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 MNI^^M ppHPHPü^nHmpi "•i" ■■-■^»^«wpwpw

N ADL-72580-2

00 TECHNICAL REPORT 2

00 TIME CORRECTION IN PASSIVE RANGING:

BREAKTHROUGH OR BOOTSTRAP?

James M. Dobble
00
3 Under Contract

N00014-70-C-0322

NR 364-025/2-12-70 (462)

Prepared For

Office of Naval Research

O Department of the Navy
Washington, D. C. 20360
Li

Prepared By
^'V f

Arthur D. Little, Inc.

Acorn Park
Cambridge, Massachusetts 02140

October 1970

"REPRODUCTION IN WHOLE OR IN PART IS PERMITTED FOR

ANY PURPOSE OF THE UNITED STATES GOVERNMENT."

(vf ^D D C
OCT 2« 'W

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TIME CORRECTION IN PASSIVE RANGING:

BREAKTHROUGH OR BOOTSTRAP?

Introduction and Summary

The concept of time correction in passive ranging is described

I
in reference (a). It was developed by Daniel H. Wagner,
Associates, who believe it to be a breakthrough in the subject
of passive ranging. The concept has been applied to the four-
1
bearings problem for which it was conceived and to the Ekelund
range estimate involving two bearing rates. A time-corrected
I
Ekelund range estimator Is being considered for part of the
target motion analysis in a new fire-control system.
I
Does the concept of time correction represent a significant
breakthrough, as claimed, or is the apparent gain an illusion?
We have examined this question carefully and have found no
support for the claims that have been made for the method. We
have concluded that the method has no special value in passive
ranging and should be discarded. Our arguments are presented
below.
iL

.■1|

■ The Time-Correction Method

Time correction, as developed in reference (a), Is a method for

the computation of a range R* at a time t* from the four bearings
B-, B» B-, B, observed at the corresponding times t.., t-, t„, t,
which are assumed to be in the order t. < t- < t < t,. We
outline the derivation of the equations below.

.
This research was supported by the Office of Naval Research
under Contract No. N00014-70-C-0322.

Arthur!) Little, Inc

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We use the standard naval coordinate system. For 1 - 1, 2, 3, 4,

let

t. - time at the 1ÜL observation

B. ■ bearing of the target from the tracking submarine
R. - range of the target from the tracking submarine
u . ■ range component of target velocity measured
outward along B.
u, . ■ normal component of target velocity measured
normal to B In the direction of positive
(clockwise) angular rotation

Also, let

Vi.. ■ distance (directed) tracking submarine moves In

^ the direction B, from time t. to time t,
k k 1 J
v - distance (directed) tracking submarine moves
^ normal to B.k from time t.1 to time t.j

It will be convenient to use

'u ■ ^ - ti' B±t' B

J ' 'i' S
IJ '8in Bir cij'C08 Bij

Assuming linear target motion It Is easy to derive the equations

R
2 S12 " ^1 ^2 " V12 (1)

R4 S34 " %3 ^4 " V34 (2)

R2 C24 " R4 " ur4 ^4 + y24 (3)

V C13 " ^3 + U
rl S13 (4)

These equations are equivalent to the equations (2-5) (2-6) (2-7)

and an unnumbered equation on page 2-18 of reference (a).
Eliminating R„, ir,, and u.- from the above equations and solving
for R,4 we obtain

R
4 " R4 + a
l url + a
2 ur4 (5)

Arthur D Little, Inc

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where

K■ (V 3
34 ' hi - C13 "U ' hi - 4u ß12) / (ß
12 " ^ (6)

a1 - S^ / (312 - &u}, a2 - t24 B12 / (0^ - 33^ (7)

ß12 - (S12/t12) (C13/C24). ß34-S34/t34 (8)

Equation (5) expresses the range R. In terms of an estimate

R, that can be computed from observed quantities, and two range
components of target velocity.

In the time-correction method the two range components, u , and

u ,, of target velocity are eliminated by Introducing two bearings,
B- and B,, at times t. and t, by using two equations of type (3)
above. They are

R,5 C.45 - R,4 + u r4. t.-

45 - y..
45 (9)

R
6 C16 " R5 C15 + U
rl ^6 " ^56 (10)

The general equation of this type Is

^ Cjk " Ri Clk + u

rk ^j " "ij W

Equations (3) (9) (10) are special cases of equation (11).

