in a network of sensors and actuators comprised of elastic wave transducers ( e . g ., piezoelectric , lead - zirconate - titanate ( pzt ) transducers ), the minimum measurement unit or building block is a single actuator - sensor path . a network of transducers , no matter how complicated in geometry , may always be decomposed into individual actuator - sensor paths defined by associated pairs of transducers . therefore , if the pod of damage of a single path at a position within the neighborhood of the path can be determined , it is possible to synthesize the pod of the entire network . more specifically , if the pod for a single path is determined first , then the pod at any position in the network can then be computed probabilistically . in practice , the pod values corresponding to a critical damage size l dmg is of central interest . the methods of the present invention can be used to directly predict the probability of detecting damage in structures that are equal or greater than the critical damage size characterized by l dmg . according to an embodiment of the present invention , a method for determining the pod of a transducer array comprises two major components : 1 ) determining the pod of a single path , and 2 ) computing the network pod . a detailed description of each of these components and example methods for carrying them out is given below . first , the concept of a “ detectable damage ” is defined . in plate - like structures , propagating elastic waves are referred to as lamb waves , in which the particle motion lies in the plane defined by the plate normal and the direction of wave propagation , and are typically characterized as “ symmetric ” or “ anti - symmetric ” in amplitude displacement about the midpoint of the plate . a symmetric mode wave , also known as an extensional mode , exhibits surface displacement on opposing sides of the midline of the plate that is substantially a mirror image displacement , as illustrated in fig1 a , which schematically illustrates a fundamental symmetric mode . by contrast , an anti - symmetric mode , also known as a flexural mode , exhibits substantially transverse displacements in the same directions on both sides of the plate , as illustrated in fig1 b . both symmetric and anti - symmetric modes may occur in successive “ orders ” of discrete modes , determined by certain boundary conditions . for simplicity , the following discussions are made in terms of the zero order of each of the symmetric and anti - symmetric modes . the anti - symmetric flexural mode has a significantly lower transverse restoring force because there is substantially no compression , as in the extensional symmetric mode , and therefore , the phase velocity of propagation is typically lower than that of the symmetric mode . thus , a pulse of a symmetric mode wave will typically exhibit a shorter time - of - arrival at a sensor transducer than that of the anti - symmetric mode thereof . for a “ baseline ” elastic wave signal , i . e ., an elastic wave propagated between two transducers in a baseline structure ( i . e ., one having no damage in it ), the respective first arrivals of the zero - order symmetric and anti - symmetric mode may each be detected within a selected time window . wave packets arriving immediately after the respective first arrivals of each type of wave may be considered as “ scattering signals ,” i . e ., as arriving from reflections from damages that are not in the direct path between the two transducers . for each of the two modes , the signal segment containing the first arrival and the corresponding scattering wave packet may first be selected as a “ featured ,” i . e ., most relevant , component of a signal . the corresponding segments ( for both the symmetric and anti - symmetric modes ) of a “ current ” signal ( i . e ., as opposed to a baseline signal ) may also be selected as the featured components of the current signal . let e b s0 denote the energy in a selected time window of the featured component of the baseline signal data corresponding to the zero order symmetric mode , and e c s0 denote that of the current signal data , respectively . then , the following damage index i s0 may be defined for the symmetric mode of each actuator - sensor path : a similar damage index i a0 may be defined for the anti - symmetric mode as : if the current signal is collected when damage does exist in the neighborhood intersecting the individual actuator - sensor path , the symmetric and / or the anti - symmetric modes of the current signal will generally be different from those of the corresponding baseline signal . where a damage is located adjacent to the direct line of a path , but does not directly interfere with ( i . e ., attenuate ) the direct signal , the elastic wave energy may scatter from the damage site and be detected at the receiving transducer ( i . e ., sensor ) by traveling along an indirect path , as indicated by the path segments r 1 and r 2 in fig2 . in such cases , a second , delayed signal may arrive in the same time window , as described above . however , the direct path signal and delayed scattered signal may then coherently interfere with each other , thereby causing a change in the energy e c s0 of the current signal in the selected time window . therefore , equations ( 1 ) and ( 2 ) remain valid without loss of generality . the amount of the differences characterized by i s0 and i a0 will be dependent on the size and severity of the damage , as well as its location with respect to the path . respective threshold values may be defined to determine whether a damage is detectable . let t s and t a be the respective thresholds defined . then , by definition , damage is deemed to have been detected if with the above definition , the pod of an individual actuator - sensor path may then be determined using one of the following methods : 1 ) by an experimental method with simulated damages on a sample structure . the simulated damage may be created by bonding stiff metal or a damping patches to the structure . each of these types of simulated damage produces a different effect on both the directly propagating and scattered signals . 