The estimation of basket willow (Salix viminalis) yield – New approach. Part I: Background and statistical description

Renewable and Sustainable Energy Reviews 65 (2016) 1118–1126

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The estimation of basket willow (Salix viminalis) yield – New approach. Part I: Background and statistical description Andrzej Żyromski a, Wiesław Szulczewski b, Małgorzata Biniak-Pieróg a,n, Wojciech Jakubowski b a Institute of Environmental Protection and Development, Wrocław University of Environmental and Life Sciences, pl. Grunwaldzki 24, 50-363 Wrocław, Poland b Department of Mathematics, Wrocław University of Environmental and Life Sciences, ul. Grunwaldzka 53, 50-357 Wrocław, Poland

art ic l e i nf o

a b s t r a c t

Article history: Received 29 October 2015 Received in revised form 15 July 2016 Accepted 21 July 2016 Available online 30 July 2016

The overview of developed so far methods for biometric measurements showed a lack of application of the uniform standards, confirmed by the appropriate statistical tools. As a consequence, biomass’ productivity models built on their basis are incomparable between each other, and their use is limited. The paper presents an innovative method of probabilistic description of biometric features of basket willow (Salix viminalis). It concerns the descriptive statistics regarding the characteristics of that plant with which it is possible to relate the diameter and length of the shoots of basket willow with its mass, introducing the concept of shoot volume index. For this purpose the results of biometric measurements, consisting in cutting randomly selected shoots in the period of vegetation and measuring their length, diameter in the middle of the length and their mass, were used. The measurements were conducted in 2011–2013, a period covering the second, third, and fourth years of extensive cultivation of basket willow. Those years were evidently distinctive as regards the weather (thermal and water) conditions. Two features of basket willow shoots were suggested for analysis: their volume index and their mass. They are statistically correctly described by a mixture of two gamma distributions, independent of plantation age and of the moment when the measurement was conducted in the period of vegetation. The number of shoots in a bush was described by shifted Pascal distribution. In each year of cultivation, this distribution statistically correctly described their number. & 2016 Elsevier Ltd. All rights reserved.

Keywords: Basket willow (Salix viminalis) biomass Biometric measurements Mixture gamma distribution Biomass yield

Contents 1. 2.

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1118 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1119 2.1. Research facility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1119 2.2. Description of the experiment and methods of measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1119 2.3. The agrometeorological conditions in individual years of the experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1120 3. Descriptive statistics of biometric features of basket willow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1122 4. Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1123 5. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1125 Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1125 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1125

1. Introduction n

Corresponding author. E-mail addresses: [email protected] (A. Żyromski), [email protected] (W. Szulczewski), [email protected] (M. Biniak-Pieróg), [email protected] (W. Jakubowski). http://dx.doi.org/10.1016/j.rser.2016.07.072 1364-0321/& 2016 Elsevier Ltd. All rights reserved.

For years now, biomass, one of the renewable sources of energy, has been of interest both for environmental protection and economic reasons. Its greatest benefit in production and use is

