Estimation of seafloor microtopographic roughness through modeling of acoustic backscatter data recorded by multibeam sonar systems

Estimation of seafloor microtopographic roughness through modeling of acoustic backscatter data recorded by multibeam sonar systems

Haruyoshi Matsumoto and Robert P. Dziak Cooperative Institute for Marine Resources Studies, Oregon State University, Hatfield Marine Science Center, Newport, Oregon 97365

Christopher G. Fox National Oceanic and Atmospheric Administration, Pacific Marine Environmental Laboratory, Hatfield Marine Science Center, Newport, Oregon 97365

(Received 25 October 1992; accepted for publication 28 June 1993)

A method is described for estimating parameters related to the power-law frequency spectrum of seafloor interface roughness from acoustic backscatter strength measured from hull-mounted multibeam sonar arrays. Scattering parameters for a sediment-free seafloor are inversely derived from backscatter data using a Kirchhoff-based interface scattering model developed by Jackson et al. [J. Acoust. Soc. Am. 79, 1410-1422 ( 1986)]. A variety of data reduction routines are employed, including slope correction, beam-pattern and pulse-width corrections [de Moustier et al., J. Acoust. Soc. Am. 90, 522-531 ( 1991 )], ping cross correlation, and an indirect method to account for the lack of system calibration. The method is applied to well-studied areas of the Juan de Fuca Ridge and the results compared to geological ground truth. Preliminary results indicate that the method may be valuable as a survey tool for routine mapping of seafloor acoustic properties.

PACS numbers: 43.30.Hw, 43.30.Pc, 43.60. Pt

INTRODUCTION

The power-law behavior of seafloor roughness spectra is known over a wide range of wavelengths from hundreds of meters to a few centimeters. 1-3 This spectrum in one dimension can be described using a simple two-parameter model. Fox and Hayes 1 developed techniques for deriving these spectral estimates from profiles based on multibeam sonar soundings. Due to limitations in soundings derived from surface sonars, such models only apply for wavelengths greater than about one hundred meters.

An alternative method for inferring seafloor roughness spectra at centimeter scales, was developed by Jackson et al., 4 who applied the Helmholtz-Kirchhoff formula for steep grazing angles to obtain an analytical equation for the surface backscattering cross section. The parameters of the Jackson et al. model can be theoretically related to the parameters of the Fox and Hayes model, although the two models apply to different spatial frequency bands. Modern multibeam echo sounders that collect both soundings and acoustic backscatter information allow the direct compar- ison of the two methods.

Dziak and Fox 5'6 attempted to apply Jackson's model to backscatter strengths obtained from the Sea Beam multibeam sounding system. The model was fit to the empirical backscatter strength as a function of angle using parameter estimation based on nonlinear regression analysis developed by van Heeswijk and Fox. 7 This algorithm is based on traditional iterative techniques for deriving regression coefficients based on the steepest descent of error. This initial study suggested that model parameters could be estimated directly from the measured backscatter strengths and used

to classify the ocean bottom. However, the simultaneous estimation of multiple parameters based on a nonlinear regression routine often resulted in low rates of conver- gence and difficulties in keeping model values within a rea- sonable range.

de Moustier and Alexandrou 8 have applied the same scattering model to data sets collected from a Sea Beam survey over a deep-sedimented seafloor (Horizon Guyot and Magellan Rise). They introduced a numerical method to obtain backscatter strength from Sea Beam data, which included corrections for theoretical beam pattern based on the design parameters of arrays and pulse-length integration to normalize the data. Sidelobe interference rejection and local slope correction routines were included, which were not applied by Dziak and Fox. 5 The Jackson et al. 4 model was then applied by fixing one estimation parameter and fitting the other. The result of this study indicated that the effect of volume scattering was deemed negligible for the data set presented.

The purpose of this study is to extend the original work by Dziak and Fox, and develop a reliable method to estimate model parameters from backscatter data using multibeam systems. Slope-correction and beam-pattern and pulse-length normalization schemes suggested by de Moustier and Alexandrou 8 are adopted. We use, however, the Gaussian beam pattern assumption to obtain a simple analytical formula to normalize for the beam pattern and pulse length. As an alternative to the nonlinear regression routine for the multiparameter estimation, a simulated- annealing algorithm 9-12 is used resulting in a more robust estimation of all model parameters. To eliminate the com-

2776 J. Acoust. Soc. Am. 94 (5), November 1993 0001-4966/93/94(5)/2776/12/$6.00 ¸ 1993 Acoustical Society of America 2776

Redistribution subject to ASA license or copyright; see http://acousticalsociety.org/content/terms. Download to IP: 128.83.63.20 On: Tue, 25 Nov 2014 23:22:00

1000.00

100.00

10.00

1.00

0.10

0.01

/C C

0.0 0.2 0.4 0.6 0.8 1.0

FIG. 1. Example of power-law based roughness spectrum IV(k) computed for a=0.51 and/•=0.02 from Eq. ( 1 ) with a cutoff wave number k½ separating small and large-scale roughness.

plications arising from sediment-volume scattering, the method is applied to data collected in nonsedimented terrains.

