Observed correlations between the sediment grain size and the high-frequency backscattering strength

Remote sensing techniques pose an important supplement to physical sample collection, as they can reduce costs and increase the efficiency of seabed characterization. The acoustic backscattering cross section per unit area per unit solid angle¹ σ_b is a key parameter for acoustic seabed characterization; it varies with the grazing angle θ_g of the incoming field as well as with frequency f, i.e., $σb≡σb(θg,f)$⁠. The backscattering strength is obtained as $Sb=10 log10(σb)$⁠. Throughout this text, “backscattering” will for brevity be denoted as “scattering.” Methods for seabed characterization using sonar include geo-acoustic inversion using a physics-based model,^2,3 a generic model that uses a parametric description of the shape of the angular response curve,⁴ a statistical analysis of the measured S_b-values,⁵ and, finally, the use of S_b-values in combination with depth data.⁶ 

In this work, a simple empirical relation for the estimation of the sediment mean grain size from the scattering strength will be presented. Snellen et al.⁵ studied the correlation between the median grain size D₅₀ and backscatter values (dB) obtained by an uncalibrated multibeam echosounder (MBES) operating at 300 kHz. D₅₀ ranged between 5$ϕ$ and $−5ϕ$⁠, where $ϕ$ refers to base-two logarithmic units.¹ The authors notice a “significant positive correlation” up to 1$ϕ$⁠, above which the backscatter values gradually reach a plateau and then begin to decrease as the grain size increases further (see Ref. 5, Fig. 15). They conclude that a one-to-one relationship between the grain size and the backscatter values does not exist for the entire grain size spectrum. In this work, the observations are pursued by investigating the correlation between the scattering strength and the normalized grain size, i.e., the grain diameter divided by the acoustic wavelength $d/λ$ (see, e.g., Boyle and Chotiros⁷ and Ivakin and Sessarego⁸). For the data presented by Snellen et al., the $d/λ$-values range between 0.006 and 3, and the positive correlation ends for $d/λ≈0.1$⁠. Here, we present S_b-estimates obtained from ten sites at Sequim Bay (Washington State, USA) by application of an MBES operating between 200 and 350 kHz. The mean grain size of the sediments ranges between 7.11$ϕ$ and 1.28$ϕ$⁠, and thus, the $d/λ$-values range between 0.001 and 0.1. The MBES, a Teledyne RESON SeaBat T50-P (Teledyne RESON A/S, Slangerup, Denmark), was subject to a limited post-calibration at the Center for Coastal and Ocean Mapping (CCOM)/Joint Hydrographic Center (Durham, NH). Thus, the results presented here are restricted to the grazing angles 45° and 75°. For $θg=45°$⁠, data found in the literature, where $d/λ$ ranges between 0.0007 and 10, are included for comparison.

Figure 1(a) shows the coordinate system of the MBES: The origin is located at the acoustic center of the transmit array, the x-axis points in the starboard direction, the y-axis points in the along-track direction, and the z-axis points upwards. The normalized beam pattern of the acoustic pressure squared in the across-track direction is given as $bxz2(θ)=(bTxz2bRCxz2bRxz2)(θ)$⁠, where θ is the across-track angle depicted in Fig. 1(b); $bTxz2$ and $bRCxz2$ represent the wide beams of the transmitter and a single receiver channel, respectively; finally, $bRxz2$ represents the narrow beam produced by the receiver beam former (see Figs. 11, 17, and 18 in Ref. 9). The corresponding combined beam pattern for the along-track direction is $byz2(ϑ)=(bTyz2bRyz2)(ϑ)$⁠, where ϑ is the along-track angle depicted in Fig. 1(c), and where $bTyz2$ represents the narrow transmit beam and $bRyz2$ the wide receive beam; $byz2$ is dominated by $bTyz2$⁠, and therefore $bRyz2$ will be neglected, i.e., $bRyz2(ϑ)≡1$⁠. As the MBES platform moves forward, a three-dimensional (3D) geometry represented by plane facets is produced. The equation of the plane of a facet is estimated by the unweighted linear least square method by the use of about 32 bottom detection points (BDP): Across the swath, the number of facets is set to 1/4 of the number of beams, which for each ping leaves about four BDPs per facet depending on the topography; along the track, BDPs from seven previous pings are included.

