Vol. 46 No. 12
SCIENCE IN CHINA (Series D)
December 2003
Thermal bidirectional gap probability model for row crop
canopies and validation
YAN Guangjian ()1, JIANG Lingmei ()1, WANG Jindi ()1,
CHEN Liangfu (
)2 & LI Xiaowen ()1,3
1. Research Center for Remote Sensing and GIS, Department of Geography and Beijing Key Laboratory for Remote
Sensing of Environment and Digital Cities, Beijing Normal University, Beijing 100875, China;
2. LARSIS, Institute of Remote Sensing Application, Chinese Academy of Sciences, Beijing 100101, China;
3. Department of Geography and Center for Remote Sensing, Boston University, Boston, MA 02215, USA
Correspondence should be addressed to Yan Guangjian (email: [email protected])
Received February 21, 2003
Abstract Based on the row structure model of Kimes and the mean gap probability model in
single direction, we develop a bidirectional gap probability model for row crop canopies. A concept
of overlap index is introduced in this model to consider the gaps and their correlation between the
sun and view directions. Multiangular thermal emission data sets were measured in Shunyi, Beijing,
and these data are used in model validation in this paper. By comparison with the Kimes model
that does not consider the gap probability, and the model considering the gap in view direction only,
it is found that our bidirectional gap probability model fits the field measurements over winter wheat
much better.
Keywords: row crop, bidirectional gap probability, thermal emission, hot spot effect.
DOI: 10.1360/03yd0550
Land surface temperature is a key parameter in monitoring the status of crop water stress by
remote sensing, and studying the water and energy balance in cropland ecosystem. The component
temperatures of crop and soil are especially significant in remote sensing drought monitoring of
crop.
What can be observed remotely is a hybrid target of soil and vegetation. The thermal emission exhibits directionality obviously[1,2]. Such directional thermal emission characteristic provides the possibility of extracting component temperatures by the combination of directional
thermal models[3].
Most of these crops are sowed row by row. Jackson et al. proposed a simple four-component
model to calculate the reflectance of wheat in 1979[4]. This model is suitable for the plane perpendicular to the row. Under the assumption that rows can be approximated by infinitely extended
solids, Kimes developed a geometric optical model suitable for any view direction, and validated
this model by the field measurements of cotton[5]. In some cases, Kimes model captures the factors
dominating the thermal directionality of cotton, i.e. the row structure and projection. Due to its
simplicity, when the ground cover of cotton is 48% in the vertical projection, Kimes used this
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model to retrieve the component radiative temperatures of vegetation and soil with accuracy of
1k and 2k respectively[5]. However, gap probability in a row is not considered in Kimes model,
and effective row width and height are used as compensation[6]. For other row crops, e.g., the
wheat, gap probability in a row is an important factor affecting the thermal directionality. From
our field measurements, we found that large errors can be introduced without considering the gap
when the leaf area index (LAI) is relatively low during the growth stage of crops, especially in the
view plane along the row. Based on Kimes model, Chen et al. developed a thermal directional gap
probability model and validated the model by the measured thermal emission data[7]. To consider
the gap probabilities in the sun and view directions, Chen et al. divided a row period into three
parts. A “hotspot” factor proposed by Kussk is used in the overlap projection area. The gaps in two
directions are treated independently of each other in the no overlap projection area. Based on the
row structural model and a concept of overlap index capable of describing the hotspot effects related to leaf inclination angle (LAD), a thermal bidirectional gap probability model is developed
for row crop in this paper. Model validation, using field measured thermal directional emission
over winter wheat, is also given in this paper.
1 Directional gap probability of vegetation
Gap probability can be defined as the probability that a photon will pass through the canopy
unintercepted. It is a function of path length for discrete vegetation[8]
Pgap ( s) = e− ksL / D = e−τ s ,
(1)
where τ = kL / D is the extinction coefficient per unit length, D is the average depth of the crown,
k is the attenuation of a unit LAI contained within a unit of canopy depth, and is determined by the
LAD, leaf transmissivity and the geometry. For a certain target, empirical formula can simplify the
model, and it is useful in inversion. We use the empirical formula proposed by Xiang et al. for
winter wheat[9]
k (θ ) = 0.676 − 0.38cosθ ,
(2)
where θ represents the sun/view zenith angle.
