US 6,246,355 B1Grant

Radar cross-section measurement system for analysis of wooden structures

Issue Date:2001-06-12

•47 Claims

•10 Drawing Sheets

Abstract

A method and system for analyzing a wooden structure, such as a utility or telecommunications pole, are provided. A location for the wooden structure is identified. A first radar signal is propagated towards the wooden structure with a radar antenna while the radar antenna is motion along a navigation path in the vicinity of the wooden structure. A reflected radar signal is received from the wooden structure, from which a determination is made whether the wooden structure contains a structural anomaly.

Metadata

Assignee

Hot/Shot Radar Inspections, LLC

Inventors

Gilbert F Miceli
Michael Parisi

Application Information

Application Number:US 09/680,745

Filing Date:2000-10-07

Priority Date:1999-12-22

Art Unit:7

Classifications

IPC:

G01S 1388G01S 1386G01S 1389

Field of Search:

342 21342 22342 25342 27342 28342 73342 89342175342189342118342147701200

Patent Drawings (10 sheets)

Description

[0002] This application claims the priority of Provisional Appl. No. 60/171,548, filed Dec. 22, 1999 and of Provisional Appl. No. 60/191,444, filed Mar. 23, 2000, both of which are herein incorporated by reference for all purposes.

Background of the Invention

[0003] The present invention relates generally to a method and system for analyzing wooden structures with radar. More particularly, the invention relates to a method and system for detecting and characterizing anomalies in wooden utility poles.

[0004] The power-utility-system infrastructure alone in North America includes approximately 150,000,000 wooden pole structures. A similarly large number of wooden poles are additionally used by the telecommunications industry. Wood remains valuable as a material for constructing power and telecommunications poles because of its cost effectiveness and reasonable durability. Such poles are, nevertheless, subject to deterioration over time, not only from climatic effects, but also from biological and mechanical assaults. For example, biological deterioration may result from the activity of decay fungi, wood-boring insects, or birds. Woodpeckers have been known to bore vertical tunnels in wooden poles greater than twelve feet in length. Mechanical damage can result from such things as vehicular collisions or shotgun impacts. Consequently, each wooden pole in the system must be inspected periodically and a determination made whether to replace the pole based on the strength of the pole. Typically, poles are inspected on a 5-9 year cycle.

[0005] Various methods currently exist for evaluating pole strength, generally requiring direct physical contact with the pole. Such methods rely primarily on sampling techniques in which the strength of the pole is deduced from an assessment of its characteristics at the sampled points. Such sampling is typically performed in the region of the pole easily accessible by a technician, i.e. between about six feet above the ground to about two feet below the ground, so that only about 10% of the pole is even within the sampling region. Crossarms, which are positioned near the tops of the poles, are rarely examined for deterioration. Current methods also tend to include significant reliance on the qualitative assessment of the technician examining the pole. Individual visits to every pole to perform the inspection additionally result in substantial costs for maintaining the pole infrastructures.

[0006] There is accordingly room for improving the reliability of wooden-pole deterioration measurements by examining substantially the entirety of the structures and automating the evaluation of their strength. In addition, the cost for assessing the structures can be reduced by using a system that eliminates the need to have each pole visited individually.

Summary of the Invention

[0007] Thus, embodiments of the invention are directed to a method and system for analyzing a wooden structure, such as a utility or telecommunications pole. In one embodiment, a location for the wooden structure is identified. A first radar signal is propagated towards the wooden structure with a radar antenna while the radar antenna is motion along a navigation path in the vicinity of the wooden structure. A reflected radar signal is received from the wooden structure, from which a determination is made whether the wooden structure contains a structural anomaly. The wooden structure may be identified by imaging the wooden structure, such as with a charge coupled device or infrared camera. In certain embodiments, longitude and latitude positions for the wooden structure are ascertained with a global positioning system. The location of the wooden structure may also be identified by reflecting a laser signal from it.

[0008] In various embodiments, a second radar signal modulated in accordance with a pulse compression scheme is propagated towards the wooden structure. The first and second radar signals may be provided by the same radar subsystem or by separate radar subsystems in different embodiments. The determination of whether the wooden structure contains a structural anomaly may be made in one embodiment from data extracted from the reflected radar signal by calculating a density distribution for the wooden structure. The calculated density distribution may be used to designate closed-volume regions within the wooden structure having a density less than a threshold density relative to a mean density for the structure thereby identifying them as possible structural anomalies.

Brief Description of the Drawings

[0009] These and other embodiments of the present invention, as well as its advantages and features are described in more detail in conjunction with the text below and the attached figures, in which similar reference numerals are used throughout the several drawings to refer to like elements. Various components of the same type may be distinguished by following the reference label with a hyphen and a second label that distinguishes among the components.

[0010] FIG. 1 is a block diagram showing the relationship between various elements of the system in one embodiment of the invention;

[0011] FIG. 2 is a schematic diagram of one possible navigational path that may be taken by the inspection vehicle when taking radar cross-section measurements of wooden poles: part (a) shows a top view of the navigational path and part (b) shows a side view of the navigational path;

[0012] FIG. 3 illustrates one configuration that may be used to equip a navigation vehicle to operate in accordance with an embodiment of the invention;

[0013] FIG. 4 is a block diagram showing the relationship between various subsystem elements used in analyzing the wooden structures;

[0014] FIGS. 5(a) and 5(b) are a schematic diagram showing the interaction of various sensor signals with wooden poles:

[0015] FIG. 5(a) shows a top view and

[0016] FIG. 5(b) shows a side view;

[0017] FIG. 6 is a representation of a cylindrical coordinate system;

[0018] FIG. 7 is a block diagram showing the analysis of collected data;

[0019] FIG. 8 is a schematic diagram of the wood-penetrating aspects of the radar analysis; and

[0020] FIG. 9 is an example of a wood inspection report that may be provided in accordance with embodiments of the invention.

Description of the Specific Embodiments

[0021] Embodiments of the invention include a radar cross-section measurement system for the analysis of wooden structures, such as utility and telecommunications poles. In accordance with one embodiment, a vehicle is operated in the vicinity of the poles to be examined, systematically making appropriate radar measurements of the poles. In one embodiment, the vehicle is a land-based vehicle such as a truck, while in other embodiments, the vehicle is an airborne vehicle such as a helicopter or other aircraft. The airborne vehicle may be preferred in circumstances where the pole is not easily accessible by land, while the land-based vehicle may be preferred in circumstances where airspace restrictions limit access by air. The measurements are used by a computational analysis system to determine the existence and location of any anomalies within any of the examined poles. A report of the results is prepared and forwarded to a client.