I. THEORY

If bottom roughness is isotropic and Gaussian, the spectral density of the seafloor may be defined as a power law given by

W(k)=13k -z, (1)

where k is the wave number, y = 2 (a + 1 ), and a and/3 are two parameters which characterize the statistical properties through the structure function D(r) [Eq. (B- 16) of Ref. 13] of the seafloor surface (Fig. 1). Jackson et al. 4 applied the power-law properties of the seafloor roughness spectrum to a Helmholtz-Kirchhoff formula at steep grazing angles, and obtained the following analytical formula for the surface backscatter cross section:

g2(rr/2) ;0 'ø exp(--qu2ø•)Jo(u)u du, Os(Og) -8rr sin 20g cos 20g where

q=sin 20g cos -2'• OgC•21-2a and

(2)

(3)

C•= [2rrl•r(2-a)2-2a]/[a(1--a)F'( 1 +a) ]. (4)

Here, cr•(0g) denotes the surface scattering cross section per unit solid angle per unit surface due to interface roughness. The grazing angle is designated 0g, which is related to the incident angle Oi by 90'-- 0i. The acoustic wave number is ka, and J0(u) is the zeroth-order Bessel function of the first kind. Here g(rr/2) is the reflection coefficient at normal incidence that is assumed to be constant over the areas

of interest, and C• is a parameter related to the structure function, 13 of the isotropic Gaussian surface, that follows a power law in the wave number spectrum.

The roughness scale valid Por Eq. (2) is constrained by the Kirchhoff criterion, 4 i.e., 2/(ka R sin 3 0•) < 1, where R is the radius of local surface curvature. At the 12.158-kHz

frequency of the Sea Beam system, this criterion corresponds to a minimum of R = 5 cm for 0s > 70*. Because of

the large impedance difference at the seafloor-water interface of a hard-rock area, the reflection coefficient g(rr/2) should be close to unity. 1•'•5 Therefore, a surface backscatter cross-section term •rs(Og) should be sufficient to model backscatter for Og > 70* from the ridge-crest area.

However, when the seafloor is heavily sedimented, the physical model requires a volume scattering term. •'•6 If we assume that both surface and volume backscatters are in-

coherent, then total backscatter strength becomes a sum of the surface and volume backscatter terms. •6 The volume scattering term of the Jackson et al. model is defined as a function of three parameters, o',,/at,, Or,, and g(Og). 4 These parameters are, respectively, the ratio of volume scattering coefficient per unit volume over the sediment attenuation coefficient (in dB), angle of refraction, and a plane-wave reflection coefficient. In sedimented areas, the volume scat-

tering term is negligibly small at near nadir angles as compared to the surface backscatter term, but as the grazing angle increases, the contribution of the volume scattering becomes significantly larger. 4'16

Although it may appear that a and/3 in Eqs. (3) and (4) are independent, roughness parameters derived from the seafloor roughness spectrum, i.e., the large-scale radius, indicate otherwise. Borrowing a criterion from Jackson et al. 4 the inverse square of the large-scale radius R -2 of the seafloor roughness spectrum is

•0 •c R-2=2rr W(k)kSdk= (2rrl•kc6-r)/(6- y), (5)

where k c is the cutoff wave number that separates large- scale from small-scale rms slope. This relation constrains a and/3 by the value of the cutoff wave number k c, and the bottom roughness parameter R -2. If the power-law behavior of the surface roughness is valid, then for a single value of the radius of curvature, there are many sets of a and/3 that would satisfy Eq. (5). However, the backscatter curves generated using these sets of a and/3 may not re- semble one another.

Figure 2 (a)-(c) illustrate the characteristics of the backscattering strength computed on a two-dimensional surface of a (0.5<a<1.0) and grazing angle ( 120' < Og < 60*) with three fixed/3 values. Figure 3 (a)-(c) are the same functions, but computed on the surface of/3 (0.005</3<0.105) and grazing angle (120'<0g<60'), with three fixed a values. All the theoretical curves were

calculated using/3 values in cgs. Note that for fixed/•s, an increase in a forces the backscatter curve to flatten. In the

case of fixed as, an increase in/3, i.e., an increase in surface roughness, also results in flatter curves. This indicates that from the measured backscatter curve (S(Oi)), a and /3 may be uniquely determined as an inverse problem,

(S(Oi) -1 ) -• cg,•. (6)

In order to estimate the surface parameters a and/3 from the measured backscatter curve (S(0i)), a simulated- annealing technique, 11'12 based on the Nelder-Mead downhill simplex method, 9'1ø is employed. The routine searches for a global minimum error (minimum energy state) between the model and observations over a multivariate pa-

2777 J. Acoust. Soc. Am., Vol. 94, No. 5, November 1993 Matsumoto et al.: Seafloor microtopographic roughness 2777


(a) (a)

i o eo. j

ß 0o91'e '

= 0.95

(b) (b)

FIG. 2. Theoretical backscatter strength computed as a function of a and the grazing angle Og for 0.5 <a <0.95 and 60 ø < Og < 120 ø at fixed/•s; in (a)/•=0.1, (b)/•=0.02, and (c)/•=0.005. An increase in/• causes the surface to flatten making the estimation of component models less accurate.

