TECHNICAL FIELD
The present invention relates generally to computing bias in range, azimuth and elevation in a radar system.

BACKGROUND
In the modern air surveillance scenario many radars cover a common global surveillance region to create a collective picture of the traffic. This collective picture of the surveillance region is presented as global tracks (estimated kinematic parameters of targets) formed using data derived from different sensors at a fusion centre. The raw data provided by a 3D radar (measurement in range, azimuth angle, elevation angle in radar coordinate system) is subject to random noise and bias. Hence in multi-sensor environments, failure in correcting the bias could result in uncorrelated sensor data at fusion centre. Legacy systems use methodologies in which bias are corrected using a pair of radars or multiple radars which could result in relative registration. Another conventional bias correction method is by using dedicated air sorties for radar calibration. This dedicated air sorties will be carrying GPS instrument and the recorded GPS measurements are taken as reference to correct the bias in radar measurements. The introduction of ADSB (Automatic Dependent Surveillance Broadcast) in passenger aircrafts provides a means to obtain the GPS measurements periodically to the ground stations through an ADSB receiver instrument. ADSB is a GPS dependent broadcast system in which aircrafts with ADSB transponder report their position to ground stations capable of decoding it. As GPS positions are usually fairly accurate, ADSB opens the possibility of significant accuracy improvements in the determination of radar measurement bias without any dedicated air sorties.
One of the conventional methods discloses a method and an apparatus for calibrating range in a radar system. The variations in temperature affect the calculated range of a target. These range errors can be corrected by detecting and accurately estimating the frequency deviation error of a radar system. Further, this method relies upon the observation that the doppler range rate is largely unaffected by frequency deviation error, and thus, is approximately equal to the actual range rate. In accordance with a first range calibration technique, the radar system measures the range, doppler range rate, and azimuth angle of a target during at least two successive time instances. If the measured data is qualified the method corrects the range using a frequency deviation correction factor, ‘K’. A second calibration technique relies upon both the observation that the doppler range rate is largely unaffected by frequency deviations and the observation that certain target tracks provide more reliable data than other target tracks. Thus, tracks with more reliable data are given more weight in calibrating the range. This invention is hardware based and focused on correcting the Range errors due to frequency deviations.
Further, another conventional method relates to an alignment in multi-sensor target tracking. It repeatedly generates estimates for sensor bias errors by minimizing a function, given on one hand by the magnitude of the discrepancy between measurements and a measuring model, where the measuring model is a function of the unknown target location and unknown bias parameters, and on the other by the bias parameters and their predetermined statistical distributions. The function minimization is performed by linearized components of the function around an approximate target position (normally obtained from the tracker 10 and around nominal (typically Zero) bias errors, and the function is subsequently minimized with respect to target positions as well as to the bias parameters. The invention is two-step process which first depends on the estimated positions from a tracker and in the second step these corrected positions are used to determine bias errors, which is suitable in a multi-sensor scenario
Further, another conventional method discloses about a radar registration algorithm that uses ADSB as the trusted reference for correcting bias in radars. The ADSB measurements are converted to radar plane using stereographic projection method resulting in computing the bias correction functions in range and azimuth. The bias computation sub-process results in a bias correction solution including range bias, azimuth bias and time related bias parameters. The quality monitoring sub-process results in an estimate of solution quality.
Furthermore, another conventional method discloses a method for registering a radar system. This method includes obtaining first values for a location of a target relative to the radar system using radar system initiated signals, obtaining geo-referenced location data for and from the target, obtaining second values for the location of the target relative to the radar system using the geo-referenced location data, computing location registration bias errors for the radar system using the first and second values, and registering the radar system using the computed location registration bias errors. The invention correlates ADSB measurements and the radar detected measurements only using the mode-S transponder response signal, thus limiting the usage of bias computation methodology to secondary surveillance radars. The bias computation of parameters is realized by averaging the error in difference of transformed geo-referenced data and radar data.
In Y Bar Shalom, X R Li, T Kirubararajan “Estimation with Applications to Tracking and Navigation: Algorithms and Software for Information Extraction”, Artech House, Norwood,2001, the Recursive Weighted Least Squares Estimation. James RClynch is discussed;” Geodetic Coordinate Conversions”,2006 discusses the transformation of Geodetic Coordinates to other coordinate systems. J Sanz Subirana, J M JuanZornoza, M H Pajares; “Transformations between ECEF and ENU coordinates”, 2011 discusses the transformation between Earth Centric Earth Fixed coordinate system and local East North Up coordinate system. Sergio Torres,MikeAbett;”: An accurate and Fast Radar Registration Algorithm”, Integrated Communications, Navigation and Surveillance Conference,2009 discusses a variant of radar bias computation method using ADS-B measurements. J.A. BesadaPortas, J. Garcia Herrero and G. De Miguel Veta, "Radar bias correction based on GPS measurements for ATC applications", IEEE Proc. Radar Sonar Navigation, 2002 discloses a variant of offline computation of Radar bias parameters using GPS measurements.
There is still a need of a better or an alternative invention which solves the above defined problems.
SUMMARY
This summary is provided to introduce concepts related generally to a method for computing bias in range, azimuth and elevation for three-dimensional (3D) radar measurements using Automatic Dependent Surveillance Broadcast (ADSB) measurements obtained from targets of opportunity. It further relates to finding the bias in 3D radars using opportunistic targets without the need of multiple radars and the expensive dedicated air sorties for calibration and thereby reducing the communication needs and the computation complexity. This summary is neither intended to identify essential features of the present invention nor is it intended for use in determining or limiting the scope of the present invention.