[0022] The overall structure of one such system is illustrated in FIG. 1 in the form of a block diagram showing in particular the flow of data through the system. The system functions centrally with a dispatcher 100 who is responsible for coordination of various other aspects of the system. In operation, a client 105 requests a report analyzing certain wooden poles 10 (not shown in FIG. 1). The dispatcher collects system information in the form of inspection data 110. Such inspection data 110 may be provided by the client or obtained from other sources to identify the poles 10 to be analyzed and the environment in which they are located. For example, to identify the environment in which the poles 10 are located, the inspection data 110 may include, among other information, maps of the region, information identifying any line crossovers that may exist proximate the poles 10, and information identifying zones where the inspection vehicle 12 (not shown in FIG. 1) is excluded, such as no-fly zones in those embodiments where the inspection vehicle 12 is an aircraft. In the United States, flight-plan data may be obtained from such sources as the Federal Aviation Administration (FAA) or the Aircraft Owners and Pilots Association (AOPA).

[0023] Relevant inspection data 110 are provided by the dispatcher 100 to an analysis system 120, which may perform various functions as necessary in the system and as described in greater detail below. As part of one such function, the analysis system 120 combines inspection data 110 with other data relevant for formulating an inspection plan 135. Such other data may also be provided by the dispatcher 100 or may be obtained directly from another source. One example of such relevant other data shown in FIG. 1 as being obtained directly from an external source is weather data 150 describing the existing and/or expected weather conditions in the region of the poles 10 to be analyzed. The invention encompasses the use of other data sources relevant to the formulation of an inspection plan 135, such as the locations of hotels, the locations of rental-car companies, the layouts of nearby airports, and others as may occur to those of skill in the art.

[0024] In this aspect of the invention, the analysis system 120 acts as a module that uses such information sources to formulate the inspection plan 135. In embodiments using an aircraft as the inspection vehicle 12, the inspection plan 135 may be equivalent to a flight plan for the aircraft. The inspection plan 135 includes such features as a proposed inspection route, including starting points, end points, possible fuel stops, and a list of known possible hazards to the vehicle 12 such as line crossovers or antennae. In addition, the inspection plan 135 may include one or more alternative routes to be followed by the inspection vehicle 12 in the event some barrier to completing the proposed inspection route is encountered. The analysis system 120 may also provide digital system maps 130 and/or a weather briefing 125, each of which may additionally be included in the inspection plan 135. The inspection plan 135 may also include other relevant information communicated by the analysis system 120 that may be useful during the inspection.

[0025] The inspection is performed by navigating the inspection vehicle 12 in the vicinity of the poles 10, such as shown in greater detail in FIG. 2 (described below). The inspection vehicle 12 is occupied by an inspection crew 145 which obtains information describing the inspection plan 135 via a satellite link 140 or equivalent communications device. The inspection crew 145 may obtain any of the additional information described above as necessary or desired during its actual navigation around the poles.

[0026] As the inspection vehicle 12 is navigated in the region of the poles 10, it performs radar cross-section measurements, described in detail below, collecting signal data that are then provided to the analysis system 120. Such signal data may be provided via the satellite link 140, although alternative methods for providing such data are within the scope of the invention, some of which are described further below. The analysis system 120 uses the received signal data to generate a final report 115, which provides information in summary format identifying the potential anomalies in the poles 10 detected by the system. The final report 115 is communicated back to the dispatcher 100, who may review it and forward it to the client 105 for action, such as making a determination of whether the reported potential anomalies warrant replacement of any of the poles 10.

[0027] While FIG. 1 and the above description depict a single analysis system 120 and a single satellite link 140, it will be understood that the multiple functions performed by these elements of the system may alternatively be performed by equivalent multiple elements without exceeding the scope of the invention.

[0028] An example of the navigation performed by the inspection vehicle 12 is shown in FIG. 2, in which the inspection vehicle 12 is depicted as a helicopter. Part (a) of FIG. 2 shows a top view of one possible navigational path that may be followed as radar measurements are made. In this example, the inspection vehicle 12 follows an inspection path 202 approximately parallel to a locus defined by the pole positions. As shown in the figure, individual poles 10 are separated by approximately 200 feet, a separation that is typical for utility poles, but the invention readily accommodates any pole separation. When the inspection vehicle detects a potential anomaly in one of the poles 10, it may deviate from the inspection path 202 to follow a verification path 204, which may include doubling back around a set of poles 10, ultimately rejoining the original inspection path to proceed to other as-yet-unexamined poles 10. While following the verification path 204, additional radar cross-section measurements are performed from different orientations with respect to an individual pole, thereby providing supplementary data from which a more accurate characterization of the potential anomaly can be made. In certain instances, the verification path 204 includes a change in relative height of the navigation vehicle 12, as may be appropriate in obtaining supplementary data used to characterize crossarms on the pole 10.

[0029] The orientation of the navigation vehicle 12 with respect to an individual pole 10 as it moves along the inspection path 202 is shown schematically in FIG. 2(b). The various distances in the arrangement are intended to be exemplary since other orientations may be used as appropriate to obtain supplementary data. In the illustrated orientation, with a pole having a height h_pof approximately 60 feet, the inspection vehicle 12 may be positioned at a height of about 500 feet. At such a height, with a distance from the pole 10 of about 100 feet, the slant range r_sbetween the inspection vehicle 12 and the pole 10 may be kept between about 135 and 550 feet, with a depression angle θ between about 30° and 80°. The underground portion 10′ of the pole 10 is preferably examined with radar signals that propagate through the wood of the pole 10 without propagating through the ground itself. Identification of anomalies with such an arrangement requires no correction for the variety of electromagnetic speeds that may exist in the ground, depending specifically on the composition of the ground where the pole 10 is located.

[0030] An example of a radar workstation that may be configured within an inspection vehicle 12 is shown in FIG. 3. The particular configuration illustrated is appropriate, for example, for a helicopter such as an MD Explorer 902 or a Bell Textron 212, 412, or 427 helicopter. The forward compartment of the vehicle includes seat positions 322 and 324 for a pilot and copilot, who navigate the inspection vehicle 12 along the inspection and verification paths 202 and 204. Such navigation is performed in accordance with instructions from an inspection technician 146 (not shown in FIG. 3) occupying seat 320 in a passenger compartment 302 of the inspection vehicle 12. The pilot, copilot, and inspection technician may constitute the inspection crew 145. The inspection technician 146 is equipped with an inspection station 304 from which he monitors results of the inspection on output interaction devices 306, 308, and 310, shown in the exemplary embodiment as computer screens, and issues instructions through input interaction devices 314 and 315, shown in the exemplary embodiment as a keyboard and mouse.