After getting the extinction coefficient, the gap probability in direction θ can be expressed
as
[8]
Pgap (θ ) =
1
e−τ s ( x , y ,θ ) dxdy,
∫
A
A
(3)
where A represents the pixel or a row period for the periodically distributed vegetation. As a result,
the key problem is to get the path length for discrete vegetation. If the shape of the vegetation is
too complex to get the path length s(x, y, θ ) at each point, but the distribution p(s) of path length
s(x, y, θ ) can be got instead, we can make the two dimensions’ integration into one dimension
only[8]
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THERMAL BIDIRECTIONAL GAP MODEL FOR ROW CROP CANOPIES & VALIDATION
Pgap = ∫
∞
0
p( s )e−τ s ds.
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(4)
What should be noticed is that (4) is still related to
the zenith angle θ even if θ is not included in the
formula. For the spherical and ellipsoidal crowns,
Li and Strahler gave the analytical expression of the
path length distribution[8]. For row crops, on the
premise that the rows can be regarded as infinite
Fig. 1. Abstraction of row crop model.
extended solids (fig. 1), Li proposed a simple mean gap probability model1)
Pgap = e−τ smax P ( smax ) + e−τ smin P( smin ) + E ( pgap ),
(5)
where smax and smin represent the maximum and minimum path lengths respectively, and P(smax)
and P(smin) are the corresponding probability distribution.
E ( pgap ) = ∫
smax
smin
e−τ s p ( s)ds =
(e−τ smin − e−τ smax ) Pr
,
τ ( smax − smin )
(6)
where E(pgap) represents the mean probability for the path length between smax and smin.
Pr = 1 − P( smax ) − P( smin ).
For single direction, (5) can give the mean gap probability accurately. However, how to get
the bidirectional gap probability based on (5) when the gaps in both sun and view directions
should be considered will be discussed in the following text.
2 Bidirectional gap probability model
2.1 Path length distribution of row crops
Based on (5), we can calculate the mean gap probabilities in the sun and view directions respectively, and then assume that they are independent of each other. Consequently, we can get a
rough bidirectional gap probability by simply multiplying these two probabilities. However, it is
obvious that
∑ ai bi ≠ ∑ ai ∑ bi .
i
i
Furthermore, the gap probabilities in two directions are not
i
independent in fact. They are completely correlated in the hotspot. As a result, if we need to consider the bidirectional gap probability, we should
know exactly the expressions of gap probabilities
in two directions at each point first, and then
make integration. Defining the direction perpendicular to the row as axis x (fig. 2), we can express the path length in a row period as a pieceFig. 2. Segmentation of a row period in calculating bidiwise function of x.
rectional gap.
Since the row is assumed to be infinitely long, similar to the solid model of Kimes, we can
1) Li, X., Direction Pgap model for rectangular row-crops, 2001.
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define an effective row angle α to make two dimensions integration into one dimension only[5]
tan α = sin ∆φ tan θ ,
(7)
where ∆φ represents the angle between the sun/view direction and the row, θ is the zenith angle.
The length of the projection over the ground can be given as
L = H tan α .
When α = 0, path length can be expressed as
 H / cosθ ,
s( x) = 
0 ,
0 x <W,
W x < S .
(8)
When α becomes larger and larger, the projection length of a row will be more than a row period,
i.e. L > S. Let us define n as integer part of L/S, then the residual projection length is Lr
Lr = L − nS ,
where n represents the light that can pass through n rows, the corresponding path length is
nW/sinβ, and sinβ = sin∆φ sinθ. What should be noticed is that β is similar to α defined in (7). We
can treat α as the effective row angle in calculating the projection length of a row, and treat β as
the effective row angle in calculating the path length. The residual path length can be expressed as
a piecewise function based on Lr in four cases
(1) LrW and LrS − W,
 x / sin β ,
 L / sin β ,

s( x) =  r
(W + Lr − x) / sin β ,
0,
0 x < Lr ,
Lr x < W ,
W x < W + Lr ,
(9)
W + Lr x < S .
(2) Lr > W and LrS − W,
 x / sin β ,
W / sin β ,

s( x) = 
(W + Lr − x) / sin β ,
0,
0 x < W ,
W x < Lr ,
Lr x < W + Lr ,
(10)
W + Lr x < S .
(3) LrW and Lr >S − W,
( Lr − S + W ) / sin β ,
 x / sin β ,

s( x) = 
 Lr / sin β ,
(W + Lr − x) / sin β ,
(4) Lr > W and Lr >S − W,
0 x < Lr − S + W ,
Lr − S + W x < Lr ,
Lr x < W ,
W x < S .
(11)
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THERMAL BIDIRECTIONAL GAP MODEL FOR ROW CROP CANOPIES & VALIDATION
( Lr − S + W ) / sin β ,
 x / sin β ,

s( x) = 
W / sin β ,
(W + Lr − x) / sin β ,
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0 x < Lr − S + W ,
Lr − S + W x < W ,
W x < Lr ,
(12)
Lr x < S .
