width of the sheet, perimeter and leaf area, and, if necessary, identify biotechnological patterns and the estimated relative parameters of the reference leaves. 8. The method according to claim 1, characterized in that the measurements and statistical modeling are carried out taking into account the time of active life from the beginning of budding of the buds to any stage of ontogenesis of the accounting leaves of the trees, while the time is calculated in days, that is, in the Earth’s circulation cycles around itself , and measurements with photographing of registration leaves through a transparent palette with 2 mm divisions are carried out at approximately the same time of the day, for example, about 15 hours, to increase the accuracy of identifying patterns in the dynamics of parameters of registration foxes b EB measurements on the median strip of the European part of the Russian Federation is carried out at least until the end of August. 9. The method according to claim 1, characterized in that each record sheet is measured without cutting from the plant and is photographed through a transparent palette for cartographic measurements, for example, by means of a cell phone with a memory for storing photographs, as one solid object, and the measurements of the parameters of each account of the sheet is carried out on a computer on photographs of the registration sheet enlarged to A4 format by cells through 2 mm on a grid near a transparent palette, and before photographing, the longitudinal axis of the plant leaf is aligned with one of the grid lines transparently mosaic about mid mosaic width. 10. The method according to claim 1, characterized in that in order to identify biotechnological patterns of dynamics of the length and width of the sheet during the growing season, a generalized two-term formula is used:

,

,

,
Where

,

- time-varying average statistical length or width of a population of at least 10 reference leaves, mm,

- a sign of logical relations “or”,

- growing time from the beginning of leaf blooming, day,

- the amplitude (half) of the vibrational disturbance of the length or width of the leaves in the dynamics of population growth of at least 10 leaves, mm,

- half period of fluctuation of length or width of at least 10 reference leaves, day,

- model parameters receiving numerical values after statistical modeling by identifying the model according to measurements of length or width for at least 10 reference leaves, with the alternation of signs:

,

take for regularity the dynamics of the average length

in a population of at least 10 reference leaves;

,

take for regularity the dynamics of the average width

in a population of at least 10 reference leaves. 11. The method according to claim 1, characterized in that in order to identify biotechnological patterns of perimeter dynamics and leaf area during the growing season, a generalized formula is recommended:

,

,

,
Where

,

- time-varying average statistical perimeter (cm) or area (cm ² ) in a population of at least 10 reference leaves,

- a sign of logical relations “or”,

- growing time from the beginning of leaf blooming, day,

- the amplitude (half) of the oscillatory perturbation of the perimeter (cm) or area (cm ² ) in the dynamics of the average population growth of at least 10 reference leaves,

- half period of vibrational disturbance when adapting to external influences of the perimeter or area of the reference leaves, day,

- model parameters that receive numerical values after statistical modeling by identifying the model according to perimeter or area measurements for at least 10 reference leaves.

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