[0031] The interior of the passenger compartment is additionally equipped with various analytical devices and instruments, which may be positioned in locations designated generally by reference numeral 312. The illustration shows one arrangement that may be used for including six individual pieces of equipment. For particular applications, various components may be substituted and the configuration changed. In one embodiment, the equipment includes the following, the operational interconnection of which is shown in FIG. 4: (1) a radar cross-section subsystem 410 including an antenna and associated hardware for propagating and receiving radar signals; (2) a laser pointing and measuring subsystem 430; (3) a differential global-positioning subsystem (GPS) 440; (4) one or more central processing units (CPUs) 450 for executing software as necessary to operate the various subsystems in combination; (5) a target recognition subsystem 460; and (6) a data-storage subsystem 470 for storing relevant data as needed to operate the various subsystems in combination. In addition, the inspection vehicle 12 may include peripheral components used to insure proper and adequate functioning of the equipment 312. Such peripheral components may include, among others, stabilization platforms for the radar subsystem antenna and laser, power conversion transformers to convert from direct to alternating current (e.g., 24 V dc to 120 V ac), and battery backups as needed.

[0032] The interconnection of these various subsystems is shown in block-diagram form in FIG. 4. The figure is divided into three primary sets: analysis elements 425, subsystem elements 465, and analysis functions 495. The analysis elements include the inspection technician 146, the Knowledgeable Observation Analysis-Linked Advisory System (KOALAS) 480, and a remote database 122 accessible by the fixed remote analysis system 120. Three functions performed as the inspection vehicle 12 follows the navigation path 202 in the illustrated embodiment include: (1) focussing 492 relevant subsystem elements on the individual structures to be analyzed; (2) extracting 494 features from each of the relevant structures; and (3) correlating 496 the location of those features both relatively with respect to the structure and absolutely with respect to ground position.

[0033] The inspection technician 146 interacts with the KOALAS system 480 through the CPU subsystem(s) 450 to control the subsystems 465 to perform such functions. For example, during a passage of the inspection vehicle 12 along the navigation path 202, the KOALAS system 480 activates the target recognition subsystem 460, which may consist of a gimbal-mounted stabilized CCD color camera or a 3-5 μm infrared thermal camera, for example. The target recognition system is configured to identify the wooden structures, i.e. poles, to be analyzed. Information provided by the target recognition subsystem is used to steer the radar antenna in the appropriate direction and to capture images of the pole structure for later use in data analysis.

[0034] Information detected by the target recognition system 460 is relayed back to the KOALAS system 480, where it can be accessed by the inspection technician 146 for modification as may be necessary. The KOALAS system 480, in conjunction with the inspection technician 146 uses that information to steer the radar antenna in a direction towards the pole. At the same time, the laser subsystem 430 is used to reflect coherent light off the structure to provide pertinent feedback data. The feedback data are used to provide the physical dimensions of each pole structure, including any cross arms that it may possess. Reflected laser light is also used to determine offset height and distances of each structure for calculation feedback to the differential GPS subsystem 440. The latitude and longitude positions of the inspection vehicle 12 are known from the GPS subsystem 440. With the height and distance information for each of the poles provided by the laser subsystem 430, the KOALAS system 480 performs the step of correlating positions 496 and thereby calculates the latitude and longitude positions of each pole studied for unique identification of those poles in the final report 115.

[0035] In order to identify anomalies in the poles, two radar schemes may be used in conjunction. In one embodiment, both radar schemes are contained in the radar subsystem 410. For example, FIG. 4 shows a pulse compression radar subsystem 420 as an active component of the radar subsystem 410. Generally, the radar cross-section subsystem 410 uses a technique in which interferometric techniques are applied to account for the motion of the inspection vehicle 12 along the navigation path 202, thereby also increasing the effective spatial resolution of the system. In combination with measurements of reflected coherent light by the laser subsystem 430, the step of focussing on an individual pole 492 is accomplished. The pulse compression radar subsystem 420 uses a technique in which short radar pulses are modulated by long ones, thereby permitting improved range resolution by removing frequency and phase modulations. The information thus obtained is used to perform the step of extracting features that describe the condition of the pole, including identification of possible anomalies. The physical arrangement of the various signals that are used may be understood more clearly with reference to FIGS. 5(a) and 5(b), which shows a top view and a side view respectively. As the inspection vehicle 12 moves along navigation path 202, signals are transmitted from a rotatable nose mount 530. For example, the radar cross-section signal 510 (long-dashed— ) is transmitted continuously as the inspection vehicle 12 follows the navigation path 202. The radar cross-section subsystem 410 sends out a broad band pulse that is compressed upon reception. This signal is also integrated into the subsequent analysis. Thus, in FIG. 5(a), the pulse compression radar signal 515 is shown as a short-dashed line (-), its frequency differing from the broad band pulse frequency. The target recognition signal 520, which may be an infrared signal, is shown as a dotted line (^{. . . . .}). Finally, the distance to a pole, its physical dimensions, and overall shape are determined by reflection off the pole of a coherent laser signal 525, which is shown as a solid line (—).

[0036] After the data have been captured and stored onboard in the data storage system 470, a preliminary data reduction may be used to filter noise and thereby control the amount of data captured. Without such preliminary data reduction, approximately one terabyte of information is collected during a typical six-hour inspection day. The filtered data are transmitted to the analysis system 120 for more complete processing. Such data may be provided to the analysis system 120 in different ways. In one embodiment, data is written to a magnetic or optical recording medium, such as a CD or tape, and is physically transported to the analysis system 120. In another embodiment, the satellite link 140 (shown in FIG. 1) is used to transmit data. In one embodiment, the on-board KOALAS system 480 includes sufficient software to make a preliminary estimate of the structural integrity of the pole. Such software is a subset of the software described below used by the fixed remote analysis system 120 in its more detailed analysis, but permits an immediate evaluation of whether there is a strong likelihood that the pole is in catastrophic condition and in imminent danger of falling over. The results of such a preliminary analysis may be provided to the inspection technician 146 in the form of a red or green light, for example. Under such circumstances, the inspection technician 146 may make a determination of whether the entire structure was captured for analysis or whether a return inspection may be necessary.

[0037] In one embodiment, the following information is provided for use by the inspection technician 146 on the output interaction devices 306, 308, and 310. On a first of the devices 310 is displayed a moving map, indicating the present position of the inspection vehicle 12. On a second of the devices 306 is displayed identification information for the particular pole then under study. Such information includes its physical dimensions, latitude and longitude positions, and any identifying number assigned to it by the client. On a third of the devices 308 is displayed preliminary results of the radar analysis, permitting the inspection technician to decide whether to make additional measurements from different angular positions or to proceed along the navigation path.

[0038] The radar cross-section measurement system makes use of interferometric analyses to improve resolution, with measurements being taken at intervals as the inspection vehicle 12 moves along the navigation path 202. The resolution characteristics of the combined broad band and pulse compression modes may be understood by considering an analysis using a cylindrical coordinate system such as shown in FIG. 6. A cylindrical coordinate system naturally matches side-looking radar operations. In the figure, the inspection vehicle 12 at point V makes a measurement of a radar signal reflected from point P. The position of P is defined by the coordinates (r_s, φ, z), with r_sbeing the slant range, φ being the look angle, and z being the azimuth (i.e. the distance between navigation path 202 and the nadir 203 projected on the surface of the earth).