(c)

rameter space. The error AEis calculated for an arbitrary set of a,/•, angle shift (50), and scaling factor (ms) to calibrate the measured backscatter curve. The angle shift (50) is included in the estimation scheme to search for the appropriate offset angle to adjust for any asymmetry of a measured backscatter curve (S(0i)). If the scaling factor (ms) to calibrate the system gain, transmit power, and reflection coefficient of the seafloor is not determined, it can be searched simultaneously, or used as a fixed value if it is known. AE for a single ping is therefore defined as

AE= • Ilog(S(0•))--log m•s(a,l•,O•+•50)l, (7) i=0

where the intent of our analysis is to find an optimum set of a,/5, 50, and ms so that AE becomes a global minimum. Note that the errors are estimated in logarithmic scale so that errors at higher angles are evaluated equally as the errors near normal incidence.

.0

ct = 0.5

FIG. 3. Theoretical backscatter strength computed as a function of/• and the grazing angle Og for 0.005 </•<0.105 and 60ø<0g< 120 ø at fixed as. In (a) a--0.95, (b) a=0.75, and (c) a=0.5. Note that increases in a cause the surface to flatten, making the estimation of component models less accurate.

The parameters, a, /•, and 50 (and ms if it is not known) are changed in accordance with a downhill simplex method, 1ø which searches for a new set of parameters by either contracting, expanding, or reflecting depending on the new and previous outcomes of the random variables E= AE-- T log P. Temperature of the annealing process, T, is decreased slowly to 0 according to an appropriate annealing schedule. Here P is the realization of a uniform random distribution. The algorithm changes the current direction of the parameter search through the opposite face of the simplex (reflection). If the error becomes smaller, then the parameter search is expanded in the same direction (expansion). However, if the error gets larger, the process chooses a parameter set that is an average of the past parameter sets and the one yielding the lowest error (contraction). The mode of the parameter search does not depend on AE alone, but rather a sum of AE and a random

2778 d. Acoust. Soc. Am., Vol. 94, No. 5, November 1993 Matsumoto et aL: Seafloor microtopographic roughness 2778


variable (- T log P). When the process becomes trapped by a local minimum, the latter term allows the process to escape from the local minimum to find the global minimum even at lower temperatures. As T slowly approaches 0, the algorithm finds a parameter set that would minimize E resulting minimization of the error AE.

II. METHODOLOGY

A. Sea Beam system

Data examined in this study were collected by the Sea Beam system, which has been used extensively as a mapping tool to produce contour maps of the seafloor since 1977. 27 The system is operated at 12.158 kHz, and trans- mits a 7-ms pulse at a transmission interval of 1 to 16 s, depending on water depth. A fan-shaped beam is transmit- ted with a width of 54 ø athwartship and 2.66 ø in the fore-aft direction. The return echoes received by the 40 hydro- phone elements are formed into 16 beams, each with a nominal 2.66 ø athwartship and a 20 ø fore-aft beamwidth. With these 16 beams, the system covers an angle range of 4-20 ø about the ship's nadir with even angle separations of 2.66 ø . More detailed descriptions of the Sea Beam system

17-20 are available from several other sources.

Processing used in this study requires ensemble averaging of 20•40 consecutive pings. Since the water depth of the study area is typically 2000 m, the areal extent over which the model is applied is a rectangle of dimensions of 1500 m X 650• 1300 m, assuming a ship's speed of 11 kn and a 6-s transmission cycle. For each ping, envelopes of the 16-beam returns were digitized at 500 Hz with 12-bit resolution. The ship's attitude and bathymetric data from each ping cycle were recorded synchronously. 2ø

B. Data reduction

Every data point of each individual beam record represents acoustic pressure backscattered from a small patch of seafloor within an area intersected by the transmit pulse bounded by the transmit beam (7 ms within 540X2.66 ø beam) and individual receive beams. Relative backscatter strengths can be calculated for each beam angle by nor- malizing: (1) the gain differences of individual receiver boards for each beam, (2) ship's roll and local slope angles, both of which affect the incident angle, (3) the pulse ensonified area bounded by the transmit and individual receive beams, and (4) the transmission loss. Several as- sumptions were made to perform the data analysis:

(i) the seafloor area covered by the 16 beams over a 20•40 pings range is homogeneous;

(ii) the reflection coefficient g(•r/2) is constant over the study area;

(iii) the sediment thickness is thin enough that volume scattering by sediment layers is negligible. Ensemble averaging through ping stacking is applied to reduce the variances of backscatter strength for each angle. The sequence of data reduction routines used in this analysis is described below.

1. Removal of the system gain

Acoustic backscatter signals are recorded from the individual outputs of "echo preamplifier board" within each beam receiver. Before the signal envelope is recorded, a time-variable system gain is applied to correct for transmission loss. However, due to the nonlinearity of the preamplifier, 2• the system gain deviates from the theoretical transmission loss. As a first step, the system gain is removed, and a theoretical transmission-loss compensation applied. The theoretical transmission loss is based on a spherical spreading law and a 1-dB/km absorption coefficient.

Another error ariseg from the unidentical gains among the individual echo amplifiers, which also tend to drift. To correct for these uneven gains, recorded system's response to a calibration signal was used at the postprocessing stage to reduce the effect of amplifier drift. •8-2ø

2. Two-dimensional angle corrections and angle-bin size

Because the incident angle of each beam is affected by the ship's roll and the local bottom slope, the ship's roll angle is subtracted from the beam pointing angle relative to the ship's centerline. The roll bias was a few tenths of a degree • in July of 1987, and it was left as a part of the angle shift 60 to be estimated. As a precaution, any ping recorded when the ship's roll angle was greater than 4- 10 ø is discarded since such records tend to have low S/N ratios on outermost beams.

