MEASUREMENTS OF MODAL ATTENUATION USING BROADBAND SOURCES IN THE NEW ENGLAND MUD PATCH

Measurements of underwater acoustic signals were made on a bottom-mounted horizontal line array during the Seabed Characterization Experiment (SBCEX) in the New England Mud Patch south of Martha’s Vineyard in about 70 m of water. The signals were generated by Signals, Underwater Sound (SUS) charges detonated at various locations in the experimental area at a depth of 18 m, during nearly-isovelocity conditions. The broadband signals were analyzed for modal arrival time and amplitude using time-frequency techniques. Ratios of modal amplitudes at the individual hydrophones were used to estimate the modal attenuation coefficients. Hence, these estimates are independent of any uncertainty in the frequency-dependent source level of the SUS charges. These coefficients are directly related to the depth-dependent sediment attenuation profile. Posteriori error analysis provides averages and standard deviations for the estimate of sediment attenuation as function of depth. The frequency bands of interest range from 30 Hz to 120 Hz for modes one through four. We compared our estimates of sediment attenuation with historical measurements. We determined the frequency exponent of mud to be 1.9 with an attenuation coefficient on the order of 10 -5 to 10 -4 decibels per meter. This work was supported by the Office of

. ......... 13 Figure 6. Illustration of crossover between generations during optimization (Tang et al.,1996 ) Figure 7. Above, the binary encoding of the parameters for each generation is shown with mutated parameters indicated by a one (Tang et al.,1996 ) (Potty et al., 1999 and2002). They estimated bottom attenuation coefficients based on modal amplitude ratios using SUS data from a single hydrophone. The use of amplitude ratios eliminates the need to know the exact source level of Signals, Underwater Sound (SUS) charges. Genetic algorithms were used to optimize the search parameters including sound wave speed through sediment. The inversions compared well to core data collected in the experimental area giving confidence in the method. Lastly, a sensitivity study was conducted to evaluate the uncertainty in the parameter contribution to the results.
Attenuation is a reduction of sound intensity as an acoustic wave propagates in range. As the sound wave propagates through a medium, some of its energy is converted into heat. This frequency dependent effect is called sound attenuation or absorption. Other factors such as scattering along interfaces and within the volume can cause additional losses. Most acoustic inversion methods estimate the total losses (effective attenuation).
For comparison purposes, the attenuation of seawater must be considered. The attenuation of seawater is also frequency dependent and important for evaluating the feasibility of sediment attenuation coefficients for this experiment. The equation for the attenuation of water is (Jensen et al., 1994) (1.1) A plot of this equation from Jensen et al. (1994) is displayed in Figure 1. We studied lower frequencies, indicated by the red circle on the plot. The surface sediment attenuation coefficient is expected to be higher than 0.00002 dB/m of seawater at ten degrees Celsius (Ainslie, 1998).
Mud at the water and sediment interface boundary is thought to have an attenuation coefficient between sand and seawater (Pierce, 2015). This thesis investigates the attenuation coefficient of the mud layer of the New England Mud Patch from an experiment conducted in March 2017. Figure 1. Attenuation coefficients plotted as a function of frequency for seawater (A') and fresh water (B'). The frequency range examined in this experiment is circled in red (Jensen et al., 1994).

Thesis Overview
This thesis consists of six chapters. Chapter 2 describes the equations and theories that are the basis of the analysis. There is a brief description of two sediment material property theories for computing intrinsic attenuation. Intrinsic attenuation is the attenuation related to damping from sediment mechanics, where effective attenuation is the overall sound reduction of a source in an environment. Normal mode theory is described in detail with a derivation for acoustic pressure. There is a section describing a way to compute the separation between source and receiver using group velocity developed by Potty et al. (2003). Genetic algorithms and their applications to optimization are introduced. Lastly, the equations for analysis a posteriori on the genetic algorithm output are presented.
Chapter 3 gives an overview of the Seabed Characterization Experiment (SBCEX) and the equipment relevant to this thesis. The environmental conditions and site composition are described.
Chapter 4 details the analysis performed using the theories from Chapter 2. A flow diagram incorporating the variables and equations of the attenuation computation is available in Figure 17 to highlight the optimization steps.
Chapter 5 presents the estimated parameters and resulting attenuation as a function of depth in tabular and graphical form. The attenuation coefficients as a function of frequency are directly compared to previous work on sandy sea bottoms.