[0039] For a radar system transmitting electromagnetic pulses of time duration τ, the sensor range resolution is Δr=cτ/2≈c/2Δ𝑓, where the time duration τ is approximately the inverse of the pulse bandwidth Δ𝑓. Use of a pulse compression ranging mode permits improved resolution by using modulated pulses. Thus, in one embodiment a chirp pulse is used, although other modulations may be used in alternative embodiments. The time dependence of the chirp pulse includes modulation with a rectangular pulse rect[t/τ] of duration τ:

[0040] A_i(t)=e^i(ωt+αt^/2)rect(t/τ),

[0041] where ω=2π𝑓 is the angular frequency associated with carrier frequency 𝑓 and α is the chirp rate related to the pulse bandwidth by ατ=2πΔ𝑓. Without loss of generality, in the cylindrical coordinate system with a radar platform moving along the navigation path 202 and localized at z=0, and a target lying at P (r_s, φ, 0) in the plane orthogonal to navigation path 202, the backscattered signal may be expressed as

A = \exp [- \frac{ⅈ}{2} (\frac{2 r}{c} + {α (t - \frac{2 r}{c})}^{2})] rect [\frac{t - 2 r / c}{τ}] .

[0042] Processing of the received waveform is performed by convolution with a range reference function

g (ct / 2) = \exp [- \frac{ⅈ {ατ}^{2}}{2} {(\frac{ct}{2})}^{2}] rect [\frac{ct}{2}],

[0043] resulting in

\hat{A} (ct / 2) = ⅇ^{- ⅈ (2 ω r / c)} rect [\frac{ct / 2 - r}{c τ}] \frac{\sin [\frac{ατ}{c} (ct / 2 - r) (1 - | \frac{2}{c τ} (ct / 2 - r) |)]}{\frac{α τ}{c} (ct / 2 - r)},

[0044] which for ct/2−r<<cτ can be written

\hat{A} (ct / 2) = ⅇ^{- 4 π i / λ} \sin c (\frac{π (ct / 2 - r)}{Δ r}),

[0045] where λ is the wavelength associated with the carrier frequency. For a continuous distribution of scatterers described by a reflectivity pattern γ(r), the received pattern is given by

\hat{γ} (ct / 2) = \int ⅆ r γ (r) \hat{A} (ct / 2 - r) = \int ⅆ r γ (r) ⅇ^{- 4 π r / λ} \sin c (\frac{π (ct / 2 - r)}{Δ r}),

[0046] with resolution of two points r₁and r₂being possible for |r₁−r₂|≧Δr.

[0047] The azimuthal resolution is dictated by the constraint that two points at a given range not be within the radar beam at the same time. Accordingly, the azimuthal resolution ΔZ is related to the radar beamwidth by the relation Δz≈rλ/L, where r is the slant range and L is the effective antenna dimension along the azimuthal direction, i.e. along the navigation direction 202 in the configuration illustrated in FIG. 6. In the radar cross-section measurement system used in embodiments of the invention, the effective antenna dimension is increased by the motion of the inspection vehicle 12 and by coherently combining the backscattered echoes received and recorded along the navigation path 202.

[0048] Thus, for 2N+1 equally spaced positions of the antenna, located at V_n(x_n=nd, r=0), and a point target P (r_s,φ, 0), and isotropic radiation by the antenna within its beam width to provide an illuminated patch X=λr/L, the azimuthal-dependent part of the backscattered signal is given by

[0049] A(nd)=e⁻ⁱ(2π/λr)(nd).

[0050] In deriving this result, the expression

\sqrt{r^{2} + {(nd)}^{2}} \approx r + {(nd)}^{2} / 2 r

[0051] has been used. The received signal is processed by summing over all antenna positions and convolving with the azimuthal reference function

[0052] g(nd)≈eⁱ(2π/λr)(nd)

[0053] to give

\hat{A} (nd) = \sum_{k = - N}^{N} ⅇ^{- ⅈ (2 π d^{2} / λ r) k^{2}} ⅇ^{ⅈ (2 π d^{2} / λ r) {(n - k)}^{2}} \approx \frac{\sin (\frac{2 π Xd}{λ r} n)}{\sin (\frac{2 π d^{2}}{λ r} n)} (nd ⪡ X) .

[0054] As for the range results, the image of a point target is spread out. In the neighborhood of the target position at z=0,

[0055] Â(nd)≈sinc(πnd/Δz),

[0056] so that a distributed target is accounted for by superposition according to the reflectivity pattern in the azimuthal direction γ(z):

\hat{γ} (nd) = \int ⅆ z γ (z) \hat{A} (nd - z) = \int ⅆ z γ (z) \sin c [π (nd - z) / Δ z],

[0057] where the azimuthal resolution is Δz=L/2. Because the spatial bandwidth of estimated reflectivity γ is determined by the sinc function to be 1/Δz, the processed signal for any position along the navigation path 202 is determined by sampling interpolation:

\hat{γ} (z) = \sum γ (nd) \sin c [π (z - nd) / Δ z] = \int ⅆ z^{'} γ (z^{'}) \sin c [π (z - z^{'}) / Δ z] .

[0058] Combining the range and azimuthal results provides the following overall image expression for radar signals reflected off an object having a two-dimensional reflectivity pattern γ (z,r):

\hat{γ} (z, r) = ⅇ^{- ⅈ (2 ω r / c)} \int \int ⅆ z^{'} ⅆ r^{'} γ (z, r) \sin c [π (z - z^{'}) / Δ z] \sin c [π (r - r^{'}) / Δ r] .

[0059] As noted, such analysis then provides a resolution capability for the radar system of Δr in the range direction and of Δz in the azimuthal direction.

[0060] A detailed overview of the analytical processing of the collected data is provided in FIG. 7. The various forms of data collected with the investigation vehicle 12, including radar cross-section measurement data 710, pulse compression radar data 720, laser data 730, target recognition data 730, and perhaps others, are subjected to a signal processing step 740 such as described in detail above. The various data are subject to data fusion 742, which is a method for combining results to increase the confidence level of the results presented in the final report 115. As an illustration, when the system makes a determination that characterizes a feature in a pole as an anomaly, the reliability of the determination is increased by calculating the product of probabilities from different sources. For example if p_kis the probability that a given feature is an anomaly based on results from technique k, then the probability P that the feature is anomalous based on the use of multiple techniques is

[0061] P=1−Π(1−p_k)

[0062] The assignments of whether features in the structures are anomalous is performed in various embodiments at step 744 with an evaluation system that has been trained to discriminate between normal and anomalous structure according to the results of the measurements, such as with an expert system or neural network configuration. Such an evaluation system may rely on knowledge of the characteristics expected in normal or anomalous wood as stored in database 122, the generation of which is further described below. For example, the expert system will have stored the density characteristics that define whether features are normal or anomalous and will have stored radar scattering signals that correspond to such densities.