Eliminating R, and R_ from equations (5) (9) (10) we obtain the

equation

R C (C
6 16 " 15/C45) (ft
4 " yt5) " A* + a
3 U
rl + a
4 U
r4 (12)

— *
where a. and a, depend on t, and t,. Let t and t be the values
of t- and tfi respectively that make a_ ■ a. - 0. Also, let B
and B be the corresponding bearings of the target at these
times, and let R be the corresponding value of R,. Then
0

Arthur D Little, Inc

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t - (ß12t2 - ß34t4) / (ß12 - ß34) (13)

t - t - S13 cos (B - ij I (012 - ß34) cos (B - B4) (14)

R cos (B - B^ - (R4 - u45) cos (B - B^ / cos (B - B4) - y^

(15)
Equations (13) (14) (15) are the same as equations (2-14) (2-18)
(2-20) respectively of reference (a) when our subscript sequence
(1, 2, 3, 4) Is replaced by the sequence (I1, 1, 2', 2) of
reference (a). (It Is convenient to retain the subscripts 5 and
4 1
6 In p,_ and w_, to avoid ambiguity.)

it it
The range R is called "the time-corrected range" and t is
called "the best range time." The time-correction method consists
■k
of using equation (15) to compute R and equation (14) to compute
* "
the corresponding time t at which it applies.

Discussion

Does the above method offer a significant breakthrough in passive

ranging? Various claims are made for the method in reference (a).
It is asserted that the basis for the method is that "every
ranging maneuver has associated with it a best range time", at
which time the error in the range component of velocity "will
have minimum effect on the accuracy of the range solution." And
*
when t is in the future, it is claimed that the four bearing
4 4
observations and the three distance measurements (VUA» ^L^*
and vicfi) "provide an accurate range at a future time with no
additional information needed." Are these claims justified?

The derivation is deceptively simple. No optimization is needed.

Only simple algebraic manipulations of well-known equations are
required to eliminate the two range components, u . and u ,, of
target velocity from the range equation (5). It is surprising
that such a simple derivation yields a range estimate that is

Arthur D Little, Inc

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superior to other range estimates, as claimed. And It Is even

more surprising that the result remained undiscovered so long,
since the derivation Involves only simple mathematics.

Since t Is not obtained by optimizing a payoff function, why

Is It called "best"? Apparently, t
Is called the best range
*
time because the corresponding range R has been written In
equation (15) In a form that does not Involve a range component
u of target velocity. Immediately following the equation, which
Is numbered (2-20) In reference (a). It Is stated that, "The
range R Is the time-corrected range. Note that all of the
terms In Equation (2-20) can be measured by the SSK. The time
*
t at which this applies Is called the best range time...

Can all of the terms In equation (15) above be "measured"?

Certainly, B and B can't be measured In the same way that B.,
B», B., B. are measured by recording the observed value at the
time of occurrence. The bearings B and B are the bearings at
the times t and t . Obviously, If t and t fall outside the
£
tracking Interval (t., t.), the corresponding bearings B and B
can't be observed. Nor can they be observed, if t and t fall
— *
inside the tracking interval. The values of t and t in
equations (13) and (14) require the values of all four bearings.
— *
Hence, t. Is the earliest time at which t and t can be computed,
— *
and it then is too late to observe B and B .

A
Hence, B and B must be "estimated" from the bearing-time plot,
— * ,
by interpolation when t and t fall Inside the tracking interval
and by extrapolation when they fall outside the tracking interval.
In either case the errors in estimating B and B may be large.
Here, Interpolation may be as inaccurate as extrapolation, since
the SSK Is maneuvering during the tracking Interval and bearing
observations usually are not made during the turns. However,
*
the resulting errors in the cosine factors, and in R , in
equation (15) would be small. Hence, the fact that B and B

Arthur D Little Inc

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must be estimated by Interpolation or extrapolation Is not a

major disadvantage. The error Is comparable to that made In
using the small-angle approximations.
4
Errors also will occur In estimating the two distances y,- and
y,,,
36
since they also must be obtained by Interpolation or
_ *
extrapolation after t (-t_) and t ("0 have been computed.
■' 0
*

These errors may produce larger errors In R than those produced

— *
by the errors In estimating B and B .