Attenuation Derived from Sediment Properties
Intrinsic attenuation is derived from sediment properties. This attenuation describes the damping of sound waves due to friction. Sound vibrates the grains and trapped water throughout marine sediments, resulting in a reduction of intensity.
There are a number of geoacoustic models available for deriving seabed acoustic properties from direct sediment measurement of physical parameters. One of the widely used models is the Biot-Stoll model. Biot deduced equations for modeling poroelastic properties as a "granular solid forming a porous skeleton, which is filled with fluid" (Badiey, 1998). Stoll specifically applied these properties to the seafloor to predict frequency-dependent attenuation. Because the voids in the sea floor sediment layers are interconnected, there is greater observed attenuation resulting from "not only intergranular friction, but also the additional losses owing to the viscosity of the fluid." The intergranular frame squeezes the local fluid, causing damping, resulting in frequency dependence (Stoll, 1985). The Biot-Stoll model takes into account thirteen different parameters to compute intrinsic attenuation. The parameters are listed in Table 1 (Badiey, 1998). The compressional wave attenuation of low frequencies for 'highly permeable sediment' was found to increase at f 2 , where f is the frequency in Hz (Buchanan, 2005).
The parameters for the Biot-Stoll model are listed in Table 1 (Badiey, 1998). Table 1. Biot-Stoll input sediment parameters for compressional wave attenuation modeling (Badiey, 1998 (Buchanan, 2005). He reported attenuation to be much more dependent on the "intergranular dissipation" between grains, loosely calling it "friction" for simplicity. The marine sediment is treated as a bulk fluid, not as the frame Stoll suggested. Particle size, roughness, and packing ability are the driving factors in addition to porosity and density that contribute to the attenuation of surface sediments. Consequently, the Buckingham model requires significantly fewer parameters than the Biot-Stoll model (Buckingham, 1997).
While these theories are not directly implemented in this thesis, the background provides a broader understanding of attenuation.

