Available at www.sciencedirect.

com

INFORMATION PROCESSING IN AGRICULTURE 6 (2019) 316–325

journal homepage: www.elsevier.com/locate/inpa

An innovative intelligent system based on remote

sensing and mathematical models for improving
crop yield estimation

Mohamad M. Awad
National Council for Scientific Research, Remote, Sensing Center, P.O. Box 11-8281, Beirut, Lebanon

A R T I C L E I N F O A B S T R A C T

Article history: There are many crop yield estimation techniques which are used in countries around the
Received 19 July 2018 world, but the most effective is the one based on remote sensing data and technologies.
Received in revised form However, remote sensing data which are needed to estimate crop yield is incomplete most
4 March 2019 of the time due to many obstacles such as climate conditions (percentage of cloud cover),
Accepted 3 April 2019 and low temporal resolution. These problems reduce the effectiveness of the known crop
Available online 5 April 2019 yield estimation techniques and render them obsolete. There was many attempts to solve
these problems by using high temporal resolution and low spatial resolution images. How-
Keywords: ever, this type of images are suitable for very large homogeneous crop fields. To compen-
Crop yield sate for the lack of high spatial resolution satellite images, a new mathematical model is
Environment created. Based on the new mathematical model an intelligent system is implemented that
Remote sensing includes the use of energy balance equation to improve the crop yield estimation. To verify
Image processing the results of the intelligent system, several farmers are interviewed and information about
Evapotranspiration their crops yield is collected. The comparison between the estimated crop yield and the
Intelligent system actual production in different fields proves the high accuracy of the intelligent system.
Ó 2019 China Agricultural University. Production and hosting by Elsevier B.V. on behalf of
KeAi. This is an open access article under the CC BY-NC-ND license (http://creativecommons.
org/licenses/by-nc-nd/4.0/).

1. Introduction in space and time. The Food and Agriculture Organization

(FAO) [2] indicated that there is need for timeliness agricul-
Roughly one third of Earth’s land is today deployed for agricul- tural statistics associated with effective monitoring system.
tural purposes, with more than ten percent used for growing Information is worth little if it becomes available too late.
crops and the reminder for pasture. The fast increase in pop- Remote sensing can significantly contribute to providing a
ulation mean more demands are being put on agriculture timely and accurate picture of the agricultural sector, as it is
than ever before [1]. Monitoring of agricultural activities faces very suitable for gathering information over large areas with
special problems not common to other economic sectors [2]. high revisit frequency.
Firstly, Agricultural activities follows seasonal patterns In remote sensing, multispectral and hyperspectral satel-
related to the crop phenology. Secondly, crop production lite images play a major role in crop management, their abil-
depends on the physical landscape, climatic parameters, ity to represent crop growth condition on the spatial and
and agricultural practices. All these factors are highly variable temporal scale is remarkable. These images can describe
the crop development, photosynthetic active radiation (PAR),
E-mail address: mawad@cnrs.edu.lb biomass accumulation (Bio), leaf area index (LAI), and actual
Peer review under responsibility of China Agricultural University. evapotranspiration (ETa).
https://doi.org/10.1016/j.inpa.2019.04.001
2214-3173 Ó 2019 China Agricultural University. Production and hosting by Elsevier B.V. on behalf of KeAi.
This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).
 Information Processing in Agriculture 6 ( 2 0 1 9 ) 3 1 6 –3 2 5 317

