!!!TBD!!!JOURNAL OF LA Maximum Likelihood Sensor Node...

!!!TBD!!!JOURNAL OF LATEX CLASS FILES, VOL. 6, NO. 1, DECEMBER 20, 2015 1

Maximum Likelihood Sensor Node Localizationusing Received Signal Strength in Multi-media with

Multi-path CharacteristicsHerman Sahota and Ratnesh Kumar, Fellow, IEEE

Abstract—Sensors are key to situation-awareness and response,and need to maintain time and position information to tag theirmeasurement-data. While local clocks can be used for time-stamping, geo-tagging can be challenging for sensors with noaccess to GPS, such as the underground environment in precisionagriculture. We study the problem of sensor node localization fora hybrid wireless sensor network for precision agriculture, withsatellite nodes located above ground and sensor nodes locatedunderground. This application is quite unique in possessingmulti-media and multi-path features. We use received signalstrength of signals transmitted between neighboring sensor nodesand between satellite nodes and sensor nodes as a means toperform the ranging measurement. The localization problem isformulated as that of estimating the parameters of the jointdistribution of the received signal strength at all nodes in thenetwork. First, we arrive at path loss and fading models forvarious multi-media and multi-path communication scenarios inour network to model the received signal strength in terms of thepropagation distance and hence, the participating nodes’ locationcoordinates. We account for various signal degradation effectssuch as fading, reflection, transmission, and interference betweentwo signals arriving along different paths. Then, we formulate amaximum likelihood optimization problem to estimate the nodes’location coordinates using the derived statistical model. We alsopresent a sensitivity analysis of the estimates with respect to soilpermittivity and magnetic permeability.

I. INTRODUCTION AND RELATED WORK

PRECISION agriculture refers to the use of informationand control technologies in agriculture. Agricultural in-

puts such as irrigation, fertilizers, pesticides, etc. are appliedin precise quantities as determined by modeling of crop growthpatterns to maximize the crop yield and to minimize theimpact on the environment. Fertilizer uptake within a fielddepends on factors such as variability in plant population,nitrogen mineralization from organic matter, water stress, soilproperties, pests, etc. These factors vary in space and time.Fertilizers must be applied according to the needs of the crop.If under-applied, the crop yield suffers while if over-applied,contamination of ground and surface water resources may takeplace leading to issues such as aquatic hypoxia. Furthermore,there is a considerable energy cost associated with the pro-duction of nitrogen based fertilizers. Also, understanding theprocess of carbon sequestration, which is thought of as amethod to control global warming, could be aided by the datacollection from soil sensing, and its further analysis.

The research was supported in part by the National Science Foundationunder the grants NSF-ECCS-0926029, NSF-CCF-1331390, and NSF-ECCS-1509420.

Precision agriculture requires the collection of higher reso-lution spatio-temporal data than is currently available by meth-ods such as remote-sensing, laboratory tests, etc. The need isalso to automate the collection of data, and sensor networksprovide a promising solution. Data about soil characteristicssuch as nitrate concentration, moisture content, etc. can besensed by densely scattered sensor nodes and collected usingwireless networking. Sensor nodes in such a network mustbe deployed below the ground so that they do not interferewith the overground agricultural activities. A proposed sensornetwork architecture is shown in Figure 1.

Fig. 1: Soil Sensor Network for Sustainable Agriculture.

Research in our group has led to the development of low-cost and portable network analyzer capable of in-situ soildielectric spectroscopy for measuring soil conductance and ca-pacitance over a spectrum of frequencies ranging from 1MHzto 40MHz with 90% accuracy [1]; see Figure 2. The sensor

Fig. 2: Dielectric Spectroscopy-based Soil Sensor.

is equipped with a small meta-materials based antenna forwireless interfacing, which also doubles as the sensing elementfor dielectric measurements. Data collected from such sensorscan then be used for making resource allocation decisions,namely, irrigation and fertilization. Research in our group


has also developed sensor networking protocols for energy-efficient operation (synchronization, scheduling and routing)enabled by multiple power modes for both transmission andreception. These schemes and the resulting energy gains arereported in [2].

Determining location information, i.e., the (x, y, z)-positioning of sensors is valuable in geo-tagging sensor data,and also for performing sensor management including syn-chronization, scheduling, routing and battery and fault man-agement. The focus of the current paper is to develop asound mathematically-grounded sensor localization strategyfor a hybrid network of sensors deployed in multiple media.Future work will then take such geo-tagged and time-stampedsensor data to combine with agro-geo-ecological models topredict crop demands and yields and enable the agriculturalinputs prescription recommendations.

