Abstract

A single detector instrument concept that collects scattered light over the full range of backscattering angles is described. Its light collection aperture is designed so as to introduce a sinθ factor into the collection probability. Hence, the instrument is exactly a bb meter; it directly measures bb, not a proxy for it. For an infinitesimal aperture to the detector, the instrument would give bb exactly; for a finite aperture (e.g., 1.26cm2), it would typically give bb to an accuracy of a few tenths of 1%. The instrumentation itself is as simple as that of the well-known fixed-angle meters—it projects a beam of light into the medium and collects backscattered light with a single detector; the differences are the position of the detector and the shape/orientation of the entrance aperture to the detector.

© 2011 Optical Society of America

1. Introduction

The volume scattering function (VSF) is β(λ,θ,φ). For natural waters, the scattering is assumed azimuthally symmetric about the incident direction and the VSF is written β(λ,θ). The integrated backscattering coefficient bb(λ) is the integral of β(λ,θ) over the solid angle (dΩ=sinθdθdφ), for the backwards hemisphere,

bb(λ)2ππ/2πβ(λ,θ)sinθdθ,
where the integration over φ has been replaced by the factor 2π.

The backscattering coefficient is one of the inherent optical properties of natural waters, i.e., it is independent of the ambient light field in the water. As such, it has a central role in many problems of optical oceanography, e.g., visibility and water clarity, remote sensing and diffuse reflectance, hydrosol composition and distributions, and even for estimates of bulk refractive index. In general, it is an important component in the characterization of natural waters [1, 2, 3, 4, 5]. It can play an important role in the modeling of ocean waters, e.g., the Fournier– Forand phase function [4, 6]. Sources contributing to the backscattering include the water molecules themselves, suspended organic and inorganic particulates, and air bubbles.

Although a measurement of the VSF can generally be a slow, difficult, and time-consuming process, a rigorous, straightforward determination of the backscattering coefficient involves measuring the VSF and then using Eq. (1) to calculate bb [7, 8, 9, 10]. Oishi et al. described an instrument to measure the entire VSF using a CCD camera [11], but simpler, single-measurement instrumentation for bb is desirable. Measurements of various optical properties of the water, together with application of an inversion algorithm [1, 12, 13, 14, 15], have also been used to obtain bb. Other approaches include an integrating cavity that collects backscattered light [2, 16] and the use of cone reflectors to select backscattered light [17, 18, 19].

But, by far the approach that has received the most widespread interest and application is measurement at a single fixed optimum angle. The optimum angle is chosen so that in a variety of typical measurements, there appears to be the tightest correlation between the scattering at that angle and the backscattering coefficient [3, 20, 21, 22, 23, 24, 25]. The optimum angle meter is, in fact, the basis of the only commercially available instruments. But again, it does not really measure bb. It measures backscattering at some specific angle that is anticipated to give a result proportional to the correct bb; this has been extensively investigated and is, in fact, expected to be generally fairly accurate to 10% [23, 26] or it “…should have a maximum uncertainty of only a few percent”[20]. But even Oishi, a major proponent of the approach, questions “…if it can hold for plankton bloom and coastal water”[19]. Situations can (and will) arise that will distort the VSF outside acceptable conditions for measuring bb with a fixed-angle meter, for example, the presence of monodisperse distributions of particles.

Instrumentation is described in the following that is as simple to employ (although the theoretical anal ysis is more complicated) as a conventional fixed- angle meter; it is, in fact, a fixed-angle meter, but one that truly measures bb. Both fixed-angle meters involve sending a beam of light into the water and using a single detector to measure backscattered light. But the instrumentation described here collects backscattered light in such a way as to measure the actual bb for arbitrary β(λ,θ); it does not rely on assumptions about scattering particle types or distributions. The insight that makes this instrumentation possible is to note that if a laser beam propagates along the z axis and a detector is mounted with the normal to its detector aperture parallel to the z axis, one will measure the integral over the detection solid angle of β(λ,θ)cosθdθ. In order to convert cosθ to the sinθ required by Eq. (1), one must have a detector aperture whose normal is perpendicular to the z axis. And that is exactly what this instrumentation does—it simply sends a well- collimated laser beam into the water and collects backscattered light that passes through an aperture whose normal is perpendicular to the z axis. It also scrambles the backscattered light in an integrating cavity to eliminate all memory of the original scattering direction before it reaches the detector. In some sense, it is a form of the Beutell and Brewer [27] instrument to measure scattering, but in reverse; it introduces the sinθ factor in the same way as the Beutell and Brewer instrument. For typical instrument parameters and due to the presence of sinθ in the integrand defining bb, most of the contributions to the observed signal come from light that has been backscattered at distances of less than a few centimeters.

The VSF—and therefore the backscattering coefficient as well—is inherently dependent on the wavelength of the scattered light. However, to simplify the readability of equations, explicit use of λ will be omitted in the following; the wavelength dependence will be considered to be implicit.

2. Instrument Theory and Design

Consider a laser beam entering a medium at z=0; see Fig. 1. The aperture that defines the directions of the scattered light to be detected is an opening in the form of a cylindrical ring of radius R extending from z=Z0 to z=+Z0. The laser beam passes through a quartz rod surrounded by a small opaque tube of radius r up to the point z=0 where it enters the scattering medium. The laser beam of cross- sectional area A and irradiance E0 propagates along the z axis.

In Fig. 1, consider a scattering volume of length dz and cross-sectional area A at position z. The power dP scattered at an angle θ into the detector aperture in the solid angle defined by θ1θθ2 is then given by

dP=2πAE0θ1θ2ecxβ(θ)sinθdθdz,
where β(θ) is the VSF, c is the attenuation coefficient, and x is the total distance a scattered photon travels in the medium, i.e., from z=0 to the scattering volume dz and then back to the detector. The total power scattered into the detector is obtained by adding (integrating) the power dP from all elements dz,
P=2πAE00dzθ1θ2ecxβ(θ)sinθdθ.

The theoretical analysis will proceed by first considering the case of small attenuation, i.e. ecx1; the analysis will then be extended to cases in which the effects of attenuation become significant. The advantage of this approach is that the small attenuation case allows an in-depth understanding of the theory of the instrument without the need for any further approximations. By then including the effects of attenuation, the efficacy of the instrument can be demonstrated at even relatively high values of attenuation.

2A. Small Attenuation

From Fig. 1, θ1 is given in terms of z. Specifically, tan(πθ1)=R/(zZ0), solving for θ1 gives

θ1(z)=πtan1(RzZ0)πcos1(zZ0R2+(zZ0)2).
Similarly, θ2 is
θ2(z)=πtan1(Rz+Z0)πcos1(z+Z0R2+(z+Z0)2).
For z=0, the two limiting values for θ are (note, 0θπ),
θ10=θ1(0)=tan1(RZ0),θ20=θ2(0)=πtan1(RZ0).

For a fixed value of z, the detector only observes light that is scattered at angles θ defined by θ1(z)θθ2(z), where θ1(z) and θ2(z) are given by Eqs. (4, 5), respectively. On the other hand, for a fixed value of θ, scattered light is only observed from points z between z1(θ) and z2(θ). These two functions are obtained by inversion of Eqs. (4, 5), respectively; or by examination of Fig. 1,

z1(θ)=Z0Rtanθ,z2(θ)=Z0Rtanθ.
Note that z1(θ) is always larger than z2(θ). Rewriting Eq. (3), setting c=0, and explicitly showing the z dependence gives
P=2πAE00dzπtan1(RzZ0)πtan1(Rz+Z0)β(θ)sinθdθ.
Reversing the order of integration in Eq. (8), i.e., integrating over z first, gives
P=2πAE0θ20πβ(θ)sinθdθz2(θ)z1(θ)dz+2πAE0θ10θ20β(θ)sinθdθ0z1(θ)dz,
where θ10, θ20, are given by Eq. (6), and z1(θ), z2(θ) are given by Eq. (7); see also Fig. 2. The integrations over z in Eq. (9) are straightforward; completing them and rearranging gives
P=(2Z0AE0){2πθ20πβ(θ)sinθdθ+πθ10θ20β(θ)sinθdθπRZ0θ10θ20β(θ)cosθdθ}.
Observed signals are proportional to the power P, and because 2Z0AE0 is just a proportionality factor that appears repeatedly, the expressions will be simplified by defining
P=P2Z0AE0;
observed signals are also proportional to P. Rewrite Eq. (10) using Eq. (11); also split the range of integration of the second integral of Eq. (10) into regions greater than and less than π/2,
P=2πθ20πβ(θ)sinθdθ+ππ/2θ20β(θ)sinθdθ+πθ10π/2β(θ)sinθdθπRZ0θ10θ20β(θ)cosθdθ.
Adding and subtracting a term identical to the second integral gives
P=2πθ20πβ(θ)sinθdθ+2ππ/2θ20β(θ)sinθdθππ/2θ20β(θ)sinθdθ+πθ10π/2β(θ)sinθdθπRZ0θ10θ20β(θ)cosθdθ.
Combining the first two integrals and rearranging gives the power observed by the detector,
P=2ππ/2πβ(θ)sinθdθ+πθ10π/2β(θ)sinθdθππ/2θ20β(θ)sinθdθπRZ0θ10θ20β(θ)cosθdθ.
Denoting the four successive integrals in Eq. (14) by I, II, III, and IV, in that order gives
P=I+II+III+IV=bb(1+ρ2+ρ3+ρ4),
where integral I is, by definition, just bb, and the ratios ρ2, ρ3, and ρ4 are
ρ2=IIbb;ρ3=IIIbb;ρ4=IVbb.

Physically, the domain of integration of integral I (i.e., the exact bb) covers the combination of the domains named A and B in Fig. 3, whereas the range of integration for the actual detected power, Eq. (3), covers the combination of domains A and C in Fig. 3. The accuracy of the instrument is given by the sum of integrals II, III, and IV; basically, it is determined by the extent to which the power detected over domain C is equal to the power not detected over domain B.

The accuracy is dominated by the choice of Z0 or, more precisely, by the choice of R/Z0. To the extent that Z0 has a small value, integrals II and III in Eq. (15) will make a relatively small contribution compared to integral I; they also have an opposite sign and will tend to cancel. Integral IV is small because cosθ is small for angles near π/2; it will also be small [even in the presence of asymmetries in β(θ) around π/2] because cosθ is an odd function over the interval of integration. Finally, not only will integral IV be small, but the symmetries are such that any contribution from integral IV will always cancel part of any remainder from the sum of integrals II and III.

To demonstrate the high accuracy possible for measurements of bb, the three ratios (ρ2, ρ3, and ρ4) in Eq. (16) and their sum (which gives the accuracy of a measurement of bb) were numerically evaluated for 11 examples covering a broad distribution of water types as well as for four examples of monodisperse suspensions of microspheres. The results are shown in Table 1. For these numerical evaluations, the radius is R=0.01m, and the three values considered for the aperture parameter are Z0=0.001, 0.002, and 0.005m. For the cases of natural water examples, the values of bb are shown; they provide an indication of the wide variability in water types being considered (for the sphere suspensions, bb depends on the number density). Table 1 also shows the conversion factor χ(θoptimum)=bb/2πβ(θoptimum) used in the discussions of fixed- angle meters. For the spheres, χ is irrelevant; e.g., for 4μm spheres, χ varies from 0.56 to 1.11 over the range 115°θoptimum120°, and it is 1.43 at 140°. Of course, conventional fixed-angle meters were never intended to work for particle suspensions of this type; but, by contrast, our instrument can provide an accurate measurement of bb for any particle suspension.

