Target motion analysis {TMA) has received enormous attention since ASW became the primary mission of attack submarines in the early 1950’s. Many, perhaps most, of the fundamentals of TMA based on three or four bearings have been innovations by naval officers, predominantly submariners, working outside their normally assigned duties. In fact, from my 38 years of experience with naval tactical analysis (as a civilian operations analyst and mathematician, not specialized in TMA), I can think of no topic that has attracted as much officer analysis work as TMA. This article reports the most successful officer TMA innovations of which I am aware. It undoubtedly has gaps, which I hope readers will fill.

The more advanced statistical processing methods, by civilian scientists, are not addressed, nor are the important and extensive officer management roles in TMA development and testing, largely by COMSUBDEVRON TWELVE (CSDS-12) and its predecessor, COMSUBDEVGRU TWO (CSDG-2). To keep the focus on officers, credit lines to civilians are suppressed. I have attempted a more comprehensive TMA history in Naval Tactical Decision Aids, Military Operations Research Lecture Notes, NPSOR-1, by Daniel H. Wagner, Naval Postgraduate School, September 1989. Contemporary expositions of TMA substance are also in Theory of Ranging and Target Motion Analysis, Draft NWP 71-1-4, in preparation, 1991, COMSUBDEVRON TWELVE, and Naval Operations Analysis (Third Edition), in preparation at USNA for Naval Institute Press, 1991.

TMA is estimation of a target’s position, course, and speed. Here we stick to linear target motion (constant course and speed).

We begin with the Lynch Plot, devised in early WWII by LT (later CAPT) Frank Lynch. He was serving on the recommissioned WWI submarine R-1. Her first sonar had been installed, and Lynch was assigned to find a method to make sound-only approaches on a surface ship, for sea trials a few days hence. Through intense pre-sail effort and excellent geometric insight, Lynch found a pivotal relationship among bearings, bearing rate, and target relative motion. In ensuing weeks, he perfected the Lynch Plot working evenings plotting geometries. He used this method throughout the war as XO HARDER under Medal of Honor winner CDR S. D. Dealey and as CO HADDO. Postwar it entered the Submarine School curriculum, remaining in use into the 1960’s.

For this account of the Lynch Plot, I am indebted primarily to David Ghen of Analysis & Technology, based on his conversations with Lynch and later his widow, and also to retired CAPT Frank Andrews. Andrews also made the interesting observation that, while it wasn’t an innovation, his Chief Engineering Officer on the K-1, LT Jimmy Carter, wrote a creditable command thesis on the bearing rate slide rule. I am unable to unearth documentation of the Lynch Plot method perhaps readers can provide this. I also invite information on the vintage and originator(s) of the strip plot.

A landmark innovation occurred with the development in 1953 of the Spiess Plot by CDR (later CAP1) Fred Spiess, USNR, in Complete Solution of the Bearings Only Approach Problem, Scripps Institution of Oceanography, 15 December, 1953. He was at the time a civilian oceanographer with the Marine Physical Laboratory (which he directed for 22 years), Scripps Institution of Oceanography. I include this work because it was inspired by his extensive combat submarine experience and a three-month West Pac SS tour in 1953, and because it is so important. Spiess showed that given bearings at three times, the locus of target position at a chosen fourth time is a computable straight line, now called a Spiess line. By intersecting the Spiess line with the bearing at the fourth time, position is found, unless the two coincide (now called a singularity). His solution also yields course and speed, which can also be done by, e.g., reversing the time sequence to find a second position. This was the first complete TMA method by bearings only. Graphic methods of solution were developed jointly by Spiess and LT (later CDR) William Liesk and introduced by Leisk to submarine officers’ classes at Fleet ASW School, San Diego.

Spiess gave an algebraic condition necessary for a singularity to occur. If own track is linear, Spiess lines are bearings lines and a singularity is inevitable. Being close to a singularity is bad, and difficulties with Spiess Plots are probably attributable to inadequate understanding of this hazard. Recently, Midshipman 2/C Frederic Nauck at the Naval Academy, who is probably a future submariner, performed an interesting investigation (see Singularities in Spiess Target Motion Analysis by Frederick E. Nauck, U.S. Naval Academy Mathematics Honors Report, May 1991 ), of loci of singularity situations, in which he developed a PC tacaid to provide guidance for avoidance of singularities.