Normal Mode Theory
Let's start with the acoustic wave equation (Urick, 1983). (2.1) Converting to cylindrical coordinates to represent spreading from a point source in a waveguide, the solution to the wave equation is the following inhomogeneous Helmholtz equation. Symmetrical cylindrical spreading is assumed about the z-axis, so the wave equation no longer requires solving for θ (Jensen, 1994).
Harmonic time-dependence is assumed for ambient environment pressure p(r,z).
The point source is represented by the Dirac delta function δ, triggered in a horizontally stratified water column of separation r from the receiver. Density ρ(z) and sound speed c(z) are a function of depth z. The variable k is the wave number in rad/m, similar to spatial frequency, and is equal to .
A solution is obtained using the separation of variables technique. The equation is broken into a solution for depth and a solution for range. The variable p(r,z) is solved in the form . The solution is unforced and therefore equal to zero. (2. 3) The range term ϕ is to the left of the plus sign and the depth term ψ is to the right in The depth equation is solved as a Sturm-Liouville problem as outlined by Jensen et al. (1994) to obtain the eigenfunctions to solve for ψ. (2.4) Please note the addition of the κ rm term and change of ψ(z) to ψ m (z). This reflects the variability of the separation constant in the modal equation required to follow the Sturm-Liouville problem. The variable ψ m (z) is an eigenfunction and describes mode shape. The modes must be orthogonal. κ rm is an eigenvalue that is the horizontal propagation constant, similar to frequency of vibration. The κ rm values are sorted in descending order and must be discrete, real, and positive. The modes form a summation that is used to solve for pressure.
The following is the range portion of the separated variable equation. (2.5) The solution for ϕ n is the Hankel function of the first kind (Jensen et al., 1994). (2.6) .
This form the Hankel function is intended for use as and more than a wavelength λ away. The wavelength equation is λ , where c is nominally 1500 m/s and frequency f ranges from 20 to 120 Hz. Any separation greater than 75 m is greater than a wavelength away from the source (Jensen et al., 1994).
The solution for pressure is a set of modes. (2.7) The resulting modal pressure is the following equation (Potty et al., 2003). (2.8) This pressure computation is used in Chapter 4 for computation of the theoretical pressure resulting from the optimized parameters of the genetic inversion.
In an ideal stratified waveguide with constant sound speed and density, perfectly reflecting pressure relief surface and perfectly reflecting hard bottom, the modes are sine waves. The first four modes in a 100 m deep ideal waveguide are shown in Figure 2.
Figure 2. The first four normal modes in an ideal waveguide of depth 100 meters. (Jensen et al., 1994).
Dispersion is the effect of phase velocity separation due to the frequency dependence of wave velocity and is only relevant to broadband sources. Each mode has a unique phase or wave front velocity. Phase velocity v m , is a function of angular frequency and is the ratio of frequency to the wave number of the mode where , is the wave number, and m is the mode number (Frisk, 1994).
Phase velocity is the speed at which the phase front of a wave propagates and group velocity is the speed at which energy travels. Group velocity can be represented as the speed at which the envelope will move as shown in Figure 3. Figure 3. The wave envelope V g , outlines group velocity propagation and v m , is the phase velocity (Frisk, 1994).
Group velocity is related to phase velocity by (2.10) Group velocity will always be less than the sound speed, where phase velocity is always faster (Frisk, 1994). This is shown in Figure 4 depicting dispersion curves of an ideal waveguide. Figure 4. Dispersion curves of modes one through three in an ideal waveguide (Jensen et al., 1994).
Modal attenuation can be found as a function of the intrinsic attenuation (Rajan et al., 1987).
(2.11) β m is the modal attenuation coefficient of mode m, α(z) is attenuation as a function of depth, k(z) is the wave number of the profile also as a function of depth, ψ m is the mode shape, and κ m is the modal horizontal propagation constant (Rajan, 1987).

Computation of Mode Travel Time Dispersion
The source-receiver range is computed using the modal travel time between the individual modes. This data is calculated from the time-frequency diagram of the broadband SUS data. Based on the group velocities calculated using Eq. 2.10, the modal travel time differences can also be predicted based on the assumed geoacoustic model. Using the observed and predicted modal travel time differences, the sourcereceiver range can be estimated (Potty et al., 2003). We assume that the ocean acoustic waveguide is stratified and range independent.
First, the times of arrival for the modal amplitude peaks are selected from the dataset to get ∆T ii and ∆T ji , differences in arrival time along the modes i and j. (2.12) In the brackets on the right hand side of Eq. 2.12, the differences between theoretical group speeds of different frequencies within the same mode are compared to a reference frequency. In this equation, i is the mode of interest for the range specified r, f H is a reference frequency different from f, and V g is the theoretical group speed from the inversion, and ∆T ii (f) is the observed difference in time of arrival for the different experimental frequencies of a single mode.
The arrival time difference and group velocity difference between two different modes, i and j is shown below. (2.13) The final step is to solve for range by comparing frequencies and times of arrival for different mode pairings. To start, the difference between the left hand side of Eq. 2.12 and Eq. 2.13 compares the travel time difference ∆T(f). (2.14) The difference between the two bracketed right side equations in Eq. 2.12 and Eq.

is K t (f).
(2.15) Combining these two equations, the relationship is linear in the form , where the slope is range r=m. (2.16)