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practically no emission of CO2 to the atmosphere as well as lower emission of sulphur or nitrogen oxides compared to fossil fuels. Another benefit of its production, compared to fossil fuels (natural gas or crude oil), is more reliable supply of raw material from Poland. Furthermore, it is also less unreliable as a source of energy compared to other renewable sources, i.e. wind or solar energy. Biomass is most often used for the production of biogas or burning in the form of pellets in power generating boilers. The possibility of using biomass at the scale of the world, Europe, and Poland is discussed by e.g. Voivontas et al. [1], Hoogwijk et al. [2], Ericsson et al. [3], Tuck et al. [4], Van Dam et al. [5], de Wit and Faaij [6], Hellmann and Verburg [7], Krasuska and Rosenqvist [8], Koçar and Civaş [9], Long at al. [10] or Vávrová et al. [11]. Those authors present various objectives, approaches and results. The expected growth in the number of plantations of energy crops in Europe will require large areas, which in turn undoubtedly can cause some competition between the production of plants for the purpose of power generation and for food. Consequently, the issues of cultivating energy crops in the context of area are addressed in many works, by De La Torre Ugarte and Ray [12] Strengers et al. [13], Verburg et al. [14], or Hellmann and Verburg [7]. According to Directive 2009/28/EC [15] the share of energy from renewable sources in the European Union countries in 2020 should be 15–20% and so its generation from e.g. biomass is highly expected. Poland has a great market potential for biomass, however, the sector of energy crops is only beginning to emerge. High expectations are related with the establishment of perennial energy crops plantations. The plants that are cultivated include basket willow (Salix viminalis), giant miscanthus (Miscanthus x giganteus), Virginia mallow (Sida hermaphrodita Rusby) and the Jerusalem artichoke (Helianthus tuberosus). Those plants are cultivated mainly on poor, fallow or resting soils as well as contaminated lands (e.g. with heavy metals), which is a real problem in many regions of Poland (e.g. [16]). The most favourable conditions for the cultivation of energy crops in the context of satisfying their needs for water are in the south and north of Poland. Such plantations should not be located in the centre of Poland (with the total precipitation below 300 mm in the vegetation period) due to low yield, especially in the years with large water deficit [17]. According to Faber [18] in Polish conditions perennial, energy crops plantations can also negatively affect the soil water balance and hydrological conditions in the water catchment areas in the case of excessive share in the cultivation structure. The energy crops whose production in Polish conditions is the most effective include basket willow and Virginia mallow, which is the subject of many works e.g. by Jurczyk et al. [19], Borkowska and Molas [20,21] or Faber et al. [22]. The climate is favourable for the cultivation of basket willow practically everywhere in Poland, however, low precipitation during the long critical period for this plant (June August) and high temperature at that time are the factors inhibiting high biomass growth. Poland also has similar and favourable conditions for the cultivation of Virginia mallow, however, in this case it is necessary to provide sufficient soil moisture at the beginning of its growth. The cooperation of planters of energy crops with the purchasers of biomass in the areas where it is grown makes it necessary to gather information on its current yield on the plantation during the whole period of vegetation. Consequently, for many years now a lot of research centres have been developing mathematical models to estimate the biomass yield from energy crops. Generally, they can be divided into empirical and mechanistic models [23]. The empirical models make use of the data resulting from direct measurements. Their objective is to find relations between the yield of the crops and selected meteorological as well as soil factors and tillage and plant care procedures. The mechanistic models, on the other hand, consist in connecting the physiological

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and morphological features which determine the plant growth. A comprehensive review of the energy crops biomass yield estimation models is presented by Nair et al in their work [23]. For this purpose the models dedicated directly to energy crops are used as well as the adapted existing models of plant yield. For instance such models as EPIC – [24], ALMANAC – [25], MISCANMOD – [26], MISCANFOR – [27], WIMOWAC – [28] or Agro-Ibis – [29] are used to calculate the biomass yield from giant miscanthus plantations, whereas the biomass yield of basket willow is calculated with the use of e.g. model 3PG or LINPAC [30–33]. Most of these models require a lot of good quality and difficult to collect input data regarding, for instance, phenological phases, dynamics of leafage growth, selected meteorological data, such as radiation rate, air temperature, precipitation, and data on tillage and plant care procedures applied, that information being often difficult to acquire. For this reason the authors of this paper, who have been dealing with the subject matter of modelling of yields of various crop plants (e.g. wheat or potato) in their earlier research [34,35], propose the innovative method of probabilistic description of biometric features of basket willow. Studies of this type can provide the basis for the creation of credible models for the estimation of its biomass.