In [3], authors use canonical duality theory for solvingsensor network localization originally formulated as a non-convex quadratic minimization problem. The dual problemturns out to be concave maximization over a convex setin symmetrical matrix space. The authors also suggestideas for solving the general NP-hard problems in casethe canonical dual problem has no solution in its positivedefinite domain. A two-objective evolutionary algorithmis proposed in [4] which takes into account localizationaccuracy as well as the topological constraints induced byconnectivity considerations. [5] proposes a self localizationsystem that selects and maintains a relative coordinatesystem for a sensor network with non-stationary nodes.The relative coordinate system is defined by a set ofstable nodes that are less likely to disappear/fail. Themethod of selection of this group of nodes is also presented.Multiple local coordinate systems are stitched together byperforming coordinate translation to arrive at a networkwide coordinate system. [6] suggests removing the anchornodes the signal from which has high measurement erroras well as those anchors that are not expected to increasethe estimate’s accuracy due to their positions. This can helpreducing the computation and communication load on theenergy constrained sensor nodes. [7] presents the imple-mentation of two methods - Gauss Newton and ParticleSwarm Optimization - for solving the optimization prob-lem for localization in micro controller. The method usesboth deployment information (obtained using a pedometerattached to a human deployment agent) and communica-tion range to arrive at location estimates. Effectiveness ofthe two approaches under different operating conditions isalso studied. In [8], authors study the challenges imposedby emerging applications (Unmanned vehicle with WSNnavigation and cyber transportation systems) on localiza-tion. [9] presents a comparative study of different variantsof Particle Swam Optimization methods used in sensornetwork localization under different node topologies. In[10], authors present the development of two approachesfor localization for a wireless sensor network deployed inan indoor environment: a model learning based approachand triangulation. The system is also tested using a realhardware implementation. [11] tackles the problem at the

level of the physical signal design. The authors presentthe development of a signaling scheme incorporating bothdata transmission and localization. The effect of bandwidthreduction and decoding errors, due to simultaneous datatransmission, on location estimates is studied.

Recent publications reveal several examples motivating theneed for self localization as a key feature to enhance thefunctionality of different layers of the sensor network system.For example, location based routing techniques proposed forwireless sensor networks require location data as an integralinput, as seen in [12], [13]. Also, in our precision agricultureapplication, the routing strategy for the wireless sensor net-work, as seen in [14], [15] and [2], chooses the next hop nodefrom neighboring nodes based on their geographical proximityto the sink. Several security schemes also rely on locationawareness of the sensor nodes. For example, in [16], authorspropose an end-to-end data security mechanism which involvesbinding the secret keys stored at a node to its location.

In [17], authors study sensor node localization from astatistical signal processing standpoint. They use statisticalmodels to describe the three common measurement technolo-gies used for ranging in sensor node localization: time ofarrival (ToA), angle of arrival (AoA) and received signalstrength (RSS). The models are used to derive Cramer-Raolower bounds on the location estimates’ variances. In [18],authors address node localization problems from a theoreticalstandpoint. They derive the conditions for unique localizabilityof a network and also study the computational complexity ofnetwork localization.

Localization approaches can be broadly categorized asanchor-free or anchor-based. Anchor-free approaches do notrely on any node in the network having prior location infor-mation. Several such localization schemes have been proposedin literature where nodes estimate their locations in a relativecoordinate system formed by a few chosen nodes in thesystem. In [19], authors formulate localization as several localoptimization problems each of which is solved at a clusterhead node for building the local coordinate system of nodes inthe cluster. Nodes estimate distances to their neighbors usingtime of arrival measurements. During the second stage, clusterheads collaborate to build a global map of the network bycombining the estimated local maps.

On the other hand, anchor-based node localization relieson a number of anchor nodes in the network that are as-sumed to have prior location information. In [20], authorspresent an anchor-based range-free localization scheme fora sensor network in an environment with obstacles. Range-free localization techniques do not use ranging measurementssuch as time of arrival, signal strength and angle of arrival.Instead, alternative data such as hop count can be used forapproximating the distance between two nodes once the hoplength is estimated using anchor nodes. As another example,centroid based schemes can be used to estimate locationcoordinates given a set of anchor nodes in proximity. In[20], authors argue that for a sensor network deployment inan environment with obstacles leading to a concave networkshape, the shortest packet delivery path distance may not beapproximated by the geographical Euclidean distance. They


propose a scheme to identify line of sight anchors fromnon-line of sight anchors in order to ignore the hop countmeasurements from the latter set of anchors. In [21], authorspropose an anchor-based scheme based on multidimensionalscaling (MDS) [22] that uses connectivity information of theneighboring nodes to localize a network of n nodes with acomputational complexity of O(n3). A distributed variant isalso presented that builds local maps of nodes and then patchesthem together to form a global map. The distributed approachis more robust for irregular networks because connectivityinformation for far away nodes is ignored taking advantageof the fact that smaller local maps are more accurate. Leastsquares optimization is incorporated to further improve theaccuracy at the expense of additional computation.

