Network baed Senor Localization in Multi-Media Application of Preciion Agriculture Part : Time of Arrival Herman Sahota IBM, Silicon Valley Laboratory Email: hahota@u.ibm.com Ratneh Kumar, IEEE Fellow Iowa State Univerity Email: rkumar@iatate.edu Abtract We tudy the problem of enor node localization for a hybrid wirele enor network deployed in a preciion agriculture farm, with node located both underground and above-ground. We conider time of arrival meaurement from unmodulated ignal tranmitted between neighboring enor node and between atellite node and enor node for ranging purpoe. While thi paper tudie the network baed localization of enor, placed in multiple media, baed on the meaurement of time of arrival, in a companion paper we tudy the ame problem baed on the received ignal trength meaurement. The localization problem i formulated with the goal of parameter etimation of the joint ditribution of the ranging meaurement. We derive the probability ditribution for the meaured time of arrival with the parameter governed by the enor node location coordinate baed on rigorou tatitical analyi. Then, we formulate a maximum likelihood optimization problem to etimate the node location coordinate uing the derived tatitical model. We alo preent a enitivity analyi of the etimate with repect to the oil complex permittivity and magnetic permeability. I. INTRODUCTION AND RELATED WORK We tudy the deign of a wirele enor network for a preciion agriculture application. Senor node are deployed in large farm-field to collect oil data to determine the optimal application of agricultural input to maximize production and minimize the impact on the environment. Location information i an integral part of the enor data thu collected. In addition, routing trategy may ue location data to chooe the mot optimal next hop when forwarding data to the ink node. In thi paper, we addre the problem of enor node localization to etimate the location coordinate of node with unknown poition uing network baed time of arrival meaurement. Localization approache can be broadly categorized a anchor free or anchor baed. Anchor free localization cheme [] etimate node location in a relative coordinate ytem formed by a few choen node in the ytem. Anchor baed node localization, on the other hand, relie on a number of anchor node in the network whoe location are known []. In [3], author preent an application of cooperative localization technique to UWB (ultra wide bandwidth) wirele network. The author quantify the performance of everal algorithm baed on the ranging model available for UWB technology. They alo preent a localization algorithm by mapping a tatitical model for graphical inference onto the network topology. In [4], author preent a equential Monte Carlo localization method for enor network with mobile node that ue both range meaurement and hop ditance. In thi paper, we preent our approach to enor node localization for uch a network in which the enor node are buried underground while the anchor node are located above ground. The following are the contribution of thi paper: We preent tatitical 3-D localization framework baed on time of arrival (ToA) meaurement for multiple media (air and oil) and multiple reflected path in a loy medium (oil). We extend the maximum likelihood etimation framework, that involve cro-correlation of tranmitted veru received ignal, to the multi-path cae to derive a tatitical model for the meaured ToA. We preent imulation reult that implement the propoed localization cheme in Python, including error analyi. We preent enitivity analyi of the etimated location coordinate with repect to variou parameter of interet. Thi i a two part paper with a companion paper [5] tudying the ame problem of enor node localization but employing received ignal trength meaurement of received ignal for ranging purpoe. Both technique were employed to undertand the trade-off between accuracy veru cot. Received ignal trength baed approach i cheaper but time of arrival baed approach i more accurate and, depending on the application, a uer can pick one over the other. Section II preent our localization framework baed on time of arrival meaurement. Section IV preent the imulation reult, including enitivity analyi in Section IV-A. Concluion are preented in Section V. II. LOCALIZATION BASED ON TIME OF ARRIVAL The time delay between the tranmiion of a beacon ignal at one node and it reception at another node can be ued to etimate the ditance between the node. Thi delay i etimated uing a matched filter at the receiving node, where a noie-free tranmitted ignal i available. The copy of the tranmitted ignal i delayed in time by a
