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Thesis for the Degree of M. S‘
MICHIGAN STATE UNWERSITY
DONALD DeWITT MOWYKE

1972

 

 

 

 

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ABSTRACT

A GROWTH MODEL FOR ASPARAGUS

BY

Donald DeWitt Moerdyke

The object of this study is to deve10p a model that
will accurately describe the growth characteristics of
asparagus spears in the field for application as a predic-
tion tool for selecting time of harvest. An equation for
spear height was derived by solving a differential equation
for growth rate. The constants of the equation were found
by least squares regression analysis and by direct calcula-
tion from the field data. The least squares regression
also indicated that growth rate is higher, by a factor of
two, for daytime growth than for night growth.

A relatively large variability was found to exist
between, and to some extent within, spears. To account for
the variability constants were selected randomly from a
cumulative probability distribution for use in height
calculations.

To verify the ability of the model to predict spear
heights, spear heights were calculated using the same

growth conditions that the measured spears underwent.

Donald DeWitt Moerdyke

Because of the variability, single spears could not be
compared directly, so distributions were used for direct
comparison of measured and calculated heights. The
calculated and measured values were compared in two ways.
First, all spears from one measurement group were "grown,"
under similar conditions, to the next measurement point.
Second, spears with initial heights within a particular
range were "grown" for 7 and 24 hours. In almost all
instances the means of the measured and calculated height
distributions differed by less than two centimeters, but
the calculated distributions were more compact than the
measured value distributions.

After verification, two examples are offered as to
how the model might be used as a prediction tool. One
uses the results of the verification calculations and the
other takes a hypothetical situation and desCribes how the

farmer might go about making a time of harvest decision.

’ I

I; -(”o =£ sor

Department‘Chairman

.Approved

 

A GROWTH MODEL FOR ASPARAGUS
By

Donald DeWitt Moerdyke

A THESIS

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

MASTER OF SCIENCE
Department of Agricultural Engineering

1972

TABLE OF CONTENTS

Page
LIST OF TABLES . . . . . . . . . . . . . iii

LIST OF FIGURES . . . . . . . . . . . . . iv

INTRODUCTION 0 O O C O O O O O O O O O O 1
REVIEW OF LITERATURE . . . . . . . . . . . 3
MATHEMATICAL MODEL . . . . . . . . . . . . 9

EXPERIMENTAL DATA . . . . . . . . . . . . 15

Field Procedure . . . . . . . . . . . . 15
Multiple Regression Analysis . . . . . . . 15
Distribution of Growth Constants . . . . . . . 18

RESULTS . . . . . . . . . . . . . . . . 24

-Verification . . . . . . . . . . . . . 24
APPLICATION OF THE MODEL . . . . . . . . . . 32
SOURCES OF ERROR . . . . . . . . . . . . . 35
SUMMARY OF RESULTS . . . . . . . . . . . . 37

Field Data 0 O O O O O O O O O O 0 O O 37

Regression Analysis . . . . . . . . . . . 37
Model Development . . . . . . . . . . . . 38
Growth Constant Distribution . . . . . . . . 38
Model Verification . . . . . . . . . . . 39

FUTURE RESEARCH . . . . . . . . . . . . . 40

REFERENCES 0 O O O O O O O O O O O O O 41

ii

LIST OF TABLES

Table . Page

1. Growth Constants for Spears Selected Randomly
from 1972 Data 0 O O I I O O O O I O 20

2. Distribution of Constant b for 1750 Values . . 22

3. Distribution Means for Measured and Calculated
Distributions at Each Measurement Group . . 27

4. Means for Height Distribution after 7 and 24
Hours 0 O O O O O O O O O O O O O 29

5. Height Distribution at Two Hour Intervals for
600 Spears Grown Under Same Conditions . . . 34

iii

LIST OF FIGURES

Figure Page
1. Growth Curve for Two Typical Spears . . . . 4

2. Three Dimensional Representation of Growth Rate
Equation . . . . . . . . . . . . 7

3. Frequency and Cumulative Distribution of Growth
Constant b for 1750 Values . . . . . . . 23

4. Means of Calculated and Measured Height
Distributions . . . . . . . . . . . 26

5. Height Distribution for Spears Grown 7 and 24

Hours from Initial Heights Between 17-20
centimeters O O O O O O O O O O O O 30

iv

INTRODUCTION

As with most fields, technology is advancing very
rapidly in the area of vegetable harvesting machinery.