2. Methods The innovative method proposed in the paper of basket willow biomass calculation on the basis of simple biometric measurements was developed on the basis of field studies conducted in 2011–2013 on the basket willow plantation founded in 2010 at the Agro and Hydrometeorology Observatory at Wrocław University of Environmental and Life Sciences. 2.1. Research facility Fig. 1 shows the location of the Observatory. The facility is located in the south-west part of Poland in Lower Silesia. It is separated from the city centre by a complex of parks and stadiums, the Odra canal, meadows and fields. It is situated at an altitude of 120 m above sea level, latitude 51°07′ and longitude 17°07′. The Observatory soils include brown soils, displaying in the surface layer the mechanical composition of weakly loamy sands. Their surface layers demonstrate poor variation and they are characterised by a high capacity for water retention. Field capacity is 217.0 mm in the 100 cm layer of soil. They have a characteristically high capillary rise. With groundwater at a depth of 100 cm, the surface layers contain approximately 18% of water in volume. Plant wilting point is approximately 5%. The mean depth of groundwater table level in this area is approximately 100 cm, and in summer time it fluctuates from 120 to 140 cm. 2.2. Description of the experiment and methods of measurements In the field experiment the plants were grown on a field of 4.9 m  22.8 m in 7 rows in 70 cm intervals, distances between plants within a row at 40 cm, which potentially gave 392 plants. The cultivation was conducted with the extensive method. The experimental plantation was founded in 2010, and the model of biomass growth was developed on the basis of results of measurements conducted on plants in 2011–2013 (2nd–4th year of the plantation). In the analysed years, various numbers of plants started their vegetation – 364 in 2011, 347 in 2012 and 348 in 2013, respectively. Measurements were made for all plants with the exception of those growing in the outer rows of the field which were considered buffer rows. In order to determine the variability of basket willow

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Fig. 1. Location of the research facility with the experimental cultivation of basket willow, Agro and Hydrometeorology Observatory at Wrocław University of Environmental and Life Sciences, Poland.

shoots three biometric features were measured in the period of vegetation. The measurements consisted in periodic trimming of randomly selected single shoots with length in excess of 0.5 m and measuring the length of the shoot cut, its diameter in the middle of its length, and then weighing to determine the mass of a single shoot. Every time the samples taken at the same intervals were of approximately the same number. During the whole vegetation period a total of 192 shoots in 2011, 166 shoots in 2012 and 175 shoots in 2013 were cut and measured in the way described above. No rules were found in the literature on the subject of shoot height at which the diameter should be taken for biometric measurements during the whole period of its vegetation. The shoot heights chosen for measurement by various researchers vary. For instance Amichew et al. [32] conducted their measurements at 30 cm from the ground, Stolarski et al. [36] at 50 cm and Sevel et al. [37] at 90 cm. In order to assess the productivity of basket willow with destructive and non-destructive methods Nordh and Verwijst [38] measured the diameters of shoots at 3 heights of 55, 85 and 105 cm, whereas Verwijst and Telenius [39] conducted their measurements at the base of the shoots. Several years of observations by the authors of this experiment revealed a high variability of shoot diameter from the base to the height of approximately 35 cm, as well as a high propensity for wilting of shoots shorter than 0.5 m. For this reason it is justified to begin biometric measurements on plants and shoots which have reached the height of at least 0.5 m. At the end of the vegetation period the number of all shoots longer than 0.5 m was counted for each plant growing on the plantation. Their total number varied from 1312 in 2011 (2nd year of vegetation) to 2948 in 2013 (4th year of vegetation). 2.3. The agrometeorological conditions in individual years of the experiment As mentioned earlier, the biometric measurements necessary as input data for the development of the current biomass