In one of the present implementation, the method recognizes the use of highly accurate ADSB data to compute the bias of radars owing to its intrinsic dependency on GPS system for positional data as well as velocity parameters. The method assumes that the radar target and the corresponding ADSB target are identified and made available. In one embodiment, the auto-bias computation process shall be integrated with radar and ADSB target association module and in another embodiment, it shall run in conjunction with the radar and ADSB target association module, wherein both communicates with each other using some communication protocol. The auto-bias computation process stores the corresponding radar and ADSB measurements, for each identified radar, till the target track goes out of the coverage of radar. It includes the manoeuvre detection to filter out straight segments of tracks to ensure better accuracy in providing the time extrapolated radar measurements to allow the comparison of position data. The latitude, the longitude and the geometric height of the target obtained from ADSB data is transformed into radar coordinate system to generate difference measurement in range, azimuth and elevation. The auto-bias computation process uses Least Squares (LS) solution to compute registration errors in range, azimuth, elevation, by minimizing the difference in measurements of radar and transformed ADSB measurements. To achieve enhanced accuracy, it may be desirable to collect large number of radar data as well as ADSB data.
In another implementation, the method computes bias of multiple radars automatically, which employs highly accurate geo referenced positional data as a reference for correcting the registration errors in radar data. The bias in radar data comprises of range bias error, range scale error, azimuth bias error, and elevation error if applicable in case of 3D radars. In one embodiment, the geo-referenced data may be from ADSB receiver and in another embodiment the geo-referenced data may be from another source like GPS receiver. The radars under consideration shall have a common coverage area with the sensor providing geo-referenced data. The auto-bias computation process may be implemented in ATC systems, air surveillance systems, defense systems and command control systems. The system may be of standalone architecture or integrated with other related system at fusion centre.
In yet another implementation, the radar bias parameters are computed for each radar periodically to obtain the corrections corresponding to range, azimuth, and elevation. The period of computation are based on configurable check parameters in conjunction with a timer for each identified radar. The part of the auto-bias computation process can be extended to other radars with limited ADSB targets in its vicinity but with overlapping coverage are with another radar with ADSB vicinity. This can be achieved by reusing the bias computation method, this time using a ADSB calibrated radar as the trusted reference source. Thus the positional accuracy of ADSB calibrated radar can be translated to the accuracy of radar without ADSB coverage. The process also identifies a metric to identify the quality of solution obtained which can be further be processed to identify the source of errors or to take a call on the usage of solution for bias correction.
BRIEF DESCRIPTION OF ACCOMPANYING DRAWINGS
The detailed description is described with reference to the accompanying figures. In the figures, the left-most digit(s) of a reference number identifies the figure in which the reference number first appears. The same numbers are used throughout the drawings for reference such as features and modules.
Fig. 1 illustrates an exemplary block diagram of an auto-bias computation module, according to an exemplary implementation of the present disclosure.
Fig. 2 illustrates a functional diagram of one cycle of operation of the auto-bias computation module, according to an exemplary implementation of the present disclosure.
Fig. 3 illustrates an exemplary functional diagram of a data storage module, according to an exemplary implementation of the present disclosure.
Fig. 4 illustrates an exemplary functional diagram of a bias pre-processing process, according to an exemplary implementation of the present disclosure.
Fig. 5 illustrates an exemplary functional diagram of a bias estimation process, according to an exemplary implementation of the present disclosure.
It should be appreciated by those skilled in the art that any block diagrams herein represent conceptual views of illustrative systems embodying the principles of the present disclosure. Similarly, it will be appreciated that any flow charts, flow diagrams, and the like represent various processes which may be substantially represented in computer readable medium and so executed by a computer or processor, whether or not such computer or processor is explicitly shown.
DETAILED DESCRIPTION
The various embodiments of the present disclosure relate generally to a method for computing bias in range, azimuth and elevation for three-dimensional (3D) radar measurements using Automatic Dependent Surveillance Broadcast (ADSB) measurements obtained from targets of opportunity. The present invention further relates to finding the bias in 3D radars using opportunistic targets without the need of multiple radars and the expensive dedicated air sorties for calibration and thereby reducing the communication needs and the computation complexity.
In the following description, for purpose of explanation, specific details are set forth in order to provide an understanding of the present disclosure. It will be apparent, however, to one skilled in the art that the present disclosure may be practiced without these details. One skilled in the art will recognize that embodiments of the present disclosure, some of which are described below, may be incorporated into a number of systems.
References in the present disclosure to “one embodiment” or “an embodiment” mean that a particular feature, structure, characteristic, or function described in connection with the embodiment is included in at least one embodiment of the invention. The appearances of the phrase “in one embodiment” in various places in the specification are not necessarily all referring to the same embodiment.
It should be noted that the description merely illustrates the principles of the present disclosure. It will thus be appreciated that those skilled in the art will be able to devise various arrangements that, although not explicitly described herein, embody the principles of the present disclosure. Furthermore, all examples recited herein are principally intended expressly to be only for explanatory purposes to help the reader in understanding the principles of the invention and the concepts contributed by the inventor to furthering the art, and are to be construed as being without limitation to such specifically recited examples and conditions. Moreover, all statements herein reciting principles, aspects, and embodiments of the invention, as well as specific examples thereof, are intended to encompass equivalents thereof.
In one embodiment herein, a system named auto-bias computation module 104 which employs a plurality of highly accurate geo-referenced positional data of one or more targets as a reference for computing the bias in radar data, is disclosed. The system comprises of a correlator 106 that associates a plurality of radar and geo-referenced positional data to collect data. Further, it provides a process to store the corresponding radar and geo-referenced data using an identification code for each radar and a process to detect manoeuvres in the targets. It further provides a process in conjunction with a timer to compute the registration errors in range, azimuth and elevation of a plurality of radars utilizing the trusted geo-referenced positional data and a processor to validate the computed registration errors.
In another embodiment, the trusted geo-referenced positional data may be from ADSB receivers, GPS receivers or any sensors of the kind. Further, the system correlates the radar and ADSB measurements with the nearest neighbour logic for the initial time instant and thereafter in conjunction with previous stored history of radar track number and ADSB unique identification code, wherein the track number is assigned for a target by some radar data processor. Further, the system stores the correlated radar measurements and ADSB measurements in a memory buffer till the target is out of the coverage of radar under observation wherein the stored data is pre subjected to a time validation check. Further, the system computes bias parameters for a plurality of radars (N radars) periodically based on a pre-defined period.