From each receive beam, σ_b-estimates can be derived from several samples within the portion of the seabed backscatter time series.¹⁰ Here, a method based on integration of the backscatter time series of the seabed is applied;¹⁰ it only produces a single σ_b-estimate per beam. Each beam is associated with a specific facet so that the grazing angle of the incoming field as well as relevant points of intersection with the facet can be found. Thus, for a given beam the scattering cross section is obtained as

where E (Pa²$·$ s) is the time series of the backscattered pressure squared integrated over the time interval spanned by the acoustic response of the seabed; E₀ (Pa²$·$ s) is the time integral of transmit pressure squared at 1 m from the source; T is the transmission factor which includes energy losses due to spreading and absorption;¹¹ finally, A is the footprint area. For the calculation of E in Eq. (1) magnitude data from a receive beam is converted from analog to digital converter (ADC) output values into sound pressure values by the use of relations obtained from tank measurements;¹¹ subsequently, the sound pressure values are squared. The transmit signal is a continuous wave (CW) pulse of duration τ shaded by a trapezoidal function with a tapering parameter of $ϵ=0.2$ [see, e.g., Ref. 11, Eq. (12)]. Thus, $E0=p02τe$⁠, where p₀ is the acoustic pressure @ 1 m from the source, and where τ_e is the equivalent pulse duration [see, e.g., Ref. 11, Eq. (16)]. Since $bTxz2$ and $bRCxz2$ vary slowly with the across-track angle they are merely weighting scalars in Eq. (1). Thus, for each beam the transmit and receive arrays are considered as rotational symmetric line arrays characterized by $bTyz2$ and $bRxz2$⁠, respectively. Therefore, for the calculation of the footprint area, the two equivalent beam widths are applied,¹² 

where θ_s is the across-track steering angle. For a given steering angle θ_s each of the two vectors,

are rotated around the y-axis by the angles $θs−Δθeq/2$ and $θs+Δθeq/2$⁠. Thus, four vectors point towards the seabed which leads to intersection points on the facet that form a quadrilateral with area A.

The standard SeaBat T50 bracket was used, and therefore, the acoustic center of the receiver was shifted from the origin of the MBES by $(Δx,Δy,Δz)=$(0, 0.192, 0.047 m). The grazing angle will be defined as 90° minus the angle between the normal vector of the seabed facet and the vector pointing from the bottom detection point up to the acoustic center of the receiver. Thus, for a plane and horizontal seabed at a depth of, say, H = 20 m, the maximum grazing angle will be $θg(max)=tan−1((H−Δz)/Δy)=89.45°$⁠. For the computation of the footprint area, the extent of the bistatic offset is considered negligible, and therefore, for the A-computation, the system is assumed monostatic.

Sequim Bay is located on the northern coast of the Olympic Peninsula in Washington State, USA. The test area was located in the northern part of the Bay and it covered an area of about 2 × 2 km², where the water depth ranges between 5 and 30 m. A CTD drop showed that the temperature and the salinity were almost constant down through the water column, that is, 8.7 ± 0.1 °C and 31.2 ± 0.1 parts per thousand (PPT), respectively. The resulting speed of sound in the water column was 1481 ± 0.4 m/s. 10 sites with different sediment characteristics were used for the experiment. Gas bubbles, sediment layers, and special species were not observed; hence, the sediments are considered relatively uncomplicated. Diver core samples were collected between March 25 and 29, 2019. Table 1 lists the mean grain size M_z (as defined by Folk and Ward¹³) sand, silt, and clay percentages, as well as the location of the diver core sample station (DCSS).

The sea trial was carried out on the APL-UW research vessel “R/V Robertson” between March 21 and 22, 2019. The MBES was mounted on a pole attached to the starboard side of the vessel. At each site, surveys of 200 m length were carried out for the frequencies 200 250, 300, and 350 kHz by the use of a CW pulse of 100 μs duration. The MBES was set to form 256 equiangular beams covering a 150° swath. At each site, pre-surveys were conducted in order to find the source level and receive gain settings that would not lead to ADC clipping of the nadir returns, but still ensure outer beam signal levels well above the quantization noise floor.