estimation model began when the plants were at least 0.5 m high. The observations made over the three years when the experiment was conducted indicated that this occurred at various times in May. That is why the description of the weather conditions in the vegetation period of basket willow was made for ten-day periods, beginning from the 1st ten-day period of May until the 3rd tenday period in October when the last biometric measurements were taken in 2011–2013. The multi-year period of 1971–2000 was adopted as the normative period. The description of the thermal conditions was made on the basis of the standard recommended by IMGW-PIB,1 applicable in Poland, and of precipitation on the basis of RPI [40]. The evaluation of the depth of the groundwater table in the years covered by the analyses was conducted in compliance with the standard developed by Biniak-Pieróg [41] on the basis of the multi-year period of 1971–2000. The description of the weather conditions during the field experiment was made for several universally applied meteorological elements. Table 1 shows the ten-day period values of air temperature, precipitation and groundwater levels from the multiyear period of 1971–2000. On the basis of the mean ten-day period values of air temperature and their deviations from the multi-year values (Fig. 2), the analysed vegetation periods in the years when the experiment was conducted can be classified as warm. 2011 and 2012 were dominated by warm and very warm ten-day periods, whereas 2013 by very warm ten-day periods. The description of the thermal conditions is further complemented with information on extreme values and the frequency of their occurrence during the vegetation season. This is the reason why it was decided to analyse the number of days from the period from the beginning of May until the end of September with the maximum daily temperature above 25 °C described as hot and with the maximum daily 1 Institute of Meteorology and Water Management – National Research Institute

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Table 1 Ten-day period values of air temperature Tp, precipitation P and groundwater levels Wg from the multi-year period of 1971–2000 adopted as normal and in the years 2011– 2013 in Wrocław-Swojec.

Tp [°C] V

1971–2000 2011 2012 2013

VI

VII

VIII

IX

X

I

II

III

I

II

III

I

II

III

I

II

III

I

II

III

I

II

III

12,2 10,2 16,1 15,2

14,4 16,0 13,1 15,9

14,6 17,9 18,0 12,9

16,6 20,5 14,4 15,2

15,9 18,6 18,5 20,8

17,2 18,2 18,8 16,9

18,0 18,1 22,4 20,6

18,3 20,3 17,4 18,7

18,7 16,4 20,2 22,0

19,1 19,3 20,3 23,1

18,3 19,4 18,0 17,9

16,4 19,2 19,5 16,3

15,0 16,7 16,6 15,8

13,3 15,8 14,2 12,6

12,4 13,9 12,9 10,2

11,0 13,1 11,3 8,6

9,2 7,2 9,3 10,4

6,6 7,9 5,4 13,1

17,2 20,3 49,2 71,9

16,5 17,4 6,5 14,0

21,4 11,7 8,0 50,0

17,4 33,6 25,7 74,1

29,9 3,1 59,5 0,0

25,3 59,2 9,5 97,6

33,3 54,7 42,8 2,7

29,1 34,7 38,7 16,0

24,7 80,9 26,7 17,6

25,2 14,1 37,6 45,8

18,9 34,9 8,7 16,5

23,0 15,8 26,9 5,9

16,4 26,9 0,9 27,7

17,1 3,4 48,0 70,2

14,6 0,0 3,3 23,5

11,3 15,0 5,7 0,4

13,1 23,3 16,8 6,0

14,5 4,3 12,9 1,4

P [mm]

1971–2000 2011 2012 2013

Wg [cm]

1971–2000 2011 2012 2013

102 119 117 60

109 122 124 68

113 127 140 96

119 134 153 61

122 138 154 62

124 142 148 56

124 123 149 80

123 116 143 102

Fig. 2. Deviation of the mean ten-day period values of air temperature ΔTp [°C] in 2011–2013 from the normal values from the multi-year period 1971–2000 at the Agro and Hydrometeorology Observatory at the Wrocław University of Environmental and Life Sciences.

temperature over 30 °C – described as very hot. October was not taken into account due to the fact that only in 2011 in the 1st decade the maximum temperature exceeded 25 °C twice. According to this classification in 2011 in the period from May until the end of September there were 58 hot days, which accounted for about 38% of the period in question. Most of them were recorded in the second ten-day period of July – 8. The first ten-day period of May was the coldest, when over three days i.e. from May 4–6 the recorded minimum air temperature was below 0 °C. There were only 5 very hot days. The year 2012 was slightly different as the recorded number of hot days in that period of five months was 39, which accounted for about 25% of the period in question. On the other hand, there were three times more very hot days than in the previous year – that is 15. 2013 was somewhat similar to the previous year in terms of the structure of hot and very hot days – 32 hot and 13 very hot days. In total, hot and very hot days accounted respectively for 20.9% and 8.5% respectively, of the whole period in question. In comparison to the preceding years, July 2013 had the highest number of hot days – 16, which accounted for 50% of all days during the five analysed months. This information