Overview of Genetic Algorithm
The genetic algorithm is a method for solving both constrained and unconstrained optimization problems that is based on natural selection, the process that drives biological evolution. The method consist of three operations: selection, genetic operation, and replacement as highlighted in Figure 5. First, a population is generated randomly within the user specified range for the model parameters. The individuals in the population are evaluated and scored with a fitness value for each generation.
Figure 5. Flow diagram of genetic algorithm optimization .
Uniform crossover means each parameter of the parent population is randomly swapped into the next generation. Figure 6 below shows the crossover points for a set of parents. Applicable to the experiment, each exchange of the parameters returns a slightly different attenuation value than the previous generation for evaluation. The fittest values are passed on, optimizing the dataset. Figure 6. Illustration of crossover between generations during optimization (Tang et al.,1996 ).
In addition to the population exchanges, the offspring can mutate as well. A binary string is generated the same length as the number of parameters being evaluated. A zero indicates there is no exchange, while a one means that parameter evolves for the next generation. Figure 7. Above, the binary encoding of the parameters for each generation is shown with mutated parameters indicated by a one (Tang et al.,1996 ).
In this analysis, each new population was generated with the same number of individual members to recreate the same population size. To accelerate the algorithm, an elitist strategy was implemented where only the fittest individuals advanced to the next generation.
To maintain diversity in the subpopulations, 20 percent of each population was migrated at random over the course of 20 generations. The migration uses a complete net topology, meaning the individuals were unrestricted in their exchanges between populations as shown in Figure 8. Figure 8. Complete net topology with unrestricted migration during genetic algorithm selection (Tang et al.,1996).

A Posteriori Analysis
During the genetic algorithm optimization, the evolving populations were recorded along with their fitness value for post processing. Based on the procedure outlined in Gerstoft et al. (1994) a posteriori statistics can be computed using the model parameter samples. The error analysis procedure is briefly outlined in this section. For a detailed discussion refer to Gerstoft et al. (1994). The probability associated with the model parameters are given by (2.17) In Eq. 2.17, σ(m i ) is probability, i is each parameter in a population of m i members sorted by fitness. Fitness is equal to .
For our analysis, i is equal to parameters 1 to 13 that were optimized in this experiment. The a posteriori mean is computed using the summation (2.18) Covariance is computed using the following equation. (2.19) The square root of the covariance matrix diagonal is used to compute the attenuation coefficient standard deviation for each layer of sediment.

Site Description
The experiment site was surveyed and cores were collected and analyzed in the prior year (Goff, 2018). Figure 9 shows an overlay of grain size on the site's coastal outline map. Water depth in this area is nominally 70 m. The area under study is known as the New England Mud Patch. This paper concentrates on the SUS data in the northeast corner of the New England Mud Patch in Figure 9.

Signals, Underwater Sound (SUS) Description
The intent of the experiment was to determine attenuation values from acoustic data collected in conjunction with the detonation of explosives in the water column.
There were two types of explosive sources, the first being Signals, Underwater Sound (SUS). SUS charges were designed as a controlled broadband source (ONR, 1975).
The 31 g MK 64 SUS charges were pressure sensitive to explode at 18 m.
Approximately 200 SUS were deployed at stations in a radial pattern about the array as shown in Figure 10. Five SUS charges were released consecutively at each drop location about 15 seconds apart. A small subset of SUS charges are the sole focus of this report. The other broadband source in the SBCEX experiment was the Combustive Signal Source (CSS). A CSS is a "high-intensity low-frequency pulse, followed by several weaker bubble pulses" (Bonnel, 2018).

Horizontal Line Array Description
Data were recorded on a horizontal line array oriented from west to east as shown in the cross section of the experiment set up in Figure 11. Locations of the three hydrophones, phone 1, 32 and 64 used in this experiment are shown in Figure 12. There was roughly a kilometer separation between hydrophone 1 and 64.