Clearly, the cases with Z0=0.001m (and R=0.01m) provide exceptional accuracy, typically <0.5% (shown in boldface); these are parameters one might use in an operational instrument. The ratio R/Z0 is 10, and the relevant integration limits are θ10=84.3° and θ20=95.7°. It should be emphasized that even when Z0=0.001m, the aperture to the detector is still large and collects an ap preciable amount of light. Specifically, the open area of the aperture is Aa=(2Z0)(2πR)=4πRZ0; for R=0.01m and Z0=0.001m, it is Aa=0.0001257m21.26cm2.

If a phase function is symmetric about a scattering angle of π/2, integrals II and III in Eq. (14), or ratios ρ2 and ρ3 in Eq. (15), will exactly cancel and integral IV (or ratio ρ4) will be zero. But most phase functions decrease with increasing angle in the vicinity of π/2; in such cases, integral II (θπ/2) has a larger magnitude than integral III (θπ/2) and they do not quite cancel, i.e., ρ2>ρ3. Because of the negative sign in front of integral IV and the fact that cosθ is an odd function around π/2, its contribution always provides further cancellation of the combination of integrals II and III (ρ2 and ρ3), e.g., see the examples in Table 1. For an uncommon phase function that increases with increasing angle in the vicinity of π/2, the integral II would have a smaller magnitude than integral III (ρ2<ρ3), but the sign of integral IV (or ρ4) would also then be changed and will still provide further cancellation. Such an uncommon phase function is demonstrated by the example for monodisperse 596nm spheres in Table 1, an example specifically included to demonstrate this case.

The bottom line is that for an infinitesimal detector aperture, this instrument provides an exact measure of bb regardless of the asymmetries in β(θ) at 90°; the question then turns to how a finite detector aperture affects measurement accuracy. Although a detailed discussion is provided regarding the balancing of the contributions due to a finite detector aperture, there is nothing unusual about this measurement process; it is typical. To illustrate with a simple and familiar example, consider a measurement of β(θ) at some angle θ. For an infinitesimal detector angular aperture, a measurement of the scattering at angle θ gives exactly the correct result. With a finite angular aperture, the change in the amount of light collected due to scattering at angles less than θ typically tends to balance the change in the amount of light collected at angles greater than θ, giving a relatively accurate result for scattering at the desired angle θ. If the shape of the phase function β(θ) in the vicinity of θ is such that these deviations do not cancel [e.g., β(θ) has a peak at θ], the deviations can still be reduced by making the angular aperture smaller. In the case of our bb instrument, the magnitude of the net deviation between the measurement and the correct bb depends on the change in β(θ) as θ increases from below 90° to above 90°. For a β(θ) symmetric around 90° between the angles θ10 and θ20 (even a symmetric peak at 90°), the deviation is zero. The maximum deviation would occur for an unphysical step function change in β(θ) at 90°, but even in this case, the deviations from the correct bb could still be reduced to any desired accuracy by decreasing the width of the detector aperture.

To summarize, the power P that enters the detector corresponds with relatively high accuracy to the backscattering coefficient bb. The proportional quantity P, Eqs. (11, 15), shows the errors due to a finite detector aperture in the form of integrals II, III, and IV. As the concentration of scattering particulates increases, these integrals increase in the same proportion that P and bb increase; thus, their ratios, Eq. (16), to bb (i.e., ρ2, ρ3, and ρ4) do not change with the particle concentration. Furthermore, these ratios can be made negligible to the level of accuracy desired by the choice of R and Z0 (e.g., Table 1). The detector provides a signal S proportional to the detected power P, or equivalently to P, i.e., S=K0P where K0 is the instrument calibration constant that converts detected power to a signal voltage. Thus, Eq. (15) gives

S=K0(1+ρ2+ρ3+ρ4)bb=Kbb,
where K is the slope of a plot of the measured signal S versus bb. The true calibration constant K0 in terms of the measured K is
K0=K1+ρ2+ρ3+ρ4.
Ideally, the systematic deviations ρ2, ρ3, and ρ4 are negligible so that K0=K; in practice, if they are large enough to introduce significant errors, they must be taken into account in a determination of the true calibration constant K0. These ratios, ρ2, ρ3, and ρ4 are evaluated in Table 1 for several cases. For example, with 203nm spheres, they sum to 3.03% when Z0=0.005m; on the other hand, they sum to 0.14% when Z0=0.001m. In fact, for Z0=0.001m, the deviations are only a few tenths of 1% across the wide variety of examples in Table 1; overall errors will typically be <0.5% for measurements with Z0=0.001m.

Further insight into the physics of the instrumentation can be obtained by examining the detection solid angle as a function of the distance z at which backscattering occurs. From Fig. 1, the solid angle subtended by the detector aperture at any point z is

Ω(z)=2πθ1(z)θ2(z)sinθdθ=2ππtan1(RzZ0)πtan1(Rz+Z0)sinθdθ=2π{Z0zR2+(Z0z)2+Z0+zR2+(Z0+z)2}.

Figure 4 is a plot of Eq. (19) for the case R=0.01m and Z0=0.001m; it shows that the detection solid angle decreases so quickly that there is very little contribution to the integral from backscattering at large distances. In fact, more than 90% of the total detection solid angle comes from distances z less than 0.021m, and more than 99% at distances less than 0.071m. This is, of course, to be expected, because the sinθ factor in the integrand of the definition of bb reduces the contributions to bb as θ approaches 180°. In fact, bb is completely independent of the value of the phase function at 180°. Because relevant distances are short, one would expect good performance even in the presence of attenuation. This sets the stage for examining attenuation effects.

2B. Effects of Attenuation

To evaluate the effects of attenuation, return to Eq. (3), in which c is the attenuation coefficient [the sum of the scattering and absorption coefficient (c=a+b)] and x is the total distance that the light travels through the medium. From the geometry in Fig. 1, the latter is

x=z+Rsin(θ),
and therefore Eq. (3) becomes
P(c)=2πAE00dzθ1θ2ec(z+Rsin(θ))β(θ)sinθdθ.
Reversing the order of integration as in Eq. (9) gives
P(c)=2πAE0θ20πβ(θ)sinθdθZ0RtanθZ0Rtanθec(z+Rsinθ)dz+2πAE0θ10θ20β(θ)sinθdθ0Z0Rtanθec(z+Rsinθ)dz.
After completing the integrations over z, Eq. (22) becomes
P(c)=2πAE0c(ecZ0ecZ0)θ20πβ(θ)sinθecR1cosθsinθdθ+2πAE0cθ10θ20β(θ)sinθecRsinθdθ2πAE0cecZ0θ10θ20β(θ)sinθecR1cosθsinθdθ.
Split the last integral into regions greater than, and less than π/2; also add and subtract the integral
2πAE0cecZ0π/2θ20β(θ)sinθecR1cosθsinθdθ.
The expression equivalent to Eq. (23) now contains six integrals; two of these integrals are combined with the first integral in analogy with the derivation of Eq. (14) for the case of no attenuation. If the proportionality constant 2Z0AE0 is also introduced as in Eq. (11), then Eq. (23) transforms into
P(c)=P(c)2Z0AE0=(ecZ0ecZ0cZ0)ππ/2πβ(θ)sinθecR1cosθsinθdθ+π{1cZ0θ10θ20β(θ)sinθecRsinθdθecZ0cZ0θ10π/2β(θ)sinθecR1cosθsinθdθecZ0cZ0π/2θ20β(θ)sinθecR1cosθsinθdθ}.

Note, there are no approximations in this expression for the measured power P(c). The first term, label it P0(c), is the detected power representing bb; it is exactly proportional to bb when c=0. The other three terms, label this combination P1(c), represent additional (but typically very small) quantities that are part of the observed power; thus, the measured power is proportional to P(c)=P0(c)+P1(c). Although in the final analysis a correction factor will be introduced to correct the measured P(c) for attenuation, useful insights can be gained through a further analysis of Eq. (25), specifically, by expanding the exponential functions in the power series.

Consider P0(c), the first integral in Eq. (25). Expanding the exponential factors, ecZ0, ecZ0 in P0(c) for small cZ0 gives

ecZ0ecZ02cZ0(1+16c2Z02+1120c4Z04).
If Z0=0.001m and c=1m1, then the first-order correction, c2Z02/6 in this expansion is less than 0.2parts per million (ppm); even for c=10m1, it is at the level of only 17ppm. Thus, the first-order correction is neglected. Using only the zeroth-order term gives
P0(c)=2ππ/2πβ(θ)sinθecR1cosθsinθdθ.
For c=0, this integral is just bb. For c0, it will be shown that the dominant reduction in the measured power P(c) due to attenuation arises from the exponential function in the integrand of this expression for P0(c); the reduction due to P1(c) is significantly smaller.

Now consider the three integrals comprising P1(c). Typically, R and Z0 are small; for the next-generation instrument, they are expected to be R=0.01m, Z0=0.001m. Consequently, even for c=10m1, cR and cZ0 are small (<0.1). Also, throughout the integration range of all three integrals, θ is close to π/2; therefore, sinθ1. Consequently, all the exponentials in the three terms of P1(c) can be expanded in the power series giving

P1(c)=P1(c)2Z0AE0={πθ10π/2β(θ)sinθdθππ/2θ20β(θ)sinθdθRπZ0θ10θ20β(θ)cosθdθ}+c{Rππ/2θ20β(θ)(1cosθ)dθRπθ10π/2β(θ)(1cosθ)dθZ0π2θ10θ20β(θ)sinθdθ+R2Z0πθ10θ20β(θ)cosθsinθdθR22Z0πθ10θ20β(θ)cos2θsinθdθ}+ε,
where ε represents higher order terms that are proportional to cj+k1RjZ0k with j and k ranging over all positive integers and zero, but with j+k=0, 1, 2, 3, 4, 5,…The term with j+k=0 is identically zero; the terms with j+k=1 and j+k=2 are shown, respectively, in the two successive pairs of curly brackets in Eq. (28). The three terms in the first curly brackets are independent of c. They are identical to the error terms given by integrals II, III, and IV in Eq. (14). They are inherent to the measurement process and are independent of the attenuation c. They also provide only a small correction as discussed previously (see Table 1). All the second-order terms (j+k=2) are proportional to c and are in the second pair of curly brackets in Eq. (28); they provide just a small correction on a small correction. These terms as well as all higher order terms (j+k>2) will, of course, be zero for c=0.

In order to demonstrate the relatively small effect of c in Eq. (27), as well as the negligible effects of the correction terms given by Eq. (28), all these terms have been numerically evaluated using the MVSM data from October 2009 in Chesapeake Bay [28, 29]. The instrument parameters used in these numerical evaluations are c=1m1, R=0.01m, and Z0=0.001m. The results are presented in Table 2 and are shown as a percentage of P0(c=0)=bb; they are calculated from the MVSM data. The sums of the correction terms are indeed negligible.