124 79 148 116

127 90 155 125

131 94 158 130

134 110 162 136

137 118 165 143

137 119 160 129

136 127 153 97

137 134 158 106

137 133 163 115

132 131 162 122

indicate the variability of the thermal conditions in specific years during the vegetation of basket willow. Precipitation is another meteorological element which is also the most important source of water. Fig. 3 shows the calculated values of RPI index from 2011 to 2013. The precipitation conditions of whole half-year summer periods with the periods of growth of energy crops in the field experiment corresponded to wet periods in 2011 and 2012 (453.8 mm and 427.6 mm, respectively), whereas that period in 2013, with precipitation level of 541.3 mm, was classified as very wet. The vegetation period of those plants in 2011 was dominated by very wet ten-day periods; the largest number of wet ten-day periods was recorded in 2012. Most tenday periods with over-normative precipitation were recorded after ten-day periods with low precipitation. 2013 was different, with the highest number of extremely wet and wet ten-day periods. Regardless of the precipitation sums in any period of time, it is also described by the number of days of its duration. Over the following years of the field experiment conducted by the authors the number of days with precipitation varied. In 2011 there were 75 days with precipitation in the half-year summer period, in 2012

Fig. 3. The variability of ten-day period values of RPI [%] index of ten-day period sums of precipitation in 2011–2013 at the Agro and Hydrometeorology Observatory at Wrocław University of Environmental and Life Sciences.

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- 93, in 2013 - 88. The analysis of daily precipitation levels demonstrated their significant variability. As high daily precipitation is usually not very effective, special attention was paid by the authors to daily precipitation levels above 10.0 mm. In 2011 there were 13 such occurrences in the half-year summer period in question, 12 in 2012, and as many as 18 in 2013. The aggregate amount of daily precipitation was 283.7 mm in 2011, 255.7 mm in 2012, and 410.5 mm in 2013. Comparing those amounts to the amounts from the whole half-year-long summer period from May through October, it can be noticed that in 2011 the amount of precipitation, which is the sum of 13 daily precipitation amounts, accounted for 62.5%, 59.8% in 2012, and as much as 75.8% in 2013. On the other hand, analysing the daily amounts above 15.0 a day, they were recorded on 10 days in 2011, 8 days in 2012 and 14 days in 2013. The precipitation was intensive so its effectiveness was poor. The short description of precipitation during the field experiment presents the potential possibility of using the precipitation by energy crops. The groundwater levels in the area of the experiments were another agrometeorological element which was analysed. The depths of groundwater table were significantly different in the half-year summer periods in the successive years under analysis and they were, on average 120 cm in 2011 (classified as moderately high,) 151 cm (classified as low) in 2012 and 122 cm (classified as high) in 2013. Fig. 4 shows the classification of tenday period groundwater levels in reference to normative values from 1971 to 2000. A detailed analysis of mean ten-day period depths of groundwater table in specific years indicated that their gradual lowering was observed in 2011 from the 1st ten-day period of May until the 3rd ten-day period of June from 119 cm to 142 cm, which corresponded to low and very low levels. In the period from the 2nd ten-day period of July until the 3rd tenday period of September the levels were moderately high and high, and then until the end of October the depth of groundwater table was at normal levels. The average ten-day period groundwater levels in the half-year-long summer period in the 3rd year of the plantation – 2012 was significantly different. In the whole period, the depth of groundwater table was below normal levels, and the period until the 2nd ten-day period of September was dominated by medium low and very high levels, whereas the next ten-day periods, until the end of October, by low levels in relation to the normative values (Table 1). In 2013, the 4th year of the plantation, over most of the halfyear-long summer period the depth of groundwater table was at very high, medium high, and high levels, from the 1st ten-day period of May until the 3rd ten-day period of July and from the

2nd ten-day period of September until the 3rd ten-day period of October. During the rest of that period those depths were at normal levels. The analysis of weather conditions in the vegetation periods of basket willow in the successive years of the field experiment indicated their change in relation to thermal conditions, precipitation, and groundwater levels.