Environmental Conditions
The nearest maintained National Oceanographic and Atmospheric Administration ( Figure   14. The mud layer was determined to be 6 m thick. The characterization of the uppermost layer is the focus of this thesis as its properties heavily influence sound propagation in the ocean. Figure 14. The density profile and thickness for the New England Mud Patch sediment boundary layer is shown above. The image is not to scale (Goff et al., 2018).
Deep layer 1, which is 15 m below the water and sediment boundary, is assumed to be sand. Attenuation values in this layer also will be computed. These values are expected to represent sandy sediments and can be used to compare the inversion results to historical data.   Each SUS detonation was extracted with surrounding data. The raw data were plotted for a quality check before being subjected to data analysis. Since there is a kilometer separation between the hydrophones examined, the difference in the pressure wave arrival time varied. Stations 49 and 54 were selected because of their location immediately west of the array for the least range dependent path. The staggering of arrival time is shown in Figure 16. This plot captures a five second span of time across the hydrophones. Hydrophone one at the top was the farthest west and receives the pressure wave first. The data was reviewed for voids during recording and excessive noise before analysis. Figure 16. Above, the difference in time of arrival is plotted for the different hydrophones. The hydrophone number is noted to the right. Figure 17 captures the data flow explained in detail in the following sections.

Data Processing Overview
The techniques and code were developed by Potty et al. (2003). Matrix Laboratory (MATLAB) software and functions were used for data analysis.
First, the experimental data scalograms were examined. The amplitudes of individual normal modes were selected from a frequency slice that focused on bands with significant modal amplitude. This is noted as the 'observed' data in the flow diagram. The sound speed, density, and frequencies of interest were assimilated into a matrix for calculating the eigenvalues and eigenfunctions. Next, an acceptable range of theoretical parameters was established, noted as 'predicted' in the flow diagram.
These variables were used to compute modal attenuation β m from Eq. 2.11 that was required to create theoretical amplitude ratios for comparison to those measured by the experiment. The difference between the experimental and observed ratios were minimized and optimized by genetic algorithms. Finally, the cost function was used to determine the optimal parameters.

Time-Frequency Analysis for Modal Amplitude Selection
Data from each SUS charge was imported individually into MATLAB for analysis. First, the data was converted to Pascal by dividing by 10 6 . The time series was then normalized by the sampling frequency. A low pass, fourth order Butterworth filter was designed using a cutoff frequency of 500 Hz by the MATLAB function butter.m. The data was filtered using the output coefficients. The data was resampled at a rate of 500 Hz.
Next, the analytic signal was derived by taking the Hilbert transform of the real portion of the filtered data signal from the SUS. Morlet wavelet analysis was computed for the signal at increments of 25 samples. Important to note with wavelet transform is reduced high frequency resolution (Wan et al., 2009). The upper and lower frequency bounds were 0.001 to 0.5 in radians. The scalogram is displayed as a color map using the MATLAB pcolor.m function. Figure 18 of the scalogram shows intensity as a function of time on the x-axis and frequency on the y-axis, and amplitude was normalized so the peak corresponds to 0 dB.
From the scalogram, the frequency cross-sections of interest were used to extract peak amplitudes in bands of significant modal analysis. The nulls were avoided as shown in Figure 19. The modes are grouped from the lower left corner, starting with mode one through mode four to the upper right in Figure 19. The higher order modes are not as evident and were not considered in this research. There may be some contribution of these higher modes in the lower modes because they were not accounted for. Figure 19. Scalogram of the fifth and last SUS deployed at Station 49, which is the same image as Figure 18. The white dotted lines note the time-frequency slices what were examined for amplitude selection.
The time series slice at each frequency, designated by a white dotted line in Figure 19, was evaluated for modal amplitude selection shown in the upper portion of Figure 20. The maximum amplitude for each mode was selected and recorded. The amplitude is the y-value in the time series in Figure 20 and noted in Table 3. These amplitude values were compared as a ratio for each hydrophone and used to evaluate attenuation.
The MATLAB function oct3dsgn.m was used to evaluate the filter coefficients based on the center frequency specified by frequency amplitude correlation on the scalogram. The decimated sampling rate and an order of three were input parameters.  Occasionally, the amplitude at hydrophone 64 is higher than the amplitude at hydrophone one although it is farther away from the source. For example, mode two for hydrophone 64 was recorded at 0.3771 at the frequency 53.1 Hz, while mode two for hydrophone 1 and 32 are lower in amplitude in Table 3. A calibration value was not applied to each hydrophone, resulting in arbitrary amplitude values. However, because the analysis computes ratios at each hydrophone individually, this discrepancy was inconsequential.