The past two decades have seen the development of machinery
which has allowed a farmer to cultivate large acreages of
vegetables. Previously, these same vegetables were grown
in smaller, hand harvested acreages. Examples of such
crops are tomatoes, lettuce, cucumbers and asparagus.

As far as asparagus is concerned, most attempts to
develop a selective harvester have proven to be impractical
or uneconomical, as stated in Stout (1967). The alterna-
tive to a selective harvester is one that cuts all spears,
i.e., nonselective. A recent development is the sled
harvester. Although this harvester is nonselective, the
high harvest rate and low initial cost enable it to operate
economically. The sled operates at speeds up to 15 mph
compared to the l to 3 mph speeds of a selective harvester
(Carpenter, 1967).

The primary problem of increased volume, aside
from physical handling, is that the time of harvest becomes
very critical. In Michigan, processors accept only 10 per
cent of the spears over 7.5 inches long before docking the
grower because taller spears have more fibrous chunks

l

than short spears. Therefore, considering the height of
the cutter bar, the grower wants to harvest when less than
ten per cent of the spears are more than ten inches (25
centimeters) high. During high temperature conditions
asparagus has been observed to grow as much as three
inches in four hours. A harvesting error of even a couple
of hours could be quite costly because the spears might
grow through the acceptable range before the harvesting is
completed. The optimum situation would allow selecting
the harvest time at the point where the duration of harvest
coincides with the period that the field grows through the
desired height range.

The objective of this study is to develop a growth
model for asparagus which can be used to aid the farmer
in predicting the Optimum time to harvest asparagus. The
model must be able to predict the height of a spear at
time T + At, given the height distribution at time t and

the average temperature during the period At.

REVIEW OF LITERATURE

To date only limited work has been done toward
modeling the growth characteristics of asparagus spears;
however, several people have investigated the elongation
of asparagus Spears. Although the objectives of their
studies differed, the parameters found to affect the growth
rate did not. The primary factors influencing growth were
temperature and height. The fact that height is involved
eliminates the possibility of a simple temperature model
since growth rate increases as the spear gets longer.

Asparagus, like many other plants, is divergent
during its early growth; that is, taller spears grow
faster than short ones, leaving a field less mature with
time. Divergence is accentuated by the fact that new
spears emerge continually during the entire growth period,
thus increasing nonuniformity. Growth rate is also sensi-
tive to changes in temperature resulting in even greater
nonuniformity at higher temperatures. The effect of
temperature, as well as height, can be seen in the two
typical growth curves shown in Figure 1.

One of the earliest studies was by Culpepper and
Moon (1939), in which they measured the elongation of

asparagus stems over a height range of O to 250 centimeters

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with temperature ranging from 45° to 95°F. They found
considerable variability in elongation rates and ultimate
heights reached. They concluded that growth rate con-
tinues to increase up to a height of about 60 to 70 centi-
meters (25 inches) which is well past the 25 centimeter
(10 inch) height of harvested spears. They also found
that within the given temperature range, growth rate
increased with increased temperature; however, little
growth took place below 40°F.

Another early study by Tiedjens (1924) recognized
the effect of increasing temperature on growth rate. A
more recent study of stem elongation by Downs (1962) con-
firms the fact that taller spears grow more rapidly than
shorter Spears during the same time period. He found that
length could be correlated with initial height in a linear
function for initial heights ranging from two to nine
inches.