3. Descriptive statistics of biometric features of basket willow The generally formulated objective of the study was to develop an innovative model of estimation of current biomass yield from a plantation of basket willow during its vegetation. That is why the authors looked for simple statistical relations describing the relationship of easy to measure features of the shoot (its length and diameter in the middle of its length) with its mass. It turned out that the introduction of the shoot volume index provides such 1 possibilities. That index was defined as follows: vi = 3 πhi di2 where hi is the length of i-th cut shoot and di is its diameter in the middle of its length. The shoot volume index was introduced instead of shoot volume because the measured mass includes both the shoot mass and its leafage. At the beginning of the study the only available measurement results were those from 2011 (2nd year of cultivation.) They were used after relevant statistical analyses have been conducted to formulate several basic assumptions on their nature:

 variability of mass Q and volume index V of the shoot of basket willow is correctly described by the gamma distribution of density function:

fα, β (x) =

 

xβ − 1 − x e α α βΓ ( β )

(1)

where α , β are distribution parameters, and independent variable x > 0 is equal to either q when the shoot mass distribution is discussed or to v when its volume index is discussed; relation between shoot mass and volume does not change its nature in the vegetation period; distribution of the number of shoots W in the plant is consistent with the shifted Pascal distribution (negative binominal):

Pr (W = k + 1) = ( − 1)k

−ν k ν p q, k

( )

k = 0, 1, 2, …

(2)

p and ν are parameters and k + 1 = 1, 2, … is the number of shoots in the plant. It was determined on the basis of relevant statistical tests that the distributions of probability (1) and (2) correctly estimate the variability of masses, shoot volume indexes, and the number of shoots in a basket willow bush. Table 2 shows the relevant p-values of χ2 test. It may seem debatable to assume that the analysed features of the shoots do not change their nature in the vegetation period. The relevant analyses which were conducted in connection with that confirmed that regardless of the moment of sample data Table 2 p-values of χ2 goodness-of-fit tests of mass and volume index of basket willow shoot and the number of shoots in a plant for data from 2011.

Fig. 4. Changes in ten-day period groundwater levels Wg [cm] from 2011 to 2013 and their classification in reference to the values in the multi-year period of 1971– 2000 at the Agro and Hydrometeorology Observatory at the Wrocław University of Environmental and Life Sciences.

Feature

Distribution

p-value

Mass Volume index Number of shoots

Gamma Gamma Pascal

0.059 0.079 0.455

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two uniform populations with gamma distribution. The first of them includes shorter and more massive shoots and the other longer shoots with evidently smaller diameter. Consequently, a hypothesis was put forward that the distribution of both volume index and mass is a mixture of gamma distributions (1):

h (x) = afα1, β1 (x) + (1 − a) fα 2, β2 (x)

Fig. 5. Goodness-of-fit of gamma distributions of mass and volume index of basket willow shoot observed in 2011.

collection they come from the same uniform population – this results from the measurement method described in the previous chapter. Fig. 5 shows the fitting of estimated distributions. The biometric characteristics of the shoots observed in the successive years of cultivation changed significantly. Apart from the uniform population observed earlier there appeared shoots with different probabilistic characteristics see Fig. 6 presenting the measured masses and the shoot volume indexes from 2012 to 2013. Such a graphic presentation demonstrates that there was no reason to assume the hypothesis of goodness-of-fit of the observed features with gamma distribution (1). The curves of the estimated distribution significantly differ from their relevant empirical distribution functions. The additional statistical tests which were conducted suggested that in this case the population of willow shoots is composed of