Many approaches have been developed to translate remote compensate for the absence of satellite data due to climatic
sensing data into crop yields, and several reviews of such factors and low temporal resolution.
approaches exist [3–5]. Some of these approaches have faced
several problems in estimating crop yield. One of these prob- 2. Data and methods
lems is the scarcity of remote sensing data suitable for use in
crop management because of climatic conditions such as Agricultural areas in different countries around the world are
clouds [6,7]. There were several attempts to solve this matter characterized by different farming practices and diversified
by replacing the lack of data with information from high tem- natural features. This diversity add to the complexity of han-
poral and low spatial resolutions images such as the method dling the problem of crop yield estimation and in turn encour-
implemented by Petitjean et al. [8]. age scientists to find one common solution that can adapt to
Most of the time, climatic conditions and low temporal climate conditions and to the availability of remote sensing
resolution are the main obstacles that prevent decision mak- data.
ers from using remote sensing data to map crops and to esti-
mate crops yield. 2.1. Study area and data type
There are several articles that use vegetation indices from
one date image or multi-temporal images to estimate crop The selected study area is located in the largest agricultural
yield. Kasampalis et al. [9] in their review of the current avail- area in Lebanon known as Bekaa valley (Fig. 1). The valley
able crop yield estimation models they concluded that the has an area size of more than 1200 km2 and the major crops
main limitations of crop growth models are the cost of obtain- in the valley are wheat and potato. It is also important to note
ing the necessary input data to run the model, the lack of spa- that two Bowen Ratio stations are installed in the area in
tial information in some cases, and the input data quality. order to measure different climate parameters which are nec-
Haig [10] conducted a study on a satellite based NDVI to essary for irrigation management and for estimating crop
predict crop yield at field level in India. He investigated the yield. The use of two Bowen Ratio one in the middle of the
relationship between NDVI calculated from satellite images valley and another in the south of the valley is sufficient since
and irrigated rice yield. The results of the study also showed in the middle the topography and climate are almost the
that the correlation between NDVI and rice yield is weak with same for a large area and this applies to the south part of
R
0.52. This is due to the fact that a long period of rice the valley.
growth is covered with water which makes NDVI index an Many studies showed success in crop yield estimation
obsolete one. when one weather station was used for a large agriculture
Prasad et al. [11] combined several parameters such as soil area with homogeneous topographic land [15].
moisture, NDVI, surface temperature, rainfall data of Iowa The two Bowen Ratio stations are located in two different
state in USA for over nineteen years for crop yield calculation areas one in an agricultural research institute while the other
and prediction using piecewise linear regression method with is located in a field owned by a potato chips manufacturer.
breakpoint. In his work, a non-linear multi-variate optimiza- The distance between both is more than 15 km. The two sta-
tion was utilized that minimizes discrepancy and errors in tions provide many valuable climate data such as net radia-
yield prediction. The method works well for large agriculture tion, wind speed and direction, temperature at different
area with homogeneous crops.
Bastiaanssen and Ali [12] used Monteith’s model [13] to
calculate the Absorbed Photosynthetically Active Radiation
(APAR), Stanford’s model for determining the light use effi-
ciency, and Surface Energy Balance Algorithm for Land
(SEBAL) to describe the temporal and spatial variabilities in
land wetness conditions. The result of the research showed
that there were gaps between the estimated and the actual
yield of about 1075 and 1246 kg/ha for wheat and rice. This
due to lack of remote sensing data and SEBAL requirements
for detailed climate data.
In this paper, the method in [12] is modified and improved
with new techniques and models. The modifications are
essential and it concerns the energy balance model such as
using another model named Mapping EvapoTranspiration at
high Resolution with Internalized Calibration (METRIC) [14].
This model depends less on climatic measurements to com-
pute actual evapotranspiration. In addition, the Monteith
model is modified with new mathematical model.
The new crop yield estimation system deploys models,
techniques, and remote sensing data that are based on using
Landsat 7 and 8 satellite images. In addition, the system
includes the development of new mathematical model to Fig. 1 – Study area.
318 Information Processing in Agriculture 6 ( 2 0 1 9 ) 3 1 6 –3 2 5