In another embodiment, the system computes bias parameters of a plurality of radars utilizing a processor that stores corresponding correlated radar and geo-referenced positional data using an identification code provided by the radar configurator 102.
In another embodiment, an auto-bias computation process which employs a plurality of highly accurate geo-referenced positional data of one or more targets as a reference for correcting the registration errors in radar data, is disclosed. Further, a method to identify a plurality of corresponding ADSB and RADAR measurements is also disclosed. It further provides a pre-processing module to select the best available correlated measurements to reduce the error in computed the bias that choose the linear segments of target trajectory.
In another embodiment, the method correlates radar and ADSB measurements using the nearness in position globally and on a consistency check on the unique identification code embedded in the track history, provided by some radar data processor, of the target. Further, the method invokes bias computation comprising Least Squares (LS) solution on the difference of transformed geo-referenced data and radar data after every time interval defined in the timer claimed in 1 and other pre-defined criteria.
In another embodiment, the method recognizes the use of highly accurate ADSB data to compute the bias of radars owing to its intrinsic dependency on GPS system for positional data as well as velocity parameters. The method assumes that the radar target and the corresponding ADSB target are identified and made available. In one embodiment, the auto-bias computation process shall be integrated with radar and ADSB target association module and in another embodiment, it shall run in conjunction with the radar and ADSB target association module, wherein both communicates with each other using some communication protocol. The auto-bias computation process stores the corresponding radar and ADSB measurements, for each identified radar, till the target track goes out of the coverage of radar. It includes the manoeuvre detection to filter out straight segments of tracks to ensure better accuracy in providing the time extrapolated radar measurements to allow the comparison of position data. The latitude, the longitude and the geometric height of the target obtained from ADSB data is transformed into radar coordinate system to generate difference measurement in range, azimuth and elevation. The auto-bias computation process uses Least Squares (LS) solution to compute registration errors in range, azimuth, elevation, by minimizing the difference in measurements of radar and transformed ADSB measurements. To achieve enhanced accuracy, it may be desirable to collect large number of radar data as well as ADSB data.
Referring now to Fig. 1, it illustrates an exemplary block diagram of an auto-bias computation module, according to an exemplary implementation of the present disclosure. The corresponding ADSB and radar measurements are identified and associated to each other by the correlator 106. The correlated data is made available to the system by a receive data unit 108 using some defined communication protocol. The data received at the system is cross checked for radar identity at a check radar ID unit 110 using radar configurator 102 that provides the database of different radars used in the surveillance system, thereby aiding in storage of data against each of the radar. The system comprises of a subsystem for a data storage module 112, a bias pre-processing module 114, a bias estimation module 116 in conjunction with a timer 118, a validation module 120. The data storage module 112 stores the record of correlated radar and ADSB measurements for a plurality of targets for each of the radar. The stored record of targets is processed by the bias pre-processing module 114 to select linear segments of target motion when the targets are out of the detection coverage of each radar. The bias of each radar is computed by the bias estimation module 116 periodically after every defined time period set by the timer 118.The computed bias is validated by the validation module 120 against the level of acceptance of accuracy of each parameter.
Fig. 2 illustrates a functional diagram of one cycle of operation of the auto-bias computation module according to an exemplary implementation of the present disclosure. A plurality of radar targets 202 and ADSB targets 204 are associated to each other and stored against each of the radar for a plurality of targets at 208 depending on the closeness of their positions after the ADSB positional data is transformed 206 to the local spherical system of the radar. The storage of data continues with a check for the target visibility 210 in the coverage of the radar. Then the execution control is transferred to 114 to select linear segments of target motion for a plurality of targets wherein a decision to discard the target data points 212 or to store the linear segment data points to the bias computation database 214. The check 216 for minimum number of points and data sufficiency criterion 218 makes the decision to call 108. The execution control is transferred to 118 wherein the bias estimation process 220 is executed. The computed bias parameters are validated for accuracy in quality monitoring 222. The validated bias parameters are sent out of a data send unit 122 to the adjoining systems, while the failure in meeting accepted level of accuracy transfers the control to 208.
Fig. 3 illustrates an exemplary functional diagram of a data storage module, according to an exemplary implementation of the present disclosure. The correlated data are checked for radar identity 308 against which it is stored. The data is stored against its unique target ID 314 or new target information is added to the data base 312 after a check 310 for corresponding target identification.
Fig. 4 illustrates an exemplary functional diagram of a bias pre-processing process, according to an exemplary implementation of the present disclosure. In accordance with Fig. 4 the consistency of ADSB address or the unique identification provided for each ADSB data is checked by 403. This may be required to ensure one to one mapping of ADSB target and radar target. The constant velocity and steady heading segments are filtered by 404, wherein the difference in successive velocity and heading values of the track data points determine the presence of manoeuvre and straight-line motion. The velocity difference greater than a threshold (MIN_VEL) and heading difference greater than a threshold (MIN_HEAD) is deemed as a manoeuvre which are ignored, lest the data samples are stored in Bias database 406. The processed tracks are deleted from the target data base 402.
Fig. 5 illustrates an exemplary functional diagram of a bias estimation process, according to an exemplary implementation of the present disclosure. The ADSB position data are first transformed into ECEF coordinates with the help of Radar location geodetic coordinates in 501 and the control is transferred to 502 where in the ECEF coordinates are transformed to local spherical coordinate system of the Radar. 503 will time project the radar measurements and propagate the measurements in spherical coordinate system. The error between radar measurements and ADSB measurements in terms of range, azimuth and elevation is the input to the bias estimation process 504. The output of 504 is the estimated registration errors 505 comprising of range bias, range dependent parameter, azimuth bias and elevation bias.
In one embodiment the auto bias computation process includes the steps of data correlation, data storage, bias pre-processing, bias computation, quality monitoring. In another embodiment, the auto bias computation process may be implemented with a sub-set of the above elements. Regardless of the implementation, Fig 2 depicts the functional flow of the auto-bias computation process. The different steps involved in the process are described below:

Data correlation
The data correlation that runs in radar and ADSB correlator 106 fulfils the association of a plurality of radar data and ADSB data to each other to identify the corresponding radar and ADSB measurements. The ADSB position data are transformed into local spherical system of the radar to facilitate the association process. The association is performed for a plurality of ADSB measurements against each of the track of the target generated by some radar data processor. The transformed ADSB measurements in the measurement gates are time aligned to the estimated position of the track to perform Global Nearest Neighbour (GNN) association for the first instance. A consistent three instants of probable association of the same ADSB target identified with a unique target address (ICAO address) confirms the association of that ADSB target with the track of the radar target. Upon confirmation of association, the transformation on ADSB are reverted and the corresponding radar measurement and ADSB measurements are made available at 104 through 108. There after the association in the subsequent scans of the radar is based on the position closeness and the identified ADSB target address.

Data storage
The correlated radar and ADSB data is received at auto-bias computation module 104 over a defined communication protocol through the gateway of the receive data unit 108.The data storage module 112 stores the received data where in the radar measurement along with the ADSB measurement are stored based on the track number allotted to the target by some RDP as indicated in Fig 3. The data storage continues for every different target with the unique track number, till the target goes out of the coverage of radar. It is desirable to store targets in different parts of the radar coverage area to enhance the accuracy of computed bias parameters.