(3)

where α1, β1, α2, β2 are estimated parameters for each of the two distributions of the mixture separately, a is the estimated parameter of the mixture and x is either the volume index v or shoot mass q. Relevant goodness-of-fit tests demonstrated at the significance level of 0.05 that the probabilistic model (3) correctly estimates the distribution of willow shoot mass and volume index both for observations from specific years 2011, 2012, 2013 and from the whole 3-year-long period. The distribution parameters (3) were estimated with the maximum likelihood method with the use of Expectation Maximisation procedure [42]. The calculations were conducted with the use of Mathematica 9 package by Wolfram Research Inc. The goodness-of-fit of the estimated distribution with the empirical data was verified with the χ2 test. Table 3 shows the calculated values of the tests. Fig. 7 also shows the good quality of fit of the data from 2011 to 2013. As mentioned above, it was assumed that the distribution of the number of shoots in a plant is consistent with Pascal distribution – negative binominal (2). The distribution was highly consistent with the assumed one every year of the cultivation (second, third and fourth) – Table (4). As expected, the number of shoots per plant in the successive years of cultivation displayed an evidently growing trend. Apart from empirical data, Fig. 8 also shows the distribution values (2) estimated with the highest credibility method for each year. The increase of variance caused by the growing variability of the number of shoots per plant is evident.

4. Discussion The research results presented in the paper were acquired on the basis of a three-year field experiment conducted in the years 2011–2013. The field experiment had been started a year earlier, but the analyses were made for results from the second to fourth years of vegetation. The authors decided that such an assumption eliminates the instability of basket willow growth and development in the first year of vegetation. The experience gained and the observations made indicate that the decision was justified, as in the first year the falling-out of a large number of plants during the vegetation period is fairly common. Taking data of this type into account in statistical analyses would make their correct interpretation difficult. A great majority of methods and models applied for the estimation of yields of crop plants, and not only energy crops, require information on the variation of numerous meteorological elements. In many cases data of that type are not readily available or acquired in indirect ways. To avoid that difficulty, the authors searched for a method which, on the basis of simple biometric features of basket willow, would allow the estimation of its Table 3 p-values of the χ2 goodness-of-fit tests of the mass and volume index of basket willow shoot (2011–2013) with the mixture of gamma distributions (3).

Fig. 6. Goodness-of-fit of gamma distributions (1) of mass and volume index of basket willow shoot observed in 2012–2013.

Feature

2011

2012

2013

2011–2013

Mass Volume index

0.138 0.991

0.181 0.247

0.167 0.197

0.202 0.563

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Fig. 7. Goodness-of-fit of the mixture of gamma distributions (3) to the mass and volume index of basket willow shoot observed jointly in 2011–2103.

Table 4 p-values of the χ2 goodness-of-fit tests of the number of shoots in the basket willow bush (2011–2013) with Pascal distribution (2). Feature Number of shoots

2011 0.455

2012 0.148

2013 0.219

Fig. 8. Goodness-of-fit of Pascal distribution (2) to the number of shoots in the basket willow bush observed in 2011–2103.

biomass. The methods of mathematical statistics applied permitted the achievement of that aim. The results obtained proved to be surprisingly good, which is confirmed by the relevant goodness-of-fit tests. To identify the relationship between the mass of basket willow shoot (destructive measurement) and selected biometric features (non-destructive measurement), the authors defined a new biometric parameter called the shoot volume index. Its value is determined on the basis of shoot length and its diameter at mid-length. Both the distribution of shoot mass and of the volume index defined in that way is a mixture of two gamma distributions, independent of plantation age and of the time of measurement in the vegetation period. The parameters are closely correlated, which allows the estimation of shoot mass (destructive measurement) on the basis of the shoot volume index (non-destructive