[0063] The radar signals reflected from a particular pole 10 may be analyzed to identify, for example, density characteristics of the pole. Information characterizing the interior structure of the wood may be obtained by using radar signals having a frequency that penetrates the wood, for example in the range between 360 MHz and 2 GHz. In addition, phase shifts resulting from the different refractive effects of the wood and air through which the radar signals are propagated permits resolution of features and their positions within the wood. The analysis is illustrated in FIG. 8, in which the pole 10 is shown schematically as including a pole surface 782 and a series of planes 786 throughout the pole. The pole is subdivided into a plurality of individual cells 788 for individual characterization.

[0064] As the inspection vehicle 12 (not shown in FIG. 8) moves along navigation path 202, the radar signal for a particular cell 788-1 is first sensed at time t₁, and last sensed at time t₂. There are two focussing functions that are used to define the particular cell 788-1. First, a planar coordinate position, shown in Cartesian coordinates (x, y) in the figure for convenience, causes a first phase shift Δψ₁that varies with the motion of the inspection vehicle 12. Second, refractive effects associated with the depth of the particular cell 788-1 within the pole 10 cause a second phase shift Δψ₂, which remains uniform with the motion of the inspection vehicle 12. Focussing for the planar coordinate position is a straightforward phase correction that accounts for path length differences between a current position of the inspection vehicle 12 and the closest approach of the inspection vehicle 12. Depth focussing is accomplished by using a wood refractive propagation model to determine the effective path length difference between the pole surface 782 and the particular plane 786 within the pole, including the propagation delay that results from the different index of refraction in wood.

[0065] Such analysis can thus provide, for example, a relative measure of the density distribution of the wood throughout the pole. By recognizing in particular closed volumes within the pole where the density is consistently less than the mean density of the pole, structural anomalies are identified, together with their location. Since the comparison is of a relative measure of the density, the method can function independent of knowing precisely what type of wood the pole is made of.

[0066] The process of drawing these conclusions by performing the model comparison 744 is essentially a pattern-recognition algorithm being conducted by the trained evaluation system. In any specific implementation of such a pattern-recognition algorithm, it is beneficial to ensure that the trained evaluation system is making reliable determinations. This may be done by preliminary training of the evaluation system with an appropriate set of certifiable data that accounts for relevant factors in making the determinations, which is then encoded before the system is used to evaluate real data. For example, measurements may be performed on a number of wooden poles, some of which are known to contain anomalies. Based on the identification of these anomalies, this information is used to train the evaluation system's pattern recognition algorithm. In particular, the preliminary training may include a pole strength assessment (perhaps expressed as a percentage probability that the pole will fail within a certain time as a result of the anomaly) determined from a complete analysis of the pole external from the radar measurements.

[0067] Using artificial-intelligence techniques, the results of subsequent tests are used continually to perform refinement of the model used in making the structural determinations (step 748). For example, in one embodiment, a neural net is used to make the structural determinations. A typical neural network includes a plurality of nodes, each of which has a weight value associated with it. The network includes an input layer having a plurality of input nodes and an output layer having a plurality of output nodes, with at least one layer therebetween. In this example, the input nodes receive the data provided by the various sensor measurements and the output nodes generate an interpretation designation. The interpretation designation may be a simple binary indication, such as described above, that a given pole is imminently likely to collapse or not. Alternatively, the interpretation designation may be a numerical percentage reflecting the pole strength assessment. In other words, given an input comprising the sensor measurements, the input is combined (added, multiplied, subtracted, etc. in a variety of combinations and iterations depending upon how the neural network is initially organized), and then the interpretation is generated accordingly.

[0068] In order to train the neural net, the output values are compared against the correct interpretation with some known samples. If the output value is incorrect when compared against such a test interpretation, the neural net modifies itself to arrive at the correct output value. This is achieved by connecting or disconnecting certain nodes and/or adjusting the weight values of the nodes during the training through a plurality of iterations. Once the training is completed, the resulting layer/node configuration and corresponding weights represents a trained neural net. The trained neural net is then ready to receive unknown sensor data and designate certain pole regions as containing anomalies. Classical neural nets include Kohonen nets, feed-forward nets, and back-propagation nets. The different neural nets have different methods of adjusting the weights and organizing the respective neural net during the training process.

[0069] The analysis system may make use of other methods for making wood-anomaly assignments on the basis of the sensor data. Such methods may be broadly categorized as falling into one of two classes. In the first class, the method begins with an initial approximation that is progressively improved using comparison feedback (step 746). For example, for a given pole, the analysis system begins with an initial structural estimate for the pole. The sensor data that would result from a pole with those precise characteristics is calculated and compared with the actual sensor data. From such a comparison, the estimated structural characteristics for the pole are refined. The process proceeds interatively, with the estimated pole structure being modified at each step to reproduce the measured sensor data more closely. When the difference between the measured sensor data and the calculated sensor data is less than a predetermined threshold, the process is deemed to have converged and the final report 115 is issued.

[0070] In the second class of methods, the system is permitted to vary essentially randomly and individual pole-characteristic representations that develop during the process are evaluated to determine which best reproduces the measured sensor data. One example of such a method is a genetic algorithm. The genetic algorithm is a model of machine learning that derives its behavior in an attempt to mimic evolution in nature. This is done by generating a population of “individuals,” i.e. pole-characteristic representations, represented by “chromosomes,” in essence a set of character strings that are analogous to the base-four chromosomes of DNA. The individuals in the population then go through a process simulated “evolution.” The genetic algorithm is widely used in optimization problems in which the character string of the chromosome can be used to encode the values for the different parameters being optimized. In practice, therefore, an array of bits or characters to represent the chromosomes, in this case the position and sizes of wood anomalies in a pole, is provided; then, bit manipulation operations allow the implementation of crossover, mutation, and other operations.

[0071] When the genetic algorithm is implemented, it is trained in a manner that may involve the following cycle: the fitness of all individuals in the population is first evaluated; then, a new population is created by performing operations such as crossover, fitness-proportionate reproduction, and mutation on the individuals whose fitness has just been measured; finally, the old population is discarded and iteration is performed with the new population. One iteration of this loop is referred to as a generation. According to embodiments of the present invention, a number of randomly generated wooden poles with various anomalies may be used as the initial input. This population of poles is then permitted to evolve as described above, with each individual pole being tested at each generation to see whether it can adequately reproduce the measured sensor data.