Clearly, the major impact of attenuation in the measurements is on P0(c), the first term in Eq. (25), or equivalently, Eq. (27) (the second entry in Table 2). As c increases, P0(c) decreases, and the change is considerably more than the change produced by all other terms combined (the latter terms were already small corrections). Further insight is possible by an examination of the effects of c in Eq. (27). Unfortunately, the exponential function in the integrand cannot be simply expanded for small cR because, for cR0 the exponent is not always small; i.e., it is at θ=π. Nevertheless, for small cR, the product of sinθ with the exponential in the integrand is still approximately sinθ for all θ; this is clearly demonstrated by the plots in Fig. 5. For example, on this scale, the curve for cR=0.001 (e.g., R=.01m, c=0.1m1) cannot be distinguished from the data points for cR=0, the latter being just a plot of sinθ. Even for an order of magnitude larger c (i.e., 1m1), the resulting curve for cR=0.01 is barely distinguishable from the sinθ data points.

The dependence on c for the expected experimentally measured value of P(c), Eq. (25), was compared with the exact value of bb=P(0), for the wide variety of natural waters considered in Table 1 as well as for a Rayleigh scattering function. Instrument pa rameters for the calculation were R=0.01m and Z0=0.001m. The calculated percent error, [P(c)bb]/bb×100 is plotted as a function of the attenuation c in Fig. 6. These results demonstrate the effectiveness of this bb meter design, even in the presence of significant attenuation. For example, from Table 2, the measured power P(c) gives bb to an accuracy of 97.67% when c=1m1; i.e., the attenuation error is only 2.33%. This is consistent with (actually, less than) the attenuation correction for commercial fixed-angle meters—e.g., the user’s manual for the WET Labs ECO BB meter states, “…attenuation error is typically small, about 4% at a=1m1” [30]. Figure 6 shows that at an attenuation of c=1m1, the direct measurement error with our instrumentation is less than 3% for the entire broad distribution of natural water samples considered in Table 1. It is interesting to note that the effects of attenuation on Petzold clear ocean, Petzold coastal ocean, and Rayleigh are nearly identical in Fig. 6, while all the others tend to cluster in a band at slightly smaller percent deviations for any attenuation c.

It is apparent from Fig. 6 that the percent error is almost linear in attenuation. Although the slopes are slightly different for the various phase functions, one should clearly be able to introduce an average attenuation-dependent linear factor that would provide a correction to the measurements observed with this bb meter; such a factor would improve the accuracy at all typical values of attenuation. The following form provides an attenuation corrected PA(c):

PA(c)=(1+2.7cR)P(c),
where c is the attenuation (m1) and R (m) is an instrument parameter (see Fig. 1).

This attenuation-dependent correction factor was used on all the examples plotted in Fig. 6. The attenuation dependence of the percent difference between bb and the attenuation corrected PA(c) is shown in Fig. 7. Of course, this correction requires an additional measurement of c, but as a result, all these data demonstrate uncertainties of less than 3% for attenuations up to 10m1; for c less than 1m1, measurement uncertainties are a few tenths of 1%.

Now consider the effect that an error δc in the perceived value of the attenuation coefficient c will have on the correction algorithm. Explicitly including δc in Eq. (29) gives

PA(c)=[1+2.7(c±δc)R]P(c)=(1+2.7cR)P(c)±2.7δcRP(c).

From Eq. (30) and setting R=0.01m, the fractional uncertainty Δ in PA(c) due to the uncertainty δc can be written

Δ=δPA(c)PA(c)=±2.7δcRP(c)(1+2.7cR)P(c)=±2.7Rδc(1+2.7cR)=±[0.0271/c+0.027]δcc.

For small c, the denominator of the coefficient in brackets is large and Δ is a small fraction of the uncertainty in c. For very large c, Δ is equal to the uncertainty in c. As an intermediate case, suppose c is 4.0m1 and δc is 1m1 (25% error in c), then the fractional uncertainty Δ in PA(c) is 2.4%. Or, for another example, suppose c=1m1 with an uncertainty of 10% (δc/c=0.10); this propagates into an uncertainty in PA(c) of 0.26%. Clearly, for c less than 10m1, the accuracy in the determination of c does not have much impact on the correction of bb for attenuation.

3. Instrument Implementation

Measurements were performed in a water tank of approximately 1.80m×0.6m×0.6m that holds up to 600 liters. It was filled with 120 liters of water that had been highly purified using a Millipore water purification system with a 0.2μm particle filter as its final stage.

Figure 8 shows a cross section of the first implementation of the bb meter; the parameters for this prototype instrument were R=0.01m and Z0=0.005m. The instrument is 0.3m in length and 0.15m in diameter; the weight is below 1kg, excluding the detector and power supply for the laser. The green laser (532nm) is a 20mW, TTL-controlled, off-the-shelf system from World Star Tech; it was operated at a fixed modulation frequency of 1kHz, and it was built directly into the instrument. The power supply for the laser (3.3Vdc, <0.5A) and the electronics to record the signals were left outside the instrument.

The window is a section of a quartz cone. The angle of the cone is 45° so that light backscattered at angles between 90° and 180° will be incident on the window at angles less than 45°. Under these conditions, the reflectance at the water/quartz interface is typically 0.18%, rising to 0.21% at 35° and 0.32% at 45°. But for a field instrument, there could be problems with fouling in the space between the aperture and the window. Some discussion of other window designs for a next-generation instrument is provided in Appendix A.

The detectors are two Hamamatsu 1P21 photomultiplier tubes (PMTs). Instead of mounting one of them directly behind the aperture, the backscattered light is diffusely reflected through an air-filled integrating cavity and is partially collected by six 0.001m diameter optical fibers that carry it outside the instrument to one PMT that provides the scattering signal. The integrating cavity employs highly reflective, compressed quartz powder with a Lambertian reflecting profile [31]. With this approach, all directionality of the backscattered light is lost and there can be no detection bias based on the backscattering angle.

However, the fibers provided an incredibly poor light transfer efficiency. Basically, the diffuse reflecting walls have essentially 100% reflection efficiency, so all light entering the cavity must leave the cavity either through the six 0.001m diameter signal fibers or back through the entrance aperture. For Z0=0.005m, the entrance aperture has area Aaperture=4πRZ0=6.3cm2. Thus, the fraction of light leav ing through the signal fibers is Afibers/(Afibers+Aaperture)0.74%.

In this regard, a comment about the next- generation instrument is in order. Specifically, decreasing Z0 to, for example, Z0=0.001m decreases the amount of light entering the aperture by a factor of 5. But, because the aperture is smaller, less light can escape through it and the fraction of light leaving the cavity through the signal fibers will increase to 3.6% from 0.74% or nearly a factor of five (almost no decrease in the total number of detected photons!). Not only would the next-generation instrument have Z0=0.001m, but instead of fibers, detectors (PMTs or p-i-n diodes) would be mounted directly in the wall of the cavity and would have an active detection area of 3cm2. This does not affect the important aspect of erasing any memory of the directionality of the photons entering through the detector aperture—that memory loss occurs at the first reflection from the diffusely reflecting walls. With detectors mounted directly in the cavity wall, the fraction of the light entering the cavity that would be detected increases to 70%, or a factor of 20 more signal.

Some stray light is produced by scattering and reflections from the laser beam inside the instrument before the light is guided through the glass rod into the water. A normalization fiber collects some of this stray laser light and carries it to the second PMT whose output provides the normalization signal.

The scattering and normalization signals produced by the PMTs are fed through separate lock-in amplifiers that only process signals at the 1kHz modulation frequency, thereby blocking any signals produced by sources other than the laser, e.g., ambient light. The signals are sent to an oscilloscope where they are read out to a PC and are simultaneously monitored in real time. The computer samples and averages the dc outputs of the lock-in amplifiers over a 4s time interval and records the average value for that time interval. This is repeated 75 times; these 75 values are then averaged to ob tain a final value for each data point (total of 5min of averaging). The 75 successive measurements improved statistics, but more importantly, they provided a backup check for drifts or systematic variations. None were observed, and the average magnitude of the deviation of these 75 measurements from their final average value was typically less than 0.5%; the maximum deviation of any of the 75 data points from their average value was typically less than 1.5%. The signals with this initial prototype are already so strong that, in practice, one of these 75 data points would generally be sufficient unless the goal is a few tenths of a percent accuracy; with the next-generation instrument and direct-mounted detectors, the signals will be even stronger. The normalized signal is the output of the first lock-in amplifier divided by the output of the second lock-in amplifier.

4. Results

4A. Calibration

To obtain a calibration, the signal (volts) was measured as a function bb for three sizes of polystyrene spheres at various concentrations. Specifically, the tank was filled with a known amount of highly purified water that had been filtered through a 0.2μm filter. The signal from this water was recorded. Then, predetermined amounts of NIST-traceable polymer microspheres [polystyrene latex (PSL)] [32] were added to successively increment the particle concentration and hence bb. The corresponding signal at each particle concentration was measured.

Because the mean particle diameter and the width of the size distribution is known, Mie calculations can be used to determine the necessary amount of the PSL suspension to add to the tank to obtain any desired backscattering coefficient bb. Rather than trying to pipette small amounts of the original suspension into the tank at each successive step, a predilution approach was used. Specifically, a set of identical suspensions were prepared, each with a desired, predetermined amount of PSL spheres corresponding to the desired increase in bb at each successive measurement. The procedure was as follows:

First, Mie calculations were used to determine the amount of the original PSL suspension needed to produce some total bb for measurements in the tank.

Second, after sonification of the PSL suspension, the total required amount of PSL suspension was pipetted from the original container into a thoroughly cleaned beaker, using a pipette of appropriate size (Eppendorf pipettes, 0.01, 0.10, and 1.0ml with single-use tips).

Third, 100ml of pure H2O and a few drops of Tween 20 surfactant were added to the beaker, and the diluted PSL suspension was thoroughly mixed by sonification.

Fourth, this suspension was distributed in equal amounts into each of a number N (5) of sample tubes as follows.

Fifth, equal amounts (1ml) of the diluted solution were pipetted into each of the N sample tubes. This was repeated until there was not enough solution to repeat for all N sample tubes.

Sixth, this remaining solution was diluted again by adding 100ml of pure H2O; it was then pipetted into the N sample tubes as before. The remaining solution was again diluted and the procedure repeated several times.

Finally, there will still be a residual amount of very highly diluted solution; it is distributed as equally as possible among the N sample tubes, but because it is so highly diluted, negligible error is introduced. This procedure was repeated with larger amounts of the initial PSL solution when larger steps in bb, or higher values of bb were desired.

The major source of error is the initial pipetting of the concentrated suspension from the original container. The estimated potential error in the concentration of PSL spheres in the tank using this procedure is approximately 3%. To keep the beads in suspension, an aquarium pump was kept turned on in the tank to keep the water circulating. Prior to each calibration run, this pump was run with pure water to ensure that it was clear of any contaminants.

In a preliminary step, a “placebo” experiment was performed; i.e., no particles were introduced, but all steps, from the filling of the tank, dilution of the samples, and the adding and mixing of those samples into the tank were done. Because no particles are added to increase the backscattering coefficient, any change in signal must be caused by unwanted contaminants. The results of this measurement are shown in Fig. 9 and are consistent with the expectation that there will be a small increase in the backscattering in the tank due to contamination from the individual samples. Using the calibration measurement (described in the next paragraph), the average increase as each sample is added corresponds to an increase in the backscattering coefficient of bb=0.00005m1, or about 6% of the backscattering coefficient of pure water. These data also illustrate the capability to measure bb for pure water.

