The Short-Range Encounter Problem
“Sonar, Conn, report the DIMUS trace bearing 030.”
“Conn, Sonar, DIMUS trace now bearing 033 designated S-24, possible submerged contact. Initial bearing rate right 7 degrees per minute.”
“Conn, Sonar, S-24 now bears 085, drawing right 15 degrees per minute.”
“Right 15 degrees rudder, steady course 060. Sonar, Conn, coming right to keep S-24 out of our baffles.”
“Conn, Sonar, S-24 faded, last bearing 118.”
A short-range encounter like this one is a confused affair, and often we walk away from one with no clear idea of what really happened. For a contact suddenly gained and lost, we can only estimate a rough solution. If this encounter had taken place in wartime, the Approach Officer would have had to choose from a poor list of options:
1. Shoot first at an extremely close target with a rough solution.
2. Shoot first at a very close target with a fair solution.
3. Shoot first at a close, faded target with a poor generated solution.
4. Shoot a snapshot down the bearing of an incoming weapon.
During the Cold War, our Submarine Force typically detected enemy submarines at long range, with plenty of time to get a good solution and drive to the preferred firing position. Now, many of our potential adversaries are so quiet that we can only hold them at short range for a short time. If our target is a quiet, capable SSN, there is a high probability of counter-detection. There may soon be a contact zig (which we may or may not detect) and torpedoes in the water (which may or may not kill us).
Instead of holding our breath and waiting for data, we could break the ice with a snapshot, but this is not always a good choice. If we hadn’t been counter·detected before, the enemy is most certainly alerted to our presence now, and there will be good counter·fire. If the range is too far, both shots will miss. If the range is too short, both submarines could end up on the bottom. There is no advantage for us in a wild exchange of weapons.
We could attempt to enter a rough solution to improve our aim, but our current methods and equipment aren’t very good at high bearing rate solutions through a very close CPA. It is easy to match bearing and bearing rate at any given moment during the encounter. On the other hand, it is hard to get a solution that matches bearing and bearing rate for more than a few seconds before it tracks off (an indication that the solution wasn’t very good in the first place).
We could try to maneuver for TMA to get a better solution. If we maneuver outside of sonar range, we get no data. If we maneuver close to the enemy, we prolong the encounter, and the time we spend on TMA gives the enemy a good chance to shoot first and evade.
If we expect to aim a torpedo and hit the target, we need a solution that is:
1. Timely. We want a firing solution right now, not during post-watch reconstruction.
2. Accurate. We want the firing solution to be close enough for an ADCAP.
3. Dependable. We want to know when the solution is close enough for that ADCAP, and more importantly, when it is not.
The Old Way to do Business: Stacking Dots
Our problem with short-range encounters arises in our combat system’s Cold War approach to TMA. When we stack dots, we are matching one Line Of Bearing after another, and we try to get a best fit for all the data. This works well for moderate and long ranges, where the bearing rates are small. The dots move into a nice vertical line, and we maneuver for another leg. The dots track off, and we tweak them back into line for a good solution.
When the bearing rate grows too fast, as in a short-range encounter, the dot stack falls apart. The bearing difference scale is either too small to contain all the data, or too big to detect and remove course, speed, and range errors from the solution. It doesn’t matter anyway, because the solution is changing so rapidly that small errors quickly grow into big errors again. This problem is one of dependability. It’s hard to tell when a short-range solution is good enough to put a weapon in the water, because a good dot stack and a bad dot stack look about the same in a short-range encounter.
As the Seawolf and Virginia class boats enter service, our Submarine Force will start to regain the acoustic advantage. The new ARCI sonar systems currently being installed in the fleet will detect targets at longer ranges, reducing the likelihood of a short-range encounter. However, for the next several years, most of our submarines will still need to deal with the short-range problem.
A Whole New Way to Look at Bearing Rates
Every submariner is familiar with the Time vs. Bearing curve. On the boat, when both own ship and a contact are steady on course and speed, we see some part of this curve. It may be stretched out on the time axis, and the bearing scale may be shifted right or left, but it is always the same kind of curve.