measurement). For this reason the authors propose this method of measurement as a standard, especially as the existing research literature does not provide accepted standards pertaining to biometric measurement of shoot. The models presented above can be used for the description of variation of the shoot volume index and the shoot mass with a mixture of two binomial gamma distributions. The undertaken attempts at the analysis of the correlation between shoot mass and its length and diameter are considerably more difficult in the probabilistic description. The distributions of shoot length and diameter are mixtures of normal distributions. In consequence, an attempt at combining the normal distribution and the gamma distribution into one trivariate distributions poses considerable calculation problems and does not lead to the achievement of better results. Analysis of measurement data accumulated in the course of the experiment and of the results of analyses conducted on their basis permitted also the formulation of a proposition of a biological character. Statistical analyses indicate that the entire population of willow shoots can be divided into two homogeneous populations described by gamma distributions. The first of the populations is made up of relatively short shoots with big diameters, and the second long shoots with notably narrower section. In consequence, only a mixture of their distributions provides a statistically correct description that applies comprehensively to the entire population. This trait became evident only in the third and fourth years of cultivation. The statistical tests conducted within the scope of the study permit the conclusion that in the second year that effect was not achieved. In that case a single gamma distribution was sufficient. The individual years included in our tests and analyses were characterised by diverse weather conditions. This is described in detail in the part containing the presentation of weather conditions in the successive years of the experiment. The main focus was on the characterisation of the pluviothermal conditions and on changes of groundwater table. The years 2011–2013 differed from one another in terms of the extreme temperature conditions that occurred during the experiment. The indicator characterising those conditions was the number of hot and sweltering days. In each of the vegetation seasons in the individual years the rainfall sums, air temperature and groundwater levels in early spring determined decisively the start of vegetation and the rate of growth and development of basket willow over the entire vegetation period. That was particularly observable in July and August in the particular years, i.e. in the period of intensive growth of the aboveground mass. In those time intervals considerable variation was observed in the influx of atmospheric precipitation, as both dry decades and extremely wet ones were noted. Likewise, dynamic changes were noted in the levels of the groundwater table. In July and August the depths of the groundwater table varied from medium high and high in 2011 to very low and low in 2012. Whereas, medium high and normal levels were noted in 2013. In spite of the large variation in such an important growth factor as the weather conditions, characterised in a large simplification by several agrometeorological factors, the mixture of distributions of shoot volume index and shoot mass permitted statistically correct description of those populations. The accumulated research material permitted estimation of the number of shoots in a bush with a shifted negative binomial distribution (Pascal). In each of the years of cultivation that distribution described their number statistically correctly. Statistical tests demonstrated that due to the large variation of that description its generalisation is not possible at that stage of the study. One can only suppose that with the passage of years the distribution of the number of shoots in plants will be subject to the process of stabilisation. This supposition finds support in the

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References

Fig. 9. The approximation with the exponential model of Pascal (2) distribution of the number of shoots in a bush in the second, third, fourth and eighth year of the plantation.

results of extrapolation of variation in time of the estimated parameters of the Pascal distribution. Simple exponential models, approximating the variation of those parameters in time, have distinct asymptotes. An example of that variation for the second, third, fourth and extrapolated eighth years of plantation are presented in Fig. 9.

5. Conclusions The results discussed in the preceding section, supported with statistical analyses, constitute an elaboration based on a severalyear field experiment. For this reason they permit the formulation of several conclusions characterising the biological features of basket willow shoots and bush: 1. The shoot volume index defined in the study, determined on the basis of basket willow shoot length and diameter at mid-length, correlates well with shoot mass and allows to propose that method of biometric measurements as a standard, as it permits the estimation of shoot mass by means of easy and non-destructive measurements. 2. The shoot volume index and shoot mass are statistically correctly described by a mixture of two gamma distributions independent of plantation age and time of measurement during the vegetation period. 3. The analyses performed permit the conclusion that the entire population of willow shoots can be divided into two homogeneous populations described by gamma distributions. That trait becomes observable only from the third year of cultivation. 4. In every year of cultivation the number of shoots in a bush is statistically correctly described by a shifted negative binomial distribution. The parameters of that distribution change with the age of the plantation. The variation is asymptotic in character. 5. The analyses conducted in the scope of the study demonstrated that the weather conditions have no effect on changes of biometric features of basket willow shoots ends bush.

Acknowledgements Research financed from Polish Budget Funds for Science in the years 2011–2014 as research project No. NN305 3835 39.

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