In any given region of the ocean the distance for detecting noise sources is highly variable, due to changes in both the conditions for sound propagation and magnitude of external noise. Previously we limited the estimates of the detection distance to the most favorable natural conditions, i.e., those that determine the maximum achieved magnitude. In the present work an attempt was made to take into account the variations in the natural conditions. Unfortunately, we did not have statistical information on the variability in losses by sound propagation in these conditions at our disposal, but we obtained information on the repeatability of the wind speed. Therefore, having this information allowed us to judge how frequently situations occur with which in principle, a particular detection range may be achieved.
In the current appendix, estimates of a submarine's detection distance were made for a wide band signal processing. As we showed earlier, using such an approach, one may attain slightly larger detection distances for submarines in shallow water than we previously received. However, the main conclusions which were made in the previous work, remains in force. Existing technical possibilities do not permit all enemy's SSNs to secretly carry out protracted, continuous tracking of Russian strategic submarines. This secret tracking may lead to a collision of nuclear submarines with unknown consequences.
I_{s}(r,f)=I_{s}(r=1 m,f) T(r,f)  (A2.1) 
(A2.2)  
(A2.3) 
DT = 5 log[D t^{1} w^{1}] 
(A2.4)

(A2.5)


I_{n0} (f)=I_{n}(f) . [H(f)]^{2} . A(f)^{1} 
(A2.6)

Therefore, the problem of calculating detection ranges is reduced to determining
the regions of the values for r where the signal level exceeds the noise
level at a given value for the detection threshold.
SN>DT 
(A2.7)

SL (r=1 m, f) = SL (r=1 m, f=1000 Hz)20 log[f/1000] 
(P2.8)

Values for SL (r=1 m, f=1000 Hz) for different SSBN designs are shown in Table A2. Therefore, estimates on the detection distance will correspond to the lowest noise mode for submarines ("ultraquiet operation").
We mention that contrary to our previous work, we will not consider "additional" losses, instead suggesting that they equal 0. The "additional" losses factor is related to losses through technical implementation of signal processing algorithms, imprecise a priori knowledge on the environment for sound propagation, and operator errors. As we previously noted, "additional losses" can amount to 5 to 10 dB. If this factor is considered then the estimated results for detection ranges will actually correspond to a noisier mode, i.e., a patrol at speed of nearly 8 knots (see Appendix 1).
Nevertheless, it is correct to suggest that with wind speeds greater than 1015 m/s, propagation losses at frequencies greater than 2 kHz will be even higher than the results in Figure A2.1. This is due to stronger damping of sound reflected from a choppy (agitated) surface. Therefore our estimates of the detection distance with great wind speeds will be somewhat too high.
Our model also assumes that the noise level was proportional to the
wind velocity squared and the form of the noise spectrum did not depend
on it [Wille, 1985].
For 200 Hz < f < 500 Hz,
NL (f) = NL (f=200 Hz, V=2.5 m/s) + 20 log [V/2.5] For f > 500 Hz

(A2.9)

We note that these estimates of the sonar amplification coefficient are somewhat higher in comparison to those which may be realized in practice. This factor will also "work" on overestimating the estimates of detection ranges.
Model of the frequency filter
The frequency characteristic of the filter, by which we achieved the
maximum value in the signal/noise ratio, may be expressed in the following
equation [Burdic, 1991]
[H(f)]^{2} = I_{s}(r,f). A(f)^{2} . I_{n}(f)^{2} 
(A2.10)

The integration time was maximized and was equal to nearly 100 s. This estimate corresponds to conditions when the distance from the source to the receiver does not exceed 1020 km. With longer integration times at such distances it is not possible to disregard: a) changes in the relative location of the source and receiver in the process of registering the signal, b) changes in the conditions of sound propagation in the environment, and c) nonstationarity of ambient noise.
In practice, it is not possible to simultaneously achieve the maximum attainable array gain and signal integration time. Therefore, the choice of integration time that was made also results in an overestimated detection range compared to that which can be achieved in practice.
It is interesting to compare these results with the estimates of the maximum submarine detection range using narrow band filtration (see Table A3). Overall, both methods give estimates of the same order of magnitude. Evidently, in shallow water wideband signal processing is preferable for detecting third generation SSBNs as it indicates a larger detection range.
Table A3. Estimates of the maximum detection
range of Russian SSBNs in their patrol regions during the most favorable
weather conditions.
design number 