It is obvious that the total path length is the sum of nW/sinβ and s(x) expressed by (9)(12).
2.2 Bidirectional gap probability and directional thermal model
Based on the expression of path length, if we assume that the gap probabilities in two directions are independent of each other, we can get the bidirectional gap probabilities
Bgap (θ i ,θ v ) =
1 S −τ (θi ) s ( x ,θi ) −τ (θ v ) s ( x,θ v )
e
e
dx.
S ∫0
(13)
Since four cases have to be considered in calculating the path length at any point x, total 16
cases should be discussed for two directions. Further, when the effective row angle θi in the sun
direction is not equal to θv in the view direction, a row period has to be separated into several
pieces for each case. This is troublesome and not necessary.
We have noticed that the temperature difference between the sunlit and shaded vegetation is
small, and the main factors affecting the thermal directionality of row crops are the proportions of
vegetation, sunlit soil and shaded soil in the sensor’s field of view (FOV). So we decompose a row
period into five pieces that are shown in fig. 2, and calculate smax, smin and the proportion in a row
period for each piece. Based on (5), in each piece, we can get the mean gap probability Ps and Pv
for the sun and view directions respectively. If they are independent of each other, the proportion
of visible sunlit soil in a piece is
Bg = Ps Pv .
(14)
Similarly, the visible shaded soil has a proportion as
Bz = Pv − Bg .
(15)
Bv = 1 − Pv .
(16)
The visible vegetation has a proportion as
Based on (14)(16), together with the effective radiation of all the components in the sensor’s
FOV, the thermal radiation received by the sensor is
L = Bg Lg + Bz Lz + Bv Lv ,
(17)
where Bg, Bz and Bv are the components in the whole FOV now, which are summed piece by piece.
Lg, Lz and Lv represent the effective components thermal emissions for sunlit soil, shaded soil and
vegetation. If we neglect the multiple scattering among the components, we can use the
component emissivities and temperatures to calculate the component radiation directly based on
Planck’s radiation law. Further, we can define an effective temperature similar to the concept
model proposed by Li et al. or the wide band model[10,11], and calculate the effective directional
emissivity considering multiple scattering in it, and get the directional thermal emission.
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considering multiple scattering in it, and get the directional thermal emission.
2.3 Overlap index and effective extinction coefficient
Due to the correlation between the sun and view directions, we cannot simply use (13) or (14)
to calculate the bidirectional gap probability. It is obvious that the correlation between the two
directions is related to their overlap projections. Li and Strahler used the overlap area of the projection along the sun and view directions to calculate the portion of visible sunlit ground in
1986[12]. Based on Boolean model in the field of statistic geometry, Strahler and Jupp proved that
the correlation between the probabilities of sunlit and view is decided by the overlap area[13]. Such
principle is also suitable for the calculation of bidirectional gap probability. As a result, we introduce a new concept of overlap index O(θ i , θ v , φi − φv ), and rewrite (13) as
Bgap (θ i ,θ v ) =
1 S −[τ (θ i ) s ( x,θi ) +τ (θ v ) s ( x,θ v )][1−O (θi ,θ v ,φi −φv )]
e
d x.
S ∫0
(18)
When the gap probability is independent in two directions, (18) should be (13). As a result,
the overlap index should be zero. When the view direction is just the hotspot, i.e. τ (θi)s(x, θi) =
τ (θv)s(x, θ v), the proportion of visible sunlit soil is equal to the gap probability in sun direction,
and the overlap index should be 0.5. It is clear that the overlap index is required to be related to
the sun and view geometries and has a range from 0 to 0.5. When the view direction is close to the
hotspot, the overlap index will tend to be 0.5.
In this paper, we assume that LAD can be described by a virtual ellipsoid. The ratio of its
vertical radius to horizontal radius represents the distribution of leaves in 3-D space. Based on this
virtual ellipsoid, overlap index can be defined as the ratio of overlap area to the total projected
area in two directions. If the projection areas of this suppositional ellipsoid are Γi and Γv in sun
and view directions respectively, and their overlap is o, we can get the overlap index O
O = o /(Γ i + Γ v ).
(19)
Further, based on overlap index O, we can get an effective extinction coefficient τ as
τ ′ = τ (1 − O).
(20)
What should be noticed is that (18) will turn to be (13) after we use τ to replace τ . As a
result, this model is suitable for describing the “hotspot effect” in thermal modelling. Further, the
correlation between the two directions is considered.