More serious questions concerning the method can be raised as

follows:

a. What Is the motivation behind the procedure? What

objective are we trying to achieve? Is It merely to
A It
find an equation for the range R at some time t In a
form that Involves only quantities that the SSK can
measure or estimate?

b. Does the procedure lead to a unique solution that has

some desirable property? If so, what?
it it
c. After we find R at time t , how will we obtain an
estimate of the range R at time t when the time t at
which we need a range estimate does not coincide with
*
t ? Will this require that we use an equation of type
(11) above that requires a range component u of target
velocity? If so, what have we gained by first computing
*
R ?

If the objective of the method Is to obtain an equation for range

In a form that Involves only quantities that the SSK can measure
or estimate, why stop at one? There are many such equations.
What special properties does the "solution" in equation (15)
have that It should be selected?

Other "solutions" of the type displayed above can be obtained

by interchanging two subscripts. This possibility seems to have
been recognized in some parts of reference (a), and used in a

Arthur D Little, Inc

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special case In chapter 4. But the Implications concerning the

validity of the general method and the uniqueness of the solution
| were not explored.
I
Some of the "solutions" obtained In this way reduce to the same
, * *
forms for t and R In the small-angle approximations, and others
I
do not. For example, If we replace equations (9) and (10) by
the equations
i
R
5 C15 " R4 C14 + U
rl ^5 " ^5
4
6 46 5 45 r4 56 56

and repeat the steps of the solution, the equations for t and
f *
R are not Identical to equations (14) and (15) but reduce to
i
the same small-angle forms that equations (14) and (15) become.
The equation for t does not do so. On the other hand, if we
Interchange the subscripts 2 and 3 in the original solution,
we get a solution for which the small-angle approximations for
* * _
t and R , as well as that for t, do not reduce to the same
small-angle forms obtained in the original solution. This
solution is written in full below:

T (0
I ' 13 H - ß24 t4) / (ß
13 " ß24) (13a)
f
s C08
t " t" - 12 ÖJ - B^ / (,&13 - ß24) cos (B - B4) (14a)
i

R cos (B - B^ - (R4 - y45) cos (B - B^ / cos (B - B4) - y^

(15a)
where now

(V
*4 " 24 / t24 " C12 Vt3 / ^3 " ^34 ß13) / (ß
13 " 324) (6a)

ß13 * (S
.13 / t13) (C
12 / C34)' ß24 = S
24 /
hk (8a)

Arthur D Little, Inc

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It is evident that we can't call the time t In equation (14)

the beet range time, since It Is neither unique nor optimal In
any demonstrated way. It may be a good range time, but the
property, If any, that makes It better than an arbitrary time
has not been displayed In reference (a) or elsewhere, to our
knowledge. The property that led to its introduction - that it
*
permits us to write an equation for the corresponding range R
In a form that Involves only quantities that the SSK can measure
or estimate - is not a sufficient reason for its use. Any
arbitrary time has this property.

Under the assumption of linear target motion, on which the

derivations above and in reference (a) are based, four bearing
observations (made on a ranging maneuver that doesn't consist
of a single linear path) are sufficient for a complete TMA.
Hence, the range R at any arbitrary time t can be written as a
function of t, the four observation pairs (t., B.), 1 - 1, 2, 3, 4,
and components of own-ship motion. It is not necessary to
— *
compute two times t and t and then estimate the corresponding
— *
bearings B and B by interpolation or extrapolation from the
bearing-time plot. The only bearings involved are B^, B., B., B,.

The TMA requires only the solution of four simultaneous linear

equations in a set of target parameters, such as x., y,, u , u ,
where u and u are the components of target velocity. The
solution has been obtained many times. In fact, Spiess' form
of the solution is discussed in chapter 5 of reference (a). By
means of the TMA we can write an equation for the range R at
an arbitrary time t, in a form that involves only quantities
that the SSK can measure or estimate.