elevations, humidity, global radiation, soil heat flux, and soil The evaporative fraction is totally influenced by soil mois-
humidity. The images of Landsat 7 ETM+ and 8 are used in ture inside the root zone [20]. The complete details for the cal-
this research because they are available to download free of culation of W are explained later in the text.
charge from USGS Web site [16].
T1 ¼ 0:8 þ 0:02Topt  0:0005T2opt
In addition, they are the only free satellite images that
1 1
have a thermal infrared band with high spatial resolution T2 ¼   
1 þ expð0:2Topt  10  Tmon Þ expð0:3 Topt  10 þ Tmon Þ
(60 m compared to 1000 m for Modis satellite). Where the long
ð6Þ
wave infrared band is used later in computing important
parameters for crop yield estimation. Availability of Landsat T1 and T2 are two different heat functions, Topt (◦C) which is
images is related to the date the crop was planted and to the mean air temperature in the month with maximum leaf
the date that was harvested. Using both Landsat 7 and 8 area index, and Tmon (◦C) is the mean monthly air tempera-
improves the temporal resolution of image availability from ture. One can notice easily that T1 depends completely on Topt
16 days to 8 days. which is in turn affected by canopy biomass. On the other
hand, T2 depends on both Topt and Tmon.
2.2. Monteith biomass model The above values can be obtained from remote sensing
images using very complex mathematical models or they
Monteith model [13] is based on the photosynthetically active can be obtained directly using Bowen Ratio stations.
radiation (PAR) (0.4–0.7 m) which is part of the short wave
solar radiation (0.3–3.0 m) that is absorbed by chlorophyll for 2.3. Enhancement and correction of the images
photosynthesis in the plants. According to [17] a value of
approximately 45–50% is generally accepted to represent the There are still problems when using Landsat 7 products such
24 h average conditions (Eq. (1)). as ‘‘no data” strips which are caused by Scan Line Corrector
PAR ¼ 0:48Kinco ð1Þ (SLC) failure. Normally, Landsat ‘‘SLC-off” data refers to all
Landsat 7 images collected after May 31, 2003, when SLC
Photosynthetically active radiation (PAR) is thus a fraction failed. These products have data gaps, but are still useful
of the incoming solar radiation (Kinco). Then a fraction of PAR and maintain the same radiometric and geometric correc-
is absorbed by the plant for carbon assimilation. tions as data collected prior to the SLC failure. To fix this prob-
Absorbed photosynthetically active radiation (APAR) in lem, a method created by Scaramuzza et al. [21] is used to fill
ðWm2 Þ can be computed directly from PAR using the follow- gaps in one scene with data from another Landsat scene. This
ing equation method is available ENVI [22] and is applied as a preprocess-
APAR ¼ fPAR ð2Þ ing step. More details about how to install and use this
method as part of ENVI environment can be found in [23].
The factor f can be estimated using Normalized Differ-
In this method, linear transform is applied to the ‘‘filling”
ences Vegetation Index (NDVI) as is explained in [12].
image to adjust it based on the standard deviation and mean
Where f ¼ 0:161 þ 1:257NDVI ð3Þ values of each band.
NDVI is the normalized difference vegetation index com- The at-surface reflectance for the visible to short-wave is
puted as the difference between near-infrared and red spec- corrected on a band-by-band basis following Tasumi et al.
trums divided by their sum. The accumulation of biomass is method [24]. It works first on correcting images at sensor
according to the Monteith model proportional to accumulated (Atmospheric correction) and then at surface correction.
APAR. Top of atmosphere correction (At Satellite) works as
tot
X follow:
Bioact ¼ e ðAPARðtÞðtÞ ð4Þ
q0 t;b ¼ Mp  Qcal þ Ap ð7Þ
tot 2
where Bioact
in (kg/m ) is the accumulated biomass in period t, 0
where q t;b = TOA reflectance, without correction for solar
e is the light use efficiency in gram per mega joules (g MJ1).
angle. Mp = Band-specific multiplicative rescaling factor,
Light use efficiency e varies, if not water short, with C3 crops
Ap = Band-specific additive rescaling factor, and Qcal = Quan-
[13]. This means that there is no needs to know exact crop
tized and calibrated standard product pixel values (DN). The
type. Some of the C3 crops vary between wheat, rice, oats,
constants Mp and Ap are extracted from the metadata file that
alfalfa, pastures, sugar beet, and potato. This fact has an
comes with the image. While Qcal value is extracted from the
important inference: conversion for most C3 crops can be
image pixel values. To correct of the TOA for the sun angle the
done with the same e. A more comprehensive global ecology
following is applied
model for computing net production was created by Field
et al. [18] where they used the following equation for light q0 t;b
qt;b ¼ ð8Þ
use efficiency e. SinðhSe Þ