Only one person has attempted to prOpose a model
for spear growth, Blumenfield (1961). In this study,
temperature and height were assumed to be the primary
factors influencing growth. Field measurements were taken
over a range of temperatures and heights and then analyzed
using multiple regression techniques. The proposed model

for growth rate was

GR = a + b SH + c T + d (SH)2 + e SHT + f (T)2 [1]

where

F-3
ll

average air temperature

SH spear height in centimeters

and
a, b, c, d, e, f are constants.

After the regression analysis, the squared and cross
product terms were eliminated because they only accounted
for four per cent of the variation while the linear terms
accounted for 80 per cent. The remaining 16 per cent was
attributed to variables such as soil moisture, fertility,
temperature or individual Spear variation. The final

equation for the growth rate was
GR = -15.25 + 0.3163 (SH) + 0.3544 (T) [2]

This equation indicates thit an increase in height of one
centimeter will cause tbs growth rate to increase 0.3163
cm/day and that an increase 3f 1‘F will increase the
growth rate by 0.3544 cm/day.

The Blumenfield equation can be represented by a
plane in three dimensional space with axes of height,
temperature and growth rate as seen in Figure 2. Of
particular interest in this plane are the temperatures at
which growth stops. These are represented by the line of

intersection between the equation's plane with the

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height-temperature plane. The line of intersection shows
that shorter spears require higher temperatures to sustain
growth. The plot represents the equation rather than
actual conditions thus explaining why unlikely situations

appear in Figure 2.

MATHEMATICAL MODEL

As stated in the introduction, the object of this
study is to develop a growth model for asparagus that will
assist the farmer in selecting the optimum time to harvest
his crop. Blumenfield's regression analysis, using height
and temperature as the significant parameters, can be
modified to yield a differential equation by defining

growth rate as

GR

(LID:
('1' :13

This definition allows Blumenfield's equation to be

written as

_ dH _ -
GR - a-E' - a + 13H 4' CT [3]
or
dH _
a-IL- - bH - a + CT [4]

where the constants have the units

a = cm/day
b = l/day
c = cm/day °F.

10

The solution of [4] consists of a complementary solution

and a particular solution. The complementary solution

_ bt
H — Coe [5]
where CO is an undetermined constant, is the solution to

the homogeneous equation

The particular solution can be obtained by using the
method of undetermined coefficients once the temperature
is known as a function of time. For the moment this will
be met by representing the temperature by the Fourier

Series

T = A + Z (A coswnt + Bn sinwnt) [6]
thus giving a general type solution of the form

n
EH - bH = a + c(T + X (A coswnt + B sinwnt))
dt ave n n

[7]

At this point two observations can be made about
the constants a, b, and c that will make handling the
particular solutions less involved. These observations
are based on both Blumenfield's regression analysis and

least squares regression of a limited amount of growth

11

data from the 1971 season. The first is that constants b
anc c are equal in magnitude but dimensionally different.
The second is that the ratio a/c represents a zero growth
temperature for an initially emerging spear; i.e., at zero
height. This relationship may be obtained by setting the
height (H) and the growth rate (GR) equal to zero in
equation [3] and solving for a in terms of c and To'

yielding

a = -cT ' [8]

The effect of these assumptions is to reduce the numerical
values in the growth equation but not the dimensional units.
In further development the factors may not appear consistent
in units but, in fact, they are because the units remain
after cancellation of magnitudes.

with the above two observations in mind the
particular solution of equation [7] may be rewritten as

dH n

5;”. bH = b<Tave - To) + b 2 (A.r1 ooswnt + Bn Slnwnt) [9]

The particular solution for the constant term is

bTave - bTo
HPl = b = TaVe " To = AT [10]

and the form of the particular solution for the Fourier

series term is

12

n

HP2 = Z (Cn coswnt + Dn Sinwnt) [11]

Where the values of Cn and Dn are obtained using the
method of undetermined coefficients. This procedure

gives the following values

-b2A + wan
n n

 

 