Athwartship bottom slopes at each beam direction were estimated using the recorded bathymetric data, which are equally spaced in angle in the athwartship direction. The depth points, however, do not correspond to the individual beam directions, and therefore the slope angles of the seafloor in the direction of individual beams had to be

estimated by interpolating the depth data. Since the fore- aft sampling interval was approximately 33 m (6-s transmission cycle at ship speed of 11 kn), a 3-point moving average was applied in the ship track direction with 20•40% weighing on the previous and proceeding depth records. No slope-angle correction was made for the areas where no depth points were available. The appropriate angle-bin size, A0, depends on the complexity of area, the depth accuracy, and the angle of incidence. The results from two previous Sea Beam surveys suggest that the rms error over the Juan de Fuca Ridge area was •4.38 m. 23 With an average athwartship horizontal sampling interval of • 100 m (water depth of 2000 m), the corresponding accuracy in the slope angle estimate is •2.5 ø [•arctan(4.38/100)]. For complex areas that contain features of a similar scale as the horizontal sampling interval, an excessive smoothing of the bathymetry by a low-pass filter does not necessarily improve the slope-angle estimates. Since the area of interest is the complex ridge-crest area, we chose a uniform angle-bin size of 2.66 ø for all incident angles. This relatively large angle-bin size helped to reduce the standard deviation of averaged backscatter strengths estimated over a small range of pings (20-40). On the other hand if the seafloor is smooth and a low-pass

2779 J. Acoust. Soc. Am., Vol. 94, No. 5, November 1993 Matsumoto et aL: Seafloor microtopographic roughness 2779


(a)

20

sep28d06

Slope Corrected

___

port sfbd

-20 ......... i ......... i ......... i ......... i ......... i .....

-30 20 10 0 10 20 30 Incident Angle (degree)

(b)

.>

20

10

-10

-20

-50

......... i ......... i ......... i ......... i ......... i .........

sep28d06

Slope Corrected

___ Without Slop• Co•,

-20 -10 0 10 20 50

Incident Angle (degree)

(c)

o

n•

.>

20

10

-10

-20

" ........ i ......... i ......... i ......... i ......... i ......

sep17d10

Slope Corrected

Without Slope Correction

port sfbd

0 -20 -10 0 10 20 50 Incident Angle (degree)

FIG. 4. Backscatter strength curves with the slope-correction applied (solid line) and without (dashed line). Examples from three different areas along the Juan de Fuca Ridge are shown: (a) smooth, broadly sloping area on the flanks of Axial Volcano, (b) smooth and fiat area within the caldera of Axial Volcano, and (c) a relatively rough area of the northern Cleft segment. Notice that although the correction is effective for smooth terrains, the correction introduces error in rough terrains. This effect relates to insufficient sounding in a complex terrain.

filter with a low cutoff frequency can be applied, the slope- angle estimate may be improved by as much as 1ø. 8

It appears that in relatively smooth terrains, either flat or gently sloping, the two-dimensional slope-correction routine works well. The data shown in Fig. 4(a) were

20

-20

-30

Averaged backscatter strength

+ + Individual data sep1710

-F -F

++

+ + +• • +. + ++++++ +++,+++ + + +'•-•" + +:I:H- •. + '•+ +

•+

++ +

port + + st• • +

-20 -10 0 10 20 30


FIG. 5. Individual backscatter strength data and the ensemble average over 20-ping range from within the caldera. A large data scatter is evident at low incident angles. The two-dimensional slope-correction process may have increased the data scatter. Due to the large fluctuations within a single ping, it is necessary to average over many pings before estimating the backscatter-model parameters.

collected over a broadly sloping area, which is inclined with a down slope of 2 ø to 3 ø toward starboard, with an average depth of about 1450 m. The morphology of this area changes little, allowing the ensemble averaging of a large number of pings. In this case, the slope-correction routine produces a backscatter curve which is continuous and symmetric about normal incidence. The data in Fig. 4(b) were also collected in a relatively smooth terrain, but in this case the area is flat and the slope correction has very little effect. In addition, the average depth was relatively shallow ( 1570 m), and consequently the bathymetry of the area is better defined.

In contrast, the slope correction routine does not necessarily improve the backscatter curves in complex terrains, which may be due to the error in depth estimates and inadequate spacing of the bathymetric soundings (alias- ing). The example shown in Fig. 4(c) is from a relatively complex terrain and greater water depth (2250 m). In this case, the slope-correction routine may introduce additional errors to the angle estimates. Some slope-correction errors may be a result of applying two-dimensional slope-angle estimates instead of three-dimensional ones. In order to

determine the three-dimensional slope angles, however, in addition to the roll and navigation record, it is necessary to acquire pitch, yaw, and heave information, which were not collected by the current logging system. Another cause of the error could be by asymmetry of the backscatter- strength curves, which may be due to inhomogeneities of seafloor properties in either the across-track and along- track directions. This error could be reduced in the along- track direction by averaging over a smaller number of pings; however, a smaller sample size would result in an increased standard deviation within samples due to the signal fluctuation caused by the Fresnel effect 24 even in a homogeneous area. Also, refraction correction was not included in the process, which biases the results in higher angles of incidence. Figure 5 shows the scatter of actual



data and a backscatter strength curve averaged for each angle bin over 20 pings. Typically, the backscatter strengths were averaged over 20-40 pings as a compromise to the problem of inhomogeneous geology and fluctuations of the backscatter levels.