Range-based localization techniques require special hard-ware at each node to measure a ranging signal and estimate thedistance to the transmitting node. The advantage is increasedlocalization accuracy compared to range-free localization. Anexample is a controlled event driven localization system wheredetectable events are generated using an event disseminator.Nodes report the detection of the events along with thetimestamps to a central location server which computes theposition of the nodes based on its own knowledge of theevent propagation delays in the sensor field. In [23], authorspresent the implementation results of such a system with lightas the detected event. The detection time at a sensor nodeis used to arrive at a spatial relationship between the nodeand the event generator. In [24], authors present the designof a localization system using uncontrolled events, which ismore suitable for an outdoor environment where controlleddissemination of events may not be readily feasible. In thismethod, a small number of anchor nodes in the field are firstused to estimate the generation parameters of the uncontrolledevents. Thereafter, controlled event localization techniques areused to estimate the nodes’ unknown location coordinates.

In [25], authors present an application of cooperative local-ization techniques to UWB (ultra wide bandwidth) wirelessnetworks. The authors quantify the performance of severalalgorithms based on the ranging models available for UWBtechnology. They also present a localization algorithm bymapping a statistical model for graphical inference onto thenetwork topology. In [26], authors present a sequential MonteCarlo localization method for sensor networks with mobilenodes that uses both range measurements and hop distance aswell as mobility information of the nodes.

Applications exist with varying degrees of localization ac-curacy needed. A trade-off is made among the localizationhardware complexity, algorithm complexity, deployment costsand localization accuracy. In [27], authors present a study ofthe error inducing parameters in sensor node localization. Theauthors derive the Cramer-Rao bound for Gaussian measure-ment error in multi-hop localization systems using distanceand angular measurements. They study the effect of theparameters such as measurement technology accuracy, nodedensity, beacon node concentration, and beacon uncertaintyon the localization error.

To the best of our knowledge, none of the existing work inliterature has addressed sensor node localization for a network

where the nodes are placed in different physical media. In thisarticle, we present our approach to sensor node localizationfor such a network in which the sensor nodes are buriedunderground while the anchor nodes (a.k.a satellite nodes) arelocated above ground.

The following are the contributions of this article:• Statistical 3-D localization framework based on received

signal strength (RSS) measurements for multiple media(air and soil) and multiple reflected paths in a lossymedium (soil).

• Rician fading models for multi-media (air to soil) andmulti-path (soil to soil) communication.

• Simulation results that implement the proposed localiza-tion scheme in Python, including error analyses.

• Sensitivity analyses of the estimated location coordinateswith respect to various parameters of interest.

Section II presents our localization framework. Section Vpresents the simulation results, including sensitivity analysis inSection V-B. Section VI presents our concluding remarks.Section VII presents the future directions for this work.

II. RSS-BASED LOCALIZATION MODELING

A wireless signal experiences fading due to a combinationof large-scale and small-scale effects [28]. The former is dueto large-scale uncertainties such as obstacles, weather etc.,whereas the latter is due to small-scale uncertainties resultingin multi-path interference. Large-scale power fading is knownto cause received power to be lognormally distributed; whereassmall-scale power faded received signal power follows a non-central χ2 distribution with 2 degrees of freedom (equivalently,the amplitude has a Rician distribution).

Accordingly, the signal power, Rmn, received at node nand transmitted from node m, has the following combineddistribution:

fRmn(rmn) =

∫ ∞0

fSRmn(rmn; pmn, κmn)

fLPmn(pmn;µmn, σmn)dpmn, (1)

where fSRmn(rmn; pmn, κmn) is the non-central χ2 pdfmodeling the effect of small scale fading with mean receivedpower pmn from the specular (line-of-sight) and scatter (non-line-of-sight) components, and κmn is the specular power toscatter power ratio; and fLPmn(pmn;µmn, σmn) is the log-normal pdf modeling the effect of large scale fading withparameters µmn and σmn. The distributions are expressed asfollows, where the notation I0(·) denotes the modified Besselfunction of the first kind and order 0:

fSRmn(rmn; pmn, κmn) =1 + κmnpmn

e−((κmn+1)rmn

pmn+κmn)

I0

(√4(κmn + 1)κmnrmn

pmn

)(2)

fLPmn(pmn;µmn, σmn) =1

pmn√

2πσ2mn

e− (ln pmn−µmn)2

2σ2mn .

(3)


Lognormal fading occurs over longer time-scales causingslower variations in the signal while multi-path fading occursover much shorter time scales [28]. In a short time framewhen the nodes communicate to gather the received signalstrength data, the large-scale fading effect can be ignored bytreating its distribution to have zero-variance (σmn = 0; so thedistribution fLPmn(pmn;µmn, 0) becomes an impulse centeredat the mean value, pmn = eµmn ). In this case, the receivedpower’s pdf simplifies to:

fRmn(rmn) = fSRmn(rmn; pmn, κmn), (4)

which is the fading model that we rely on for the short timeperiods when the localization measurements are taken.

