Simultaneous Conjugate Gradient Search Applied to Target Motion Analysis

This thesis discusses a mathematical approach to solving the bearings only target motion analysis problem. The problem of solving a sonar contact's tract can be reduced to a two variable minimization problem of finding the values of initial and final ranges to contact that minimize the sum of squares of error between the actual sonar bearings and the computed bearings given the estimated initial and final ranges. Because a submarine can use specific tactics when tracking a contact and because maximum detection ranges can be estimated, the general shape of the sum of squares of error function and its orientation are known. This research develops a search technique in which two conjugate gradient searches converge simultaneously from opposite sides of the optimum. Rules are developed to determine starting points which guarantee that the searches remain on opposite sides of the optimum in both variables. The stopping criterion for the search is the distance between the searches after each iteration. A measure of the progress of the search in terms of maximum distance from the optimum is guaranteed because either search is no further from the optimum at any iteration than the distance between the two searches. The

iii      x LIST OF FIGURES (CON'D) This can be accomplished using active sonar which provides both range and bearing to target. Submarines, however, usually prefer to refrain from using active sonar in order to remain as quiet as possible and thus minimize the probabi 1 i ty of being counterdetected. Consequently, submarines usually patrol using passive sonar only. Passive sonar, which 1 is tens for sounds from other submarines, provides a bearing to the sound source but does not provide range information.
Although passive sonar provides bearings only information (as compared to active sonar which provides bearing and range to target), it is still possible to estimate the sonar contact's track using bearings only information (1). This procedure is called passive target motion analysis.
Developing and improving the efficiency and accuracy 1 of procedures that solve the passive bearings only target motion analysis problem has been a priority of the U.S. Navy for over forty years (2,3 (4), maximum likelihood estimator (5), and regression techniques (6) have been developed. Some algorithms require hand held calculators (7,8) while others require desk top computers (9). This research, however, is limited to examining the conjugate gradient search as a technique for solving the passive bearings only target motion analysis probl em.
The following example describes the problem. A tracker, holding contact on a target submarine, travels east for five time steps and then turns left 135 degrees and travels two additional time steps to the northwest (see figure 1-1).
At each time step, the tracker holds a sonar bearing to A . target (Bi solid arrows). RINIT   The advent of multi-processors and co-processors in small computers (12,13) (15,16).
A number of constraints are imposed on the theoretical problem of finding the minimum of a contour by operational considerations and the physics of underwater acoustics.
These factors constrain both the shape of the function and also the area in R 0 ,Rn space in which the function is defined. Section 2 discusses some of the constraints imposed by the physical conditions and the implication of these constraints to the solution of the problem. It also discusses the form of the conjugate gradient search used for this analysis and some of the parameters selected for the algorithm.

OPERATIONAL CONSIDERATIONS
The primary rationale for the submarine service is the unique ability of the submarine to operate in a clandestine These can be divided into active and passive search. In active search, the searcher projects sound into the water.
This sound travels to the target, is deflected off the t a r get a n d is echoe d back t o t h e sea rch e r (pr o v id ed the t a r get i s wit h i n de t ec tion ran ge ). Th e so n a r s y s t e m on th e s ea r c h e r c omp ut es th e time delay be for e t he echo i s rec e ived a nd a l so the bea rin g on which th e ec h o i s r e ceiv e d. Because th e spee d o f so un d in water is known, it is possible to det e rmin e both r ange and bearing t o t a rget u s in g active search.
Targe t motion analysis using a ctive s e arch is relatively e asy. Knowing own ship cours e a nd speed as well as rang e and bearing t o target at two or more time steps, it is possible to estimate the course and speed of the target.
For a s ubmarine to detect a contact at 10 nautical miles on active sonar, sound must travel 20 nautical miles, 10 in each direction. The liability of active sonar is that the target can hear the active pinging a t twice the distance that the searcher can detect the target.
Passive search, on the other hand, for noise from the target submarine.