Calculation of Experimental Modal Amplitude Ratios
The modal amplitudes entered into Table 3 were copied into the calcAmpRatio.m script listed in Appendix B to compute the amplitude ratios. These experimental values were compared to the predicted data generated by the genetic algorithm optimization. The amplitude ratios were computed at a single hydrophone, which made the computations range independent. The modal amplitude ratios used in the computations for each SUS were the following:  The ratio of Mode 1and Mode 2.
 The ratio of Mode 2 and Mode 3. The mode ratio sets are d obs in Figure 17.

Parameter Descriptions for Genetic Algorithm Computation
The atten_opt.m function in Appendix B, establishes the parameters for the genetic algorithm computations. Genetic algorithm optimization was performed using the standard settings of the Matrix Laboratory Genetic and Evolutionary Algorithm (MATLAB GEA) Toolbox compiled by Hartmut Pohlheim (2006). A description of genetic algorithms is available in Chapter 2.
The inputs into the genetic algorithm were the upper and lower bounds of the parameters under evaluation in Table 4. For the first population generation, a matrix of random variables from zero to one was multiplied by a uniform distribution between the upper and lower bounds of each variable. This created a randomized starting population generation that was evaluated for fitness, where only the best individuals were passed to the next generation. The mutation probability setting was for the 13 parameters being evaluated.
In this thesis, the A and n parameters were optimized to compute the attenuation profile as a function of depth α(z) for each sediment layer. The effective attenuation coefficient is represented by the relation (4.1) where A and n are depth dependent constants, f is frequency in kHz, computed for each frequency utilized during experimental data amplitude ratio selection, and α(z) is in nepers per meter. To convert α from nepers per meter to decibels per meter, the following equation is used where A 0 is the root mean square amplitude at x = 0 (Jensen, 1994).

Calculation of Theoretical Modal Amplitude Ratios
Recall the compressional wave pressure Eq. 2.8 from Chapter 2 for a single hydrophone. This equation sums the normal modes M to represent the far field acoustic pressure, where r is range, z is the depth at the receiver, is the depth of the source, ρ is the density at the source, is the mth mode shape, is the horizontal wave propagation constant, and is the modal attenuation coefficient from Eq. 2.11.
The modal attenuation coefficient relies on the optimized parameters from the genetic algorithm in Section 4.4 for computation of effective attenuation α. A theoretical amplitude pressure needs to be computed for each mode compared in the amplitude ratios of Section 4.3. The theoretical modal amplitude ratio is the following. (4.3) In Eq. 4.3, m and n are two different normal modes at the same hydrophone, ρ is medium density, r is the source to receiver range, z is the receiver depth, z s is the source depth, is the mode shape for the mode under examination for modes m and n, κ is the horizontal propagation constant of each mode, and β is the intrinsic modal attenuation coefficient obtained by inverse methods. The amp_ratio.m script in the Appendix B computes the predicted modal amplitude ratio.
The computation of the variables required to obtain the theoretical amplitude ratio are described in the remainder of this section.
The mode shapes ψ m are computed first and substituted into the β m from Eq.
2.11. Code called zmode.m from the Ocean Acoustic Library archive was used (Ocean Acoustic Library). The Ocean Acoustic Library script solves for the following system equation. Lastly, the range r between the array hydrophone and the SUS charge is computed using the script sw_dist.m (Morgan ,1992). The first input to the function is the SUS latitude as provided by the waypoints in Appendix A and the hydrophone latitude. The second inputs are the respective longitudes. The output is separation in kilometers.