3. Sidelobe level reduction by cross correlation

The adaptive noise cancellation technique proposed by de Moustier 8 cannot be used here, since only the signal envelope was stored. As an alternative, a technique based on a cross correlation and coherent-ping stacking was developed. The jth ping record at the kth angle bin, i.e., sj,k(t), was cross correlated to a reference ping record Sn,k(t) to find a time lag Atj, k at which the maximum correlation occurs. This can be rewritten as

max(•Sn,k(t)Sj,k(t+Atj,k) ). (8) The time lag was then added to the corresponding records sj,•(t) to shift them in time and allow them to have a maximum overlay in energy. The mean square envelope of the new records, from J0 to jo+M pings, was determined

1 Jø+M •(t)=• .Z. •,k(t+Ati,•). (9)

J =J0

Figure 6 shows examples of 5 records within a 15 ø- to 20ø-incident angle range, and the mean-square envelope resulting from coherent stacking of a 20-ping range. Notice that only the coherent return energy in successive pings is enhanced, while the effect of sidelobes is reduced. Further- more, the method results in a more consistent detection of peak energy than by estimating peaks from individual ping records without the adaptive noise-cancellation technique. How well the records align depends on how well the reference ping Sn,•(t) represents the area from which the ensemble average of backscatter was taken. Therefore, the ping whose total energy is closest to the average of all the pings within a given ping range is selected as the reference.

The method proposed here works well to align multiple records relative to the center of the beams over a short range of pings. However, it does not reduce the sidelobe interference buried within the main lobe, particularly near the nadir, where the return from sidelobes comes back almost simultaneously. In this angle region, a bias may be introduced to the angular dependence curve.

4. Beam-pattern and pulse-width corrections

Following the notation of Urick, 25 the backscattering cross section •s(0) can be defined as

1 oR Ls/lOr 410 2•' r/10

?'s(O) -- f •Iob( O, q9)b' (O, q9)dA ' (10)

where is R Ls is the echo level in dB at the receiver, r is the range relative to 1 m, a' is the absorption coefficient of water, I 0 is the axial intensity of the projector evaluated at 1 m, b(0,q•) is the transmission beam pattern, and b'(O,q•) is the receive beam pattern. The total energy on the backscattering surface can be obtained by integrating the inci-

dent intensity I 0 over the pulse width, weighted by the beam pattern of a two-dimensional scattering surface of area "A." 0 is the athwartship angle relative to the nadir, and q• is the azimuthal angle. Note that Eq. (10) is an average of the true backscatter cross section ss(O) over the area of "dA." In near nadir region, where a true backscatter strength changes rapidly within the beam width, particularly for the data from the smooth seafloor, this integration yields a bias. 26'27

Although a theoretical beam pattern of Sea Beam array is available, 19 the actual beam patterns of the Sea Beam array have never been measured. We used the Gaussian beam-pattern assumption to correct for the echo intensity, which is a convenient form to solve Eq. (10) analytically,

27 and could be fitted to the main-lobe with good accuracy. By assuming a Gaussian beam pattern and taking the integration limits imposed by the pulse width and across- track beam width, the surface integral in Eq. (10) can then be approximated by

fxø+Axd2 •_ ' ( --(X--Xm )2 ) exp lob(O) J x0-ax•'2 y' 2(0.8489 AXb) 2

X exp 2(0.8489 Ayb)2 dx dy, (11)

where Ay b and Ax b is the half-beam width on the x-y plane in the ship track and the athwartship directions, respectively, and x m is the offset between the beam center and the pulse center. b(0) is the athwartship transmission beam pattern. The fore-aft half-beam width on the x-y plane, Ayb is approximately Aq•D/(2 cos 0), where Aq• is the beam width in the ship-track direction in radian, and D is an apparent depth normal to the plane extended from the sloped bottom. The athwartship upper and lower integration limits of Eq. (11 ) are Xo+ Ax,/2 and Xo--Axe/2, respectively, where x0 is the center of a pulse-covered area, and Ax• is the athwartship width covered by a pulse of Ar seconds.

In high incident angles, where D/cos Oi--c Ar/4>D, utilizing a simple geometry, the athwartship width covered by the pulse, Axe, can be defined as

Ax,= •/( D/cos Oi-}-C At/4) 2--/92

-- •/( D/cos 0 i-- C At/4) 2 _ 192, (12a)

where 0 i is an incident angle relative to the sloped bottom at athwartships horizontal distance Xo. The area ensonified by a short pulse coincides with the intersection of a circular annulus 8'24 with the center directly below the ship, and the area limited by the fore-aft transmit beam width. The athwartship integration limits are, therefore, the leading and trailing edges of the transmit pulse in a circular shape on the x-y plane (pulse-limited region). The scattering area is approximately a trapezoidal shape, but Ay, is much smaller than the radius of the circular annulus. We use, therefore, an average of fore-aft beam widths on the x-y plane at the trailing and leading edges of the pulse for Ay,.