We can describe the entire Time-Bearing curve by the time, bearing, and bearing rate of CPA. If we know the CPA, we can calculate the bearing and bearing rate for ~ time before or after CPA. We submariners can measure bearing rate pretty well, and we know a short-range CPA when we see one, but we can’t tell the exact bearing of CPA any closer than about ± 10° in a high bearing rate situation.
Now examine the graph of Bearing Rate vs. Bearing on a semi-logarithmic scale.
The entire curve can be described mathematically by the bearing and bearing rate of the highest point. That is the point of maximum bearing rate, or the Closest Point of Approach. The bearing rate at any point on the curve is equal to the CPA bearing rate times the square of the cosine of the angular distance from CPA.
This graph looks very simple. The curve is concave down, and it is symmetric. It has no inflection points. If we were to choose any two points on this curve, we could determine the angular difference between the two, and the ratio of the two bearing rates. These two values together are unique for any pair of points on the curve. We can use a pair of bearings and bearing rates to calculate an accurate bearing and bearing rate of CPA. We can likewise calculate the time of CPA from the time of either data point.
This is the key to solving the short-range TMA problem. The bearing and bearing rate at any two points on a single leg can give us the exact time, bearing, and bearing rate of the CPA or any other point on that leg.
Using the Relative Motion Triangle
When looking at a maneuvering board plot, we can see that the bearing of CPA is always perpendicular to the relative course (DMhr) (Editor’s Note: Other fire control terms defined in the attached Mathematical Basis.) by definition. Suppose that own ship is traveling north at eight knots, and that the bearing of CPA has been calculated as 070. The relative motion plot then looks like this.
The vectors in the lower right quadrant represent just three out of an infinite number of possible target solutions.
To derive an accurate single-leg solution using this method, it is necessary to provide some input besides the time, bearing, and bearing rate at two points. For example, we can usually estimate a contact’s speed based on classification, intelligence, or sonar data. If we can guess the target speed within two knots, we can determine target course within about 10° , with a range error of 15 percent or less. An exact speed input will give a near-perfect solution.
If we have no idea what speed our contact is making, we can still get important information about his behavior. For instance, we can obtain the target’s minimum speed by setting his course equal or reciprocal to the CPA bearing. This also means that the minimum speed equals the target speed in the line of sight (yDMht) at CPA, whatever the target’s actual course and speed may be.
The relative speed DMhr is unique for every possible solution, which means the CPA range (proportional to DMhr/DBy at CPA) is also unique for each solution, as is the range at any other point in time. This means that sonar Range of the Day can be used to narrow the choice of possible target solutions, as can any other ranging information at any point in time. If we maneuver own ship for TMA, we can bacldit any new data to the first leg to refine the solution.
Even better, if we can obtain another pair of bearings and bearing rates on the second leg, we can obtain a near perfect solution without any other supporting data. Here is how:
Draw a relative motion plot with own ship’s first leg course/speed vector and the relative course line. On the same plot, draw our second leg course/speed vector and the new relative course line. The intersection of the relative course lines marks the precise contact solution.
Zig Detection
Sometimes it is hard to call a short-range zig by looking at the Time-Bearing plot. We can make a new procedure to detect a target zig very easily. This TMA method assumes that own ship
and the target are steady on course and speed in the interval between measurements, thus we have constant relative motion. The bearing and bearing rate at two points will mathematically define the expected time interval from one point to the other, because there is only one way to change the bearing by a given amount and change the bearing rate by a certain factor. What if the target zigs? If our actual (measured) time interval between two points does not equal the expected time interval, that means a possible target zig and counter-detection.
If the contact were to zig during own ship’s maneuver between legs, the zig may become apparent when the relative course lines fail to intersect, or if they indicate some unrealistic target speed. It is also possible to fuse the two legs of data using our advance and transfer to see if the bearings and ranges match before and after our maneuver. If they do, we have a good solution. If they don’t, there has been a zig.