In general, the received signal has in-phase as well asquadrature-phase envelopes Xin and Xquad both of which areGaussian random variables. This follows from the applicationof the Central Limit Theorem, as the received signal is thesuperposition of a large number of scattered components trav-eling along a multitude of paths [29]. Thus, the overall signalenvelope X = Xin+ jXquad is complex gaussian distributed.If the signal travels over a direct and a reflected path, thereceived signal envelope, X +X(r), is also complex gaussiandistributed; where X(r) = X

(r)in + jX

(r)quad corresponds to the

reflected path. Then, the overall average power, pmn, of thereceived signal is given by Equation (8), where (5) → (6)follows from assuming that X and X(r) are independent;VXe

jφX and VX(r)ejφX(r) represent the specular (line-of-sight) amplitudes of X and X(r), respectively, in polar form.(8) follows from the following: the specular component power,V 2X is related to the total average power PX by the Rician

factor (κ) as: V 2X = κ

κ+1PX . Similarly, V 2X(r) =

κ(r)

κ(r)+1PX(r) .

E[|X +X(r)|2] =E[(X +X(r))(X +X(r))∗]

=E[XX∗] + E[X(r)X(r)∗] + E[XX(r)∗]+

E[X(r)X∗] (5)

=PX + PX(r) + E[X]E[X(r)∗]+

E[X(r)]E[X∗] (6)=PX + PX(r)+

VXejφXVX(r)e−jφX(r) +

VXe−jφXVX(r)ejφX(r)

=PX + PX(r) + VXVX(r)ej(φX−φX(r))+

VXVX(r)e−j(φX−φX(r))

=PX + PX(r) + 2VXVX(r)

cos (φX − φX(r)) (7)=PX + PX(r)+

2

√PXPX(r)

κ

κ+ 1

κ(r)

κ(r) + 1

cos (φX − φX(r)), (8)

Note that while the received signal envelope X + X(r) iscomplex Gaussian, its amplitude has a Rician distribution andits power has a non-central χ2 distribution with 2 degrees offreedom. In the special case that there is no line of sight path

(so the received signal has no “specular” component and onlythe “scatter” component), the amplitude becomes Rayleighdistributed (as a special case of Rician) and the power becomesexponentially distributed (as a special case of non-central χ2),given as follows:

fRmn(rmn) = fSRmn(rmn; pmn, 0) =1

pmne−pmnrmn . (9)

III. EXPRESSING RSS PARAMETERS USING LOCATIONS

In this section we express the parameters of the probabilitydistribution of the received signal power in terms of the loca-tion coordinates of the sensor nodes in order to formulate theestimation problem for the location coordinates. The receivedpower pdf given by Equation (4) has two parameters; theaverage received power pmn and the Rician K-factor κmn.Existing works study the variation of the Rician K-factor withdistance [30] and its estimation [31]. However, we assume it tobe a known constant for the purposes of localization for ourapplication. Thus, we express pmn in terms of the locationcoordinates of the sender and receiver nodes.

It is known that the mean signal power pmn = eµmn decayswith distance along a single path within a single losslessmedium following the power law [32]:

pmn(d) = ηd−k (10)

where d is the distance between the two nodes m and n, η isa constant and k is the path loss exponent, both of which arefunctions of the medium. For example, in free space:

k = 2 and η =pmGmGnλ

2

(4π)2, (11)

where pm is the sender power, Gm and Gn are the sender andreceiver antenna gains, and λ is the wavelength. For simulationpurposes, we use η = pmλ

2/(4π)2 (ie, Gm = Gn = 1)and k = 2. A lossy dielectric medium such as soil hasadditional attenuation due to conductivity losses [32]. Thewave propagation equation for such a medium is given by:

E(r, t) = E0e−αr cos(ωt− βr), (12)

where E is the electric field at time t at a distance r fromthe source which transmits at amplitude E0. The complexpropagation constant of such a lossy dielectric medium isα+ jβ, where:

α = ω

√√√√√µε′

2

√1 +

(ε′′

ε′

)2

− 1

, (13)

and

β = ω

√√√√√µε′

2

√1 +

(ε′′

ε′

)2

+ 1

, (14)

where µ is the permeability, and ε′ and ε′′ are the real andimaginary parts of the complex permittivity of the medium.

Hence, in the more general setting where α 6= 0, Equation(10) takes the form:

pmn(d) = ηd−ke−2αd. (15)


In the following discussion we extend this calculation tothe case of multiple paths as well as multiple media. In thenotation that follows, we introduce the superscripts a and s,respectively, when referring to air and soil as the medium.