50.0 100.0 ! 50.0

TIME

z

L• ' n-- 0.0 50.0 100.0 150.0

TIME

50.0 100.0 150.0

TIME

O:: 0.0 50.0 100.0 150.0

TIME

50.0 I00.0 150.0

TIME

20'0.0

20•0.0

20•0.0

•o'o.o

20•. 0

J n

rl-

z

MERN SOURRE ENVELOPE

O. 0 100.0 150.0 200.0

TIME

1

50.0

FIG. 6. Raw backscatter envelope records from five consecutive pings within the 150-20 ø range of incidence angle. After cross correlating each record to a reference record, they are realigned and coherently added. The resulting mean-squared envelope shows reduced sidelobe interference and produces a more consistent representation of the seafloor backscatter curve.

In the near-nadir region, where D/cos Oi--c A•-/4<D, however, the radius of the pulse annulus becomes larger than the fore-aft beamwidth, i.e.,

Axe=2 •( D/cos Oi+c A½/2)2-- D 2. (12b) A factor of 2 is necessary to take into account the fact that the near-nadir receive beam is sensitive to both port and starboard return energy, and both contribute to the echo intensity. In this region, the athwartship beam width [lst exponent term of Eq. (11)] is narrower than the width defined by the integration limits. Therefore the integration result is constrained by the beam pattern function rather than the athwartship integration limit factor Ax• (beam

limited), and the actual shape of the scattering area becomes elliptical in this region.

The analytical expression of the surface integral Eq. (11) is

(0.8330y') lob(0)0.7206 Ax0 Ayo •r erf Ayo

erf(0'8330(-xmd-Ax•/2) ) X Axo

--eft • ß (13)



6OOO

Echo envelope Correction factor

o -10

o -20

-30

\

\

\

alpha =0.51300

beta =0.008845

alpha =0.54260

• alpha =0.50750 : • beta =0.042040 \ ..

\ ...

0.00 0.05 0.10 0.15 0.20 0.25

Frequency (1/degree)

FIG. 7. Echo envelopes from individual starboard-side beams (beam number 1 and 8 correspond to center and outermost, beam respectively). Correction curves assuming Gaussian beam and a 7-ms pulse are overlaid on the corresponding beam records. Individual beam envelopes were sam- pled at 500 Hz, after low-pass filtering (f½=64 Hz).

For the selection of fore-aft integration limits y', y'= Ay o should be appropriate to include most of the energy within the beam. Figure 7 shows the actual signal envelopes of one ping from beam one to eight on starboard side overlapped with the correction curves based on Eq. (13). The relative backscattering cross section, •s(Oi), is calculated for each sample within the angle bin, assuming that the peak of the coherently stacked record over the pings coincides with the center of the respective beams, and is then divided by Eq. (13). The results are then averaged over a consecutive ping range to obtain the ensemble average (Ys(0i)) for the individual angle bin.

C. Model fitting

After completion of the preprocessing steps described above, the resulting backscatter strength versus incidence- angle data are fit with the model of Jackson et al. 4 Only the a and/• parameters in Eqs. (3) and (4) are required for characterization of seafloor roughness; however two other parameters were estimated simultaneously to yield inter- pretable results. First, the estimated backscatter curves are not necessarily symmetric about nadir, due to signal fluctuations, roll bias, noise, inhomogeneous geology, and a residual error after a two-dimensional slope correction in preprocessing. Therefore, it is necessary to estimate an angle-shift correction term (60) independently.

A more critical term required for proper fitting of the model in rough terrains is the scaling of the backscatter strength. As illustrated in Fig. 2 (a) where the value of/• is the largest, the backscatter-strength curves are almost identical in shape regardless of the value of a. Similarly, for high a-value ranges [Fig. 3 (a)] the shapes of the curves are similar, independent of the/• values. Spectra of backscatter curves (Fig. 8) obtained by FFT reveal these characteristics more explicitly. When the/• value is very high (dashed lines in Fig. 8), the spectrum is narrow and the

FIG. 8. Spectra of theoretical backscatter curves. Each curve represents spectral density of the backscatter curve for various a-/• value sets. Back- scatter curves with the lowest a and/• values (thick solid line) exhibit a wide spectral characteristic than those with higher a or/• values (dashed lines). This suggests that for a smooth terrain the parameters of backscatter curves, including the scaling parameter, can be estimated reliably. On the other hand, backscatter curves with high/• (rough terrain) or a values tend to be flat [Figs. 2 (a) and 3 (a)], and its spectral density tends to be concentrated in lower frequencies. Therefore for a rough terrain it is necessary to obtain the scaling parameter separately and to calibrate the backscatter curves prior to the annealing process in order to reliably estimate a and/•.

curve is skewed toward the low frequency region. This indicates that for backscatter curves in which a or/• are high, the modeling results can be ambiguous without a proper scaling factor (corresponds to a dc term in the spectrum) to calibrate the backscatter curves. Therefore, if neither the source level nor system gain of the system was measured, the measured backscatter levels are relative, the estimates of a and/• of a rough terrain are not reliable.