Bias Pre-Processing
In accordance with Fig 1 the bias pre-processing module 114 selects quality track segments which can be used in developing input to bias estimation module 116. The bias pre-processing module 114 selects the straight-line sections of track, using the velocity and heading of the target. The velocity and heading are estimated for each track by some RDP. The track segments with nearly constant velocity and nearly steady course are filtered out from the tracks under process.

Bias Computation
The bias computation is executed in bias estimation module 116 in conjunction with the timer 118. The timer 118 invokes bias estimation module 116 of each of the radar, which houses the bias computation process, after every defined time period. The bias computation process is executed if there are data points greater than some minimum pre-defined number MIN_DATA in the bias database of each of the radar. In accordance with Fig 5, the bias computation process implemented using recursive Weighted Least Squares solution on the difference in measurement of radar as well as the ADSB measurements stored for bias computation, after bias pre-processing. It is because of the fact that the radar bias parameters are constant over the period of computation. The radar measurement model assumed in the bias computation process is given in equations (1), (2) and (3).
R_mp=R_mGps+KR_mGps+ +n_R (1)
?_mp=?_mGps+ +n_? (2)
f_mp=f_mGps+ +n_f (3)
Equations (1),(2),(3) can be formed based on the fact that ADSB is the trusted reference data making it the true position available at the particular instant of time. The difference measurement in range, azimuth, elevation can be derived using (1),(2), (3) as given in (4) and (5).
(4)
Where (5)
Equations (4) and (5) show the necessity of the ADSB positional data viz latitude, longitude, geometric height to be transformed to local radar coordinate system to generate difference measurement. The accurate transformation from geodetic to radar coordinate system requires an intermediate transformation to earth centric earth fixed (ECEF) coordinate system. This is carried out by sub-process 501. The mathematical operations involved in 501 are given in equations (6)-(11). Any point in geodetic datum can be converted to ECEF datum by the equations mentioned in “Geodetic Coordinate Conversions”
(6)
(7)
(8)
(9)
=0.08181919 (10)
(11)
(12)
where the parameters defined in world geodetic system-84 for modelling earth are the semi-major axis RE a, the ?attening factor f, the semi-minor axis RE b, the ?rst eccentricity e, the meridian radius of curvature ME, and the prime vertical radius of curvature as NE
The transformation from ECEF to radar coordinate system is carried out by the sub-process 502. The mathematical operations pertaining to 502, appears in the publication “Transformations between ECEF and ENU coordinates”. However they are described below in the light of present disclosure.
Let be the ECEF coordinates of the GPS measurement and be the ECEF coordinates of the radar location, then the line of sight vector (dx,dy,dz)between the target and the radar location is given by equation (13)
(13)