C =
n b2 + (nw)2
and
2
wnbA - b B
D = n n
n b2 + (nw)2

If the complementary solution [5] and the two particular
solutions [10] and [11] are combined the result is the

total solution for the height

n
H = Coebt - AT + X (Cn coswnt + DD sinwnt) [12]

Using the boundary conditions H = Hi at t = 0, the constant

CO may be determined as

n
C = H. + AT - 2C [13]
o 1 n

Combining [12] and [13] gives the general equation for

spear height after time t as

13

H = (Hii-AT - gcn)ebt - AT-+§(Cn coswnt + Dn sinwnt) [14]
When a digital computer is available, the Fourier

solution does not appear too unwieldly; however, to
enable practical application a simpler solution is
preferable. The most obvious simplification is to repre-
sent temperature by a constant; the average temperature
over the growth period. This procedure also has merit
because it represents the type of information available to
growers. For the case of constant temperature all of the

summation terms can be neglected and [14] reduces to

H = (HO + AT)ebt - AT [15]
where the AT terms contain the factor b/c = l cm/°F to
make the units consistant with height. The growth rate
equation [3] from which [15] was derived can be described
in three dimensional space just as Blumenfield's equation
was in Figure 2. In comparing the general shape of
Figure 2 with what [15] represents it becomes apparent
that the AT term is affected by spear height. The tempera-
ture at which growth stops becomes lower as the spear
' becomes taller. The relationship between the zero growth
temperature and height may be determined by resolving
[3] using the conditions H = Hi and T = TO and GR = 0. The

result is that

14

T = - — - —-H. [16]

AT=T -T=(T -%)+—H. [17]

If this equation is substituted into [15] the resulting
equation for height is
bt b

a a
+ a] e [EH1 + Ta + a] [18]

H = [(l+-13)H. +T
C 1 a ve

V8

With the establishment of the usable model the
task remains to define the constants and verify that the
model will calculate spear heights within an acceptable

range of accuracy.

EXPERIMENTAL DATA

Field Procedure

 

Height and temperature data were taken at Michigan
State University Horticulture Research Center. The plants
used were five years old and of the California 711 variety.
Height measurements were taken twice daily at 0900 and
1600 hours from May 8 through June 6, 1972. Numbered
pieces of wood, held in the ground by a nail, were used for
identification and as a base for measurement. Measurements
were made on an individual spear until the spear exceeded
33 centimeters (13 inches), then it was cut to keep the
field clean. A total of 3920 measurements were made on
519 spears. Hourly air temperature values were taken from
a continuous recorder located at the research center. The
hourly values were used to calculate the average temperature

for each growth period.

Multiple Regression Analysis

 

The measurement data were divided into two groups.
Sixty per cent of the data was used in the least squares
regression analysis and for the determination of the growth
constant. The remaining 40 per cent was used for compari-
son with height distributions calculated using the
mathematical model. To distribute the data uniformly

15

16

between the two groups the data were sorted using the last
digit of the identification number. The 60 per cent
consisted of all spears with the last digit 1, 3, 4, 6, 8,
and 9; and the 40 per cent those with the last digits of
2, 5, 7, and 0.

The height and time data were converted to growth
rate and a multiple regression analysis was performed
using height and temperature as the important parameters
influencing growth rate. The day and night growth periods
were analyzed separately to determine any differences in
the constants.

The first regression performed was a repeat of
Blumenfield's original equation [1] which included the
squared and cross product terms of temperature and height.
The coefficients for the squared and cross product terms
appeared considerably smaller than the linear terms in both
the day and night periods. Another reason for neglecting
the squared and cross product terms is that they did not
contribute much to explaining the variability. The per-
centage increase in explaining variability due to the
squared and cross product terms was four and one per cent
for the day and night equations respectively. Included
in the regression analysis was the determination of the
significance of each coefficient. The analysis that
included the square and cross product terms yielded

several coefficients with significance at a level greater

17

than 0.10, while the analysis with only the linear terms
gave all the coefficients significant at the 0.0005 level.
This fact was also used to make the decision to neglect
the nonlinear terms. The results for both analysis can

be seen in equations [19] through [22].