An alternative method is to estimate the scaling factor indirectly applying the same algorithm to the backscatter data from a relatively smooth area where both a and/• are low and the corresponding backscatter curve is steep and has a high peak [such as Fig. 10(a) and 10(d) ]. The spectra of such backscatter curves are wider (a thick line and dotted line of Fig. 8), which allows use of a wider spectrum content between low to medium frequency ranges, which contain the level as well as the general shape of the curves.

Model fitting without knowing an exact scaling parameter works reasonably well for low a and/• values. Thus as a first step, four parameters (a,/•, 60, and m s) are estimated simultaneously by applying simulated-annealing algorithm to a relatively flat region (in our case, an area covered with a sheet of lava flow) within the area surveyed. A dotted line in Fig. 8 is the spectrum of the backscatter curve, and one of data used to estimate a mean scaling factor (ms). By assuming this fixed scaling parameter (ms) is representative of the system response and the reflection coefficient for the area surveyed (covered by various types of lavas), the remaining three parameters, (a,/•, and 60) can be estimated for the rest of the area. The simulated-

annealing algorithm used in fitting the model of "n" parameters is initiated by selecting "n+ 1" sets of simplex




•' 20• alpha =0154260' • i beta =0.01 4832

_c: i shifting- -0.4318 c- 10 '- scaling

aa _10 f

-30 -20 -10 0 10 20 30


(a)

•' 20 p"dlp6a .... --'0•50750' ' •' E beta =0.042040

_c: I shifting- 0.3736 :c• 10- scaling = 2.18500

m -10

'• I port r• -20 ......... • ........ • ........ • ........

-,50 -20 -10 0 10


sfbd

2O ,5O

(b)

•- 10

o

a: -10

"ali2h a ..... --'0'•691 0 .................. beta =0.0546,50

shifting= 0.6445 scaling = 2.18500

ß > port sfbd

n,- -20

-30 -20 -10 0 10 20 30


:c• 10

= 0

m -10

._

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20 "81i:;1•' 0 ..... --"0:51',500 ............................... beta =0.008845

shifting= 0.2078

scaling = 0.85090,. o port stbd

-30 -20 -10 0 10 20 30


FIG. 10. Relative backscatter strengths (diamonds) and derived models (solid line) for several example areas, with corresponding parameter estimates. A factor of 10 8 had to be multiplied to the scaling value shown in each figure in order to fit the data to the model. (a) Results from a site within the caldera of Axial Volcano known to contain relatively smooth lavas. The resulting model shows a rapid falloff with incident angle consistent with a smooth seafloor and indicated by the low B value. Scaling factors were estimated from this area for fitting data from areas of rough terrain. (b) Results for the Vance Segment, where the surface is heavily fissured and faulted by tectonic activity. The rougher terrain is reflected by a flatter scattering curve and higher value of B. (c) Results for the northern Cleft Segment, where the surface is heavily fissured and faulted by tectonic activity similar to the Vance Segment site shown in (b). The derived scattering curve and B parameter are also similar. (d) Results from a heavily sedimented area on the Cascadia Abyssal Plain. The results appear to indicate a smooth seafloor area similar to Axial Caldera, which may be geologically correct; however, the scattering model as implemented does not account for the effect of sediment volume scattering and may produce inaccurate estimates.

nificantly lower than the value obtained from the sediment- free areas in Fig. 10(a)-(c). Taking into account the large acoustic impedance difference between sediment and lava and the resulting difference in the plane wave reflection coefficients, the discrepancy in the scaling factors between soft sediment and lava flows is expected.

Although the annealing algorithm converged reasonably well without including the volume scattering term, this does not imply that volume scattering is insignificant. If the effect of volume scattering is significant, the curve falls off more slowly toward higher incidence angles, caus- ing the values of rn s and • to be higher. This results in a certain misinterpretation of the interface roughness as rougher than the corresponding interpretation for

sediment-free areas. In areas of sediment cover, it is critical to include the additional volume scattering terms in the Jackson et al. model.

Figure 11 summarizes the modeling results from numerous samples within the study areas and data collected in an area of thick sediment on the Cascadia Abyssal Plain near 45ø19'N and 127ø22'W. The majority of the a-/• values from the northern Cleft and Vance segments are scattered in the low-a and high-/3 regions as compared to the results from within the caldera of Axial Seamount. In general, a-B values are scattered around the regression line (solid), calculated from Eq. (5), with R = 28 cm and kc=0.25 cm -1. The large-scale radius of seafloor features within the tectonically dominated northern Cleft and



0.100

cr• 0.010

[] Northern Cleft Yt Vance Segment X Axial Volcano A Pelagic Sediment

0.001

0.:• 0.4 0.6 0.8 1.0

FIG. 11. Summary of modeling results from several geologically distinct areas of the Juan de Fuca Ridge. Solid line is the best fit for all data from sites dominated by tectonic deformation and corresponds to a large-scale curvature of 28 cm and a cutoff wave number of 0.25 cm-•. The dashed line represents the best fit to samples from volcanically dominated areas and corresponds to a large-scale curvature of 40 cm and a cutoff wave number of 0.25 cm- •. The larger curvature estimate is consistent with the observed smoothness of the area.

Vance segments appears to be 1.5 times less than the large- scale radius of features within the caldera of Axial Vol-

cano. In all three areas, the a-/• distributions seem to be correlated through different values of .the roughness parameter.

