Aerospace and Electronic Systems - June 2019 - 7
Visalini et al.
Figure 1.
Optimal sensor placement in precision farming.
adds another dimension to the already complex
problem.
(C2) Variance of the sensor measurement depend on the
distance between the sensors.
Solutions to the problems stated above and overcoming challenges (C1)-(C2) are investigated in the rest
of this paper. We have the following assumptions:
(A1) Sensor locations are not constrained by farming
related operations.
(A2) RS information has low resolution, which means
that the range of measurements is in few meters.
Yet, there are four five patches of the land where
variables of interest are provided by the data.
(A3) Wireless communication vulnerabilities and data
integrity issues are negligible.
(A4) The sensor model and underlying process are
assumed to be linear.
Notation: In the following analysis, bold-faced letters are
used to denote vectors or matrices as x. In addition, '0 and
'1 denote the zero norm (cardinality) and the one-norm,
respectively.
Fc ðxÞ ¼ FðxÞ þ "ðxÞ;
x2V
(1)
where "ðxÞ denotes the error in estimation of Fc ðxÞ. Further, we assume "ðxÞ has an upper bound given by
"ðxÞ "ðxÞmax . Here, Fx denotes the average value of
the physical field and is obtained from RS information.
The measurement model of the sensor is given by
sk ¼ F ðxk Þ þ hk ; xk ¼ ðxk ; yk Þ 2 Vs & V; k ¼ 1; 2; . . .; m
(2)
where hk is the measurement noise of the k-th observation
with "ðxk Þ ) hk . Our aim is to estimate "ðxÞ with x 2 V
to compute Fc ðxÞ. We propose an optimization approach
for computing the physical field from the sensor measurements by denoting the error at locations xk as ek and from
(1) and (2), we have
SENSOR PLACEMENT ALGORITHM WITH RANGE
CONSTRAINTS
CALIBRATION OF THE FIELD
To perform sparse sensor placement with range
constraints, we first try to reconstruct the physical field
JUNE 2019
from measurements. To this extent, we use the prior
knowledge about the variance of the field variable across
the farm and the average value provided by the RS
information as the starting point. Then, M sensors are
deployed at different locations in the farm to obtain measurements physical field denoted by Fc ðxÞ ¼ ðx; yÞ 2 V
with V being the domain of interest. The continuous field
is obtained by the knowledge of variance across the farm
from the remote sensed data. In addition to the RS information, M sensors are deployed in the farm to obtain
measurements about the field.
Suppose that the RS information is given by FðxÞ to
estimate the field over V. The continuous field can be
modeled as
ek ¼ Fc ðxk Þ À sk ¼ "ðxk Þ À hk % "ðxk Þ 1
IEEE A&E SYSTEMS MAGAZINE
k
m: (3)
7
Aerospace and Electronic Systems - June 2019
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