certain amount and it correlation with the received ignal i meaured for a certain obervation time. The delay value at which thi output i maximum give an etimate of the propagation delay between the tranmitter and receiver. However, the etimate i uncertain owing to noie in the received ignal. Furthermore, multi-path interference may caue additional uncertainty in the etimate. For background, we introduce the ingle-path cae firt. Fig. : Senor node. node m filter decribed above, we proceed by dicretization and then take the limit to get back to the continuou domain. Accordingly, let u conider the oberved data coniting of n + equally paced ample of r (t ), at time intant t k k t,where t, t n n t T, and k {,,,...,n}. The individual ample r k are Gauian random variable with mean A k and variance N N δ[], where k i the ampling of the ignal (t ) at time t k τ, defined a k (t k τ) t. The oberved data-vector r ha a multivariate-gauian ditribution, where the joint-pdf i driven by the noie pdf. The covariance matrix Φ of r i (n + ) by (n + ), with {j,k } t h element: z x y node n (a) Air to oil communication between two node m and n. φ j k δ[j k ], () where φ j k i the covariance between r j and r k. The mean of r equal the ample of the tranmitted ignal: E [r] A : A[... k... n ] T, (3) where T denote vector tranpoe operation. Thu, the jointpdf of r i: p(r;τ) exp (π) (n+) Φ (r A)T Φ (r A), (4) n+ (r k A k ) exp. (5) (π ) (n+) k where Equation 5 follow from Equation 4 by uing Equation, and ubtituting φ j k N δ[j k ] for additive white Gauian noie. z x y b a c node m node n Noting that the quared term r k and (A k ) um up to received and attenuated tranmitted ignal power, which are independent of the delay of communication, the loglikelihood function, ignoring the term that do not depend on τ, i given by: n+ logλ(τ r) k r k A k., (6) Under the limit a t (and n ) Equation 6 become: (b) Soil to oil communication between node m and n: Direct v. reflected path A. Variance of the MLE of ToA for ingle path We aume that the tranmitted ignal (t ) i corrupted by additive white Gauian noie n(t ) upon reception at the receiver node a r (t ), after a ditance and medium dependent attenuation A and delay τ: r (t ) A (t τ) + n(t ), () The auto-covariance function of the noie i [n(t )n(t )] N δ(t t ), where δ(.) i the dirac-delta function and N i the noie pectral denity. r (t ) i oberved over an interval [, T ] at the receiver node. To how that MLE baed on ToA reduce to a matched logλ(τ r (t )) r (t )A (t τ)d t. (7) Thu, the log-likelihood function reduce to the correlator function between r (t ) and (t ) given a certain delay between the two, which i the output of a matched filter. Hence, ˆτ that maximize thi correlator function between the tranmitted and received ignal turn out to be the maximum likelihood etimate of the delay τ. The mean of ˆτ, denoted τ, i determined by the ditance between the ender and receiver node and the propagation peed of the ignal in the medium it travel. The variance of ˆτ, being an MLE, equal the Cramer-Rao lower bound (CRLB) and i the invere of the Fiher information matrix, given by [6]: σ logλ(τ r (t )) logλ(τ r (t )) (8)
The next expreion follow from the derivation given in [6, p. 64-65]. σ A A (t τ) (t τ)d t (9) (πf ) S(f ) d f () 8π p T β, () 8π T Bβ SNR () where S(f ) i the Fourier tranform of the ignal (t ), and A i the ignal power attenuation due to propagation in the repective medium, o that p A S(f ) d f i received ignalpower, β f S(f ) d f S(f ) d f T i a function of the ignal, B i the p B bandwidth of the ignal and SNR i the ignal to noie power ratio. Thu, we have etablihed that the variance of the time of arrival a etimated by the receiver matched filter, which i alo the maximum likelihood etimate, i given by Equation (). Since the MLE i aymptotically normal, the etimated time of arrival ˆτ can be approximated to be a Gauian ditributed random variable: p ˆτ (τ) πσ e (τ τ) σ. (3) In the next ection we derive the relation that etablih the dependence of τ and σ on the location coordinate of the enor node for our preciion agriculture application. In thi way, the pdf of ˆτ can be parametrized in term of the location coordinate in order to etimate the latter from the oberved time of arrival for ignal tranmitted between neighboring pair of enor node and between the enor node and the atellite node with known location coordinate. B. Mean and variance of time of arrival in term of location coordinate ) Multi-path extenion: oil to oil communication: The received ignal in time domain i given by: r (t ) A (t τ) + A