e ¼ e0 T1 T2 W ð5Þ where qt;b is the corrected TOA reflectance, hSe = Local sun ele-
vation angle. Next step is to correct with respect to at surface
where e0 is the maximum conversion element for above
reflectance which is done as follow:
ground biomass when the environmental conditions are opti-
qt;bCb ð1sin;b Þ
mal, it is equal to 0.29 g/MJ for C3 crops [19]. W is a function of qs;b ¼ ð9Þ
sin;b sout;b
the effective fraction of the available soil moisture.
 Information Processing in Agriculture 6 ( 2 0 1 9 ) 3 1 6 –3 2 5 319

 
C2 Pair C3 Wp þ C4 in ðWm2 Þ, RL" is the emitted longwave radiation in ðWm2 Þ,
sin;b ¼ C1 exp  þ C5 ð10Þ
Kt SinðhSe Þ SinðhSe Þ and Eo is the surface thermal emissivity.
  ðTa  Ts Þ
C2 Pair C3 Wp þ C4 H ¼ qCp ð19Þ
sout;b ¼ C1 exp  þ C5 ð11Þ rH
Kt Cosð0Þ Cosð0Þ
H is the sensible heat flux in ðWm2 Þ, Ta and Ts are the air and
Wp ¼ 0:14ea Pair þ 2:1 ð12Þ
surface temperatures in (◦C), q is the air density (kg m3) and
Rh es Cp heat capacity of air (Kj kg1) are constants, and rH is the
ea ¼ ð13Þ transfer resistance (s m1) depends on wind speed and sur-
100
face characteristics.
ð17:27Tair  
es ¼ 0:6108  expð Þ ð14Þ G0 0:05 þ 0:10e0:521LAI Rn ðLAI  0:5Þ ð20Þ
Tair þ 237:3
where qs;b is at-satellite reflectance for band b, and Cb is a band G0 ¼ maxð0:4H; 0:15RnÞ ðLAI < 0:5Þ ð21Þ
b specific given constant. sin;b and sout;b are narrowband trans-
where G0 is the soil heat flux in ðWm2 Þ and LAI is the leaf
mittances for incoming solar radiation and for surface
area index unit less. For more details about the above equa-
reflected shortwave radiation. Kt is a unit less ‘‘clearness”
tions one may refer to the paper of Allen et al. [29].
coefficient
1.0 where Kt = 1.0 for clean air and Kt = 0.5 for
extremely turbid, dusty or polluted air, Pair is air pressure
2.5. Creating the new mathematical model
(kPa), Wp is perceptible water in the atmosphere (mm), ea is
the actual vapor pressure (kPa), es is the saturated vapor pres-
Since the estimation of crop yield depends on few satellite
sure (kPa), Rh is the relative humidity in percent, and Tair is the
images there will be some periodic gaps between the date of
air temperature. Air pressure is calculated using a form of the
planting and harvesting. For this reason, the information
universal gas law equation as standardized by [25,26].
obtained from each image is used to create a mathematical
 5:26
293  0:0065z model to compensate for missing data.
Pair ¼ 101:3 ð15Þ
293 Mathematical models analyze the observation from the
real world such as satellite images in order to predict unfore-
seen incidents, behavior or productivity. Here they are used to
2.4. Using METRIC to compute evaporative fraction
enhance data availability and to predict missing data in a
specific information process such as biomass yield estima-
The computation of the evaporative fraction is based on using
tion. The model works on interpolating missing data from
Eq. (16) which in turn depends on the energy balance Eq. (17):
existing ones with minimum error.
kE
W¼ ð16Þ Normally, nonlinear curve-fitting (data-fitting) problems
Rn  G0
are solved using least-squares method [30]. However, the data
kE ¼ Rn  G0  H ð17Þ extracted from satellite images represent specific crop bio-
mass which is considered a complex non polynomial (NP)
where W is the evaporative fraction unit less, H is the sensible type of problems. For this reason, there is a need for a more
heat flux in ðWm2 Þ, kE latent heat flux in ðWm2 Þ, Rn is the reliable method that can solve the problem and which can
net radiation in ðWm2 Þ, and G0 is the soil heat flux in provide an optimal solution.
ðWm2 Þ. The equation can be solved using METRIC which The objective is to fit a curve to the crop yield data using
was developed by Allen et al. [14]. METRIC is a sort of ‘‘hybrid” unconstrained nonlinear optimization algorithm such as
between pure remotely sensed energy balance and weather- Trust-Region Methods for Nonlinear Minimization [31].
based evapotranspiration methods. Where energy balance is Trust-region methods are efficient, and can solve easily ill-
calculated from satellite image which delivers spatial infor- conditioned problems. The basic idea is to approximate a
mation that includes the available energy, and the sensible function F with a simpler function Q, which reasonably
heat fluxes for a large area. reflects the behavior of function F in a neighborhood N
In this research, METRIC is used as part of the intelligent around the point x. This neighborhood is the trust region, a
system to calculate evaporative fraction and actual evapo- trial step s is computed by minimizing over N
transpiration. Both are compared with the values computed  
( sMin fQ ðsÞ; s 2 Ng). This is the trust-region sub-problem,
from the Bowen Ratio stations measurements. METRIC foun- mathematically the Trust-Region sub-problem is typically
dation, principles and techniques are based on SEBAL [27,28]. stated as:
METRIC works on solving Eq. (16) by calculating each variable