[0072] Still further methods that may occur to those of skill in the art, involving such techniques as simulated annealing or various fuzzy logic systems, may be used alternatively or supplementally to perform the analysis of the measured sensor data to generate the final report 115. An example of the format of the final report 115 and the type of information included on it is shown in FIG. 9. The final report 115 may comprise of a plurality of such depictions as shown in FIG. 9, one for each pole examined, and may include information in more summary form such as in a table.

[0073] In the report format shown in FIG. 9, preliminary information is used to identify the name of a client 802 who commissioned the investigation of the wooden structures and the date 804 the inspection was performed. Specific information identifying the individual pole for the report may take the form of providing a pole number 806 and line name 808; in this example the report is for pole 17 of 354 poles on the Rio Osa—Table Mountain 69 kV line. The specific location 810 of the pole, determined as described above, is provided in a format specifying longitude and latitude to facilitate identification of the pole should remedial measures be warranted and/or desired. The report includes information reporting the results of the analysis. Such information may be in the form of a graphic 814 showing the general size and shape of the pole, with an indication of where detected anomalies lie. It may also include a textual description 812 of the location of the anomalies, using ground level and the pole centerline as reference points. The report may further include a quantitative evaluation 816 of the effect on pole strength caused by the various detected anomalies. The identity of the inspection technician may also be included.

[0074] In the example shown, the system has detected three anomalies in the pole, which has a single crossarm. The first anomaly is approximately at ground level in the vertical pole having a size of about 1.3098 ft³. Based on the analysis system, using a trained evaluation system such as an expert system or neural network, this anomaly is estimated to reduce the strength of the pole by 35.7% from its strength without the anomaly. The second anomaly is larger and located about 50 ft above ground level and the third anomaly, which is smaller, is located in the crossarm. The estimated effect of each of these anomalies on the strength of the pole is included in the report. This information may then be used by the client to decide whether to take corrective action based on its own criteria, such as to replace any wooden structure suffering from a strength reduction greater than 40%.

[0075] Having described several embodiments, it will be recognized by those of skill in the art that various modifications, alternative constructions, and equivalents may be used without departing from the spirit of the invention. Accordingly, the above description should not be taken as limiting the scope of the invention, which is defined in the following claims.

Claims

What is claimed is:

1. A method for analyzing a wooden structure, the method comprising:

identifying a location for the wooden structure;

propagating a first radar signal towards the wooden structure with a radar antenna while the radar antenna is in motion along a navigation path in the vicinity of the wooden structure;

receiving a reflected radar signal from the wooden structure; and

determining whether the wooden structure contains a structural anomaly from the reflected radar signal.

2. The method according to claim 1 wherein the first radar signal has a frequency between 360 MHz and 2 GHz.

3. The method according to claim 1 wherein identifying the location for the wooden structure comprises imaging the wooden structure.

4. The method according to claim 3 wherein imaging the wooden structure is performed with a charge coupled device.

5. The method according to claim 3 wherein imaging the wooden structure is performed at infrared electromagnetic wavelengths.

6. The method according to claim 3 wherein propagating a radar signal towards the wooden structure with the radar antenna comprises steering the radar antenna in a direction determined by imaging the wooden structure.

7. The method according to claim 1 wherein identifying the location for the wooden structure comprises ascertaining longitude and latitude positions for the wooden structure with a global positioning subsystem (GPS).

8. The method according to claim 1 wherein identifying the location for the wooden structure comprises reflecting a laser signal from the wooden structure.

9. The method according to claim 8 further comprising ascertaining a shape and physical dimensions of the wooden structure with the reflected laser signal.

10. The method according to claim 1 further comprising propagating a second radar signal towards the wooden structure, wherein such second radar signal is modulated in accordance with a pulse compression scheme and wherein the reflected radar signal includes signal components originating from both the first and second radar signals.

11. The method according to claim 10 wherein the second radar signal has a different frequency than the first radar signal.

12. The method according to claim 1 wherein determining whether the wooden structure contains a structural anomaly comprises:

calculating a density distribution for the wooden structure by manipulating data extracted from the reflected radar signal; and

designating closed-volume regions within the wooden structure having a density less than a threshold density relative to a mean density for the wooden structure.

13. The method according to claim 1 wherein determining whether the wooden structure contains a structural anomaly is performed by a trained evaluation system.

14. The method according to claim 13 wherein the trained evaluation system comprises a neural net.

15. The method according to claim 13 wherein the trained evaluation system comprises an expert system.

16. The method according to claim 1 further comprising propagating the first radar signal towards the wooden structure with the radar antenna while the radar antenna is in motion along a verification path in the vicinity of the wooden structure, in response to a preliminary determination from the reflected radar signal that the wooden structure contains a structural anomaly, wherein such verification path differs from such navigation path.

17. The method according to claim 1 wherein determining whether the wooden structure contains a structural anomaly comprising ascertaining a location of any such structural anomaly within the wooden structure.

18. The method according to claim 17 further comprising generating a report indicating the position of any such structural anomaly within the wooden structure.

19. The method according to claim 1 wherein the wooden structure is a wooden pole.

20. The method according to claim 19 wherein the wooden pole is a utility or telecommunications pole.

21. A method for analyzing a wooden pole, the method comprising:

imaging the wooden pole to determine a location for the pole;

steering a radar antenna in a direction towards the determined location of the wooden pole;

propagating a first radar signal towards the wooden pole with the radar antenna while the radar antenna is in motion along a navigation path in the vicinity of the pole, wherein the first radar signal has a frequency between 360 MHz and 2 GHz;

receiving a reflected radar signal from the wooden pole;

calculating a density distribution for the wooden pole from data provided by the reflected radar signal; and

determining whether the wooden pole contains a structural anomaly based on the calculated density distribution.

22. The method according to claim 21 wherein imaging the wooden pole is performed at infrared electromagnetic wavelengths.

23. The method according to claim 21 wherein imaging the wooden pole is performed with a charge coupled device.

24. The method according to claim 21 further comprising ascertaining longitude and latitude positions for the wooden pole with a global positioning system (GPS).

25. The method according to claim 21 further comprising reflecting a laser signal from the wooden pole to ascertain a shape and physical dimensions of the wooden pole.

26. The method according to claim 21 further comprising propagating a second radar signal towards the wooden pole, wherein such second radar signal has a different frequency than the first radar signal and is modulated in accordance with a pulse compression scheme and wherein the reflected radar signal includes signal components originating from both the first and second radar signals.

27. The method according to claim 21 further comprising propagating the first radar signal towards the wooden pole with the radar antenna while the radar antenna is in motion along a verification path in the vicinity of the wooden structure, in response to a preliminary determination from the reflected radar signal that the wooden pole contains a structural anomaly, wherein such verification path differs from such navigation path.