r (t τ r ) + n(t ), (4) where A and A r are the attenuation along the two path and τ and τ r are the propagation delay along the two path. Following the tep in Section II-A for the received ignal given by Equation (4), the joint log-likelihood function for the parameter τ [τ τ r ] T i given by: logλ(τ r (t )) r (t ) A (t τ) + A r (t τ r ) d t. (5) The etimate of τ that maximize Equation (5) i the maximum likelihood etimate. Hence, the covariance matrix of the etimate i given by the invere of the Fiher information matrix. The (i, j ) th element of the covariance matrix Σ i, then, given by: σ i j logλ(τ r (t )) logλ(τ r (t )), (6) θ i θ j where i, j {, } correpond to the line-of-ight and reflected path, θ τ and θ τ r. The next expreion follow from the derivation given in [6, p. 64-65]. A i A j σ i j (t θ i ) (t θ j )d t (7) θ i θ j A i A j (πf ) S(f ) d f (8), (9) pi p j 8π T β 8π T Bβ SNR i SNRj, () where SNR i pi and SNR B j p j are the ignal to noie B power ratio for the two path i and j. Note that for a ender-receiver pair (m,n), p p and p p r, are obtained uing the path lo equation, with the latter containing an additional multiplicative factor of reflection contant ρ modeling the power lo due to reflection from the ground urface [5]. Thu, we have p p η(d ) k e α d m n and p p r η(d ) k e α d m n ρ. For ender-receiver pair (m,n), the mean value τ [ τ τ r ]T i determined by the propagation ditance and the peed of the ignal: τ d, τ r c d (r ), () c where the line of ight propagation ditance d and the reflected path propagation ditance d (r ) are derived uing Figure b; and the peed of light in oil c i given by: c, () µ ε ( + ( σ ) + ) ωε where ε, µ and σ are, repectively, the real part of permittivity, permeability and the conductivity of oil. Following the ame argument a in Section II-A, ˆτ i aymptotically ditributed a a bivariate Gauian random variable with the pdf: p ˆτm n (τ ) (π) Σ / exp (τ τ ) T Σ (τ τ ). (3) ) Multi-media extenion: air to oil communication: For ignal propagation between a atellite node m and a enor node n, the propagation medium i partly air and partly oil, ee Figure a, where d (a ) and d ( ), the propagation ditance in air and oil, repectively, were derived in [5].
Therefore, we obtain an expreion for the mean value of the propagation delay between the atellite and enor node: τ d (a ) c a + d ( ), (4) c where c a c i the ignal peed in air that approximately equal the peed of light in vacuum, and c i given by Equation (). Thu, from Equation (4) we obtain an expreion for the mean of the time of arrival in term of the location coordinate of the ender and receiver node, a the firt tep in expreing it pdf parameterized by the location coordinate. Next, we obtain an expreion for the variance of the meaured time of arrival in term of the location coordinate. From Equation (): σ 8π p (a ) T β 8π T Bβ SNR, (5) where p (a ), the average received power for air to oil communication, wa derived in [5] in term of the location coordinate of the atellite node m and enor node n. III. MAXIMUM LIKELIHOOD PROBLEM FORMULATION Now we formulate a maximum likelihood problem to etimate the location coordinate of the node given the meaurement of the time of arrival at the node from their neighboring underground enor node a well a above-ground atellite node. Let N and N a denote the et of oil-to-oil and airto-oil node pair that communicate to gather the time of arrival data for oil to oil ignal propagation and air to oil propagation, repectively. The log-likelihood of the time of arrival ditribution parameter for all the ender-receiver pair (m,n) N N a can be expreed a follow: L(Θ T ) ln(σ ) + (τ τ ) (m,n) N a (m,n) N σ ˆτ ln Σ + (τ τ ) T Σ (τ τ ) (6) where T repreent the time of arrival data {τ (m,n) N a,τ [τ τ r ]T (m,n) N } between all pair of node that communicate with each other to gather localization data. σ, τ, Σ and τ [ τ τ r ]T were expreed in term of the location coordinate of the enor node and the known location coordinate of the atellite node in Section II-B and II-B for the different ignal propagation cenario in our application. Thu, the maximum likelihood etimate of the location coordinate are obtained a the value that maximize the log likelihood function given by Equation (6). IV. SIMULATION RESULTS Following a imilar approach to [5], we imulate a network of 5 enor node in a quare field of ize 5 m 5 m. The enor node are randomly deployed within Fractional volume of water ε ε %.36.966 % 5.86.44 3% 6.4.368 4% 4.485 3.857 TABLE I: Real and imaginary part of relative permittivity of oil for different moiture content. Parameter Symbol