GRD = -106.098-—l.O78H+-3.232T-0.003H2-0.023T2
+0.026H T [19]
GRN = -48.449-0.228H+-1.557T+-o.003H2-0.012T
+ 0.006H T [20]
GRD = -34.668 + 0.506H + 0.606T [21]
GRN = -l6.741 - 0.259H + 0.297T [22]

An advantage derived from neglecting the second order
terms is the many-fold reduction in effort required to
solve the differential equation.

Two important results are obtained from the above
regression equations. First, the ratios of the constants
obtained from the 1972 season appear consistant with both
the previous season's and Blumenfield's values. Since it
is the ratios a/c and b/c of the constants and not their
magnitudes that are used in the final equation, the
previous assumptions of their values are not critical.

If these two ratios are compared between the day and night
equations they are very close even though the individual

magnitudes differ by a factor of two. Secondly, the

18

difference in magnitude points out that the spears grow
more slowly during the night. This fact agrees with
another finding of Blumenfield's (1961) that 60 to 80 per
cent of the growth took place between 7 AM and 7 PM. The
difference in growth rates also means that the day and
night growth periods must be handled separately in further
analysis. Graphing height equation [21] or [22] would
yield a plane in three-dimensional space similar to the

plot in Figure 2.

Distribution of Growth Constants
The growth constant b was determined by first
fixing the ratios b/c and a/c at 0.835 and -57.20

respectively, yielding as the equation for height

 

 

H =[(l.835)H. +T - 57.201ebt- (0.835)H.
1 ave 1

-+T - 57.20

ave
This equation was solved for b
1n[Hn+ (0°835H(n-l) + Tave - 57.20)]
l‘835H(n-l) + Tave - 57.20
b = [23]
tn - t(n-l)

The height, temperature and time data used in the
regression analysis were used to calculate a value of b

for each measurement interval.

19

After evaluating the results it was concluded that
the values represented a truncated normal distribution with
a mean of 0.1500 and a standard deviation of about 0.05.
If the objective is to grow an average spear, then the
average value of the growth constant could be used, but
this would not give an indication of the distributions of
Spear height. In looking at the calculated constants, the
values for a particular spear are somewhat arbitrary
within the distribution with respect to the other Spears.
There is also some degree of randomness within a spear
when compensation is made for the day-night differences.
The constants for several spears selected randomly are
shown in Table 1. The measurement groups referred to in
Table 1, and again in Table 3, represent the interval
between measurements. A group does not necessarily repre-
sent a particular height range because a range of heights
were selected as initial measurement points. However, the
higher the group number the higher the average height of
the spears in the group. All initial measurements were
made in the morning so that the first and all even
numbered groups are always the daytime growth periods and
the odd numbers are the night growth periods.

In order to Simulate the dispersion of the growth
constants a procedure was used in which values could be
selected randomly from a particular distribution. The

distribution of the calculated values of the growth

20

 

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21

constant, after dividing the day values by two, is given
in Table 2 and Shown graphically in Figure 3. The distri-
bution used for selecting constants was the cumulative

probability curve, also shown in Figure 3.

22

TABLE 2.--Distribution of Constant b for 1750 Values.

4

 

. . . Cumulative Cumulative
D1V151on Frequency Frequency Per Cent
0.0000-.0099 3 3 .0017
.0100-.0199 2 5 .0029
.0200-.0299 12 17 .0097
.0300-.0399 12 29 .0166
.0400-.0499 16 45 .0257
.0500-.0599 28 73 .0417
.0600-.0699 33 106 .0606
.0700-.0799 42 148 .0846
.0800-.0899 54 202 .1154
.0900-.0999 69 271 .1549
.1000-.1099 85 356 .2034
.1lOO-.ll99 99 455 .2600
.1200-.1299 109 564 .3223
.1300-.l399 133 697 .3983
.1400-.l499 135 832 .4754
.1500-.1599 153 985 .5629
.1600-.1699 138 1121 .6406
.1700-.l799 109 1230 .7029
.1800-.1899 87 1317 .7526
.1900-.1999 69 1386 .7920
.2000-.2099 57 1443 .8246
.2100-.2199 55 1498 .8560
.2200-.2299 49 1547 .8840
.2300-.2399 47 1594 .9109
.2400-.2499 33 1627 .9297
.2500-.2599 30 1657 .9469
.2600-.2699 28 1685 .9629
.2700-.2700 28 1713 .9789
.2800-.2899 21 1734 .9909
.2900-.2999 10 1744 .9966
.3000-.3099 6 1750 1.0000