28. A method for analyzing a wooden pole, the method comprising:

imaging the wooden pole to determine a location for the wooden pole;

reflecting a laser signal from the wooden pole to ascertain a shape and physical dimensions of the wooden pole;

ascertaining longitude and latitude positions for the wooden pole with a global positioning system (GPS);

steering a radar antenna in a direction towards the determined location of the wooden pole;

propagating a second radar signal towards the wooden pole with the radar antenna while the radar antenna is in motion along the navigation path, wherein the second radar signal is modulated in accordance with a pulse compression scheme and has a different frequency than the first radar signal;

receiving a reflected radar signal from the wooden pole;

calculating a density distribution for the wooden pole from data provided by the reflected radar signal; and

determining whether the wooden pole contains a structural anomaly based on the calculated density distribution.

29. A system for analyzing a wooden structure, the system comprising:

a target-recognition device configurable for connection with a vehicle;

a rotatable radar antenna configurable for connection with the vehicle, the rotatable radar antenna being adaptable to emit and receive electromagnetic signals;

an arrangement of at least one computer system in communication with the target-recognition device and rotatable radar antenna and configured to accept instructions from an operator and to operate the target-recognition device and rotatable radar antenna in accordance with the following:

identifying a location for the wooden structure with the target recognition device;

propagating a first radar signal towards the wooden structure with the radar antenna while the radar antenna is in motion along a navigation path in the vicinity of the wooden structure;

receiving a reflected radar signal from the wooden structure with the radar antenna.

30. The system according to claim 29 wherein the target-recognition device comprises an imaging device.

31. The system according to claim 30 wherein the target-recognition device comprises a charge coupled device.

32. The system according to claim 30 wherein the target-recognition device comprises an infrared camera.

33. The system according to claim 30 wherein the arrangement of at least one computer system is configured such that propagating the first radar signal towards the wooden structure with the radar antenna comprises steering the radar antenna in a direction determined by imaging the wooden structure with the target-recognition device.

34. The system according to claim 29 further comprising a global positioning subsystem (GPS) in communication with the arrangement of at least one computer system and configurable for connection with the vehicle, and

wherein the arrangement of at least one computer system is configured such that identifying the location for the wooden structure comprises ascertaining longitude and latitude positions for the wooden structure with the GPS.

35. The system according to claim 29 further comprising a laser subsystem in communication with the arrangement of at least one computer system and configurable for connection with the vehicle, and

wherein the arrangement of at least one computer system is configured such that identifying the location for the wooden structure comprises reflecting a laser signal from the wooden structure with the laser subsystem.

36. The system according to claim 29 wherein the arrangement of at least one computer system is further configured in accordance with propagating a second radar signal towards the wooden structure with the radar antenna, wherein such second radar signal is modulated in accordance with a pulse compression scheme and wherein the reflected radar signal includes signal components originating from both the first and second radar signals.

37. The system according to claim 36 wherein the second radar signal has a different frequency than the first radar signal.

38. The system according to claim 29 wherein the arrangement of at least one computer system is further configured for determining whether the wooden structure contains a structural anomaly from data provided by the reflected radar signal.

39. The system according to claim 38 wherein the arrangement of at least one computer system is configured such that determining whether the wooden structure contains a structural anomaly comprises:

calculating a density distribution for the wooden structure by manipulating data extracted from the reflected radar signal; and

designating closed-volume regions within the wooden structure having a density less than a threshold density relative to a mean density for the wooden structure.

40. The system according to claim 38 wherein the arrangement of at least one computer system includes a trained evaluation system configured for determining whether the wooden structure contains a structural anomaly.

41. The system according to claim 40 wherein the trained evaluation system comprises a neural net.

42. The system according to claim 40 wherein the trained evaluation system comprises an expert system.

43. The system according to claim 38 wherein the arrangement of at least one computer system is configured such that determining whether the wooden structure contains a structural anomaly comprises ascertaining a location of any such anomaly within the wooden structure.

44. A system for analyzing a wooden pole, the system comprising:

an imaging device configurable for connection with a vehicle;

a laser subsystem configurable for connection with the vehicle;

a global positioning system (GPS) configurable for connection with the vehicle;

a rotatable radar antenna configurable for connection with the vehicle, the rotatable radar antenna being adapted to emit and receive electromagnetic signals; and

an arrangement of at least one computer system in communication with the imaging device, laser subsystem, GPS, and rotatable radar antenna and configured to accept instructions from an operator and to operate the imaging device, laser subsystem, GPS, and rotatable radar antenna in accordance with the following:

imaging the wooden pole with the imaging device to determine a location for the wooden pole;

reflecting a laser signal from the wooden pole to ascertain a shape and physical dimensions of the wooden pole;

ascertaining longitude and latitude positions for the wooden pole with the GPS;

steering the radar antenna in a direction towards the determined location of the wooden pole;

propagating a first radar signal towards the wooden pole with the radar antenna while the radar antenna is in motion along a navigation path in the vicinity of the wooden pole;

receiving a reflected radar signal from the wooded pole with the radar antenna.

45. The system according to claim 44 wherein the arrangement of at least one computer system is further configured for determining whether the wooden pole contains a structural anomaly from data provided by the reflected radar signal.

46. The system according to claim 45 wherein determining whether the wooden pole contains a structural anomaly comprises:

calculating a density distribution for the wooden structure by manipulating data extracted from the reflected radar signal; and

designating closed-volume regions within the wooden pole having a density less than a threshold density relative to a mean density for the wooden pole.

47. The system according to claim 45 wherein the arrangement of at least one computer system includes a neural net configured for determining whether the wooden pole contains a structural anomaly.

Patent Citations (4)

Patent	Date	Inventor	Cited By
US5512834	1996-04-01	McEwan	Examiner
US5883591	1999-03-01	McEwan	Examiner
US5905455	1999-05-01	Heger et al.	Examiner
US6091354	2000-07-01	Beckner et al.	Applicant

Non-Patent Literature (22)