The results of measurements with 203, 596, and 3005nm diameter polystyrene spheres are shown in Fig. 10. Linear fits to these data were made, and the fitted equations are also shown in Fig. 10; their slopes are

K203=304.3±1.7,K596=298.2±2.9,K3005=308.5±3.7,
where the error bars are from the statistical fits to the straight lines; the total error for each slope is larger and is effectively determined by the error in the initial sampling of the concentrated suspension from the original container, estimated at ±3%.

Because the Z0 for these data is relatively large (Z0=0.005m), the true calibration constant K0 (the instrumental ratio that gives the conversion from optical power to signal voltage) must be obtained by adjusting for the known systematic shifts in each of the experimentally determined slopes. This is achieved using Eq. (18) together with the three ratios ρ2, ρ3, and ρ4 (listed in Table 1 for each diameter of polystyrene spheres). The results, derived from the fitted K values listed in Eq. (32), are K0=295, 313, and 299 for spheres with diameters of 203, 596, and 3005nm, respectively. The average of these three values is taken as the K0 for the instrument:

K0=303±5.5%,
where the error is based on the ±3% error in the particle concentrations and the standard deviation of the fitting parameter for each slope. However, in order to encompass systematic shifts such as are shown in Table 1, the error in K0 for this prototype instrument should be increased to approximately ±10%. Of course, just reducing the aperture param eter from Z0=0.005m to Z0=0.001m for the next-generation instrument would reduce the systematic shifts to a few tenths of 1%, and overall errors of a few tenths of 1% would also be expected.

4B. Absorption Effects

To investigate the effects of absorption on the signal, measurements were taken while keeping bb constant and adding dye (McCormick black food color, which has no particle scattering and no fluorescence) to the tank to increase the absorption. Because small errors in attenuation are relatively insignificant, a simple transmission measurement was used to measure the attenuation in the tank after each addition of dye. A small detector measures a signal proportional to the transmitted laser power P(c) at four to six distances where the detector package is inserted into the water. A plot of ln[P(c)/P(0)] versus distance for the four to six data points is fitted to a straight line whose slope is the attenuation coefficient. A frosted glass plate whose area is approximately 5cm2 covers the aperture to the detector. This large size was chosen to ensure that the full transmitted beam will always be detected (neither the bb instrument nor the detector are rigidly mounted in the tank). Of course, measurements taken with this detector will always slightly underestimate the total attenuation in the tank because very small angle forward scattered light will be detected and counted as transmitted light.

The results of the experiment to study attenuation are shown in Fig. 11. At the beginning of this experiment, a reference signal P(0) is measured with only pure water and PSL spheres in the tank at a concentration so as to give bb0.01m1. This level for bb was chosen so as to ensure that the small amount of contaminants invariably introduced into the water sample when adding dye would have a negligible effect on the backscattered signal (e.g., as in the placebo experiment, Fig. 9). The relative signal is then the measured signal P(c) at different attenuations divided by the reference signal P(0). Thus, the relative signal is by definition 100% at the relative attenuation data point labeled 0m1.

The theory shown in Fig. 11 is not a fit to the data; it is a calculation of P(c) using Eq. (25) with the phase function β(θ) for 4.3μm PSL spheres; the curve plotted is P(c) divided by P(0), i.e., the calculation with c=0.

5. Summary and Conclusions

In conclusion, after decades of effort by many investigators to develop and characterize instruments to measure bb, an instrument is described that, for the first time to our knowledge, provides a direct measurement of bb rather than some other quantity that will hopefully approximate bb. The instrumentation can be constructed (by appropriate choice of instrument parameters R and Z0) to provide measurements of bb to arbitrarily high accuracy, i.e., 5%, 1%, 0.1%,…, etc. It is instrumentation that for the first time provides a direct and accurate measurement of bb regardless of the shape of the phase function; it works for any suspension of particles and does not rely on the hope that the sample being measured is consistent with some characteristic natural water type.

Finally, although this instrument concept might be viewed as a version of a fixed-angle meter, it is in some ways actually simpler than the conventional fixed-angle meters that only measure a proxy for bb. Specifically, it has the detector and light source colocated on the z axis rather than having a detector set off to the side at an appreciable angle. That separation of the light source and detector in conventional fixed-angle meters leads to a very complicated scattering volume with undesirable different sensitivities to positions and scattering directions in the observation volume. On the other hand, our instrument might appear more complicated because the detector aperture is unconventional—i.e., the normal to the detector aperture is perpendicular to the z axis rather than looking directly at the scattering volume, but it is still just an aperture to the detector. Overall, the instrument is extremely simple and robust; it is basically a small package that sends a laser beam out into the water and collects backscattered light in an appropriate way. Many improvements in the initial prototype version of the instrument are possible. Two of the most important improvements are (i) for high accuracy and precision, a practical instrument should have parameters Z0=0.001m and R=0.01m and (ii) to dramatically increase the signal strength, the detectors should be mounted directly in the integrating cavity rather than using optical fibers to collect and transmit the light.

A calibration using particles with different VSFs confirms that the theory presented correctly explains the operation of the instrument. The effects of attenuation were investigated, and a simple algorithm was presented that will provide some correction for the effects of attenuation in the measurements. For attenuations less than 1m1, the corrected measurements will have errors of less than a few tenths of 1%. At higher attenuations (up to 10m1), errors can be a few percent.

Appendix A: Window Considerations

The aperture to the detector must be sealed with a transparent window. At the same time, it is crucial that the window not introduce a significant bias into the angular detection sensitivity. Figure 12 shows the front portion of the instrument with three window possibilities. Figure 12a is reproduced from Fig. 8 and is the implementation used in our prototype instrument.

Figure 12b shows a window in the form of a cylindrical quartz ring whose cross section is a right triangle with the inner acute angle designated ψ. The window surface in contact with the water is the inner face of the cylindrical ring. Light backscattered at angles from 90° to 180° will be incident on this surface at angles from 0° to 90°, respectively. For unpolarized light, the reflectivity at this water/quartz interface increases from 0.18% for light incident at 0°, to 0.32% at 45°, to 100% at 90°. To evaluate the seriousness of these reflections, first recall Fig. 4. It shows the observation solid angle as a function of the distance z at which backscattering occurs. Now, the angle of incidence on the window is given by θ90° where θ, the backscattering angle, is related to z by z=R/tanθ. Hence, the fraction of light lost due to reflection at the input window can be included as an effective reduction in the observation solid angle as a function of the distance z; this reduction as a function of z is shown in Fig. 13.

The maximum reduction in observation solid angle due to reflection loss at the window actually occurs at normal incidence (when z=0). At normal incidence, the decrease is 0.0023sr, which is only a 0.18% decrease. As the angle of incidence increases, the percent decrease will monotonically increase, but the actual net decrease in the observation solid angle never exceeds the decrease at 0°. At angles of incidence greater than 70° (z=0.027m), the actual decrease in the solid angle goes into a steady decline to zero when the angle of incidence is 90° (where reflection is 100%). Basically, the inherent decrease in the observation solid angle dominates the reflection losses. Variations in the reflection losses at the second surface of the quartz ring in Fig. 12b can be minimized (2%) by appropriate choice of the angle ψ. However, perhaps the best way to eliminate angle-of-incidence dependencies is by frosting this second surface. Frosting reduces transmission, but there is plenty of signal and the resulting diffuse transmitted light is perfectly acceptable for input to the integrating cavity.

Finally, the window example shown in Fig. 12c is perhaps optimum. Although light transmission will be less, the Teflon diffuser at the input surface eliminates angle-of-incidence dependencies. The use of Teflon for such purposes has been tested [33]. Again, the back surface of the quartz plate can be frosted.

We gratefully acknowledge support from the Robert A. Welch Foundation under grant A-1218, the U.S. Army/REDCOM Edgewood Chemical and Biological Center Aberdeen Proving Ground under contract W911SR-08-C-0019, and the George P. Mitchell Chair in Experimental Physics. We also thank George Kattawar, Deric Gray, and Yu You for many helpful discussions and comments. Finally, we thank Jim Sullivan for some very valuable suggestions on the final manuscript.

Tables Icon

Table 1. Deviations from bb That Would Be Observed for Measurements Made with This Instrumentation in Natural Waters or Suspensions of Microspheresa

Tables Icon

Table 2. Numerical Evaluations of Eq. (25, 27), and the Various Terms in Eq. (28) Using MVSM Data for β(θ) Obtained during October 2009 in Chesapeake Bay [28, 29]a

 figure: Fig. 1

Fig. 1 Conceptual sketch of instrument (cylindrical symmetry around z axis).

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 figure: Fig. 2

Fig. 2 The functions θ1(z), θ2(z) and the functions z1(θ), z2(θ) are the same curves, respectively, i.e., for the latter, z is read off the horizontal axis as a function of θ on the vertical axis. The light gray and dark gray areas correspond to the domains of integration of the first and second integral in Eq. (9).

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 figure: Fig. 3

Fig. 3 Domains of integration for the integrals in Eq. (14).

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 figure: Fig. 4

Fig. 4 Detection solid angle Ω(z) with R=0.01m and Z0=0.001m.

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 figure: Fig. 5

Fig. 5 Coefficient of β(θ) in the integrand of Eq. (27).

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 figure: Fig. 6

Fig. 6 Dependence on attenuation c of the percent deviation from bb that would be observed in a measurement of P(c), Eq. (25).

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 figure: Fig. 7

Fig. 7 Percent deviation from bb of a measurement of P(c) after multiplication by the attenuation correction factor (1+2.7cR) to obtain PA(c).

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 figure: Fig. 8

Fig. 8 Instrument implementation—cylindrical symmetry around laser beam (except for fibers).

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 figure: Fig. 9

Fig. 9 “Placebo” experiment showing a slight signal increase as samples are added.

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 figure: Fig. 10

Fig. 10 Measured data for aqueous suspensions of polymer microspheres with diameters d=203, 596, and 3005nm. The least-squares fits of the data to the straight lines are also shown.

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 figure: Fig. 11

Fig. 11 Attenuation effects on the measured signal.

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 figure: Fig. 12

Fig. 12 Three possibilities for the window at the aperture to the detector.

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 figure: Fig. 13

Fig. 13 Effective reduction in the observation solid angle as a function of z due to reflection at the quartz window in the aperture of Fig. 12b.