 

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RESULTS

Verification

 

The first approach to verification is to compare
the height distributions of field data with a height dis-
tribution calculated using-identical growing conditions.
Equation [18] was employed to calculate the spear heights.
The field data used was the 40 per cent of the measured
data which was not used in the regression analysis.

To calculate heights under identical conditions
required careful selection of values for the parameters
involved; initial height, average temperature, growth time
and a growth constant. The initial height for each growth
period was defined as the measured height at the start of
the growth period. The average temperature was taken to
be the same as that determined for the measured growth
period. The growth time was taken to coincide with each
interval in the measured data. Therefore, for each cal-
culated spear there is a real spear with the same initial
height; growing at the same average temperature for the
same length of time. The fourth parameter, the growth
constant, was selected from the distribution generated in
the previous chapter. Using existing computer programs a

random number between zero and one was generated and a

24

25

growth constant value was selected from the cumulative
probability distribution shown in Figure 3. A new growth
constant was generated for each growth period rather than
retaining the same constant for the entire Spear. The
growth constant was multiplied by two for the day growth
periods.

Both the measured and calculated heights were sorted
and simple statistics were calculated. Figure 4 shows the
mean values of the height distributions at the end of each
measurement period. Table 3 gives the same information in
numerical form. There was no difference between the means
of groups five, seven, eight, and nine at the five per
cent probability level when compared using the Student's-
t test. Although no tolerance limits have been clearly
defined, it is questionable whether the rejection limit
of about one centimeter truly reflects field tolerances.

It is interesting to note that the means of the
calculated values reflect an underestimate for the shorter
spears with the best accuracy occurring near 28 centi-
meters. For application purposes it would be desirable to
lower the crossover point to the region of interest. It
appears that for taller spears the initial height terms
become more significant. Also, close evaluation of the
growth constant determination could yield greater

accuracy in the region below 25 centimeters.

26

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A second approach to verification was to compare
height distributions of measured and calculated values
that have initial values within particular ranges. The
idea is to begin with a distribution that would be similar
to values found in a field sample. The intervals selected
were 14-17, 17-20, 20-23, 23-26 centimeters. They were
chosen because they represent initial height groups that
approach the 25 centimeter height value in increments of
approximately one inch. The initial conditions were those
of the field data and the growth constants were selected
using the same procedure as described for the previous
calculations. In the present situation the spears were
only grown through two measurement intervals thus simu-
lating an early morning, 0800 hours, initial observation
and two subsequent observations, one at about 1600 hours
and another 24 hours after the initial observation. The
initial height for the second time period was the calcu-
lated result of the first interval rather than the
measured distribution of the second group. The means of
the distribution are given in Table 4. Figure 5 Shows
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height group for both the calculated and measured data at
1600 hours and at 0900 hours the next day. All but one
of the calculated means are within two centimeters of the
means of the measured distributions. The distributions

of the calculated values were more compact than the

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31

measured values even though the means appeared to be
quite close. This situation means that in the present
state, with limited data, the model does not handle the
distribution tails as well as it should. One possible
explanation is that constants for Short spear heights
were included in the distribution from which the growth
constants were selected. This may have made the distri-
bution less effective for taller spears. Closer evalua-
tion of the growth constant distribution could possibly

correct the deviation.