The solution for R is simple when t coincides with one of the

observation times. For example, when t ■ t,, the solution
becomes

Arthur D Little, Inc

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 mm, —• -• - ■■"""

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V S V S + V S
R . 14 ^24 '34 23 - 24 ^14 ^34 13 34 ^14 ^24 12
4 S S S S
'13 '24 12 34 ^12 '34 13 24 (16)

where, as before,

v.. ■ distance SSK moves normal to B. In the Interval

S - sin (B - Bj)

'li ■ V'I

We do not recommend the use of the estimator (16) for R.. However,
equation (16) Is a simple equation for the range R, In terms of
the four (t., B.) pairs and three components of own-ship motion.
Also, It Is exact, that is, It will yield the exact value for
the range at time t. In the absence of errors In the measured
quantities. And, certainly, the SSK can measure all the
quantities In equation (16). Do these facts make t, a "good"
range time? If not, what property does t possess that makes
It better? Will any gain that is obtained from using t and R
if
be lost in computing R, from R , if t- is the time at which a
range estimate is needed.

It is surprising that nothing is said in reference (a) about the

if ^
problem of converting from the estimate R at time t to an
estimate of the range at the time at which it Is needed. How is
this conversion to be made? By solving for the TMA? If so,
won't we finish with the same deterministic relationships, but
in a more circuitous form, with many more chances for errors in
the measurements? Will the estimate of P. by such a method be
more accurate than that obtained from equation (16)?

A possible way of avoiding these questions is to state that the

if
submarine commander can control the value of t by his choice of
*
ranging maneuvers and hence can make t equal to, or close to,
the desired time, say t,. In fact, this point is offered in

Arthur D Little, inc

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reference (a) as a major advantage of the time-correction method.

We contend that control of the value of t Is not a practical
solution to this problem. The commander of own submarine has
only partial control over t and that control Is rather tenuous.
The commander of the target submarine must cooperate by
maintaining constant course and speed. Even when he does so, the
problem of the selection of maneuvers and times t-, t?> t.( t,
that will make t fall at t. Is not an easy one to solve, since
the value of t can't be computed until the last bearing B.
has been recorded. It Is the type of problem that can be solved
after the fact but Is very difficult to solve in real time and
sequence. Even if it can be done, there remains the problem of
how to assure linear target motion in the relevant time Interval.

Criteria For Selection of a Range Estimator

The equations derived above have been based on two Important

assumptions, as follows:

(1) The target submarine maintains constant course and

speed throughout the relevant time Interval, which is
the maximum interval spanned by the times involved in
the computation.

(2) There are no errors in the measurements of bearings,

including those obtained by interpolation or extra-
polation, or in the measurement of own-submarine
motion.

Both assumptions are questionable, at best, and we should consider

the effects of departures from these assumptions.

The argument so far has been based solely on formula accuracy

*
under these assumptions. The main argument for the use of R
in equation (15) seems to be that equation (15) is exact and
contains only quantities that the SSK can measure or estimate;

10

Arthur D Little, Inc

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whereas some other equations for a range estimator are Inexact,

such as equation (6) as an estimator for R,; or they contain
quantities that the SSK can't measure or estimate, such as
equation (4). More precisely, equation (4) contains two factors,
u ^ and u ,, that the analyst hasn't taken the trouble to express
In terms of quantities that the SSK can measure or estimate.
If this had been done, equation (5) would have reduced to
equation (16) when simplified.

Formula accuracy Is only one characteristic of a range estimator,

and not the dominant one In most applications. The main
characteristics of a range estimator are:

(a) Formula accuracy

(b) Ruggedness to target maneuvers
(c) Sensitivity to error measurements

It Is not likely that an estimator will dominate all other

contenders In all respects. For example, the estimator for R,
In equation (16) Is exact, and hence superior In formula
accuracy to the estimator R, for R, In equation (6), and to the
estimator R- obtained by dropping the y.. term In equation (6),
and to the Ekelund estimator R_K (a limiting form of K„), since
R,, R_,, and R_,K are biased estimators. However, R, In (16)
probably Is not as rugged to target maneuvers as some of the
other estimators, and undoubtedly Is more sensitive to random
errors than R., R^, or R-,^. It Is Impossible to estimate the
ruggedness and sensitivity, or even the formula accuracy, of an
it
estimator for R4 that starts with R In equation (15), since It
*
depends on the procedure used to calculate R, from R .