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1. H. R. Gordon, M. R. Lewis, S. D. McLean, M. S. Twardowski, S. A. Freeman, K. J. Voss, and G. C. Boynton, “Spectra of particulate backscattering in natural waters,” Opt. Express 17, 16192–16208 (2009). [CrossRef]   [PubMed]  

2. M. Kim and W. D. Philpot, “Development of a laboratory spectral backscattering instrument: design and simulation,” Appl. Opt. 44, 6952–6961 (2005). [CrossRef]   [PubMed]  

3. R. Maffione and D. Dana, “Instruments and methods for measuring the backward-scattering coefficient of ocean waters,” Appl. Opt. 36, 6057–6067 (1997). [CrossRef]   [PubMed]  

4. C. D. Mobley, L. K. Sundman, and E. Boss, “Phase function effects on oceanic light fields,” Appl. Opt. 41, 1035–1050 (2002). [CrossRef]   [PubMed]  

5. M. S. Twardowski, E. Boss, J. B. Macdonald, W. S. Pegau, Andrew H. Barnard, and J. Ronald V. Zaneveld, “A model for estimating bulk refractive index from the optical backscattering ratio and the implications for understanding particle composition in case I and case II waters,” J. Geophys. Res. 106, 14129–14142 (2001). [CrossRef]  

6. G. R. Fournier and J. L. Forand, “Analytic phase function for ocean water,” Proc. SPIE 2258, 194–201 (1994). [CrossRef]  

7. G. Kullenberg, “Observed and computed scattering functions,” in Optical Aspects of Oceanography, N. G. Jerlov and E. S. Nielsen, eds. (Academic, 1974), pp. 25–49.

8. M. E. Lee and M. R. Lewis, “A new method for the mea surement of the optical volume scattering function in the upper ocean,” J. Atmos. Ocean. Technol. 20, 563–571 (2003). [CrossRef]  

9. T. J. Petzold, “Volume scattering functions for selected ocean waters,” (Scripps Institution of Oceanography Visibility Laboratory, 1972).

10. J. E. Tyler and W. H. Richardson, “Nephelometer for the measurement of volume scattering function in situ,” J. Opt. Soc. Am. 48, 354–357 (1958). [CrossRef]  

11. T. Oishi, H. Tan, R. Doerffer, and R. Heuermann, “Development of new spectral scattering function meter,” presented at Ocean Optics XVII, Fremantle, Australia, October 25–29 2004, paper 004.pdf.

12. G. C. Boynton and H. R. Gordon, “Irradiance inversion algorithm for estimating the absorption and backscattering coefficients of natural waters: Raman-scattering effects,” Appl. Opt. 39, 3012–3022 (2000). [CrossRef]  

13. G. C. Boynton and H. R. Gordon, “Irradiance inversion algorithm for absorption and backscattering profiles in natural waters: improvement for clear waters,” Appl. Opt. 41, 2224–2227 (2002). [CrossRef]   [PubMed]  

14. H. R. Gordon and G. C. Boynton, “Radiance-irradiance inversion algorithm for estimating the absorption and backscattering coefficients of natural waters: homogeneous waters,” Appl. Opt. 36, 2636–2641 (1997). [CrossRef]   [PubMed]  

15. H. R. Gordon and G. C. Boynton, “Radiance-irradiance inversion algorithm for estimating the absorption and backscattering coefficients of natural waters: vertically stratified water bodies,” Appl. Opt. 37, 3886–3896 (1998). [CrossRef]  

16. A. Bricaud, A. Morel, and L. Prieur, “Optical efficiency factors of some phytoplankters,” Limnol. Oceanogr. 28, 816–832 (1983). [CrossRef]  

17. T. Oishi, H. Tan, T. Hosaka, A. Tanaka, and R. Heuermann, “Realization of new bb meter and its quality control,” presented at Ocean Optics XVI, Santa Fe, New Mexico, November 18–22 2002, paper 042.pdf.

18. J. Suzuki, T. Oishi, K. Ura, H. Tan, and T. Hosaka, “Numerical evaluation of new bb meter,” presented at Ocean Optics XVI, Santa Fe, New Mexico, November 18–22 2002, paper 041.pdf.

19. H. Tan, T. Oishi, and R. Doerffer, “Analysis of measured spectral backward scattering coefficient,” presented at Ocean Optics XVII, Fremantle, Australia, October 25–29 2004, paper 006.pdf.

20. J. M. Sullivan and M. S. Twardowski, “Angular shape of the oceanic particulate volume scattering function in the backward direction,” Appl. Opt. 48, 6811–6819 (2009). [CrossRef]   [PubMed]  

21. E. Boss and W. S. Pegau, “Relationship of light scattering at an angle in the backward direction to the backscattering coefficient,” Appl. Opt. 40, 5503–5507 (2001). [CrossRef]  

22. N. G. Jerlov, “Particle distribution in the ocean,” in Deep Sea Expedition, H. Pettersson, ed. (Swedish Natural Science Research Council, 1953), pp. 71–98.

23. T. Oishi, “Significant relationship between the backward scattering coefficient of sea water and the scatterance at 120°,” Appl. Opt. 29, 4658–4665 (1990). [CrossRef]   [PubMed]  

24. J.-F. Berthon, E. Shybanov, M. E.-G. Lee, and G. Zibordi, “Measurements and modeling of the volume scattering function in the coastal northern Adriatic Sea,” Appl. Opt. 46, 5189–5203 (2007). [CrossRef]   [PubMed]  

25. M. Chami, E. Marken, J. J. Stamnes, G. Khomenko, and G. Korotaev, “Variability of the relationship between the particulate backscattering coefficient and the volume scattering function measured at fixed angles,” J. Geophys. Res. 111, 1–10 (2006). [CrossRef]  

26. D. R. Dana and R. A. Maffione, “Determining the backward scattering coefficient with fixed-angle backscattering sensors-revisited,” presented at Ocean Optics XVI Santa Fe, New Mexico, November 18–22 2002, paper 212.pdf.

27. R. G. Beutell and A. W. Brewer, “Instruments for the measurement of the visual range,” J. Sci. Instrum. 26, 357–359 (1949). [CrossRef]  

28. D. J. Gray (personal communication, 2011).

29. D. J. Gray and A. Weidemann, “Volume scattering function effects on underwater imaging systems,” presented at Ocean Optics XIX, Tuscany, Italy, October 6–10 2008, paper 481.pdf.

30. WET Labs, “ECO BB user’s guide (BB)” (WET Labs, Inc., 2010).

31. E. S. Fry, J. Musser, G. W. Kattawar, and P.-W. Zhai, “Integrating cavities: temporal response,” Appl. Opt. 45, 9053–9065 (2006). [CrossRef]   [PubMed]  

32. Duke Scientific, 0.203μm (Cat. No. 3200A, No. Lot 36926), 0.596μm (Cat. No. 3600A, No. Lot 36446), and 3.005μm (Cat. No. 4203A, No. Lot 36453).

33. M. S. Twardowski, “An integrated inherent optical property sensor for AUVs” (WET Labs, Inc., 2006).

References

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  1. H. R. Gordon, M. R. Lewis, S. D. McLean, M. S. Twardowski, S. A. Freeman, K. J. Voss, and G. C. Boynton, “Spectra of particulate backscattering in natural waters,” Opt. Express 17, 16192–16208 (2009).
    [Crossref] [PubMed]
  2. M. Kim and W. D. Philpot, “Development of a laboratory spectral backscattering instrument: design and simulation,” Appl. Opt. 44, 6952–6961 (2005).
    [Crossref] [PubMed]
  3. R. Maffione and D. Dana, “Instruments and methods for measuring the backward-scattering coefficient of ocean waters,” Appl. Opt. 36, 6057–6067 (1997).
    [Crossref] [PubMed]
  4. C. D. Mobley, L. K. Sundman, and E. Boss, “Phase function effects on oceanic light fields,” Appl. Opt. 41, 1035–1050(2002).
    [Crossref] [PubMed]
  5. M. S. Twardowski, E. Boss, J. B. Macdonald, W. S. Pegau, Andrew H. Barnard, and J. Ronald V. Zaneveld, “A model for estimating bulk refractive index from the optical backscattering ratio and the implications for understanding particle composition in case I and case II waters,” J. Geophys. Res. 106, 14129–14142 (2001).
    [Crossref]
  6. G. R. Fournier and J. L. Forand, “Analytic phase function for ocean water,” Proc. SPIE 2258, 194–201 (1994).
    [Crossref]
  7. G. Kullenberg, “Observed and computed scattering functions,” in Optical Aspects of Oceanography, N.G.Jerlov and E.S.Nielsen, eds. (Academic, 1974), pp. 25–49.
  8. M. E. Lee and M. R. Lewis, “A new method for the measurement of the optical volume scattering function in the upper ocean,” J. Atmos. Ocean. Technol. 20, 563–571(2003).
    [Crossref]
  9. T. J. Petzold, “Volume scattering functions for selected ocean waters,” (Scripps Institution of Oceanography Visibility Laboratory, 1972).
  10. J. E. Tyler and W. H. Richardson, “Nephelometer for the measurement of volume scattering function in situ,” J. Opt. Soc. Am. 48, 354–357 (1958).
    [Crossref]
  11. T. Oishi, H. Tan, R. Doerffer, and R. Heuermann, “Development of new spectral scattering function meter,” presented at Ocean Optics XVII, Fremantle, Australia, October 25–29 2004, paper 004.pdf.
  12. G. C. Boynton and H. R. Gordon, “Irradiance inversion algorithm for estimating the absorption and backscattering coefficients of natural waters: Raman-scattering effects,” Appl. Opt. 39, 3012–3022 (2000).
    [Crossref]
  13. G. C. Boynton and H. R. Gordon, “Irradiance inversion algorithm for absorption and backscattering profiles in natural waters: improvement for clear waters,” Appl. Opt. 41, 2224–2227 (2002).
    [Crossref] [PubMed]
  14. H. R. Gordon and G. C. Boynton, “Radiance-irradiance inversion algorithm for estimating the absorption and backscattering coefficients of natural waters: homogeneous waters,” Appl. Opt. 36, 2636–2641 (1997).
    [Crossref] [PubMed]
  15. H. R. Gordon and G. C. Boynton, “Radiance-irradiance inversion algorithm for estimating the absorption and backscattering coefficients of natural waters: vertically stratified water bodies,” Appl. Opt. 37, 3886–3896 (1998).
    [Crossref]
  16. A. Bricaud, A. Morel, and L. Prieur, “Optical efficiency factors of some phytoplankters,” Limnol. Oceanogr. 28, 816–832 (1983).
    [Crossref]
  17. T. Oishi, H. Tan, T. Hosaka, A. Tanaka, and R. Heuermann, “Realization of new bb meter and its quality control,” presented at Ocean Optics XVI, Santa Fe, New Mexico, November 18–22 2002, paper 042.pdf.
  18. J. Suzuki, T. Oishi, K. Ura, H. Tan, and T. Hosaka, “Numerical evaluation of new bb meter,” presented at Ocean Optics XVI, Santa Fe, New Mexico, November 18–22 2002, paper 041.pdf.
  19. H. Tan, T. Oishi, and R. Doerffer, “Analysis of measured spectral backward scattering coefficient,” presented at Ocean Optics XVII, Fremantle, Australia, October 25–29 2004, paper 006.pdf.
  20. J. M. Sullivan and M. S. Twardowski, “Angular shape of the oceanic particulate volume scattering function in the backward direction,” Appl. Opt. 48, 6811–6819 (2009).
    [Crossref] [PubMed]
  21. E. Boss and W. S. Pegau, “Relationship of light scattering at an angle in the backward direction to the backscattering coefficient,” Appl. Opt. 40, 5503–5507 (2001).
    [Crossref]
  22. N. G. Jerlov, “Particle distribution in the ocean,” in Deep Sea Expedition, H.Pettersson, ed. (Swedish Natural Science Research Council, 1953), pp. 71–98.
  23. T. Oishi, “Significant relationship between the backward scattering coefficient of sea water and the scatterance at 120°,” Appl. Opt. 29, 4658–4665 (1990).
    [Crossref] [PubMed]
  24. J.-F. Berthon, E. Shybanov, M. E.-G. Lee, and G. Zibordi, “Measurements and modeling of the volume scattering function in the coastal northern Adriatic Sea,” Appl. Opt. 46, 5189–5203 (2007).
    [Crossref] [PubMed]
  25. M. Chami, E. Marken, J. J. Stamnes, G. Khomenko, and G. Korotaev, “Variability of the relationship between the particulate backscattering coefficient and the volume scattering function measured at fixed angles,” J. Geophys. Res. 111, 1–10(2006).
    [Crossref]
  26. D. R. Dana and R. A. Maffione, “Determining the backward scattering coefficient with fixed-angle backscattering sensors-revisited,” presented at Ocean Optics XVI Santa Fe, New Mexico, November 18–22 2002, paper 212.pdf.
  27. R. G. Beutell and A. W. Brewer, “Instruments for the measurement of the visual range,” J. Sci. Instrum. 26, 357–359 (1949).
    [Crossref]
  28. D. J. Gray (personal communication, 2011).
  29. D. J. Gray and A. Weidemann, “Volume scattering function effects on underwater imaging systems,” presented at Ocean Optics XIX, Tuscany, Italy, October 6–10 2008, paper 481.pdf.
  30. WET Labs, “ECO BB user’s guide (BB)” (WET Labs, Inc., 2010).
  31. E. S. Fry, J. Musser, G. W. Kattawar, and P.-W. Zhai, “Integrating cavities: temporal response,” Appl. Opt. 45, 9053–9065(2006).
    [Crossref] [PubMed]
  32. Duke Scientific, 0.203 μm (Cat. No. 3200A, No. Lot 36926), 0.596 μm (Cat. No. 3600A, No. Lot 36446), and 3.005 μm (Cat. No. 4203A, No. Lot 36453).
  33. M. S. Twardowski, “An integrated inherent optical property sensor for AUVs” (WET Labs, Inc., 2006).