AV 611 



629 



658 



667 A 



667 B 



667 BD 



667 BDR 



941 



667 BDRM 



SSBN yr. 2000 



Figure A2.3 gives statistical data on observations made over many years on the repeatability of wind speeds during winter in the Barents sea area (this was limited to the 73^{o} and 77^{o} N and 25^{o} and 55^{o} E) [Wind and waves, 1974]. The repeatability of external noise levels and of the corresponding maximally achieved detection ranges were calculated using this data.
The meaning of the graphs, which are shown in Figure A2.4 (a) and (b),
is sufficiently straightforward. The xaxis is the maximum detection range
which corresponds to a given wind speeds V. The number on the yaxis is
the probability that the wind speed does not exceed V. For example, based
on Figure A2.4 (a), we can conclude that the detection range for the strategic
submarine design 667 B (Delta I) in shallow waters under favorable conditions
for sound
propagation will not exceed 40 km  in 15% of the naturally occurring
situations, 30 km  in 40% of the naturally occurring situations, and 20
km  in 92% of the naturally occurring situations.
To carry out secret tracking, an enemy nuclear submarine must constantly be in some "corridor" of distances to the target. The upper limit, obviously, is actually the detection range and is located below our estimates. The lower limit of this "corridor" is determined by two factors:
In this connection we stress the importance of the circumstances which will not depend on the enemy.
Firstly, during the tracking process, both the conditions for sound propagation and the level of external noise will change. Shallow water regions are characterized by fluctuations in propagation losses which reach up to 5 dB. In particular, an average detection range of 5 km will result in the upper limit "corridor" fluctuating from 23 to 89 km (model A). The tracking submarine will hardly be able to respond adequately to such changes during long periods of weeks and months.
Secondly, the strategic submarine may also change the mode of its movement. We propose that it will change to another depth. As a result the algorithm for the signal filtration will stop being optimal and the tracking submarine may lose its target. Frequently submarine collisions result from one of them changing their operating mode.
At small distances, in which tracking takes place, the integration time (the time needed for accumulating information on the signal in order to decide that a target is detected) is very important. Our calculations assumed an integration time of 100 seconds. Even if the tracked submarine moves at a minimum speed of 5 knots (2.5 m/s), during the time it takes to accumulate the signal, it has moved 250 m before changes in the target's behavior are detected. Obviously, with such a method for signal processing, attempting to tail at short distances is simply dangerous. Decreasing the accumulation time to 10 s results in the "dead distance" being decreased to 25 m, but by doing this the detection threshold is raised by 5 dB (see formula A2.4) and this leads to a reduction in the maximum achieved detection range. If in our computation the detection distance is 10 km, then with a 10 second accumulation, it is lowered to 6 km (model A). If it reached 5 km, then an increase in the detection threshold by 5 dB is equivalent to a decrease in the maximum distance to 23 km.
One of the biggest problems that occurs during tracking is localizing the target with the necessary accuracy. At short distances (less than 5 km), the precision for determining the distance to the target must be no worse than 0.20.5 km. To provide such accuracy the enemy will have to have good knowledge of the conditions for signal propagation in a specific region of military activity within a given period.^{(32)} Horizontal inhomogenities in the environment will lead to fluctuations in the conditions for signal propagation so that the lower limit of the "corridor" will also fluctuate because of changes in the accuracy of localizing the target.
The reasons cited permit the claim that an estimate of the detection distance at 10 km is the limit at which it is technically possible to realize continuous tracking, while ensuring covertness and guaranteeing the potential of avoiding collisions. If this is true then it is possible to see (see Figure A2.2a, A2.4a), that even in the most favorable conditions for sound propagation the "Los Angeles" class submarine can not execute this task.
[Miasnikov, 1993] E.V. Miasnikov, "Can Russian strategic submarines survive at sea? The fundamental limits of passive acoustics", Science and Global Security, 1994, vol. 4, p.213251
[Akal, 1980] Tancay Akal, "Sea Floor Effects on ShallowWater Acoustic Propagation," Bottom Interacting Ocean Acoustics, Ed. by Kuperman and F.B. Jensen, Plenum press, New York, 1980, pp. 557575
[Burdic, 1991] William S. Burdic, Underwater Acoustic System Analysis, Prentice Hall, Inc., New Jersey, 1991, pp. 428433
[Wille, 1985] Peter C. Wille, "Ambient Noise: Characteristics
of the Noise Field" Adaptive Methods in Underwater Acoustics, Ed.
by H.G. Urban, D. Reidel Publishing company, 1985, pp.1336
^{29)} See Burdic (1991) for details.
^{30)} Data were taken from Akal (1980).
^{31)} Large detection distances occur in open ocean due to so called convergence zones (zones, at which sound rays of a given source are focused and its audibility sharply increases). A term "far zones of acoustic illumination (DZAO)" is frequently used in the Russian literature as well. Typically, the convergence zones are located at intervals of 55 to 65 kilometers. The width of the first zone is roughly several kilometers, the second zone is two times wider than the first, and so on until eventually, at ranges of several hundred kilometers, the zones overlap and become indistinguishable.
^{32)} Incidently, one of
the goals of systematic visits by foreign SSNs to the seas around Russian
territory is gathering hydrophysical data on specific areas and the seasonal,
temporal, and spatial variability of this data.