1) Multi-path extension: soil to soil communication: Whenthe sender node m as well as the receiver node n are within soilas shown in Figure 3, there exist the direct and the reflectedpaths and consequently, following Equation (8), the averagereceived power is given by:

pmn =pmn(dmn) + pmn(d(r)mn)ρ+

2

√pmn(dmn)pmn(d

(r)mn)ρ

κmnκmn + 1

κ(r)mn

κ(r)mn + 1

cos2π

λ(s)(d(r)mn − dmn), (16)

where

ρ =

(√µa/ε′a −

√µs/ε′s√

µa/ε′a +√µs/ε′s

)2

, (17)

is the reflection coefficient, and dmn and d(r)mn are the distancesof the direct and reflected paths, respectively, which are givenby (see Figure 3):

dmn =√

(xm − xn)2 + (ym − yn)2 + (zm − zn)2, and(18)

d(r)mn =

√[(xm − xn)2 + (ym − yn)2]

z2m(zm + zn)2

+ z2m

+

√[(xm − xn)2 + (ym − yn)2]

z2n(zm + zn)2

+ z2n.

(19)

node−n

xy

z

b a c

node−m

Fig. 3: Soil to soil communication between nodes m and n:Direct vs. reflected paths

Also, the parameter κmn measuring the ratio of the specularto scatter power in the received signal can be derived by notingthat the total average received power is pmn as calculated in

Equation (16), while the total average received scatter poweris simply the sum of the scatter powers for the two paths:pmn(dmn)κmn+1 +

pmn(d(r)mn)

κ(r)mn+1

. Since specularscatter = total−scatter

scatter =

totalscatter − 1, we have:

κmn =pmn

pmn(dmn)κmn+1 + pmn(d

(r)mn)

κ(r)mn+1

− 1. (20)

Next, we derive the average received power for signal propa-gation from a satellite node in air to a sensor node buried insoil.

node−m

xy

z

node−n

Fig. 4: Air to soil communication between two nodes m andn.

2) Multi-media extension: air to soil communication: Whenthe sender node m is in air while the receiver node n is insoil, the average received power is given by:

pmn = η(d(a)mn)−k(a)(d(s)mn)

−k(s)e−2α(s)d(s)mnτ, (21)

where τ = 1 − ρ is the transmission coefficient (ρ is givenby Equation (17)), and d

(a)mn and d

(s)mn are, respectively, the

distances traveled in the air and soil, which are computed asthe solutions of the following two equations:

d(a)mn√

(d(a)mn)2 − z2m

√(d

(s)mn)2 − z2nd(s)mn

(=

sin θ(a)

sin θ(s)

)=λ(a)

λ(s)(22)

√(d

(a)mn)2 − z2m +

√(d

(s)mn)2 − z2n =√

(xm − xn)2 + (ym − yn)2, (23)

where λ(a) and λ(s) represent the wavelengths in air and soil,respectively. The above result follows from the application ofthe Snell’s law of refraction.

Because of the high refractive index of the soil compared tothe air, the signal travels almost vertically downwards in thesoil so as to minimize its sojourn time in the soil (this fact isalso demonstrated by the small value of the Brewster’s anglefor the soil-air interface [33]). Accordingly, we can obtain thefollowing approximations:

d(s)mn ≈ zn, (24)

d(a)mn ≈√

(xm − xn)2 + (ym − yn)2 + z2m. (25)


Thus, we have expressed the parameters of the distributionof the received signal strength for both signal propagationscenarios in our application in terms of the distance and hence,the coordinates of the respective sender and receiver nodes.

IV. MLE-BASED LOCALIZATION FROM RSSMEASUREMENTS

Now, we formulate a maximum likelihood problem toestimate the location coordinates of sensor nodes given themeasurements of the received signal powers at the nodesfrom their neighboring underground as well as above-groundsatellite nodes.

Let N denote the set of node pairs (m,n) that communicateto gather the received signal power measurements. The log-likelihood of the measured received signal strength for allthe sender-receiver pairs (m,n) ∈ N can be expressed usingEquation (2) as follows:

L(Θ|R) =−∑

(m,n)∈N

ln(pmn)− ln(1 + κmn)+(rmn

κmn + 1

pmn+ κmn

)

− ln

I0√4(κmn + 1)κmnrmn

κmn

(26)

where R is the set of received signal strength measurements({rmn|(m,n) ∈ N}) and Θ is the set of unknown parametersto be estimated (i.e., the location coordinates). Also, thereceived signal strength rmn is measured and hence known,whereas the set of unknown parameters consists of pmn andκmn which in turn are functions of the unknown locationcoordinates that must be estimated.