Error Minimization
The predicted amplitude ratios are then compared to the observed amplitude ratios for each frequency. The difference between predicted and observed is the error and is given by In this equation, e is the error vector, is the observed amplitude vector by frequency, and is the predicted amplitude vector by inversion matched to the observed frequencies.
The error e for each ratio from the different SUS was assimilated into an overall error to evaluate the current population computed by the genetic algorithm.
The cost function, also known as the objective function, was used to quantify the fitness of the parameters under evaluation by minimizing the cumulative sum of the square of the residuals.
Each station was combined into a joint optimization problem. Both experiment Station 49 and Station 54 were evaluated as a whole using the cost function, summing the squares of the five SUS charges and the resulting five sets of amplitude ratios at a single hydrophone. The best and average of the sets of individual parameters are compared in a posteriori analysis and to historical data in Section 5.3.

RESULTS
The goal of this research is to establish a depth dependent attenuation coefficient of the mud layer in the New England Mud Patch. Equation 4.1 is evaluated using the estimated A and n parameters output by the genetic algorithm as a function of depth.

A Posteriori Analysis of Depth Dependent Attenuation
Throughout the genetic algorithm optimization, the population of the evolving generations was saved. These were then used to calculate the a posteriori mean and standard deviations of the model parameters as outlined in Section 2.5. Table 5 summarizes the individual parameter results and computed standard deviations. Table 5. Summary of a posteriori statistical analysis for each parameter on all SUS for Stations 49 and 54.

Parameter Description Mean Value and Standard Deviation
Unit 1 A of layer 1 (mud) 0.0054 ± 0.01 n/a 2 n of layer 1 (mud) 1.9 ± 0.14 n/a 3 A of layer 2 (sand) 0.0283 ± 0.01 n/a 4 n of layer 2 (sand) 1.4 ± 0.22 n/a 5 A of layer 3 (deep layer 1) 0.0190 ± 0.01 n/a 6 n of layer 3 (deep layer 1) 1.4 ± 0.16 n/a 7 Depth of source 17.1 ± 0.2 m 8 (hydrophone 1) Receiver depth increment 70.7 ± 0.6 m 8 (hydrophone 32) Receiver depth increment 70.5 ± 0.5 m 8 (hydrophone 64) Receiver depth increment 70.7 ± 0.5 m 9 A of layer 4 (deep layer 2) 0.0309 ± 0.02 n/a 10 n of layer 4 (deep layer 2) 1.5 ± 0.18 n/a 11 A of layer 5 (basement) 0.0357 ± 0.02 n/a 12 n of layer 5 (basement) 1.6 ± 0.17 n/a 13 α of layer 5 (basement) 0.0002 ± 0.0001 dB/m The results of depth dependent attenuation are plotted for 50 Hz as shown in Figure 21 and Figure 22. Figure 21 displays the best genetic algorithm estimate in that they have the lowest misfit for the three locations on the array. Each line on the plot is an independent hydrophone inversion. The average results in Figure 22 correspond to the a posteriori mean estimate and are consistent with the best estimate, merely dampened. Since each hydrophone trace follows a similar shape among the group, there is confidence in the results. The standard deviation is roughly ten percent of the attenuation coefficient, indicating a good estimate as well. Figure 23 and Figure 24 demonstrate the frequency dependence of the sediment profile. The effective attenuation coefficients at the frequencies 30, 75, and 120 Hz are compared on the same axis. The standard deviation is included to the right. Figure 23 displays the best genetic algorithm outputs and Figure 24 is the a posteriori mean computed as discussed in Section 2.5. The depth attenuation coefficient profiles are available in Appendix C for Station 49.
Within each of the depth dependent attenuation figures, the first 6 m are expanded in the inner box in red, representing the mud layer. The mud has low attenuation that could not be read on the larger range. Figure 21. Above are the genetic algorithm lowest misfit parameters and standard deviation for each layer. The mud layer is expanded in the red box. Figure 22. Above is the a posteriori average of the 50 lowest misfit parameters for each layer and standard deviation. The mud layer is expanded in the red box. Figure 23. Above are the genetic algorithm best output parameters demonstrating attenuation dependence on frequency. The mud layer is expanded in the red box. Figure 24. Above are the a posteriori average output parameters demonstrating attenuation dependence on frequency. The mud layer is expanded in the red box.