The detailed calculations for overlap area and projected area of an ellipsoid can be found in
refs. [14, 15].
3
Model validation
A large satellite-aircraft-ground synchronous remote sensing experiment was conducted from
April to May, 2001 in Shunyi, Beijing. One of the important missions is observing the directional
thermal emission from winter wheat. To validate our model, we compare the field measurements
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1247
with our bidirectional gap model, the Kimes model (no gap)[5], and the Pgap model that only considers gap in view direction1). Two groups of field-measured thermal emissions captured on April
12 are used in validation.
In the experiment, a thermal infrared radiometer (814 µm) with an FOV of 8.2° was fixed
on a multiangular viewing equipment to capture the radiative temperature on the scale of row. A
radiative thermometer was used to measure the sunlit soil, shaded soil and vegetation synchronously. The measured component temperatures are listed in table 1.
Table 1 Synchronously measured component radiative temperatures
Local time
Vegetation/
Sunlit soil/
Shaded soil/
11351139
20.8
33.1
19.6
1200
23.2
35.1
21.6
Row structural parameters were also measured in the same day. The width, height and space
of the row are 9.8, 9.8 and 20 cm respectively. LAI over the row is 1.5. Row orientation is 177°
from north. Based on the field measured row structural parameters and component radiative temperatures, the model simulation results are compared with the field measured radiative temperatures in fig. 3.
Fig. 3. Model comparisons with the field measurement captured on April 12. “Pgap model” represents the single direction gap probability model, “Bgap model” is our thermal bidirectional model. (a) and (c) represent the plane perpendicular to the row, the local time is 1145 and 1214 respectively. (b) and (d) represent the plane along the row, the
local time is 1136 and 1205 respectively. , Measurement; |, Kimes model; , Pgap model; ------,
Bgap model.
1) See footnote on page 1243.
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From fig. 3, we can see that the results obtained by our bidirectional gap model match the
field measurements in most of the angles. We can notice that the orientation of sun is very near the
row’s orientation at 1205. As a result, the observation plane along the row is just the principle
plane now. We can find that the model-predicted “hotspot” matches the measurement very well.
On the contrary, we can also find great difference among the results obtained by Kimes model, the
single direction gap probability model and the measurements. Further, we can even notice that the
Kimes model and the single direction gap model give out the wrong shape in the plane along the
row compared with the measurements. This just illustrates that bidirectional gap probability is
important and should be considered in the modeling of directional thermal emission during the
growth stage of winter wheat.
4 Discussion and conclusion
We propose a thermal bidirectional gap probability model in this paper, and introduce a new
concept of overlap index to consider the hotspot effect related to the leaf angle distribution. Because the bidirectional gap probability is considered in this model, and overlap index is used to
describe the correlation between the sun and view directions, our model can describe the basic
characteristic of the directional thermal emission from row crop, and is also capable of describing
the “hotspot” of such directional thermal emission. This is clear from the results we presented in
model validation. As a result, our model is distinct from the Kimes model and the single direction
gap probability model. The yield of winter wheat is mainly determined by the growth stage from
turning green to covering uniformly. It is very important to manage and monitor the winter wheat
during its growth stage. The leaf area index is relatively small in this period. As a result, it is necessary to consider bidirectional gap probability in modeling.
In addition to the row structural parameters and component temperatures, only one more parameter, LAI, is included in bidirectional gap probability model compared with Kimes model.
Because the number of parameters in this model is less than that in the general radiative transfer
models, this model is simple and easy to find solutions. Reasonable inversion results can be obtained if prior knowledge can be used further[16].
Because this bidirectional gap probability model assumes that the cross section of a row is a
rectangle that may have a little difference from the real crop, this model may not be suitable for all
of the row crops. However, the basic idea of this paper is still reasonable, i.e. introducing the concept of overlap index to consider the correlation between the gaps in two directions.
In model validation, some errors may exist in sun and view angles calculation, in ground
truth collection such as the row structure, LAI and component temperatures measurements. As a
sequel of this study, we can analyze their sensitivity and uncertainty in the future. Thus, we can
not only validate the model further, but also prepare model inversion.
Acknowledgements This work was subsidized partially by the National Natural Science Foundation of China (Grant No.
40101020), Special Funds for Major State Basic Research Project (Grant No. G2000077900) and the National High Technology
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THERMAL BIDIRECTIONAL GAP MODEL FOR ROW CROP CANOPIES & VALIDATION
1249
Research and Development Program (Grant No. 2001AA131030).
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