separately such that net radiation is calculated based on the 1
Min sT S þ ST X such that jDs 6 w ð22Þ
following Eq. (18) 2
Rn ¼ RS # aRS # þRL # RL " ð1  E0 ÞRL # ð18Þ
where X is the gradient of a given function F at the current
2
where Rn is the net radiation in ðWm Þ where RS; is the point x, ø is the Hessian matrix (the symmetric matrix of sec-
incoming shortwave radiation in ðWm2 Þ, a is the surface ond derivatives), D is a diagonal scaling matrix, w is a positive
albedo (dimensionless), RL; is the arriving longwave radiation scalar, and k . k is the 2-norm.
320 Information Processing in Agriculture 6 ( 2 0 1 9 ) 3 1 6 –3 2 5

X
N
NewBio ¼ Qðxi Þ ð23Þ
i¼1

where NewBio is the new total estimated biomass in kg/ha, N

is the number of days, xi is the day i during the crop growth
progress, and Qðxi Þ is the new mathematical model which
can be created from the estimated crop yield of the available
Landsat image or any other remote sensing images. The
advantage of the new method is the ability to show the com-
plete crop growth period even with the lack of complete
remote sensing images. Fig. 2 shows in detail the steps
needed to create the mathematical model from few Landsat
images.

2.6. The intelligent system

The decision makers, scientists, and famers need a reliable

system that can estimate crop yield accurately. The system
should be available all the time this means that no cause
whether it is natural or manmade should prevent it from pro-
viding the needed information.
The system is normally made of different components
that interact between each other to complete a specific task.
Fig. 2 – The new mathematical model. It should be able to take decision and perform specific tasks
accordingly. The following schema (Fig. 3) shows the compo-
nent of the intelligent system (IS) and the different tasks it per-
forms. The intelligent system can be considered as a first step
After calculating the biomass data for each satellite image toward estimating the yield for all crop types. The role of each
by using Monteith model, the biomass data with day of the component in the intelligent system is essential for the success
year for each image are used in the optimization process to of the process. The IS starts its processes by running the
create the new mathematical model. acquisition of data component, then checking the data integ-
At the end a new biomass yield equation is obtained (Eq. rity and completeness.
(23)) which can compute the total accumulated biomass from Then another component recompenses for the missing
the plantation phase until crop maturity phase. data using a model based on the optimization of the final

Fig. 3 – Intelligent system for estimating crop yield.

 Information Processing in Agriculture 6 ( 2 0 1 9 ) 3 1 6 –3 2 5 321

Fig. 4 – Pilot area in Bekaa Valley 8th and 24th of June (a, d) Original images (b, e) Actual evapotranspiration maps and (c, f)
Evaporative fraction maps.