Value Tranmit power (Satellite node) P (a ) 3 dbm Tranmit power (Senor node) P ( ) 3 dbm Thermal noie - dbm Path lo factor (air) k a Path lo factor (oil) k Relative permeability (oil) µ.84 Frequency f 433 MHz Wavelength λ.7 m Number of reading per node pair (RSS) N RSS Number of reading per node pair (ToA) N ToA Signal duration T S m TABLE II: Parameter ued in localization imulation. the quare field at a depth between 3 5 cm. Our localization model dicued in Section II depend on the oil permittivitie which in turn are function of the moiture content of the oil. We aume a clay loam oil with a clay content of %. For thi oil type, the relative real and imaginary value of the permittivity are computed [7] for variou value of volumetric water content a given in Table I. Other parameter ued in the imulation are ummarized in Table II. The relative magnetic permeability of oil i cloe to for oil not containing ignificant iron. Thermal noie i aumed to be dbm baed on tandard receiver enitivity. Thu, the underground tranmiion range i approximately 34 m for dry oil at a tranmiion power level of 3 dbm leading to an average of 3 neighbor per enor node, wherea the air to-oil tranmiion range i approximately m [5]. Figure preent the localization etimate for oil with moiture content of 3% (ε 6.4, ε.4). A viual inpection of the reult how that our method uccefully etimate the location coordinate of the enor node, with the error margin a decribed below., We ue a minimal etup of our network to perform variance analyi of our location etimate: three enor node deployed randomly within the field but within the radio communication range of each other and four atellite node o that we can cloely capture the real world deployment cenario where each enor node ha 3 neighboring enor node. Then, we compute the ample tandard deviation of the location etimate of one of the enor node a hown in Figure 3. Time of arrival baed localization ha almot ten order of magnitude better error performance than received ignal trength baed localization. The increaed accuracy of time of arrival baed localization come at the cot of clock ynchronization among all the node in the network. But clock ynchronization i built into our network a it i alo needed for cheduling [8]. The tandard deviation in X and Y etimate i relatively inenitive to change in the oil permittivity. However, the Z etimate ha the leat tandard
deviation for ε 4.5 and ε 3.86, correponding to a moiture content of 4%. The tandard deviation of the Z coordinate etimate decreae with the increae in the moiture content of the oil. The imulation reult demontrate that our technique i efficient, calable and reliable. Although the reult are preented for a field ize of 5 m 5 m, the technique i equally applicable to larger field. Standard deviation (m).5 9.45.4.35.3.5 X etimate error Y etimate error Y-coordinate Z-coordinate 4 8 6 4 4 6 8 4 X-coordinate.5.45.4.35.3 4 6 8 4 X-coordinate Actual Initial Etimated ɛ 6.4.37 α ().66 N/m Actual Initial Etimated ɛ 6.4.37 α ().66 N/m Standard deviation (m). 4.5 4. 3.5 3..5..5. ɛ.36.967 5. ɛ.36.967 ɛ 5.9.44 ɛ 5.9.44 ɛ 6.4.37 ɛ 6.4.37 ɛ 4.5 3.86 ɛ 4.5 3.86 Z etimate error Fig. 3: Sample tandard deviation of etimate for ToA localization. Sample ize. Fig. : Localization uing time of arrival model. A. Senitivity analyi Senitivity analyi i ued to determine the level of error that may be introduced due to uncertainty in variou parameter employed in the localization model. A an example, the oil permittivity i a parameter of our ranging model ued in localization. Hence, localization performance depend on the accuracy with which the oil permittivity value (ε and ε ) are meaured by our oil enor. The uncertainty in their meaurement may caue the mean of the etimator to be hifted relative to the actual coordinate value. We tudy the change in the etimated coordinate a a function of the difference between the meaured µ, ε and ε and their true value from Table I and II. We ue the ame etup for enitivity analyi a ued in variance analyi. That i, a et of three enor node are randomly deployed in the enor field within the radio communication range of each other along with the four atellite node. Figure 4a, 4b and 4c preent the enitivitie of the etimate (the lope of the etimate with repect to parameter of interet), baed on time of arrival localization to ε, ε and µ, repectively. Time of arrival localization baed etimate enitivitie with repect to ε and µ are independent of the oil moiture content. However, the enitivity with repect to ε varie with the moiture content, the etimate being leat enitive to change in ε for dry oil. The