The average powers (pmn’s) are expressed in terms of thelocation coordinates of the nodes m and n following Equations(16 -19) and (21 - 25), while the Rician factors (κmn’s) areexpected to be constants, independent of the locations, over theduration of the round and can be estimated at the beginningof each round. An MLE approach will proceed by writing theparameters pmn and κmn appearing in the likelihood functionof Equation (26) as the corresponding functions of the locationcoordinates of the sensor and satellite nodes as derived inSection III, and then optimizing the likelihood function withrespect to the location variables.

Method of Moments based Approximation: Alternatively,one can use the approach based on the Method of Momentsthat avoids solving the optimization problem of maximumlikelihood for non-central χ2 distribution, and instead reducesthe parameter estimation problem to solving a set of algebraicequations in which the first few moments of the distributionat hand are simply equated to their respective sample-basedvalues.

Fractional volume of water ε′s ε′′s0% 2.36 0.096610% 5.086 0.44130% 16.410 2.36840% 24.485 3.857

TABLE I: Real and imaginary parts of relative permittivity ofsoil for different moisture contents.

Parameter Symbol ValueTransmit power (Satellite node) P (a) 30 dBmTransmit power (Sensor node) P (s) 30 dBmThermal noise N0 -110 dBmPath loss factor (air) ka 2Path loss factor (soil) ks 2Relative permeability (soil) µs 1.0084Frequency f 433 MHzWavelength λ 0.7 mNumber of readings per node pair (RSS) NRSS 100Number of readings per node pair (ToA) NToA 100Signal duration TS 1ms

TABLE II: Parameters used in localization simulations.

In our setting, following the properties of non-central χ2

distribution, the first and second moments of Rmn are givenby [34]:

E[Rmn] = pmn, (27)

E[R2mn] = p2mn

(1 +

1 + 2κmn(1 + κmn)2

). (28)

These are equated to the corresponding sample-based esti-mates of the moments, and then we can algebraically solve theequations to obtain the estimates of the unknown parameterspmn’s and κmn’s, and consequently the location coordinates.The demonstration of method-of-moments based approachis a direction for future research as noted in the Conclusionsection.

V. SIMULATION RESULTS

We simulate a network of 25 sensor nodes in a square fieldof size 150 m×150 m to validate and evaluate the performanceour localization framework. The sensor nodes are randomlydeployed within the square field at a depth between 3 − 5cm. Our localization model discussed in Section II dependson the soil permittivities which in turn are functions of themoisture content of the soil. We assume a clay loam soil witha clay content of 20%. For this soil type, the relative real andimaginary values of the permittivity are computed [35] forvarious values of volumetric water contents as given in TableI. Other parameters used in the simulations are summarizedin Table II. The relative magnetic permeability of soil is closeto 1 for soils not containing significant iron [36]. Thermalnoise is assumed to be −110 dBm based on standard receiversensitivity, which is also the case for our receivers [37].Thus, applying Equations (11), (15), (16) and (21), with theparameters of Tables I and II, the underground transmissionrange is approximately 34 m for dry soil at a transmissionpower level of 30 dBm leading to an average of 3 neighborsper sensor node, whereas the air–to-soil transmission range isapproximately 1000 m.


We use a hierarchical and iterative approach for estimatingthe location coordinates of the sensor nodes, applying firstthe air-to-soil models on the signals transmitted between thesatellite nodes and sensor nodes to arrive at a first set ofestimates, followed by the combination of air-to-soil and soil-to-soil models, adding signal data among neighboring sensornodes to arrive at more refined and accurate estimates ofthe coordinates. The second stage is applied iteratively witheach new iteration initialized with estimates obtained fromthe preceding iteration, until the estimates converge within atolerance. The convergence tolerance for X and Y estimatesis chosen to be 5 cm, while for Z estimate it is chosen to be1 cm.

Figure 5 presents the localization estimates obtained usingthe received signal strength model for a Rician K-factor (κ) of20. We have also verified the localization framework for othervalues of κ (40, 60, 80, 100). Each simulation for estimatingthe nodes’ coordinates for the network of chosen size tookabout 30 minutes on a laptop computer with a 2.4 GHz dual-core processor and 8 GB 1067 MHz DDR3 DRAM. A visualinspection of the results shows that our method successfullyestimates the location coordinates of the sensor nodes, withthe error margins as described below.

0 20 40 60 80 100 120 140

X-coordinate

0

20

40

60

80

100

120

140

Y-co

ordi

nate

ε′s = 16.4

ε′′s = 2.37

α(s) = 2.66 N/m

κ = 20

ActualEstimated

0 5 10 15 20 25

Node-id

0.0

0.1

0.2

0.3

0.4

0.5

Z-co

ordi

nate

ε′s = 16.4

ε′′s = 2.37

α(s) = 2.66 N/m

κ = 20

ActualEstimated

Fig. 5: Localization using received signal strength model forRician factor κ = 20.