involves listening
When a target is detected on a passive sonar system, · a signal is received on a particular bearing. The searcher knows that a target is within maximum detection range on a particular bearing but has no other informati-0n as to the ran g e to target. In ge neral, t he intensity of sound decreases with distance from the sou rce (17). For this r eason , th e r e is a max imum distance at which a so nar can detect a targ e t in a given environment.
Acoustic range prediction computer models are avai lable which predict propagation loss as a function of distance from source. The primary input to these models is a profile of the temperature of the ocean as a function of depth. A device called an expendable bathythermograph is used to measure ocean temperature as a function of depth as it is dropped through the ocean. The computer prediction model uses the temperature profile and other information such as salinity, bottom type, wave height and frequency to predict the loss in intensity of a sound source versus distance from source.  algorithm u sed in this analysis , it i s acc e ptable to u se t he lon ges t possible detection r ange as an initial estimate.
Because the maximum range at which a submarine can det ec t a target can be es tim a ted, the minimum and maximum range to tar ge t at the beginning and end of the trackin g maneuver are also predicted.
Only values between zero and the maximum range are possible. This information is important to solving the target motion analysis problem because the region in which the problem is defined is limited to the area of positive R 0 ,Rn space between zero and the maximum

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The on l y r efe rences in the classif i e d lite r ature pertain to th e SURFLOC algo rithm ( 9 ,1 8 ), which runs on a HP 9825 desk top comput e r. Some problems a r e i de nti fied with SURFLOC, specifically that it seems to work for some s t ar t ing points but not for others. 21 cent method can be greatly improved by mod i fy in g it in to a conjug ate ( or deflected) g radien t me thod. It has been s hown (15) that any minimization method t ha t makes us e of th e conjug ate d ire ctions is quadratically co n ve r ge n t . The co njugate g radi e nt procedur e s ets up a n ew sea r ch direction as Because of the shape of the function, it is safely assumed that the function has one minimun in the direction of the search. Functional evaluation are made at X2 and X3.
If the function value at X3 is less than at X2, then th e area from Xl to X2 is eliminated because the functi o n is increasing from X2 to Xl and the mJnimum, therefore, is to the right of X2. If X2 is less than X3, then X4 is elimi- Sample computer runs demonstrate that this algorithm finds the minimum of the function for all the tracking geometries tested (see figure [2][3][4][5][6][7][8][9]. Note that once the search is within 0.06 nautical miles in both variables, it converges very slowly to the true minimum (9.00, 7.724).
For figure 2-9, the search is allowed to continue to 15 iterations after a solution is found.

Target Motion Analysis Geometries
Fourteen specific geometries are chosen for this analysis. In all cases the tracker uses a two legged tracking maneuver, transiting thirty minutes to the east and then fifteen minutes to the northwest at 7 knots. The target always remains on a constant course and speed from the west to the east (but not parallel to the tracker's course). The search algorithm assumes that the target remains on a constant course and speed.
If it is determined that the target has changed course or speed, it is necessary to res tart the tracking sequence.
Although an infinite number of tracking geometries is theoretically possible, the geometries are constrained by a number of factors. As is discussed in section 2.2, detection ranges are limited and thus the tracker must keep the target within detection range. The speed at which the tracker can search is limited because of the r eq uir emen t of minimizing own ship noi se that can r ed uce s onar performanc e and increase the target's ability to counterdetect t he tracker and realize that it is being tracked (1). It i s necessary that the tracker either lead or lag the target.
If bearing to target remains a constant over the first le g of the track, no information is obtained (21). The target could be on any of a number of tracks (see figure 2-10).
The target on tracks 1, 2 and 3 are maintaining the same speed as the tracker while the target on tracks 4, 5 and 6 are faster than the tracker.

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Convergence zone tracking is especially difficult.
First it is difficult to maintain detection in a narrow convergence zone. Secondly, bearing rate is genera lly low at convergence zone ranges (see figure [2][3][4][5][6][7][8][9][10][11]. A contact at a 35 nautical mile range moves a maximum of 1.6 degrees as it transits 1 nautical mile, whereas a contact at 6 nautical miles moves 9.5 degrees as it transits one nautical mile in the same amount of time.
It is possible that bearing error will overwhelm a bearing rate at convergence zone ranges.
It is often necessary to close a low bearing rate contact in order to track in a geometry that is more conducive to passive target motion analysis.
Because of the above mentioned cons train ts, fourteen specific geometries are chosen for the analysis (see table   2-1). Four of these geometries are depicted with their respective contours later in this chapter. In all cases, tracker course maneuver remains constant. These geometries provide a representative sample of tracking geometries ineluding variations in target speed, target course, and initial bearing to target.

Plots of the Contours
The first computer program written for this analysis is called CONTOUR (see appendix b). Out put of CONTOUR --Tru e Data for a Geo me try .   Because the function appears to be a well behaved function, it should be possible to search with the conjugate gradient search from two points, one below the minimum and one above the minimum in both variables. If the two searches converge on the minimum from opposite directions, then the distance between the two searches provides valuable information on the success of the search. Either search can be no further from the minimum than the distance between the two searches.
In order for the two searches to converge from oppo-

TESTING THE CONJUGATE GRADIENT SEARCH
For each geometry, the conjugate gradient search is used to search from ten low starting points and ten high starting points. The 20 starting coordinates are listed in     In some instances, (for example, . geometry 9), the high search is often much better than the low search. Even though low starting points are more efficient statistically, they are not necessarily more efficient for a particular geometry.