Six weeks after the Sequim Bay experiment, the MBES was subject to a control calibration in CCOM's Engineering Tank, an indoor freshwater test tank, 18 m long, 12 m wide, and 6 m deep. The water temperature was around 20 °C. The MBES was mounted on a rotatable pole. A tungsten carbide sphere was positioned along the centerline of the MBES at a distance of 14.2 m. Due to limited available time at the facility, measurements were limited to rotor angle intervals of $−1°$ to 1° in 0.1° steps around the rotor angles $θr=±45°$ and $θr=±15°$⁠. Thus, only beams around these angles were calibrated, but for all four frequencies. For the MBES, the target strength is estimated as $TS(MBES)=10 log10(pb2/(p02T bTxz2bRCxz2))$⁠, where p_b is the backscattered sound pressure from the sphere, and p₀ is the sound pressure of the transmitted signal at 1 m from the source. The theoretical Target Strength $TS(Theory)$ is found from a partial-wave series solution¹⁴ by the use of the material parameters of the sphere. Consequently, for the four frequencies in ascending order, the corrections, $TS(MBES)−TS(Theory)$⁠, are $(0.6,1.7,1.9,2.1)$ dB for $|θr|=15°$⁠, and $(−0.6,0.9,0.9,1.4)$ dB for $|θr|=45°$⁠.

For a flat seabed, the calibrated beams will correspond to the grazing angles 45° and 75°. Therefore, the S_b-analysis will be limited to these two angles. Consequently, S_b-values will not be included when the seafloor has a significant across-track slope; for example, when $θg=45°$ and the S_b-value has been obtained from an uncalibrated beam such as, e.g., 57°. Since the vessel was manually controlled, the paths varied between the different lines of a site, and the maximum distance between the vessel and the DCSS reached up to 25 m. Only the 40 pings closest to a DCSS were included for the analysis. A DCSS located in the proximity of a transition between two sediment types, and where MBES data include both types, is inspected manually to ensure that only data from sediments that correspond to the DCSS are used.

Here, the mean grain diameter will be given in units of meters, that is, $d=10−3×2−Mz$⁠. For $θg=75°$⁠, Fig. 2(a) shows the scattering strength as a function of frequency (left) and normalized grain size (right). For each sediment group (DCSS 1–10, Table 1), S_b decreases with frequency. Earlier observations in the 200–350 kHz interval for grazing angles above 70° show almost no frequency dependence (Wendelboe,¹¹ Fig. 29, $θg=85°$⁠), or they show slightly weak negative, as well as weak positive, trends (Fezzani et al.,⁴ Fig. 10, $θinc=20°$⁠). Conversely, S_b increases with the normalized grain size. Furthermore, the negative frequency dependency within a sediment group gets stronger as the grain diameter increases. Finally, it is unclear whether the upward bends of about 0.5–1 dB at 350 kHz are caused by the sediment or by changes in the MBES characteristics. For site 10, the 200 kHz data point seems to be an outlier. For $θg=45°$⁠, S_b increases with frequency [Fig. 2(b), left]. Moreover, the S_b-values almost collapse into a straight line when plotted as a function of $d/λ$ [Fig. 2(b), right]. This result seems to represent a clear-cut case for simple geo-acoustic inversion, but the question is to what extent it can be expanded to other sites and to $d/λ$-values outside the 0.001–0.1 interval. Thus, in the following our results will be compared with data found in the literature for $θg=45°$ (see Table 2). Figure 2 outlines the chosen references and the associated frequencies and grain sizes required for the estimation of the (⁠$d/λ,Sb$⁠) pairs. For the references where the water sound speed has not been stated, a value of 1500 m/s will be used; since the sound speed typically varies between 1480 and 1530 m/s, the resulting error on $d/λ$ will be less than 2%. Figure 3 shows the scattering strengths from Sequim Bay and the references. The overall picture shows a scattering strength that increases linearly with $log10(d/λ)$ up to $d/λ≈0.1$⁠, above which it seems to fluctuate around a constant level. The point of change is consistent with the findings by Snellen et al.⁵ as mentioned in Fig. 1. For the 0.02–0.1 interval (very fine to medium sand for frequencies between 100 and 350 kHz), the data from Goff et al.,¹⁹ Simons and Snellen,¹⁶ and Williams et al.¹⁷ altogether show a trend that is different from the Sequim Bay data, including S_b-levels 2–4 dB lower than those of the Sequim Bay. For Goff et al.,¹⁹ the fraction of large particles, i.e., particles for which $d/λ>0.25$⁠, is 2.6%; the value corresponds to the median of all the data sets from Goff et al. used here. For the Sequim Bay data, the corresponding value is only 0.7%. Thus, a higher fraction of large particles (for example shell fragments) will lead to a larger mean grain diameter, but S_b may not necessarily increase with $d/λ$ with the same rate as the same sediment with a low fraction of large particles. For the other reference data in the 0.0005–0.1 interval, a rough assessment concludes that six out of eight data points from Weber and Ward,¹⁵ three out of five from McKinney and Anderson,²¹ one out of three from Pouliquen and Lyons,²² and finally, two out of three data points from Haris et al.²³ are in the region of being well-aligned with the Sequim Bay data. The remaining six points are located around 0.02, where the two (⁠$▹$⁠)-points exceed the Sequim data by 6 dB, and in the interval 0.0005–0.002, where five points exceed the Sequim data by 10 dB. It is unclear what the reasons are, but air bubbles produced by bioturbation might be an explanation.