APPLICATION OF THE MODEL

How can this model be used to predict optimum time
of harvest? The values in Table 4 offer an example. If
the majority of taller spears in the field are in the
14-17 centimeter range it can be predicted that the field
would not be ready for harvesting because most of the
spears are around 20 centimeters after 24 hours. When
the 17-20 centimeter group is considered, the field should
be harvested before the next morning because more than half
of the spears will be over the desired height at that
time. Growth of the 20-23 centimeter spears indicates that
harvesting should probably be completed by the end of the
same day. Although the above example is not based on the
same conditions for each spear, it gives some idea of how
the model can be used.

Another example illustrating how the model can be
used is to consider the hypothetical Situation of a grower
who samples his field at 0900 and finds that his maximum
Spear height is around eight inches (20 cm). He also
knows that the day's average temperature will be about
70°F. The farmer is faced with a decision whether to

harvest today or wait until the following morning.

32

33

Information about the height of the eight inch
spears at some future time can be obtained by growing this
spear a large number of times. Table 5 shows the height
distributions, at two hour intervals, for 600 spears grown
at 70°F from an initial height of 20 centimeters (8 inches).
After four hours there are two per cent of the spears
over 25 centimeters, but after Six hours 26.8 per cent of
the spears are over 25 centimeters. If the crop is left
to grow ten hours, to 1900 hours, 75.5 per cent of the
Spears will be over 25 centimeters. It is obvious that the
farmer will have to harvest today and he Should start
sometime after lunch, i.e., after four hours.

The growth of 600 Spears, 20 centimeters high,
starting at 1600 hours and growing at 60°F is also given
in Table 5. After 16 hours the distribution of heights
appear equivalent to about seven hours or less than half of
the day's growth for the same period. The difference is
caused both by the difference in the day and night growth
constants and by the lower growth temperature. If the
farmer had the above situation, he could let the spears

grow over night.

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SOURCES OF ERROR

When dealing with natural systems there are always
a number of factors that can influence the results. Such
parameters as soil moisture and fertility, soil tempera-
ture, temperature gradients, and inherent plant variation
may contribute substantially to asparagus growth. Blumen-
field (1961) attributed 16 per cent of the variation in the
growth rate to the above factors, but chose to neglect them
and focus on height and temperature.

It has been hypothesized, but not proven, that a
particular distribution of growth constants, b, is particu-
lar to the location and variety. This fact would indicate
that it may be possible to compensate for location dependent
parameters such as soil fertility and temperature gradients
by using the random procedure. The same thought applies
to inherent plant variations. This type of compensation
would necessitate calibrating a field by taking limited
data and generating the distribution for the growth con-
stants.

Another possible source of error was the use of
air temperature rather than the temperature at the growing
region which, according to Culpepper and Moon (1939), is

primarily at the spear tip. Depending on the absorptive

35

36

and reflective factors of the soil surface, the temperature
in the first ten centimeters could be i8°F from the average
air temperature, according to Geiger (1965). However, this
difference is less pronounced above 15 centimeters where
the interest of the model lies. It was also thought that
average air temperature is a parameter easily obtained by
the growers.

Physical limitations in measurement might also be
responsible for some error introduction. Heights were
measured to the nearest 0.1 centimeter, and in the case of
low temperatures, daily growth was only several tenths of
a centimeter which could result in large percentage errors
in the growth rate. Measurement was also hindered by the
fact that many Spears tended to grow curved or at an angle

from vertical, making consistent referencing difficult.

SUMMARY OF RE SULT S

The objective of this study was to obtain a model
that could accurately describe the growth characteristics

of asparagus for use in predicting time of harvest.

Field Data

 

During the 1972 season more than 500 spears were
measured twice a day until they exceeded 30 centimeters
thus giving from 5 to 12 measurements per spear and 3920
measurements over all. Temperature was recorded continually
for the entire measurement period. These data were
divided; 60 per cent being used for model development and

40 per cent for verification.