2011 (1)

D. J. Gray (personal communication, 2011).

2010 (1)

WET Labs, “ECO BB user’s guide (BB)” (WET Labs, Inc., 2010).

2009 (2)

2007 (1)

2006 (3)

M. Chami, E. Marken, J. J. Stamnes, G. Khomenko, and G. Korotaev, “Variability of the relationship between the particulate backscattering coefficient and the volume scattering function measured at fixed angles,” J. Geophys. Res. 111, 1–10(2006).
[Crossref]

E. S. Fry, J. Musser, G. W. Kattawar, and P.-W. Zhai, “Integrating cavities: temporal response,” Appl. Opt. 45, 9053–9065(2006).
[Crossref] [PubMed]

M. S. Twardowski, “An integrated inherent optical property sensor for AUVs” (WET Labs, Inc., 2006).

2005 (1)

2003 (1)

M. E. Lee and M. R. Lewis, “A new method for the measurement of the optical volume scattering function in the upper ocean,” J. Atmos. Ocean. Technol. 20, 563–571(2003).
[Crossref]

2002 (2)

2001 (2)

E. Boss and W. S. Pegau, “Relationship of light scattering at an angle in the backward direction to the backscattering coefficient,” Appl. Opt. 40, 5503–5507 (2001).
[Crossref]

M. S. Twardowski, E. Boss, J. B. Macdonald, W. S. Pegau, Andrew H. Barnard, and J. Ronald V. Zaneveld, “A model for estimating bulk refractive index from the optical backscattering ratio and the implications for understanding particle composition in case I and case II waters,” J. Geophys. Res. 106, 14129–14142 (2001).
[Crossref]

2000 (1)

1998 (1)

1997 (2)

1994 (1)

G. R. Fournier and J. L. Forand, “Analytic phase function for ocean water,” Proc. SPIE 2258, 194–201 (1994).
[Crossref]

1990 (1)

1983 (1)

A. Bricaud, A. Morel, and L. Prieur, “Optical efficiency factors of some phytoplankters,” Limnol. Oceanogr. 28, 816–832 (1983).
[Crossref]

1974 (1)

G. Kullenberg, “Observed and computed scattering functions,” in Optical Aspects of Oceanography, N.G.Jerlov and E.S.Nielsen, eds. (Academic, 1974), pp. 25–49.

1972 (1)

T. J. Petzold, “Volume scattering functions for selected ocean waters,” (Scripps Institution of Oceanography Visibility Laboratory, 1972).

1958 (1)

1953 (1)

N. G. Jerlov, “Particle distribution in the ocean,” in Deep Sea Expedition, H.Pettersson, ed. (Swedish Natural Science Research Council, 1953), pp. 71–98.

1949 (1)

R. G. Beutell and A. W. Brewer, “Instruments for the measurement of the visual range,” J. Sci. Instrum. 26, 357–359 (1949).
[Crossref]

Barnard, Andrew H.

M. S. Twardowski, E. Boss, J. B. Macdonald, W. S. Pegau, Andrew H. Barnard, and J. Ronald V. Zaneveld, “A model for estimating bulk refractive index from the optical backscattering ratio and the implications for understanding particle composition in case I and case II waters,” J. Geophys. Res. 106, 14129–14142 (2001).
[Crossref]

Berthon, J.-F.

Beutell, R. G.

R. G. Beutell and A. W. Brewer, “Instruments for the measurement of the visual range,” J. Sci. Instrum. 26, 357–359 (1949).
[Crossref]

Boss, E.

C. D. Mobley, L. K. Sundman, and E. Boss, “Phase function effects on oceanic light fields,” Appl. Opt. 41, 1035–1050(2002).
[Crossref] [PubMed]

M. S. Twardowski, E. Boss, J. B. Macdonald, W. S. Pegau, Andrew H. Barnard, and J. Ronald V. Zaneveld, “A model for estimating bulk refractive index from the optical backscattering ratio and the implications for understanding particle composition in case I and case II waters,” J. Geophys. Res. 106, 14129–14142 (2001).
[Crossref]

E. Boss and W. S. Pegau, “Relationship of light scattering at an angle in the backward direction to the backscattering coefficient,” Appl. Opt. 40, 5503–5507 (2001).
[Crossref]

Boynton, G. C.

Brewer, A. W.

R. G. Beutell and A. W. Brewer, “Instruments for the measurement of the visual range,” J. Sci. Instrum. 26, 357–359 (1949).
[Crossref]

Bricaud, A.

A. Bricaud, A. Morel, and L. Prieur, “Optical efficiency factors of some phytoplankters,” Limnol. Oceanogr. 28, 816–832 (1983).
[Crossref]

Chami, M.

M. Chami, E. Marken, J. J. Stamnes, G. Khomenko, and G. Korotaev, “Variability of the relationship between the particulate backscattering coefficient and the volume scattering function measured at fixed angles,” J. Geophys. Res. 111, 1–10(2006).
[Crossref]

Dana, D.

Dana, D. R.

D. R. Dana and R. A. Maffione, “Determining the backward scattering coefficient with fixed-angle backscattering sensors-revisited,” presented at Ocean Optics XVI Santa Fe, New Mexico, November 18–22 2002, paper 212.pdf.

Doerffer, R.

H. Tan, T. Oishi, and R. Doerffer, “Analysis of measured spectral backward scattering coefficient,” presented at Ocean Optics XVII, Fremantle, Australia, October 25–29 2004, paper 006.pdf.

T. Oishi, H. Tan, R. Doerffer, and R. Heuermann, “Development of new spectral scattering function meter,” presented at Ocean Optics XVII, Fremantle, Australia, October 25–29 2004, paper 004.pdf.

Forand, J. L.

G. R. Fournier and J. L. Forand, “Analytic phase function for ocean water,” Proc. SPIE 2258, 194–201 (1994).
[Crossref]

Fournier, G. R.

G. R. Fournier and J. L. Forand, “Analytic phase function for ocean water,” Proc. SPIE 2258, 194–201 (1994).
[Crossref]

Freeman, S. A.

Fry, E. S.

Gordon, H. R.

Gray, D. J.

D. J. Gray and A. Weidemann, “Volume scattering function effects on underwater imaging systems,” presented at Ocean Optics XIX, Tuscany, Italy, October 6–10 2008, paper 481.pdf.

D. J. Gray (personal communication, 2011).

Heuermann, R.

T. Oishi, H. Tan, R. Doerffer, and R. Heuermann, “Development of new spectral scattering function meter,” presented at Ocean Optics XVII, Fremantle, Australia, October 25–29 2004, paper 004.pdf.

T. Oishi, H. Tan, T. Hosaka, A. Tanaka, and R. Heuermann, “Realization of new bb meter and its quality control,” presented at Ocean Optics XVI, Santa Fe, New Mexico, November 18–22 2002, paper 042.pdf.

Hosaka, T.

J. Suzuki, T. Oishi, K. Ura, H. Tan, and T. Hosaka, “Numerical evaluation of new bb meter,” presented at Ocean Optics XVI, Santa Fe, New Mexico, November 18–22 2002, paper 041.pdf.

T. Oishi, H. Tan, T. Hosaka, A. Tanaka, and R. Heuermann, “Realization of new bb meter and its quality control,” presented at Ocean Optics XVI, Santa Fe, New Mexico, November 18–22 2002, paper 042.pdf.

Jerlov, N. G.

N. G. Jerlov, “Particle distribution in the ocean,” in Deep Sea Expedition, H.Pettersson, ed. (Swedish Natural Science Research Council, 1953), pp. 71–98.

Kattawar, G. W.

Khomenko, G.

M. Chami, E. Marken, J. J. Stamnes, G. Khomenko, and G. Korotaev, “Variability of the relationship between the particulate backscattering coefficient and the volume scattering function measured at fixed angles,” J. Geophys. Res. 111, 1–10(2006).
[Crossref]

Kim, M.

Korotaev, G.

M. Chami, E. Marken, J. J. Stamnes, G. Khomenko, and G. Korotaev, “Variability of the relationship between the particulate backscattering coefficient and the volume scattering function measured at fixed angles,” J. Geophys. Res. 111, 1–10(2006).
[Crossref]

Kullenberg, G.

G. Kullenberg, “Observed and computed scattering functions,” in Optical Aspects of Oceanography, N.G.Jerlov and E.S.Nielsen, eds. (Academic, 1974), pp. 25–49.

Lee, M. E.

M. E. Lee and M. R. Lewis, “A new method for the measurement of the optical volume scattering function in the upper ocean,” J. Atmos. Ocean. Technol. 20, 563–571(2003).
[Crossref]

Lee, M. E.-G.

Lewis, M. R.

H. R. Gordon, M. R. Lewis, S. D. McLean, M. S. Twardowski, S. A. Freeman, K. J. Voss, and G. C. Boynton, “Spectra of particulate backscattering in natural waters,” Opt. Express 17, 16192–16208 (2009).
[Crossref] [PubMed]

M. E. Lee and M. R. Lewis, “A new method for the measurement of the optical volume scattering function in the upper ocean,” J. Atmos. Ocean. Technol. 20, 563–571(2003).
[Crossref]

Macdonald, J. B.

