Benchmarking Results 

 


   In this chapter we present  the results of  benchmarking SWAMP against the two most used computer models for sound propagation in underwater acoustics.  The first model, the Range-dependent Acoustic Model  (RAM)  by  Collins,  is based on the split-step Padè solution of the parabolic equation.   The split-step Padè algorithm employs a rational function approximation of the differential operators and is claimed to be the most efficient PE algorithm that has been developed.  It allows larger range steps in comparison with  other PE models for a marching-in-range procedure that gives it  much better computational stability and performance in time.  The second model,  KRAKEN by Porter,  represents the normal mode model with a finite-difference approach for solution of the depth-separated modal equation.   KRAKEN computes the solution of range-dependent problems in the coupled or adiabatic approximation.

The first benchmark problem contains range-dependent bathymetry and a sound speed profile, which varies with depth.  The ocean bottom profile (bathymetry) is shown in  Figure 4.1. It includes a range-independent environment with a 1000 meter deep water-column from 0 to 35 km and a relatively steep up-sloping  ocean bottom from 35 to 45  km with  depth changes from 1000 m to 200 m.  The sound speed profile is shown in Figure 4.2. The environment has a bottom duct. The receiver depth is 30 m. The source is  placed at depths of  25 m and 400 m. The test frequency is 100 Hz.

Transmission loss () is the standard normalized measure of the change in signal strength with range in underwater acoustics. It is defined as the ratio in decibels between the acoustic intensity at a receiver point and the intensity at 1-m distance from the source:

                  (1)

In Eq. (1) the fact that the intensity is proportional to the square of the pressure amplitude has been used.  Usually, the reference pressure at 1-m distance is determined by  the approximation for a  source in free space:

   ,                                                (2)

because all  waveguide effects start at distances not less than the wavelength. Sometimes a minus sign is put in front of the logarithm function in (1), so the transmission loss becomes a  positive value for the far field.  All the transmission plots in this paper have been calculated using formula (1), so that smaller values of the transmission loss (large negative values) correspond to smaller signals, i.e., larger loss in the propagation  to the given point. 

Figures 4.3(a) and 4.3(b) show plots of the transmission loss versus range obtained with SWAMP (solid line) and RAM (dashed line).  The two parts of Fig. 4.3 - (a) and (b) - represent the same continuous calculations but are separated into the range-independent part of the propagation path (up to 35 km) and  the range-dependent one (35-50 km) for better visibility. The  source depth is 25 m. The number of  Padè terms np in the rational function approximation of the differential operator in the PE-model RAM is 4.  Figure 4.3 (a) shows no visible discrepancies in the  match between the transmission loss predicted by SWAMP and RAM for the  range-independent  bathymetry starting around 8 km. The later plots of benchmarking SWAMP with the other propagation model, KRAKEN,  show a good agreement in the near field. Thus, mismatching between SWAMP and RAM results in the near zone (up to 8 km) can be  probably related to the details of a new self-starter for a point source implemented in RAM.  RAM employs an indirect method for taking the continuum modal spectrum  into account, which is important in the near field. SWAMP does not take the continuum modal spectrum into account.  It can also contribute into the observable difference in the transmission loss prediction in the near zone.  The benchmark results for the range-dependent bathymetry shown in Fig. 4.3(b) are in quite good agreement,  which can not  be expected to be  perfect because of the different types of implemented approximations and a fair number of ambiguous input parameters required for any PE model, such as a number of  the Padè terms, the type of  stability constraints, the reference sound speed, the false bottom depth, the vertical sampling mesh, etc.. Both models have used the same minimum marching step size in range - 10 m.  It is important to note that the steep up-slope bottom problems are considered  to be among the most difficult problems for numerical simulation because high-order modes become lost.  The loss of  modes is equivalent to the energy transfer into the bottom layers.  This effect can be accurately accounted only by including the continuum modal spectrum. The benchmark results for the down-slope bottom and  more gradual up-slope bottom cases show the much better agreement between the two models. 

Fig. 4.4 shows the use of more Padè terms in  RAM, namely np=8, gives better agreement for the up-slope  propagation as  is indicated in Fig. 4.4 (b).  The phenomenon responsible for a more oscillatory interference pattern for the PE results is probably related to the so-called Gibbs oscillations, since different quantities are conserved across horizontal  and vertical interfaces in PE modeling with sloping interfaces.  The basis of the problem is the choice of the vertical interface boundary conditions for range-dependent problems. It was shown that solutions which match pressure alone or only particle velocity show serious deficiencies.  Fairly accurate solutions for most PE problems in ocean acoustics can be obtained by conserving   , which is energy conserving in a forward sense for horizontally propagating sound. But one still has to be aware of Gibbs oscillations. More numerical modeling with different RAM input parameters is required to understand the correct mechanism causing these additional wiggles in the interference pattern.

The computational time taken by SWAMP to do the calculations presented in Fig. 4.3 and 4.4 on a Pentium-133 is 53 seconds.  For RAM, the computational time is very dependent on the number of Padè terms.  It takes  2 minutes 8 seconds to perform the calculations with np=4 (Figure 4.3) and  4 minutes 14 seconds with np=8 (Figure 4.4).

Wide-angle propagation is illustrated in Fig. 4.5(a),(b), when the source is placed at 400 m depth and the transmission loss is measured at 30 m depth.  The fit between the two models is amazingly good.

                                                                   

 

The benchmark results for SWAMP and the coupled normal mode model KRAKEN are  presented in Fig. 4.6(a) and (b).  As  was pointed out before, we can see good agreement for the near zone which  was not the case for RAM.  All the KRAKEN results were obtained on an  SGI computer.  Therefore I can not make any reasonable  computational time comparisons for the range-independent case. KRAKEN uses a finite-difference scheme for the solution of the modal equation, which  slows it down considerably relative to SWAMP.  KRAKEN was not intended for range dependent calculations with the coupled mode approach, so it is very difficult to run for  range-dependent cases.  One has to create  environmental input files for as many range-independent subregions as one wants to have. There is  no literature on an optimal procedure for the determination of optimal step-size in range for this model. The adiabatic approximation for the KRAKEN model produces very poor results for this type of environment after proceeding  half-way through the sea mount region (after 42 km) as can be seen in  Fig. 4.7.