A very interesting innovation in TMA theory was observed in 1953 by LT John Kettelle, USNR, now CEO of Ketron, Inc .. If own track is linear, then three distinct bearings determine a parabola tangent to bearings at all times and to all possible target tracks consistent with the three bearings. At the time Kettelle’s active duty on GUITARRO interrupted his graduate work in mathematics. This observation was published by Kettelle in Parabolic Envelope of Bearings-Only Tracks, Journal of Underwater Acoustics, 11 October 1961, and independently by a NEWRES civilian in 1960. The first documented proof was by the latter in 1970. It can be shown in Naval Operations Analysis (Third Edition) that even if own track is not linear, three bearings determine a parabola tangent to the Spiess lines and the tracks. Also, the axis of the parabola is parallel to the direction of relative motion.

In 1954 LT (later CAPT) John F. Fagan, in his Command Theses, A Mathematical Method for Solving the Sonar Fire Control Problem, derived a four-bearing TMA solution from a system of three transcendental equations. To make this computable by slide rule, he assumed own motion during three bearings was approximately zero, which was probably satisfactory for diesel operations.

Probably the most famous TMA method is Ekelund ranging, devised in 1958 by LT (later RADM) John J. Ekelund, as an instructor at the Submarine School. The Ekelund range estimate is the difference between own speeds across line of sight, before and after own tum, divided by the bearing rate difference in reverse order. After deriving this theoretically, Ekelund tested it in lunch hours on the attack trainer, assisted by fellow instructor LT (later CAPT) Roy Goldman. Ekelund’s report through channels was bounced for revision multiple times, so he submitted it directly for publication in the COMSUBLANT Quarterly Information Bulletin in the summer of 1958 From that dissemination it was picked up in the Aeet and eventually gained widespread use in several navies. I believe it would be very difficult for contemporary dissemination of this nature to gain attention amid the pressures on Aeet personnel to absorb existing technology.

Ekelund ranging is convenient and can be quite useful. It does assume idealizations that can introduce serious errors. Ekelund gave attention to maneuvers to reduce such errors, and investigation of this issue was carried much further by CAPT Fagan. The powerful method of time correction can greatly reduce such errors by finding best range times at which range estimation is insensitive to target speed in line of sight. Also, the Spiess line at a best range time is perpendicular to a particular bearing used to find the best range time. Fagan’s work is credited as being an important precursor to time correction.

The important classified innovation called FLIT was developed by ENS Lyle Anderson in 1970. Much of the fundamentals had been given independently by an Electric Boat (EB) civilian in 1969. Anderson was a surface officer assigned to CSDG-2 awaiting nuclear power school. His work was quickly taken up by CSDG-2, and with supplementary implementation work by EB it was tried successfully at sea in a few months. He drew high praise from CSDS-2 management and civilian mathematicians and was awarded the Navy Commendation Medal. CSDG-2 held him over and steered him into submarine duty.

The final innovation I’ll mention is the following observation of LT Jerry Gullick at CSDS-12 in 1980: If own and target tracks are linear and L is the bearing line between them at some time, then the intersection of the other bearing lines with L moves at a constant rate. This came to be known as “Gullick’s theorem” and is not hard to prove. Gullick used it as a basis for a four-bearing TMA method — W. J. Browning of Applied Mathematics, Inc. supplied me with informal documentation. Of more interest is the fact that Gullick’s theorem can be used as a step in a proof of the parabola theorem.

That naval officers have provided abundant TMA innovations should not be surprising. What has been surprising to me is that after four decades of extensive investigations in TMA by large amounts of expertise, very interesting and useful innovations, which I am not trying to report here, based on a few bearings, are still being obtained. Future officer initiative and success in this area are to be expected.