M. S. Twardowski, E. Boss, J. B. Macdonald, W. S. Pegau, Andrew H. Barnard, and J. Ronald V. Zaneveld, “A model for estimating bulk refractive index from the optical backscattering ratio and the implications for understanding particle composition in case I and case II waters,” J. Geophys. Res. 106, 14129–14142 (2001).
[Crossref]

Maffione, R.

Maffione, R. A.

D. R. Dana and R. A. Maffione, “Determining the backward scattering coefficient with fixed-angle backscattering sensors-revisited,” presented at Ocean Optics XVI Santa Fe, New Mexico, November 18–22 2002, paper 212.pdf.

Marken, E.

M. Chami, E. Marken, J. J. Stamnes, G. Khomenko, and G. Korotaev, “Variability of the relationship between the particulate backscattering coefficient and the volume scattering function measured at fixed angles,” J. Geophys. Res. 111, 1–10(2006).
[Crossref]

McLean, S. D.

Mobley, C. D.

Morel, A.

A. Bricaud, A. Morel, and L. Prieur, “Optical efficiency factors of some phytoplankters,” Limnol. Oceanogr. 28, 816–832 (1983).
[Crossref]

Musser, J.

Oishi, T.

H. Tan, T. Oishi, and R. Doerffer, “Analysis of measured spectral backward scattering coefficient,” presented at Ocean Optics XVII, Fremantle, Australia, October 25–29 2004, paper 006.pdf.

T. Oishi, H. Tan, R. Doerffer, and R. Heuermann, “Development of new spectral scattering function meter,” presented at Ocean Optics XVII, Fremantle, Australia, October 25–29 2004, paper 004.pdf.

T. Oishi, H. Tan, T. Hosaka, A. Tanaka, and R. Heuermann, “Realization of new bb meter and its quality control,” presented at Ocean Optics XVI, Santa Fe, New Mexico, November 18–22 2002, paper 042.pdf.

J. Suzuki, T. Oishi, K. Ura, H. Tan, and T. Hosaka, “Numerical evaluation of new bb meter,” presented at Ocean Optics XVI, Santa Fe, New Mexico, November 18–22 2002, paper 041.pdf.

T. Oishi, “Significant relationship between the backward scattering coefficient of sea water and the scatterance at 120°,” Appl. Opt. 29, 4658–4665 (1990).
[Crossref] [PubMed]

Pegau, W. S.

E. Boss and W. S. Pegau, “Relationship of light scattering at an angle in the backward direction to the backscattering coefficient,” Appl. Opt. 40, 5503–5507 (2001).
[Crossref]

M. S. Twardowski, E. Boss, J. B. Macdonald, W. S. Pegau, Andrew H. Barnard, and J. Ronald V. Zaneveld, “A model for estimating bulk refractive index from the optical backscattering ratio and the implications for understanding particle composition in case I and case II waters,” J. Geophys. Res. 106, 14129–14142 (2001).
[Crossref]

Petzold, T. J.

T. J. Petzold, “Volume scattering functions for selected ocean waters,” (Scripps Institution of Oceanography Visibility Laboratory, 1972).

Philpot, W. D.

Prieur, L.

A. Bricaud, A. Morel, and L. Prieur, “Optical efficiency factors of some phytoplankters,” Limnol. Oceanogr. 28, 816–832 (1983).
[Crossref]

Richardson, W. H.

Shybanov, E.

Stamnes, J. J.

M. Chami, E. Marken, J. J. Stamnes, G. Khomenko, and G. Korotaev, “Variability of the relationship between the particulate backscattering coefficient and the volume scattering function measured at fixed angles,” J. Geophys. Res. 111, 1–10(2006).
[Crossref]

Sullivan, J. M.

Sundman, L. K.

Suzuki, J.

J. Suzuki, T. Oishi, K. Ura, H. Tan, and T. Hosaka, “Numerical evaluation of new bb meter,” presented at Ocean Optics XVI, Santa Fe, New Mexico, November 18–22 2002, paper 041.pdf.

Tan, H.

H. Tan, T. Oishi, and R. Doerffer, “Analysis of measured spectral backward scattering coefficient,” presented at Ocean Optics XVII, Fremantle, Australia, October 25–29 2004, paper 006.pdf.

T. Oishi, H. Tan, R. Doerffer, and R. Heuermann, “Development of new spectral scattering function meter,” presented at Ocean Optics XVII, Fremantle, Australia, October 25–29 2004, paper 004.pdf.

J. Suzuki, T. Oishi, K. Ura, H. Tan, and T. Hosaka, “Numerical evaluation of new bb meter,” presented at Ocean Optics XVI, Santa Fe, New Mexico, November 18–22 2002, paper 041.pdf.

T. Oishi, H. Tan, T. Hosaka, A. Tanaka, and R. Heuermann, “Realization of new bb meter and its quality control,” presented at Ocean Optics XVI, Santa Fe, New Mexico, November 18–22 2002, paper 042.pdf.

Tanaka, A.

T. Oishi, H. Tan, T. Hosaka, A. Tanaka, and R. Heuermann, “Realization of new bb meter and its quality control,” presented at Ocean Optics XVI, Santa Fe, New Mexico, November 18–22 2002, paper 042.pdf.

Twardowski, M. S.

J. M. Sullivan and M. S. Twardowski, “Angular shape of the oceanic particulate volume scattering function in the backward direction,” Appl. Opt. 48, 6811–6819 (2009).
[Crossref] [PubMed]

H. R. Gordon, M. R. Lewis, S. D. McLean, M. S. Twardowski, S. A. Freeman, K. J. Voss, and G. C. Boynton, “Spectra of particulate backscattering in natural waters,” Opt. Express 17, 16192–16208 (2009).
[Crossref] [PubMed]

M. S. Twardowski, “An integrated inherent optical property sensor for AUVs” (WET Labs, Inc., 2006).

M. S. Twardowski, E. Boss, J. B. Macdonald, W. S. Pegau, Andrew H. Barnard, and J. Ronald V. Zaneveld, “A model for estimating bulk refractive index from the optical backscattering ratio and the implications for understanding particle composition in case I and case II waters,” J. Geophys. Res. 106, 14129–14142 (2001).
[Crossref]

Tyler, J. E.

Ura, K.

J. Suzuki, T. Oishi, K. Ura, H. Tan, and T. Hosaka, “Numerical evaluation of new bb meter,” presented at Ocean Optics XVI, Santa Fe, New Mexico, November 18–22 2002, paper 041.pdf.

Voss, K. J.

Weidemann, A.

D. J. Gray and A. Weidemann, “Volume scattering function effects on underwater imaging systems,” presented at Ocean Optics XIX, Tuscany, Italy, October 6–10 2008, paper 481.pdf.

Zaneveld, J. Ronald V.

M. S. Twardowski, E. Boss, J. B. Macdonald, W. S. Pegau, Andrew H. Barnard, and J. Ronald V. Zaneveld, “A model for estimating bulk refractive index from the optical backscattering ratio and the implications for understanding particle composition in case I and case II waters,” J. Geophys. Res. 106, 14129–14142 (2001).
[Crossref]

Zhai, P.-W.

Zibordi, G.

Appl. Opt. (12)

M. Kim and W. D. Philpot, “Development of a laboratory spectral backscattering instrument: design and simulation,” Appl. Opt. 44, 6952–6961 (2005).
[Crossref] [PubMed]

R. Maffione and D. Dana, “Instruments and methods for measuring the backward-scattering coefficient of ocean waters,” Appl. Opt. 36, 6057–6067 (1997).
[Crossref] [PubMed]

C. D. Mobley, L. K. Sundman, and E. Boss, “Phase function effects on oceanic light fields,” Appl. Opt. 41, 1035–1050(2002).
[Crossref] [PubMed]

G. C. Boynton and H. R. Gordon, “Irradiance inversion algorithm for estimating the absorption and backscattering coefficients of natural waters: Raman-scattering effects,” Appl. Opt. 39, 3012–3022 (2000).
[Crossref]

G. C. Boynton and H. R. Gordon, “Irradiance inversion algorithm for absorption and backscattering profiles in natural waters: improvement for clear waters,” Appl. Opt. 41, 2224–2227 (2002).
[Crossref] [PubMed]

H. R. Gordon and G. C. Boynton, “Radiance-irradiance inversion algorithm for estimating the absorption and backscattering coefficients of natural waters: homogeneous waters,” Appl. Opt. 36, 2636–2641 (1997).
[Crossref] [PubMed]

H. R. Gordon and G. C. Boynton, “Radiance-irradiance inversion algorithm for estimating the absorption and backscattering coefficients of natural waters: vertically stratified water bodies,” Appl. Opt. 37, 3886–3896 (1998).
[Crossref]

T. Oishi, “Significant relationship between the backward scattering coefficient of sea water and the scatterance at 120°,” Appl. Opt. 29, 4658–4665 (1990).
[Crossref] [PubMed]

J.-F. Berthon, E. Shybanov, M. E.-G. Lee, and G. Zibordi, “Measurements and modeling of the volume scattering function in the coastal northern Adriatic Sea,” Appl. Opt. 46, 5189–5203 (2007).
[Crossref] [PubMed]

J. M. Sullivan and M. S. Twardowski, “Angular shape of the oceanic particulate volume scattering function in the backward direction,” Appl. Opt. 48, 6811–6819 (2009).
[Crossref] [PubMed]

E. Boss and W. S. Pegau, “Relationship of light scattering at an angle in the backward direction to the backscattering coefficient,” Appl. Opt. 40, 5503–5507 (2001).
[Crossref]

E. S. Fry, J. Musser, G. W. Kattawar, and P.-W. Zhai, “Integrating cavities: temporal response,” Appl. Opt. 45, 9053–9065(2006).
[Crossref] [PubMed]

J. Atmos. Ocean. Technol. (1)

M. E. Lee and M. R. Lewis, “A new method for the measurement of the optical volume scattering function in the upper ocean,” J. Atmos. Ocean. Technol. 20, 563–571(2003).
[Crossref]

J. Geophys. Res. (2)

M. S. Twardowski, E. Boss, J. B. Macdonald, W. S. Pegau, Andrew H. Barnard, and J. Ronald V. Zaneveld, “A model for estimating bulk refractive index from the optical backscattering ratio and the implications for understanding particle composition in case I and case II waters,” J. Geophys. Res. 106, 14129–14142 (2001).
[Crossref]

M. Chami, E. Marken, J. J. Stamnes, G. Khomenko, and G. Korotaev, “Variability of the relationship between the particulate backscattering coefficient and the volume scattering function measured at fixed angles,” J. Geophys. Res. 111, 1–10(2006).
[Crossref]

J. Opt. Soc. Am. (1)

J. Sci. Instrum. (1)

R. G. Beutell and A. W. Brewer, “Instruments for the measurement of the visual range,” J. Sci. Instrum. 26, 357–359 (1949).
[Crossref]

Limnol. Oceanogr. (1)

A. Bricaud, A. Morel, and L. Prieur, “Optical efficiency factors of some phytoplankters,” Limnol. Oceanogr. 28, 816–832 (1983).
[Crossref]

Opt. Express (1)

Proc. SPIE (1)

G. R. Fournier and J. L. Forand, “Analytic phase function for ocean water,” Proc. SPIE 2258, 194–201 (1994).
[Crossref]

Other (13)

G. Kullenberg, “Observed and computed scattering functions,” in Optical Aspects of Oceanography, N.G.Jerlov and E.S.Nielsen, eds. (Academic, 1974), pp. 25–49.