Table 1 – Actual ET (ETa) calculated using remote sensing and Bowen station data.

Date Location Bowen ratio Remote sensing Absolute percentage error

ET (mm) ETa (mm)

8/6/2015 Qab-Elias 7.54 6.4 15

8/6/2015 Taanayel 5.49 4.6 16
24/6/2015 Qab-Elias 8.29 6.88 17
24/6/2015 Taanayel 6.37 5.6 12
MAPE 15

solution. Finally, estimating crop yield is accomplished by but majority of farmers plant potato in the end of winter
either using known or created method depending on the and harvest in mid-summer. The potato leafs appears almost
availability of data. 20 days after seeding. Because the weather in Lebanon during
winter and spring is mainly rainy and cloudy, it was very hard
3. Experimental results to get images during the months of March (the date potato
was planted) and April. Seven images are collected between
The experiments are conducted based on data type, crop type, the months of April, May, June, and July from the date of
and season length. In case of any gap in the needed data, the potato sowing (20–30 days after) and near the end of the
intelligent system can be used to prove its efficiency and growth stage (10–30 days before harvesting). Landsat 7 was
robustness estimating crop yield. In this paper it is decided subject to corrections to remove no data stripes as was
to estimate potato crop yield which has two different cycles, explained before in data and method section. Moreover,
322 Information Processing in Agriculture 6 ( 2 0 1 9 ) 3 1 6 –3 2 5

Table 2 – Evaporative fraction values for the 8th of June

computed using energy balance.

Fig. 5 – Potato above ground biomass progress graph.

Table 3 – Evaporative fraction values for the 24th of June (actual values) and the predicted values from METRIC model.
computed using energy balance. The accuracy is computed based on the Mean Absolute Per-
centage Error (MAPE) [33].
Table 2 shows the computed evaporative fraction Wvalues
for the local time between 9:00 and 14:00 on the 8th of June
2015. The value of the predicted W = 0.74 computed by
METRIC model and compared with the computed one from
Bowen Ratio station at time 11:00 shows that the absolute per-
centage error is 5%.
The estimated evaporative fraction results for day 24 of
June 2015 are also verified. The following data collected by
Bowen Ratio station (Table 3) shows the different parameter
at-surface reflectance is calculated by applying atmospheric values used by the energy balance equation. The value of
correction to the at-satellite and at-surface reflectance. Potato the evaporative fraction W computed from data provided by
map of Bekaa valley for spring 2015 is created using the the Bowen Ratio station (highlighted row in Table 3) is com-
method in [32]. pared with the value of W computed by METRIC model
(value = 0.7). The result indicates that the absolute percentage
3.1. Evaporative fraction calculation error is 8%.
The results are promising ones and it proves that W com-
Evaporative fraction and actual evapotranspiration (Eta) maps puted by METRIC model is of high accuracy and can be used to
are created using METRIC model as explained in the method solve crop yield estimation problem.
section. The maps covers seven days in April, May, June and
July. 3.2. Estimating potato crop yield
The values of the evaporative fraction is less than 1 except
in few cases when net radiation is very large compared to To estimate potato yield several Landsat 7 and 8 images are
sensible and soil heat fluxes (Rn  H and Rn  G0 ) or when collected. However, the number of images and their temporal
H is zero because air and surface temperature are equal or coverages are not sufficient because a biomass model
the difference is negligible. requires information about the complete crop growth stage.
To validate the accuracy of METRIC model, some recorded The problem can be solved if daily satellite images such as
climatic data from two Bowen Ratio stations are used to eval- Modis are obtained, however the extracted information would
uate actual evapotranspiration and evaporative fraction esti- be useless due to the coarse spatial resolution of this satellite
mated maps. Fig. 4a–f show the satellite images, the actual (from 250 m to 1000 m) especially thermal long infrared
evapotranspiration maps and evaporative maps for a large image. So the solution is to create a mathematical model that
area in Bekaa valley in the 8th and 24th of June 2015. can help in compensate for missing data.
The following data (Table 1) shows the actual evapotran- After creating the potato map, some statistical informa-
spiration values obtained from the Bowen Ratio stations tion related to biomass are extracted from the seven

Table 4 – Estimated potato biomass production for different periods of time in spring 2015.