Range Verification
Range verification using group velocity was used to independently verify the assumption of range independence and bottom model. This technique is described in Section 2.3.
First, the times of arrival for the modal amplitude peaks are selected as shown in the scalogram (left panels of Figure 25 and Figure 26) for the first SUS at Station 49 and 54, respectively. The selections are shown as a white asterisk for input into Eq.2.14. Similar to the frequency slice technique in Chapter 4.3 where nulls were avoided, the modes were noted for time of arrival and frequency at the peak amplitudes in two vectors.
The plot of Eq. 2.15 versus Eq. 2.14, results in the right panels of Figure 25 and Figure 26, where the range r is the slope of the best fit line. of the receiver will also be reflected in the discrepency.

Comparison to Historic Data
We computed effective attenuation coefficients α(z) for a frequency range of 30 to 120 Hz and overlaid the results on Figure 27. Figure 27 is a compilation of attenuation values of sand published by Zhou et al. (2009). Attenuation coefficient estimates corresponding to deep layer 1 are shown in yellow triangles and these values match the published data well. Deep layer 1 is assumed to be sand based on core data (Goff et al., 2018). Although the deep layer 1 slope is slightly different from the historical data, it is consistent with the previous inversions from the nearby Primer experimental location, shown in black triangles (Potty et al., 2003). The frequency exponent of the attenuation equation for deep layer 1 (sand) was 1.4 in Table 5. The attenuation coefficient for sand was approximately 10 -3 (dB/m), varying slightly with frequency.
The mud layer attenuation, shown in brown triangles, is an order of magnitude lower than sand. The frequency exponent of the attenuation equation for mud averaged about 1.9 as shown in Table 5. The mud attenuation coefficient values range from 10 -5 to 10 -4 (dB/m) depending on the frequency.
Included on Figure 27 are a few reference points for seawater at 0.5, 1, and 2 kHz depicted as blue triangles. The mud layer split the difference between the sand and water, but is slightly closer to sand. The average attenuation coefficient is computed for the entire 6 m layer of mud, meaning some of the more dense sediment at the bottom of the layer is averaged with the more fluid upper mud layer. Figure 27. SBCEX average attenuation coefficient α(z) for Station 49 and 54 overlaid published experimental results. The seafloor surface mud layer is shown in cyan and the deep layer one is in magenta (Zhou et al., 2009).

Sensitivity Analysis
A sensitivity study of attenuation with respect to depth was conducted a posteriori. Figure 28 shows the effect of a ten percent perturbation in α(z) at various depths on the cost function that was then normalized. (5.1) In Eq. 5.1, E is the original cost function corresponding to the un-perturbed model parameters, E' is the result with a modified (perturbed by 10%) set of model parameters and ∆E is the difference. The resulting sensitivity is the normalized ∆E plotted in Figure 28.
The interpretation of this plot indicates the deeper layer was as sensitive to changes in effective attenuation α as the topmost mud layer. Figure 28. This plot examines the sensitivity of attenuation as a function of depth. A 10% perturbation of attenuation was applied at various depths (Potty et al., 2018).