Regression Analysis

 

Sixty per cent of the field data was analyzed using
a least squares regression analysis. The linear, squared,
and cross product terms of air temperature and spear height
were the dependent variables and growth rate the independent
variable. Preliminary analysis showed that the square and
cross product terms could be neglected and the model based
on the linear terms. Values for constants and their ratios

were established. A difference was recognized between the

37

38

day and night values. Even though the difference was a

factor of two it did not affect the ratios.

Model Development

 

By defining growth rate as the change in height
over the change in time the regression equation was written
as a differential equation. Solution of the differential
equation yielded an equation for Spear height at a time At
as a function of initial height, average temperature and a
characteristic growth constant. It was recognized, from
the regression analysis, that differences existed between
day and night growth constants. It was further determined
that a single value for the growth constant could not
simulate accurately the Spread in heights of Spears which

were initially the same height.

Growth Constant Distribution

To handle the problem of variability between Spears
existing computer programs were utilized to generate a
random number between zero and one which in turn was used
to select a growth constant value from a cumulative prob-
ability distribution. The cumulative distribution for the
growth constant b was generated by calculating the value
using equation [23], for each measurement period. Vari-
ability within a spear was handled by selecting a new
constant for each growth period rather than retaining a

single value throughout the spear.

39

Model Verification

Using the above procedure, and the remaining 40
per cent of the field data, heights were calculated using
equation [18]. The resulting height distributions were
compared. The first approach took all the Spears from each
measurement group and calculated the height at the next
measurement point. The second approach selected Spears
within a particular height group and calculated heights
for the next two measurement points; 7 and 24 hours later.
In both comparison distributions the means of the calculated
data, around the area of interest, differed by less than
two centimeters from the means of the measured height
distributions. Although the means were close, the calcu-
lated values were more compact than the measured values.
The model does not adequately handle the distribution tails
as well as it should for use as a prediction tool. The
model; however, is realistically accurate when considering
all of the factors that might influence the growth of

asparagus.

FUTURE RESEARCH

A closer evaluation of the distribution of the
growth constants is needed. Possibilities would be to
evaluate temperature and height effects to see if a
refinement of the distribution would produce better pre-
dictions in the tail regions of the height calculations.
The model needs to be field tested by selecting a sample
of spears, calculating the heights based on forecasted
temperatures, and compare the calculated results with what
really exists in the field. Further research could also
focus on the development of a series of tables or graphs

that, after field calibration, could be used by a grower.

4O

REFERENCES

41

REFERENCES

Blumenfield, David; K. W. Meinkin and S. B. LeCompte (1961).
A field study of asparagus growth. Proc. Amer.
Soc. Hort. Sci. 77:386-392.

Carpenter, H. E. and R. G. Latimer (1967). ASparagus
harvester evaluation--l967. Cooperative Extension
Service, College of Agriculture and Environmental
Science, Rutgers--The State University, New
Brunswick, New Jersey.

Culpepper, C. W. and H. H. Moon (1939). Effect of tempera-
ture upon the rate of elongation of the stems of
asparagus grown under field conditions. Plant
Physiology 14:225. -

Culpepper, C. W. and H. H. Moon (1939). Changes in the
composition and rate of growth along the developing
stem of asparagus. Plant Physiology 14:677-698.

Downs, J. D., D. P. Watson and R. R. Dedolph (1962). Stem
elongation of ASparagus Officinalis L. Michigan
State University Agricultural Experiment Station
Quarterly Bulletin 44:773.

Geiger, Rudolf (1965). The Climate Near the Ground.
Cambridge, Harvard University Press, pp. 85, 88.

 

Stout, B. A.; C. K. Kline and C. R. Hoglund (1967).
Economics of Mechanical asparagus harvesting
systems. Research Report 64--Farm Service.
Michigan State University Agricultural Experiment
Station, East Lansing.

Tiedjens, V. A. (1924). Some physical aspects of Asparagus
Officinalis. Massachusetts Agricultural Experiment
Station, Waltham, Mass. Proc. Amer. Soc. Hort.
Sci. 20:129-140.

42

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