Transition points,⁵ i.e., unique maxima, for which S_b begins to decay for increasing $d/λ$-values, are observed for the data from Greenlaw et al.²⁰ for $d/λ≈0.2$⁠; and a weaker maximum is observed for the Eleftherakis et al.⁶ data for $d/λ≈1.3$⁠. The data from Ivakin and Sessarego⁸ fluctuate from −19 dB up to about −6 dB. It is unclear whether the large fluctuations are caused by flattening and degassing the sand, but the remote data point at $(0.1,−35$ dB $)$ may be a consequence of this: A similar drop in scattering strength with decreasing normalized grain size for flattened and degassed sand is presented by Boyle and Chotiros⁷ (Fig. 2, Group 1, laboratory sand). The data from Simons and Snellen,¹⁶ Weber and Ward,¹⁵ McKinney and Anderson,²¹ Williams et al.,¹⁷ and, finally, from Sequim Bay at the gravelly site indicate the existence of a plateau of about –15 dB. The S_b-levels obtained by Elefterakis et al.⁶ appear too low compared to the other S_b-values; since their MBES was calibrated by the use of the seabed and the corresponding grain size information, and not a well-defined calibration setup, the estimated S_b-levels are uncertain. For $θg=45°$ and $d/λ≤0.1$⁠, we therefore obtain the following regression equation which is based on the Sequim Bay data only:

where $c1=9.63$ dB, and $c2=−8.42$ dB, and where the fit has been obtained with a coefficient of determination of $r2=0.98$⁠. When expressed as a function of M_z we obtain

Hence, simple geo-acoustic inversion of the mean grain diameter can be conducted for $θg=45°$⁠, when the scattering strength is less than the threshold of –18 dB. If S_b is greater than the threshold, it can only be stated that $d>λ/10$⁠. For example, if the frequency is 350 kHz and $Sb>−18$ dB, the minimum mean grain size will be 1.2 $ϕ$⁠, i.e., medium sand; thus, the sediment can be anything between medium sand and, for example, pebble or cobble. However, if the frequency is 200 kHz, and $Sb>−18$ dB, the minimum mean grain size will be 0.42 $ϕ$⁠, i.e., coarse sand; thus, the sediment can be anything between coarse sand and stones. Hence, by lowering the frequency from 350 to 200 kHz the ambiguity interval can be reduced by one Udden-Wenworth class. Expressions similar to Eq. (4) have also been found for the beams not subject to the control calibrations. For grazing angles between 35° and 70°, fits have been obtained with r²-values that range between 0.98 and 0.99, and the potential for classification exists on a wider range of grazing angles.

It should be noted that mechanisms controlling seafloor scattering strength are usually angular dependent. The two grazing angles considered here are both well above the critical angle for the sediment, which may make the volume scattering mechanism more pronounced, while below the critical angle, the roughness scattering can become comparable and in many cases (such as for sand sediments at high frequencies) larger than the volume scattering (see Ref. 1 for more details). We believe the correlation between S_b and the $d/λ$-ratio may indicate the importance of scaling effects of the sediment granular structure, mentioned for instance in Ref. 8. However, modeling of these effects is beyond the scope of this express letter and will be subject to future research and more detailed papers.

For $θg=75°$⁠, the scattering strength increases with rising grain diameter but decreases with frequency (Fig. 2). Thus, a simple linear relation between $log10(d/λ)$ and S_b for the frequency interval 200–350 kHz has not been found. For $θg=45°$⁠, the scattering strength increases with rising grain diameter as well as with increasing frequency. The mean grain size can be derived from S_b and λ by the use of Eq. (5) provided that S_b is less than −18 dB or $d/λ<0.1$⁠. For larger S_b-values, it can only be stated that the mean grain diameter is $d>λ/10$⁠. Expressions similar to Eq. (5) can be found for grazing angles between 35° and 70°. Although it requires extended calibrations, it opens up for producing maps of the grain size in real time.

This work was supported by the U.S. Department of Defense's Strategic Environmental Research and Development Program (SERDP), Project No. MR-2229. Many thanks to CCOM for the use of the Engineering Tank and for the expert help needed to conduct the control calibrations.