NUMBER OF ITERATIONS REQUIRED TO REACH A SOLUTION
If the criterion for stopping the conjugate gradient search is n iterations, it is necessary to determine the Probability that an acceptable solution is reached in n iterations.  thus theoretically performing two searches in the same time as a single sea rch (12). There i s some loss of efficiency with a co-processor ( g enerally one iteration requires more functional evaluations than the other) and some loss occurs with the need for the two processors to communicate (13).
In general, however, a computer with co-processing capability should be able to perform two searches in significantly less time than a comparable single processor machine can perform two searches.

STOPPING AND EVALUATION CRITERIA
As is discussed in section 3, the conjugate gradient search is an effective tool to solve the bearings only target motion analysis problem. Solving the problem in real time applications, however, requires invoking some stopping criteria.
Because of the canoe shape of the curve, two solutions with the same sum of squares of error are not 75 nec essarily equally close to the optimum.
The simult aneous conjugate g radi ent search provides some absolute information on the success of the search in that it provides at every it eration a maximum er ror, namely th e distance between both searche s . This type of absolute information on the success of the sea rch is not available from a single search.
One of the major difficulties in evaluating the success of the simultaneous conjugate gradient search is the lack of information on which search is closer to the optimum. In general, one search is likely to be closer to the optimum than the other as is discussed in section 2. Although the distance between the two searches provides a limit on the maximum error, it does not identify which search is farther from the optimum.

Stopping Criteria
Clearly it is possible to continue the two conjugate gradient searches until they are within 300 yards in both            It is noted, however, that this technique produced 100 percent success in five geometries. No other technique tested provided 100 percent success for as many geometries. The common trend in these five geometries is that most searches in these geometries converge in 5 or less iterations, that is, the two searches tended to converge quickly and simultaneously, the conditions under which using the average between two searches tends to provide a good solution.
I\ e s u 1 t s of Se t G -S 1 mu 1 t a n e o us Conj u g a t e     It is not necessary that every individual attempt at solving the problem be successful. As is discussed in section 2, target motion analysis is a continuous process.
As a new bearing is added to a full set of bearings, the earliest bearing is dropped and a new solution estimated. This research has identified a number of areas in which the conjugate gradient search must be tuned to the particular problem. Otherwise there is a probability that the search will either stall or move unacceptably slowly toward the optimum.

RECOMMENDATIONS
Although this research demonstrates the effectiveness of the simultaneous conjugate gradient search in solving the bearings only target motion analysis problem in the representative geometries, the resu 1 ts of this ana 1 ys is are not necessarily conclusive. One of the more important findings of this research is that minor changes to the starting positions for the search or to the contour can cause significant differences in the ability of the search to converge quickly on the minimum of the function.
The next step in this analysis is to test and fine tune the algorithm to solve the problem using typical bearings that are generated on sea tests. A number of variables are included in sea test data that affect the contour.
Among these are random bearing error and random error on the time steps. Another important concern is · the issue of lost data points. A bearing can be lost at a given time step for a number of reasons including temporary loss of detection by the sonar. Bearing readings also must be rejected at times if a bearing seams to be clearly erroneous. If the target is on a constant course and speed (as the algorithm assumes), then the bearing rate is fairly constant between time steps. A student's t or other suitable test is used to determine if a new bearing is acceptable and to be added to the algorithm or if it is to be rejected. Because all these factors affect the shape of a contour, it is important to tune and test the algorithm with data taken from sea tests 108 I or at l e ast with simulated sea t e st data that include random bearing e rror.
Th e geometries used in this study assume an environment with a 15 nautical mile direct path/bottom-bounce detection range.
It is important to test the technique both in other environments and with different tracking tactics.
Lastly the conjugate gradient search itself must be tested and optimized. Various conjugate gradients need to be tested as well as the optimal ratio of conjugate gradient to gradient directions.
The results of this analysis on theoretical tracking geometries and theoretical bearings suggest that a properly tuned conjugate gradient search is an effective tool for solving the passive target motion analysis problem. The technique of using two simultaneous conjugate gradient searches converging from opposite directions provides both an effective search and also criteria for evaluating the success of a search at any iteration.