PoleCalc, Pamphlet, 1995, New York State Electric & Gas Corporation, Binghamton, NY.Applicant
Gouri S. Bhuyan, (from book entitled) 1998 Ieee, Esmo by IEEE Power Engineering Society, Condition Based Serviceability and Reliability Assessment of Wood Pole Structures, pp. 333-339.Applicant
Nelson G. Bingel, III et al., (from book entitled) Ieee Esmo by IEEE Power Engineering Society, Utility Pole Computerized Maintenance Management Systems, pp. 340-345.Applicant
R. Jerry Jost, Airborne Underground Imaging-Radar System Definition Final Report, pp. 1-5, System Planning Corporation, Arlington VA 22207; Feb. 20, 1995.Applicant
S.I. Tsunoda et al., Lynx: A high-resolution synthetic aperture radar, pp. 1-8, SPIE Aerosense 1999, vol. 3704.Applicant
Bob Butera, Welcome to the Osmose Utilities Division, Osmose, Inc., Buffalo, NY, www.osmos.com/utilities/Utilities.htm, no date.Applicant
Providing wood pole remedial treatment in partnership with Samtech Inc., PoleCare International Inc., www.guelphpole.com/polecare.html, no date.Applicant
Introduction, Guelph Utility Pole Company Limited, www.guelphpole.com/, no date.Applicant
Wood Testing, NDT Equipment Sales, 1998, www.ndt.com.au/wood.htm.Applicant
Monitoring Wooden Poles, EDF, 1999, www.edf.fr/der/html/produits/services/diagnostic/surveil-poteaux.en.htm.Applicant
Control and Measuring Instruments for Trees, Timber Structures and Utility Poles, IML, 1998, www.imlusa.com/pcare.html.Applicant
Complete Pole Network Asset Management, Deuar Pty Ltd., 1999, http://finddit.cowleys.com.au/clients/deuar.htm.Applicant
GPR for Structural Assessment, www.sensoft.on.ca/struc.htm.Applicant
SAR Theory/Interpreting Images, Alaska SAR Facility's Homepage, 1995, www.glue.umd.edu/-gsun/radar/sar-theory.html.Applicant
SAR FAQ—Frequently Asked Questions, 1998, www.asf.alaska.edu/user-serv/sar-faq.html.Applicant
An economic extension to every company's resources, Activities at ERA, www.era.co.uk/act/act.htm, no date.Applicant
Passive millimetre-wave imaging, DERA, 1999, ww.dra.hmg.gb/html/products/electron/esrdpmwi.htm.Applicant
Microradar Sensors for the Next Millennium, McEwan Technologies, LLC, 1999, www.getradar.com/about-us.shtml.Applicant
A digital Ground Penetrating Radar System, RAMAC/GPR 1998-2000, www.teraplus.com/grdetails.htm, no date.Applicant
Sandia and General Atomics Developed New Synthetic-Aperture Radar, SPACEDAILY, 1999, www.spacedaily.com/spacecast/news/radar-99c.html.Applicant
Sandia, General Atomics unveil new fine resolution synthetic-aperture radar system, News Releases 1999, www.sandia.gov/media/NewsRel/NR1999/Lynx.htm.Applicant
Lynx Synthetic Aperture Radar (SAR), General Atomics, www.generalatomics.com/atg/SAR/lynxSAR.html, no date.Applicant

Abstract

Description

Background of the Invention

Summary of the Invention

Brief Description of the Drawings

[0010] FIG. 1 is a block diagram showing the relationship between various elements of the system in one embodiment of the invention;

[0012] FIG. 3 illustrates one configuration that may be used to equip a navigation vehicle to operate in accordance with an embodiment of the invention;

[0013] FIG. 4 is a block diagram showing the relationship between various subsystem elements used in analyzing the wooden structures;

[0014] FIGS. 5(a) and 5(b) are a schematic diagram showing the interaction of various sensor signals with wooden poles:

[0015] FIG. 5(a) shows a top view and

[0016] FIG. 5(b) shows a side view;

[0017] FIG. 6 is a representation of a cylindrical coordinate system;

[0018] FIG. 7 is a block diagram showing the analysis of collected data;

[0019] FIG. 8 is a schematic diagram of the wood-penetrating aspects of the radar analysis; and

[0020] FIG. 9 is an example of a wood inspection report that may be provided in accordance with embodiments of the invention.

Description of the Specific Embodiments

[0040] A_i(t)=e^i(ωt+αt^/2)rect(t/τ),

A = \exp [- \frac{ⅈ}{2} (\frac{2 r}{c} + {α (t - \frac{2 r}{c})}^{2})] rect [\frac{t - 2 r / c}{τ}] .

[0042] Processing of the received waveform is performed by convolution with a range reference function

g (ct / 2) = \exp [- \frac{ⅈ {ατ}^{2}}{2} {(\frac{ct}{2})}^{2}] rect [\frac{ct}{2}],

[0043] resulting in

\hat{A} (ct / 2) = ⅇ^{- ⅈ (2 ω r / c)} rect [\frac{ct / 2 - r}{c τ}] \frac{\sin [\frac{ατ}{c} (ct / 2 - r) (1 - | \frac{2}{c τ} (ct / 2 - r) |)]}{\frac{α τ}{c} (ct / 2 - r)},

[0044] which for ct/2−r<<cτ can be written

\hat{A} (ct / 2) = ⅇ^{- 4 π i / λ} \sin c (\frac{π (ct / 2 - r)}{Δ r}),

[0045] where λ is the wavelength associated with the carrier frequency. For a continuous distribution of scatterers described by a reflectivity pattern γ(r), the received pattern is given by

\hat{γ} (ct / 2) = \int ⅆ r γ (r) \hat{A} (ct / 2 - r) = \int ⅆ r γ (r) ⅇ^{- 4 π r / λ} \sin c (\frac{π (ct / 2 - r)}{Δ r}),

[0046] with resolution of two points r₁and r₂being possible for |r₁−r₂|≧Δr.

[0049] A(nd)=e⁻ⁱ(2π/λr)(nd).

[0050] In deriving this result, the expression

\sqrt{r^{2} + {(nd)}^{2}} \approx r + {(nd)}^{2} / 2 r

[0051] has been used. The received signal is processed by summing over all antenna positions and convolving with the azimuthal reference function

[0052] g(nd)≈eⁱ(2π/λr)(nd)

[0053] to give

\hat{A} (nd) = \sum_{k = - N}^{N} ⅇ^{- ⅈ (2 π d^{2} / λ r) k^{2}} ⅇ^{ⅈ (2 π d^{2} / λ r) {(n - k)}^{2}} \approx \frac{\sin (\frac{2 π Xd}{λ r} n)}{\sin (\frac{2 π d^{2}}{λ r} n)} (nd ⪡ X) .

[0054] As for the range results, the image of a point target is spread out. In the neighborhood of the target position at z=0,

[0055] Â(nd)≈sinc(πnd/Δz),

[0056] so that a distributed target is accounted for by superposition according to the reflectivity pattern in the azimuthal direction γ(z):

\hat{γ} (nd) = \int ⅆ z γ (z) \hat{A} (nd - z) = \int ⅆ z γ (z) \sin c [π (nd - z) / Δ z],

\hat{γ} (z) = \sum γ (nd) \sin c [π (z - nd) / Δ z] = \int ⅆ z^{'} γ (z^{'}) \sin c [π (z - z^{'}) / Δ z] .

[0058] Combining the range and azimuthal results provides the following overall image expression for radar signals reflected off an object having a two-dimensional reflectivity pattern γ (z,r):

\hat{γ} (z, r) = ⅇ^{- ⅈ (2 ω r / c)} \int \int ⅆ z^{'} ⅆ r^{'} γ (z, r) \sin c [π (z - z^{'}) / Δ z] \sin c [π (r - r^{'}) / Δ r] .