We use a minimal setup of our network to perform varianceanalysis of our location estimates: three sensor nodes deployedrandomly within the field but within the radio communication

ranges of each other and four satellite nodes so that we canclosely capture the real world deployment scenario where eachsensor node has 2 − 3 neighboring sensor nodes. Then, wecompute the sample standard deviation of the location estimateof one of the sensor nodes. Figure 7 shows the sample standarddeviation of the estimates for a sample size of 1000 forreceived signal strength based localization, which we chooseas a metric for the error margin of the estimates. We also studythe variation of the standard deviation of the estimates with thechange in the moisture content (leading to change in ε′s andε′′s as per Table I). The figures are omitted from this paper forspace considerations. X and Y coordinates are estimated witha Rician K-factor (κ) level of 20 (which amounts to standarddeviation of about 0.4 m). Note as κ increases to 100, thestandard deviation reduces to about 0.18 m. Similarly, thestandard deviation of the Z coordinate estimate decreases withan increase in κ. The standard deviation in X and Y estimatesis relatively insensitive to change in the soil permittivity.However, the Z estimate has the least standard deviation forε′s = 24.5 and ε′′s = 3.86, corresponding to a moisture contentof 40%. The standard deviation of the Z coordinate estimatedecreases with the increase in the moisture content of the soil.This behavior can be explained as follows: with the increasein the moisture content the attenuation of the signal increases,decreasing the average received power. From Equations (27)and (28), the variance of the received signal strength alsoreduces, reducing the variance of the estimate.

The simulation results demonstrate that our technique is effi-cient, scalable and reliable. Although the results are presentedfor a field size of 150 m × 150 m, the technique is equallyapplicable to larger fields.

A. Simulation Software

Numpy Random modulesignal strength valuesObserved received Scipy Optimize module

Estimates node locationcoordinates

Data gathering phase Estimation phase

Fig. 6: Simulation framework.

Our modeling framework requires sampling of thereceived signal strength values measured between pairsof sender and receiver nodes. Since there is no explicitneed for measuring the timing of various events for thesimulations (although in deployment, scheduling of thevarious transmissions needs to be performed to ensurethere are not collisions between different signals), we donot require an elaborate simulation approach such as thatof event driven simulations. We model signal strengthvalues by sampling from their probability distribution


20 40 60 80 100κ

0.15

0.20

0.25

0.30

0.35

0.40St

anda

rdde

viat

ion

(m)

ε′s = 16.4ε′′s = 2.37α(s) = 2.66 N/m

X estimate errorY estimate error

20 40 60 80 100κ

0.00025

0.00030

0.00035

0.00040

0.00045

0.00050

0.00055

0.00060

Stan

dard

devi

atio

n(m

)

ε′s = 16.4ε′′s = 2.37α(s) = 2.66 N/m

Z estimate error

Fig. 7: Sample standard deviation of estimates for RSS local-ization. Sample size = 1000.

as derived using our approach. For this purpose we usethe random module from the numpy package of python.Next, during the optimization phase, we initialize thenode locations to random values within the constraintsof the field boundaries and use the observed values ofthe received signal strength (as sampled in the first phaseof the simulations), to minimize the maximum likelihoodfunction as derived in Equation (26) to estimate the truevalues of the nodes’ coordinates. For this purpose we usethe optimize module from the scipy package of python.

B. Sensitivity analysis

Sensitivity analysis is used to determine the level of errorsthat may be introduced due to uncertainty in various parame-ters employed in the localization model. As an example, thesoil permittivity is a parameter of our ranging model usedin localization. Hence, localization performance depends onthe accuracy with which the soil permittivity values (ε′s andε′′s ) are measured by our soil sensors. The uncertainty intheir measurement may cause the mean of the estimators tobe shifted relative to the actual coordinate values. We studythe change in the estimated coordinates as a function of thedifference between the measured µs, ε′s and ε′′s and theirtrue values from Tables I and II. We use the same setup forsensitivity analysis as used in variance analysis. That is, a setof three sensor nodes are randomly deployed in the sensor fieldwithin the radio communication ranges of each other alongwith the four satellite nodes.

Figures 8, 9 and 10 present the sensitivities of the estimates(the shift in the estimate from its true value) to ε′s, ε

′′s and

µs, respectively. The x axes in these figures represent the

difference between the observed value of the respectiveparameter and its true value expressed as a percentage ofthe true value. The sensitivities with respect to the differentparameters vary. A 10% error in the measured ε′′s caused thelargest shift in the estimated location of 0.024 m; whereas thesame error in ε′s or µs causes a shift of 0.012 m.