Practical Uses
This TMA method is excellent for a quick solution on a high bearing rate contact. Under ideal conditions, the solution will be accurate enough to support a covert launch of a quiet weapon from the preferred firing position. To obtain the best results, follow these guidelines:
1. Get a sonar tracker on the contact and send the data to the fire control system as soon as possible.
2. If the sonar tracker tracks off, adjust track and buzz the sonar bearings manually. Remember, garbage in equals garbage out.
3. Allow at least ten degrees of bearing change to get a precise CPA calculation.
4. Measure the bearing rates as accurately as possible using the fire control system.
5. Use a spreadsheet program to handle the calculations quickly.
6. Practice this TMA method on surface contacts. A cooperative merchant will allow own ship to drive several legs for training, and can be easily tracked by more traditional methods to compare solutions.
Future Developments
In time, this new TMA method could be fully automated to give real-time solution updates and zig detections. The necessary elements would be:
1. Data filters to reject bad bearings from a wandering tracker. Right now, the best filter is a trained operator looking at the Time-Bearing plot and the sonar display simultaneously.
2. Direct measurement and input of time, bearing, and bearing rate at intervals as short as twenty seconds.
3. Own ship course and speed input, and automatic advance and transfer adjustments between legs.
4. Logic instructions to detect zigs and adjust the solution accordingly.
5. Interfaces with other automated TMA methods to combine the data for the best overall solution.
6. A decision aid that optimizes weapon tactics, updates ballistics, and recommends the best launch time.
This new TMA method is just one example of how computers can make us better submariners. Over the next ten years, commercial off-the-shelf processors and software will vastly improve our ability to analyze and interpret the thousands of signals our sensors collect every second at sea. Our submarines will become much more powerful and effective combat ships.
Mathematical Basis
I . Submariner lnputs
Co Own ship’s course in degrees true
DMho Own ship’s speed in knots
DMht Target speed in knots (assumed)
T 1 Time of first darn point
By 1 Target bearing at first data point, in degrees true
DBy1 Target bearing rate at first data point, in degrees/minute
T 2 Time of second data point
By2 Target bearing at second data point
DBy2 Target bearing rate at second data point
2. Simplifications
TCI = T2 – Tl
In minutes
o = By2 – By1 For this example, assume o > 0 (right bearing
drift)
R = Dby1 I DBy2
Assume R < 1 (increasing bearing rate, such that the CPA is to the right of By1, but not necessarily By2) u = 2 (1 - R cos2 o) v = ./R2 sin2 2o - 4R2 cos2 o + 4R + 4R2 cos 4 o -4R cos2 o w = R sin 2o P1 = tan · 1 [u / (v + w] in degrees P2 = P, -o in degrees 3. Outputs
Bytpa = By, + P.
Target bearing at CPA, in degrees true
Dby tpa = Dby 1 I cos
2 P1
Target bearing rate at CPA, in degrees/minute
Tprct1 = (180 / II DBycpa)
v'(l/cos2 p,) + (1/cos2 PJ – (2 cos o) I (cos P. cos PJ
In minutes. To be compared to !>. T ac1 to detect a zig.
Cr= By + 90° tpa
Relative course in degrees true
To determine target course for an assumed target speed, use the trigonometric identities in combination with the Law of Cosines, which states: c 2 = a2 + b2 -2ab cos B, where a, b, and c are the sides of a triangle and 8 is the angle between sides a and b.
For our purposes, we can consider the triangle parts to be defined as follows:
Side a = DMho, side b = DMht, and side c = DMhr (relative speed)
6 =Ct- Co
It is helpful to define another angle 11> =- Co + 180° – Cr to solve the problem. By rearranging formulas, we can calculate Ct and DMhr for an assumed target speed DMht. We can also solve for Ct and DMhr graphically on a maneuvering board plot.
Moving on,
Rh = K Dmhr I DBytpl
CPA range in yards; K = 1934 °yd-hr/NM-radian
Rh1 = Rh / cos P1
Range at time 1 in yards
Rh2 = Rh / cos P2
Range at time 2 in yards
We now have time, bearing, range, course, and speed for data points 1 and 2, and the CPA. The solution is complete.