etimate i le enitive to change in ε when compared with the enitivity with repect to ε and µ, with a % error in the parameter cauing a hift of only.35 m in the etimate. Equivalent error in ε and µ caue hift of.5 m and.7 m, repectively. V. CONCLUSION AND FUTURE DIRECTIONS Node localization i a deirable property of a wirele enor network for our preciion agriculture application in which a number of enor node are placed underground while a handful of atellite node with known coordinate are located above ground. We propoe method for three dimenional localization to compute the X, Y coordinate along with the depth Z of the enor node. We ue two type of ranging meaurement (received ignal trength
Ditance of etimate from true value (m).4...8.6.4.. 4 6 8 Percentage hift in meaured ɛ ɛ.36,.967 ɛ 5.9,.44 ɛ 6.4,.37 ɛ 4.5, 3.86 (a) Senitivity analyi of time of arrival baed localization w.r.t. ε Ditance of etimate from true value (m).8.7.6.5.4.3... 4 6 8 Percentage hift in meaured ɛ.36,.967 ɛ 5.9,.44 ɛ 6.4,.37 ɛ 4.5, 3.86 (b) Senitivity analyi of time of arrival baed localization w.r.t. ε Ditance of etimate from true value (m).6.5.4.3... 4 6 8 Percentage hift in meaured µ ɛ.36,.967 ɛ 5.9,.44 ɛ 6.4,.37 ɛ 4.5, 3.86 (c) Senitivity analyi of time of arrival baed localization w.r.t. µ Fig. 4: Senitivity analyi. and time of arrival) to etimate the inter-node ditance and, conequently, the coordinate of the underground enor node. Time of arrival meaurement provide a more accurate ranging olution than the received ignal trength baed approach. The meaured time of arrival i found to be Gauian ditributed with mean governed by the internode ditance, and the variance further governed by ignal hape, duration, bandwidth and SNR. We poe the localization a a maximum likelihood etimation problem. The improvement in localization accuracy come at a cot of clock ynchronization which, however, i in-built in our et up that require clock to be ynchronized for communication cheduling. Our localization reult are baed on the modeling accuracy for multi-path fading and path lo effect. We have modeled the mean ignal trength value while keeping the pecular to catter ignal power ratio (κ) a contant for a ignal propagation path. However, literature reult have indicated that the Rician K-factor doe change with the propagation ditance [9]. Accurately meauring κ i key to localization accuracy during field deployment. The catter power correpond to the variance in an AWGN model (ued in ToA baed localization), dicounting the thermal noie. Similarly, our localization model require parameter calibration uing field experiment. Extenive field tet are alo needed to validate and calibrate the performance evaluation model. An improvement in the running time complexity of the localization problem may be achieved by implifying the optimization problem ued for the etimation. For example, formulating the etimation a a method-of-moment problem i expected to peed up the computation of the etimate. ACKNOWLEDGMENT Thi work wa upported in part by the National Science Foundation under the grant NSF-ECCS-969 and NSF- CCF-3339. REFERENCES [] Nianka B. Priyantha, Hari Balakrihnan, Erik Demaine, and Seth Teller. Poter abtract: anchor-free ditributed localization in enor network. In Proceeding of the t international conference on Embedded networked enor ytem, SenSy 3, page 34 34, New York, NY, USA, 3. ACM. [] Bin Xiao, Lin Chen, Qingjun Xiao, and Minglu Li. Reliable anchor-baed enor localization in irregular area. Mobile Computing, IEEE Tranaction on, 9():6 7,. [3] H. Wymeerch, J. Lien, and M.Z. Win. Cooperative localization in wirele network. Proceeding of the IEEE, 97():47 45, feb. 9. [4] Bram Dil, Stefan Dulman, and Paul Havinga. Rangebaed localization in mobile enor network. In Wirele Senor Network, page 64 79. Springer, 6. [5] Herman Sahota and Ratneh Kumar. Network baed enor localization in multi-media application of preciion agriculture part : Received ignal trength. ubmitted to IEEE ICNSC 4, Nov 4. [6] Carl W. Heltrom. Statitical theory of ignal detection. Pergamon pre, econd edition, 968. [7] V. L. Mironov. Spectral dielectric propertie of moit oil in the microwave band. In Geocience and Remote Sening IEEE International Sympoium, 4. [8] Herman Sahota, Ratneh Kumar, and Ahmed Kamal. A wirele enor network for preciion agriculture and it performance. Wirele Communication and Mobile Computing, ():68 645,. [9] L. Thiele, M. Peter, and V. Jungnickel. Statitic of the ricean k-factor at 5. ghz in an urban macro-cell cenario. In Peronal, Indoor and Mobile Radio Communication, 6 IEEE 7th International Sympoium on, page 5, 6.