0 2 4 6 8 10Percentage shift in measured ε′s

0.000

0.002

0.004

0.006

0.008

0.010

0.012

0.014

0.016

Dis

tanc

eof

esti

mat

efr

omtr

ueva

lue

(m)

κ = 100

ε′s = 2.36, ε′′s = 0.0967

ε′s = 5.09, ε′′s = 0.441

ε′s = 16.4, ε′′s = 2.37

ε′s = 24.5, ε′′s = 3.86

Fig. 8: Sensitivity analysis of received signal strength basedlocalization w.r.t. ε′s

0 2 4 6 8 10Percentage shift in measured ε′′s

0.000

0.005

0.010

0.015

0.020

0.025

0.030

Dis

tanc

eof

esti

mat

efr

omtr

ueva

lue

(m)

κ = 100

ε′s = 2.36, ε′′s = 0.0967

ε′s = 5.09, ε′′s = 0.441

ε′s = 16.4, ε′′s = 2.37

ε′s = 24.5, ε′′s = 3.86

Fig. 9: Sensitivity analysis of received signal strength basedlocalization w.r.t. ε′′s

0 2 4 6 8 10Percentage shift in measured µs

0.000

0.002

0.004

0.006

0.008

0.010

0.012

0.014

Dis

tanc

eof

esti

mat

efr

omtr

ueva

lue

(m)

κ = 100

ε′s = 2.36, ε′′s = 0.0967

ε′s = 5.09, ε′′s = 0.441

ε′s = 16.4, ε′′s = 2.37

ε′s = 24.5, ε′′s = 3.86

Fig. 10: Sensitivity analysis of received signal strength basedlocalization w.r.t. µs

VI. CONCLUDING REMARKS

We study node localization for a hybrid wireless sensornetwork system deployed across multiple physical media. As


in most applications of wireless sensor networks where senseddata is collected from different locations, location informationis an integral part of the data for the application. Moreover,network layer routing can benefit from such information byrouting data more efficiently to a destination node multiplehops away. Location information is also useful for networkservicing operations such as battery replacement. We proposea cost-effective method based on receive signal strength mea-surements for three dimensional localization to compute theX,Y coordinates along with the depth Z of the sensor nodes.A group of nodes with known location coordinates, calledsatellite nodes, serve as the anchor nodes in our scheme.Received signal strength of the signal between neighboringunderground sensor nodes and between satellite nodes andsensor nodes are used for estimating the inter-node distances.

To the best of our knowledge, past research has not consid-ered localization in a sensor network where nodes are locatedin different physical media or on modeling the signal strengthmeasurements for transmission across such multiple media.We present a model for the variation in received signal strengthdue to path loss for transmission in multiple media, multi-pathfading, and reflection and refraction across soil-air interface.We use the non-central χ2 distribution to model the multi-pathfading effects for two transmission scenarios: air to soil andsoil to soil. The parameters of the distribution are governed bythe distance between the receiver and the transmitter nodes,divergence and loss exponents, and Rician factor. We applymaximum likelihood optimization to the localization problemin a hierarchical and iterative manner to efficiently computethe location coordinates even for very large sensor networks.

Fig. 11: Model-based Data Assimilation and Site-specificPrescription.

VII. FUTURE DIRECTIONS

Our localization results are based on the modeling accuracyfor multi-path fading and path loss effects. We have modeledthe mean signal strength value while keeping the specular toscatter signal power ratio (κ) a constant for a signal propaga-tion path. However, literature results have indicated that theRician K-factor does change with the propagation distance.Accurately measuring κ is key to localization accuracy duringfield deployment. Extensive field tests are also needed tovalidate and calibrate the performance evaluation models.

An improvement in the running time complexity of thelocalization problem may be achieved by simplifying theoptimization problem used for the estimation. For example,formulating the estimation as a method-of-moments problemis expected to speed up the computation of the estimates.However, the improvement in running time may come at thecost of the accuracy of the estimates. An error analysis ofthe estimates obtained using the method-of-moments approachshould be performed.

Aside from developing accurate and efficient methods forresolving the proposed statistical estimation approach for sen-sor localization, we plan to develop a model-based frameworkfor assimilation of sensor data towards agricultural prescriptionfor irrigation and fertilization decision-making. Our approachwould be to develop a framework (see Figure 11) which willallow us to integrate sensor data into site-specific fertilizerresponses and to optimize the application of site specificfertilizer. Besides the sensor moisture and nutrient measure-ments, soil temperature and moisture (also from sensors),weather data (historical and forecasts) and crop management(planting, tillage, rotation, irrigation, etc.) will be assimilatedinto the model-based simulator APSIM (Agricultural Produc-tion System Simulator, www.apsim.info) to predict the yieldresponses. The model, as constrained by the data, will predicttwo important processes: nitrogen leaching (losses) versusorganic-N mineralization (for crop uptake) and using these willdetermine yield-response curves to agricultural inputs. Theeconomic optimum resource application rate will be achievedwhen the marginal cost for resources matches the marginalreturn on the yield.

ACKNOWLEDGMENT

The authors would like to thank Dr. Ahmed E. Kamal forhis generous contribution in the technical discussions and forreviewing the work.

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