2 ) by an experimental method with real damages on a sample structure . this is a more destructive method , as the sample structure is not reusable , but may more accurately represent that which may be obtained in actual implementation . using the above methods , a pod database can then be established as follows . it may be noted first that , as illustrated in fig2 , the path between the two transducers s 1 and s 2 on the test plate 200 is symmetric , and as a result , in either of the above methods , it is necessary to simulate or create damages in only a quarter of the effective region of a path ( e . g ., the quadrant 202 indicated in fig2 ). in implementation , damages of different sizes may be created or simulated at each cell of the grid a selected number of times , for example x times . however , damage detection , as defined by equation ( 3 ), may result in identifying a damage at that location only y times . the signal detection pod for a damage at each location may then be computed as the total number of detected damages divided by the total number of simulated or created damages , i . e ., the pod = y / x , expressed as a percent . in implementation , transducers may be placed at various distances , the grid may be sparse and the cell size can be roughly set to the same order of the damages to be monitored , e . g ., 1 inch ( 1 ″) to 3 ″ for impact - type damages . the pod at locations where no damage is created may be obtained by interpolation . thus , a database can be established that provides a probability of detection between two transducers defining an actuator - sensor path for damages of different types and sizes at various locations on a structure in relation to that actuator - sensor path . thus , to generate the pod of a sensor network , the individual path pod for different path lengths may first be determined , as above , where , for example , path lengths may vary from 4 ″ to 15 ″, in 1 ″ increment . as in traditional non - destructive testing ( ndt ) processes , this process of pod generation needs a considerable amount of testing . however , once created , the pod results for different path lengths and damage locations may then be used to form a standard database . this database may be used for evaluation of the pod of various designs of transducer networks , and for transducer layout optimization for various applications . individual path pod test results for different structure materials and thicknesses may also be included to further improve the database . the accuracy of the sensor network pod will be improved by continually adding new individual path pod results to the database . to determine the probability of detection ( pod ) of damage of a shm transducer network on a structure , i . e ., a network comprising a plurality of transducer paths , a grid is first defined over the area to be monitored by the network . the cell size of the grid may be set in the same way as that for the individual path pod above . two types of pods may be defined for the shm sensor networks : 1 ) the pod at a specific cell of the grid , and 2 ) the pod of the entire monitored area of the structure . once the respective pods at all of the cell locations of the monitored structure are obtained , the pod of the entire monitored area of the structure may be defined simply as the average of the pods at all of the cell locations . as illustrated in fig3 , each cell may have a number of neighboring paths ( e . g ., six paths in the example embodiment of fig3 ). after the signal detection pods of the individual paths are obtained as above , the pod of the network of damage at a specific cell may be obtained by computation using certain rules . the following are two examples of the rules that can be used : 1 ) use the maximum signal detection pod of all the neighboring paths of a particular cell as the damage detection pod of the network at that cell . 2 ) use the second maximum signal detection pod ( according to equations ( 1 ), ( 2 ) and ( 3 )) above of all the neighboring paths of a particular cell as the damage detection pod of the network at that cell . this means that damage is said to be detectable by a network at a certain pod value if at least two neighboring paths can detect this damage . compared with rule 1 ) above , this rule may reduce the false positive detection by reducing the sensitivity . the computational method described above is merely an example thereof , and different computational rules may be defined , depending on the particular application at hand . after the pod at all of the cell locations of the monitored structure are computed , the pod of the whole network may then be defined as the average of the pods at all the locations . in planning the layout of transducers for a shm system , it is often desirable to know the relationship between the transducer density and the detectable damage size at a desired value of pod ( e . g ., 90 %). it may also be desirable to determine the system cost , based on the transducer density required to detect the minimum detectable damage size . the transducer density can be specified as the number of transducers per unit area for a given layout . finding the relationship between the transducer density and the minimum detectable damage size is a direct application of the pod curve of the layout , which can be carried out as follows : step 1 : specifying a structure on which a network of transducers is to be placed ; step 2 : specifying a minimum desired value of pod of damage detection , denoted by pod d ( for example , pod d may be a 90 % probability of detecting damage ); step 3 : generating a plurality of transducer position layouts for the network , each layout being determined by the number and position of the transducers forming the network ; step 4 : generating the pod curve for each sensor layout using the pod determination method described above ; step 5 : finding from the pod curve of each sensor layout the damage size that corresponds to pod d , denoted by l k , for k = 1 , 2 , . . . , k , where k denotes the total number of the sensor layouts ; and , step 6 : generating the sensor density layout curve as a function of l k using the results of step 5 . more specifically , fig4 is a process flow diagram of a method 500 for determining the transducer density on a structure for a given desired minimum probability of detection , according to an embodiment of the present invention . the method 500 begins by specifying the structure to be monitored ( step 510 ). the specification includes at least the geometry and the area of the structure , and may include other structure information , such as the area and location of cutouts , stiffeners , regions with varying thickness , and the like . next , the minimum desired network pod ( pod d ) is specified ( step 520 ). for example , the desired pod d may be set at 90 %, 95 %, and so on . then , for the specified structure defined in block 510 , a plurality of sensor layouts c k , where k = 1 , 2 , . . . , k , is defined ( block 530 ). a number m of damage sizes d m is then selected ( step 540 ), where m = 1 , . . . , m . for each sensor layout c k , k = 1 , 2 , . . . , k , the pod is calculated for all of the m damage sizes ( step 540 ) using the methods described above , such as , for example , relying on a single path pod in a database and applying it to the sensor layout . from the pod curve generated above for the sensor layout c k , where k = 1 , 2 , . . . , k , the damage size corresponding to pod d , denoted by l k , is picked out ( step 550 ). the sensor density of c k , k = 1 , 2 , . . . , k , is calculated as the number of transducers divided by the structure area . denote by d k the sensor density for c k , where k = 1 , 2 , . . . , k . then , the plot of d k versus l k , where k = 1 , 2 , . . . , k , is created , which shows the relationship between the transducer density and the detectable damage size at the desired pod level ( step 560 ). although the present disclosure has been described with reference to certain specific exemplary embodiments thereof , it will be understood by those skilled in the art that a variety of modifications and variations may be made to these without departing from the spirit and scope of the present disclosure , as defined in the appended claims and their functional equivalents .