If there Is no dominant estimator, how do we select one? We

believe that this decision requires a careful examination of the
use that will be made of the range estimate and the selection of
a suitable performance measure or payoff function. We then

11

Arthur D Little, Inc

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compute the payoff function for possible target maneuvers and

error distributions, select the dominant estimator, If It exists,
or use a weighted mean otherwise.

In many applications formula accuracy Is relatively unimportant,

particularly when likely target maneuvers or measurement errors
will produce range errors that are much larger than the Inherent
bias In the formula. Under such conditions It Is absurd to select
an estimator on the criterion of formula accuracy, alone or
predominantly.

Comparisons of range estimators, or complete IMA estimators,

often (usually?) have been made on a partial basis. We believe
that all characteristics of the estimator should be considered.
It should be possible to analyze and compare leading contenders
In respect to the major characteristics listed above. When
this has been done, and done Impartially and completely, we
will have available for the first time the facts that are
necessary for the Intelligent selection of an estimator. And
the selection should be made In consideration of the use that
will be made of the estimate and the characteristics of the
estimator that are Important for that application.

In many applications of passive tracking the entire TMA is

needed. For this reason we believe that the analysis should be
extended to TMA estimators, as well as range estimators.

A complete analysis of the type described above seldom has been

attempted, and perhaps never has been done satisfactorily. An
analysis of several range estimators (the Ekelund and a new
three bearing-rate estimator that removes the bias in the Ekelund
estimator) was made in reference (b). Reference (b) also Includes
analyses of several TMA estimators (CHURN and a new one developed
by the writer), using acquisition probability as the payoff
function. While these analyses cover all the major characteristics

12

Arthur D Little, Inc

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of the estimators, they are not extensive enough to be

considered complete.

References;

(a) COMSUBDEVGROUP TWO Report No. 2-69, Passive Ranging

Manual (U), Volume III THEORY, Dec. 1969 Confidential.

(b) ADL Report NUWRES #12, "Control Modes and Acquisition

Probabilities for Torpedo MK 48 (U), "Contract No.
N00140-68-C-0278, January 1970, Final Report Confidential,
Technical Appendices (bound separately) Unclassified.

I
13

Arthur D Little, Inc

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Sacurity CUttiflcatton

DOCUMENT CONTROL DATA .R&D

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ONiaiNATlNa ACTIVITY CComeraMautllerj M. RIPORT IICURITV CLASSIFICATION
Arthur D. Little, Inc. UNCLASSIFIED
Acorn Park Mb. OROUR
Cambridge, Massachusetts 02140
RKRORT TITL«

Time Correction In Passive Ranging: Breakthrough or Bootstrap?

4. OCSCRIRTIVK NOTCS (Tfp* ol npmt anrf JltcilMiV* itel«c>

Technical Report 2
S- AuTHONiSI (KtSi mm», mlddl» MIUI, !••( nmrnm)

James M. Dobble

• ■ RKRORT OATK 7«. TOTAL NO- OF 7b, NO. OF RKFS

October 1970 13
S«. CONTRACT OR «RANT NO. M. ORIttlNATOR'S RKRORT NUNnSCRISI

N0001A-70-C-0322 ADL-72580-2
6. RROJCCT NO.

Sb. OTHKR RKRORT NO(SI (Any «Öl« nunban (hat aiar ba aaaifnad
Ihi« import)

Technical Report 2
10. DISTRIBUTION STATKMKNT

Each transmlttal of this document outside the agencies of the U.S. Government
must have prior approval of the Office of Naval Research Code 462 or the Naval
Underwater Systems Center/Newport.
II. SURRLKMKNTARV NOTKS It. SRONSORINO MILITARY ACTIVITY

Naval Analysis Programs

Office of Naval Research
Dept. of Navy, Washington P.C. 20360
I». ABSTRACT

A review is made of the method of time correction in passive ranging. It is

concluded that there is little, if any, support for the claims made for the
method.

f%f% FMM 41 M "T<i RKRLACKS DO

RKRLACK« OO FORM UTS. I JAM S«. «MICM It
DD • MTSSIA/O OMOLKT« FOR ARMY USK
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Time Correction
Passive Ranging

UNCLASSIFIED
tocnritv CUsslfleatlaii

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