10- March 5-Apr 7-May 8-Jun 16-Jun 24-Jun 10-July 26-July

Days 1 25 57 79 87 95 111 127

Mean 0 1.4 67.0 252.4 275.1 320.0 435.7 80.5
Min 0 0.5 30.0 103.8 210.0 275.0 295.4 15.3
Max 0 5.0 140.9 285.9 307.0 374.0 583.2 179.3
 Information Processing in Agriculture 6 ( 2 0 1 9 ) 3 1 6 –3 2 5 323

Fig. 6 – Maps of (a) Selected crops in Bekaa crop (b) above ground crop biomass and (c) Potato tuber.

processed satellite images. Table 4 lists the minimum, maxi- where a1 = 489; b1 = 111.1; c1 = 14.92; a2 = 283.1; b2 = 80.96;
mum and average biomass production kg/pixel (225 m2) col- c2 = 28.64.
lected from these images for several potato fields. Although Based on the new model and Eqs. (23) and (24), a new crop
the plantation dates of spring potato is variable, but the yield model is obtained shown in Eq. (25).
majority of farmers start planting in different dates of March XN  2 !
Xdatai  b1
month especially in the first two weeks. In addition, most of NewBio ¼ a1 exp  þ a2
i¼1 c1
the farmers harvest potato in July. Eq. (24) is the result of fit- !
 2
ting a curve the given data in Table 4 using the Trust region Xdatai  b2
 exp  ð25Þ
algorithm The Parametric fitting of the given data involves c2
finding coefficients (parameters) for one or more models that
Fig. 5 shows the progress of the potato biomass after about
fit to data.
three weeks from plantation and few weeks before the end of
 2 !
Xdata  b1 the growth stage. The graph was created using the new yield
FðXdataÞ ¼ a1 exp  þ a2
c1 model (Eq. (25)).
 2 ! In addition, a map is created using the new crop yield
Xdata  b2
 exp  ð24Þ method (Fig. 6a–c). They show the crop map, estimated above
c2
ground crop biomass, and potato dry matter (for the whole
where Xdata is the day number from the beginning of the crop season of spring 2015). The graph was verified using real bio-
season to harvesting, a1, b1, c1, a2, b2, and c2 are constants mass data collected periodically from the potato fields after a

Table 5 – Potato crop yield for several farmers in different areas in Bekaa valley.
Farmer # % Area (ha) Production (tons/ha) Type of potato Estimated (tons/ha) Error

1 50 30 Spunta 28.5 0.05

2 50 30 Spunta 29 0.03
3 15 35 Agria 33 0.06
4 75 40 Spunta 38 0.05
5 20 40 Spunta 38.5 0.04
6 10 44 Spunta 40.5 0.04
7 10 36 Spunta 35 0.03
8 50 38 Spunta 39 0.03
9 300 25 Agria 25 0
10 100 35 Agria 33 0.06
11 15 30 Spunta 31 0.03
12 20 35 Agria 34 0.03
13 100 25 Spunta 24 0.04
14 70 30 Agria 29 0.03
15 10 15 Spunta 14.5 0.03
324 Information Processing in Agriculture 6 ( 2 0 1 9 ) 3 1 6 –3 2 5

month from sowing day and every 10 days. The comparison [3] Gallego J, Carfagna E, Baruth B. Accuracy, objectivity and
showed an agreement of about 96% as the graph shows. efficiency of remote sensing for agricultural statistics. In:
To gain confidence in the estimated crop yield results, sev- Benedetti R, Bee M, Espa G, Piersimoni F, editors. Agricultural
survey methods. UK: John Wiley & Sons; 2010. p. 193–211.
eral potato farmers are interviewed. Table 5 shows the real
[4] Zhiwei J, Jia L, Zhongxin C, Liang S. A review of data
potato yield and the estimated for several farmers (collected assimilation of crop growth simulation based on remote
from the map according to each farmer’s location). The survey sensing information. In: Proc the third international
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