T. Oishi, H. Tan, R. Doerffer, and R. Heuermann, “Development of new spectral scattering function meter,” presented at Ocean Optics XVII, Fremantle, Australia, October 25–29 2004, paper 004.pdf.

T. J. Petzold, “Volume scattering functions for selected ocean waters,” (Scripps Institution of Oceanography Visibility Laboratory, 1972).

T. Oishi, H. Tan, T. Hosaka, A. Tanaka, and R. Heuermann, “Realization of new bb meter and its quality control,” presented at Ocean Optics XVI, Santa Fe, New Mexico, November 18–22 2002, paper 042.pdf.

J. Suzuki, T. Oishi, K. Ura, H. Tan, and T. Hosaka, “Numerical evaluation of new bb meter,” presented at Ocean Optics XVI, Santa Fe, New Mexico, November 18–22 2002, paper 041.pdf.

H. Tan, T. Oishi, and R. Doerffer, “Analysis of measured spectral backward scattering coefficient,” presented at Ocean Optics XVII, Fremantle, Australia, October 25–29 2004, paper 006.pdf.

Duke Scientific, 0.203 μm (Cat. No. 3200A, No. Lot 36926), 0.596 μm (Cat. No. 3600A, No. Lot 36446), and 3.005 μm (Cat. No. 4203A, No. Lot 36453).

M. S. Twardowski, “An integrated inherent optical property sensor for AUVs” (WET Labs, Inc., 2006).

N. G. Jerlov, “Particle distribution in the ocean,” in Deep Sea Expedition, H.Pettersson, ed. (Swedish Natural Science Research Council, 1953), pp. 71–98.

D. J. Gray (personal communication, 2011).

D. J. Gray and A. Weidemann, “Volume scattering function effects on underwater imaging systems,” presented at Ocean Optics XIX, Tuscany, Italy, October 6–10 2008, paper 481.pdf.

WET Labs, “ECO BB user’s guide (BB)” (WET Labs, Inc., 2010).

D. R. Dana and R. A. Maffione, “Determining the backward scattering coefficient with fixed-angle backscattering sensors-revisited,” presented at Ocean Optics XVI Santa Fe, New Mexico, November 18–22 2002, paper 212.pdf.

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Figures (13)

Fig. 1
Fig. 1 Conceptual sketch of instrument (cylindrical symmetry around z axis).
Fig. 2
Fig. 2 The functions θ 1 ( z ) , θ 2 ( z ) and the functions z 1 ( θ ) , z 2 ( θ ) are the same curves, respectively, i.e., for the latter, z is read off the horizontal axis as a function of θ on the vertical axis. The light gray and dark gray areas correspond to the domains of integration of the first and second integral in Eq. (9).
Fig. 3
Fig. 3 Domains of integration for the integrals in Eq. (14).
Fig. 4
Fig. 4 Detection solid angle Ω ( z ) with R = 0.01 m and Z 0 = 0.001 m .
Fig. 5
Fig. 5 Coefficient of β ( θ ) in the integrand of Eq. (27).
Fig. 6
Fig. 6 Dependence on attenuation c of the percent deviation from b b that would be observed in a measurement of P ( c ) , Eq. (25).
Fig. 7
Fig. 7 Percent deviation from b b of a measurement of P ( c ) after multiplication by the attenuation correction factor ( 1 + 2.7 c R ) to obtain P A ( c ) .
Fig. 8
Fig. 8 Instrument implementation—cylindrical symmetry around laser beam (except for fibers).
Fig. 9
Fig. 9 “Placebo” experiment showing a slight signal increase as samples are added.
Fig. 10
Fig. 10 Measured data for aqueous suspensions of polymer microspheres with diameters d = 203 , 596, and 3005 nm . The least-squares fits of the data to the straight lines are also shown.
Fig. 11
Fig. 11 Attenuation effects on the measured signal.
Fig. 12
Fig. 12 Three possibilities for the window at the aperture to the detector.
Fig. 13
Fig. 13 Effective reduction in the observation solid angle as a function of z due to reflection at the quartz window in the aperture of Fig. 12b.

Tables (2)

Tables Icon

Table 1 Deviations from b b That Would Be Observed for Measurements Made with This Instrumentation in Natural Waters or Suspensions of Microspheres a

Tables Icon

Table 2 Numerical Evaluations of Eq. (25, 27), and the Various Terms in Eq. (28) Using MVSM Data for β ( θ ) Obtained during October 2009 in Chesapeake Bay [28, 29] a

Equations (33)

Equations on this page are rendered with MathJax. Learn more.

b b ( λ ) 2 π π / 2 π β ( λ , θ ) sin θ d θ ,
d P = 2 π A E 0 θ 1 θ 2 e c x β ( θ ) sin θ d θ d z ,
P = 2 π A E 0 0 d z θ 1 θ 2 e c x β ( θ ) sin θ d θ .
θ 1 ( z ) = π tan 1 ( R z Z 0 ) π cos 1 ( z Z 0 R 2 + ( z Z 0 ) 2 ) .
θ 2 ( z ) = π tan 1 ( R z + Z 0 ) π cos 1 ( z + Z 0 R 2 + ( z + Z 0 ) 2 ) .
θ 10 = θ 1 ( 0 ) = tan 1 ( R Z 0 ) , θ 20 = θ 2 ( 0 ) = π tan 1 ( R Z 0 ) .
z 1 ( θ ) = Z 0 R tan θ , z 2 ( θ ) = Z 0 R tan θ .
P = 2 π A E 0 0 d z π tan 1 ( R z Z 0 ) π tan 1 ( R z + Z 0 ) β ( θ ) sin θ d θ .
P = 2 π A E 0 θ 20 π β ( θ ) sin θ d θ z 2 ( θ ) z 1 ( θ ) d z + 2 π A E 0 θ 10 θ 20 β ( θ ) sin θ d θ 0 z 1 ( θ ) d z ,
P = ( 2 Z 0 A E 0 ) { 2 π θ 20 π β ( θ ) sin θ d θ + π θ 10 θ 20 β ( θ ) sin θ d θ π R Z 0 θ 10 θ 20 β ( θ ) cos θ d θ } .
P = P 2 Z 0 A E 0 ;
P = 2 π θ 20 π β ( θ ) sin θ d θ + π π / 2 θ 20 β ( θ ) sin θ d θ + π θ 10 π / 2 β ( θ ) sin θ d θ π R Z 0 θ 10 θ 20 β ( θ ) cos θ d θ .
P = 2 π θ 20 π β ( θ ) sin θ d θ + 2 π π / 2 θ 20 β ( θ ) sin θ d θ π π / 2 θ 20 β ( θ ) sin θ d θ + π θ 10 π / 2 β ( θ ) sin θ d θ π R Z 0 θ 10 θ 20 β ( θ ) cos θ d θ .
P = 2 π π / 2 π β ( θ ) sin θ d θ + π θ 10 π / 2 β ( θ ) sin θ d θ π π / 2 θ 20 β ( θ ) sin θ d θ π R Z 0 θ 10 θ 20 β ( θ ) cos θ d θ .
P = I + I I + I I I + I V = b b ( 1 + ρ 2 + ρ 3 + ρ 4 ) ,
ρ 2 = I I b b ; ρ 3 = I I I b b ; ρ 4 = I V b b .
S = K 0 ( 1 + ρ 2 + ρ 3 + ρ 4 ) b b = K b b ,
K 0 = K 1 + ρ 2 + ρ 3 + ρ 4 .
Ω ( z ) = 2 π θ 1 ( z ) θ 2 ( z ) sin θ d θ = 2 π π tan 1 ( R z Z 0 ) π tan 1 ( R z + Z 0 ) sin θ d θ = 2 π { Z 0 z R 2 + ( Z 0 z ) 2 + Z 0 + z R 2 + ( Z 0 + z ) 2 } .
x = z + R sin ( θ ) ,
P ( c ) = 2 π A E 0 0 d z θ 1 θ 2 e c ( z + R sin ( θ ) ) β ( θ ) sin θ d θ .
P ( c ) = 2 π A E 0 θ 20 π β ( θ ) sin θ d θ Z 0 R tan θ Z 0 R tan θ e c ( z + R sin θ ) d z + 2 π A E 0 θ 10 θ 20 β ( θ ) sin θ d θ 0 Z 0 R tan θ e c ( z + R sin θ ) d z .
P ( c ) = 2 π A E 0 c ( e c Z 0 e c Z 0 ) θ 20 π β ( θ ) sin θ e c R 1 cos θ sin θ d θ + 2 π A E 0 c θ 10 θ 20 β ( θ ) sin θ e c R sin θ d θ 2 π A E 0 c e c Z 0 θ 10 θ 20 β ( θ ) sin θ e c R 1 cos θ sin θ d θ .
2 π A E 0 c e c Z 0 π / 2 θ 20 β ( θ ) sin θ e c R 1 cos θ sin θ d θ .
P ( c ) = P ( c ) 2 Z 0 A E 0 = ( e c Z 0 e c Z 0 c Z 0 ) π π / 2 π β ( θ ) sin θ e c R 1 cos θ sin θ d θ + π { 1 c Z 0 θ 10 θ 20 β ( θ ) sin θ e c R sin θ d θ e c Z 0 c Z 0 θ 10 π / 2 β ( θ ) sin θ e c R 1 cos θ sin θ d θ e c Z 0 c Z 0 π / 2 θ 20 β ( θ ) sin θ e c R 1 cos θ sin θ d θ } .
e c Z 0 e c Z 0 2 c Z 0 ( 1 + 1 6 c 2 Z 0 2 + 1 120 c 4 Z 0 4 ) .
P 0 ( c ) = 2 π π / 2 π β ( θ ) sin θ e c R 1 cos θ sin θ d θ .
P 1 ( c ) = P 1 ( c ) 2 Z 0 A E 0 = { π θ 10 π / 2 β ( θ ) sin θ d θ π π / 2 θ 20 β ( θ ) sin θ d θ R π Z 0 θ 10 θ 20 β ( θ ) cos θ d θ } + c { R π π / 2 θ 20 β ( θ ) ( 1 cos θ ) d θ R π θ 10 π / 2 β ( θ ) ( 1 cos θ ) d θ Z 0 π 2 θ 10 θ 20 β ( θ ) sin θ d θ + R 2 Z 0 π θ 10 θ 20 β ( θ ) cos θ sin θ d θ R 2 2 Z 0 π θ 10 θ 20 β ( θ ) cos 2 θ sin θ d θ } + ε ,
P A ( c ) = ( 1 + 2.7 c R ) P ( c ) ,
P A ( c ) = [ 1 + 2.7 ( c ± δ c ) R ] P ( c ) = ( 1 + 2.7 c R ) P ( c ) ± 2.7 δ c R P ( c ) .
Δ = δ P A ( c ) P A ( c ) = ± 2.7 δ c R P ( c ) ( 1 + 2.7 c R ) P ( c ) = ± 2.7 R δ c ( 1 + 2.7 c R ) = ± [ 0.027 1 / c + 0.027 ] δ c c .
K 203 = 304.3 ± 1.7 , K 596 = 298.2 ± 2.9 , K 3005 = 308.5 ± 3.7 ,
K 0 = 303 ± 5.5 % ,

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