Then the modulo of vector difference between the target and the radar location is the range of the GPS measurement with respect to the radar location and is given by equation (14)
(14)
The GPS measured azimuth is computed as the angle between local north axis and the projection over the local radar horizontal plane. The GPS measured azimuth can be obtained as given in equation (15),
(15)
where and are the local north and local west unitary vectors, obtained from geodetic radar position in an offline manner as given in equation(16),

(16)
The elevation angle in local coordinates with respect to the radar is given by equation (17)

(17)

Where,
(18)

The ADSB data are asynchronous in nature, based on GPS derived positions viz latitude, longitude and geometric height, as compared to radar data. However, least squares solution could be effective when comparison can be made with positions corresponding to same time. The time aligned positions are obtained using sub-process 503. The radar measurements are time aligned with GPS transformed measurements the kinematics of track. The radar measurements are linear interpolated using Constant velocity (CV) motion kinematics.

Recursive Weighted Least Square Estimation
The Weighted Least Square Estimation (WLSE) is the variant of least squares adopted to compute the bias parameters of each of the radar. The recursive implementation of WLSE mentioned in “Estimation with Applications to Tracking and Navigation: Algorithms and Software for Information Extraction” is computationally cost effective when compared to batch WLSE, as the latter requires matrix inversions of higher order. The mathematical steps in 504 are described below.
Let Z(k+1), H(k+1),( x) ^(k+1) denotes the difference measurements for bias computation, the measurement matrix, and estimated registration errors at instant k+1.
(19)
(20)
x(k+1)=[?R_s,K,??_s,?f_s ]^T (21)

The Recursion for covariance update:
The recursion for Estimated Error covariance can be represented by the equation (22)
………… (22)
Where (k+1), P(k), R(k+1) denotes the posterior estimated error covariance, prior estimated error covariance and bias measurement error covariance respectively.
If we denote the matrices for residual covariance by S(k+1) given by the equation (23)and the parameter update gain as W(k+1) given by the equation (24)

S(k+1)=H(k+1)P(k) ?H(k+1)?^T+R(k+1) (23)
W(k+1)=P(k) ?H(k+1)?^T ?S(k+1)?^(-1) (24)
The Recursion for the Covariance:
With equations (23),(24),(22) the recursion for covariance can be rewritten more compactly as in equation (25)

that is,
(25)

The Recursion for the Estimate
The recursive parameter estimates updating equation the recursive LS estimator, written as:
(26)

The new (updated) estimate is therefore equal to the previous one plus a correction term. This correction term consists of the gain W(k+1) multiplying the residual difference between the observation z(k+1) and the predicted value of this observation from the previous kth instant measurements.

Quality monitoring
The estimated bias parameters are checked for accepted levels of accuracy in quality monitoring 222. The quality of the estimated Radar bias parameters can be measured by the Mean Square Error (MSE) or Goodness of fit. The goodness of fit can be approximated as a chi-square statistic. The chi-square statistic is a measure of data samples following the measurement model. The goodness of fit, as described in “Estimation with Applications to Tracking and Navigation: Algorithms and Software for Information Extraction”, of the estimated parameters can be evaluated as given in the equation (27)
(27)
Where , , are the measurements, estimated vector, Measurement noise covariance respectively of a Least Squares problem.
If the measurement noise is assumed to be Gaussian distributed, then is Chi-square distributed with degrees of freedom, where , are the dimensions of measurements and estimated vector respectively.
The estimated vector is valid [8] or follows the bias measurement model if
(28)
Where is obtained from chi-square table such that the probability of a Chi-square random variable with degrees of freedom exceeding it is (usually 1% or 5%)
If the goodness of fit is acceptable two other following tests, explained in “Estimation with Applications to Tracking and Navigation: Algorithms and Software for Information Extraction” and “An accurate and Fast Radar Registration Algorithm”, are carried out.
Test for reasonableness
Let, s_(?R_s ), s_(??_s ) , s_(?f_s ), denote estimated standard deviation in range bias, azimuth bias, elevation bias respectively and , , denotes the standard deviations in differential measurements.
If (29)
Then discard the solution for .
Similar tests are done on ??_s as well as ?f_s.
Test for significance of parameters :
Parameter significance determines whether the estimated parameter really contribute to the Bias measurement model.
If |?R_s |/s_(?R_s )