'Figure 12 shows a-/• values estimated from backscatter data collected within the caldera of Axial Volcano, where lava types have been categorized 28 based on photographs collected from Alvin and camera-tow surveys, and geological maps compiled. 3ø'31 The area contains both jumbled flows (analogous to Hawaiian Aa) with very rough surface textures and lobate flows which appear smoother at short wavelengths. Other lava types are mixed in nearly all samples. It appears that samples dominated by jumbled lavas have smaller aS and larger/•s than those dominated by lobate flows. This implies a flatter seafloor roughness spectrum with higher relative energy in short wavelengths for jumbled lava flows consistent with observation. It is obvious that the clusters of a and/• values are correlated [as required by Eq. (5)] with a large-scale radius of R =40

0.100

crz 0.010

A

A Jumbled/Pillow <> Jumbled/Ropy + Pi!low/Lobate/Jumbled X Jumbled/Lobate [] Lobate/Sediment/Fissur • Lobate/Pillow 0.001

0.:• 0.4 0.6 0.8 1.0

FIG. 12. Summary of modeling results from a limited area of diverse lava flows within caldera of Axial Volcano. Each symbol represents different types of lava as identified by photographic images and submersible observations. Solid line fits the data cluster, which represents large-scale curvature of 40 cm and the cutoff wave number of 0.25 cm-•. Although the samples represent composites of differing lava forms, the results suggest the potential for differentiating bottom types based on acoustic backscatter properties.

cm, and a cutoff wave number kc=0.25 cm-1 for the rel- evant acoustic wave number ka=0.509 cm-1.

IV. CONCLUSIONS

Using the techniques described in this study, it appears possible to estimate seafloor acoustic backscattering parameters directly from data collected with hull-mounted multibeam sonar systems. A variety of corrections must be applied in order to use the record properly in model fitting, and the mathematical form of the model requires the use of methods (such as the simulated annealing algorithm) to produce global solutions. Other scattering models, such as those based on Gaussian spectra that estimate the rms roughness and correlation length, 32'33, could utilize the same correction techniques to produce additional seafloor parametrizations.

The preliminary results from the application of the described method to the Juan de Fuca Ridge indicates that mappable units are produced by the analysis. In many cases, the geological features vary over much smaller spatial scales than the sample size required to produce a stable model estimation, and consequently the results Fepresent some composite measurement of the component geological properties.

The method could be further improved with advances in sonar system technology. The a horizontal sampling of the Sea Beam system ( 16 beams) appears to be insufficient to remove the slope effect for complex terrains such as ridge or caldera areas. A more accurate system, with beam spacing narrower than 2.66 ø , would significantly reduce this error. In addition, it appears that a three-dimensional treatment (pitch, yaw, heave, and accurate navigation) of the bottom surface is necessary to correct for the slope angle. Equation (13) is an approximation based on the Gaussian beam pattern, and the assumption of constant backscatter strength within the beamwidth near the nadir may explain some of the data misfit. Side-lobe interference can contaminate backscattering strength samples, and the collection of complete quadrature information from the system would be ideal for the reduction of this source of error. 34 Sample picking based on centroid or CDF may reduce the standard deviation of the backscatter strength estimates. 8. The availability of calibrated sensors and pro- jectors would allow direct estimation of the scaling parameter, rather than requiring the less-reliable indirect methods described in this study. Caution must be used, since the a,/•, and scaling factor values estimated in this study are inevitably biased due to the numerous errors discussed above. The simulated-annealing algorithm used in this analysis converged reliably and provided good model estimates; however, other methods, such as those based on neural network protocols, 35 may provide certain advan- tages.

Finally, the availability of wide-swath, narrow-beam systems may allow the direct estimation of bathymetric roughness parameters using the surface spectral estimate techniques. • The 16 beams available from Sea Beam are insufficient to produce a reliable spectrum from a single swath, and composites of several swaths must be combined



to produce model parameter estimates. Modern systems with over 120 beams will allow the estimation of spectral parameters closer to the roughness scale used for backscatter strength modeling. Since the model parameters are theoretically related, TM though applied to different spatial scales, the resulting estimates may be compared. Ulti- mately, an operational system could be envisioned in which seafloor roughness parameters are derived over broad spatial scales, based on spectral roughness ::n. odel estimates of soundings and scattering characteristics obtained from acoustic backscatter.

ACKNOWLEDGMENTS

The authors wish to thank J. Miller and P. Lemmond

(University of Rhode Island) for data collection; S. Byrne (University of Rhode Island) for providing data formats for the URI logging system; J. Cappell (SeaBeam Instru- ments) for the information on the system gain; and T-K. Lau and M. van Heeswijk (Oregon State University) for mathematical support. We benefited from discussions with and comments from D. R. Jackson (University of Wash- ington) and C. de Moustier (Scripps Institution of Ocean- ography). The data were collected through a grant from NOAA's National Undersea Research Pr6gram. The analysis was supported by the Tactical Oceanographic and Warfare Systems Program of the Naval Research Labora- tory under contract of N68462-92-MP-20102: Additional support was provided by NOAA's VENTS Research Pro- gram. PMEL Contribution No. 1402.

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Estimation of seafloor microtopographic roughness through modeling of acoustic backscatter data recorded by multibeam sonar systems