## 1. Introduction

Acoustic Doppler current profiler (ADCP) echo intensity data are usually recorded as a by-product of the velocity measurements. Echo intensity reflects the backscattering strength of the water, which is due to the presence of backscatterers such as solid particles, bubbles, or living organisms. The level of echo intensity is also very dependent on the acoustic frequency of the ADCP, because the ratio between the acoustic wavelength and the size of the scatterers partially governs the backscattering strength. These data can provide useful information on sediment transport (Gartner 2004; Wall et al. 2006), plankton activity (Plueddemann and Pinkel 1989), or surface wave field intensity (Zedel 2001). In physical oceanography, the displacement of the scatterers advected by the flow helps in visualizing the flow structure (van Haren 2009). Moreover, by taking advantage of the angular separation between the (four) different beams of the ADCP, additional information on the flow can be obtained that infers direction of fronts or any other nonlinear structure (van Haren 2007a).

To be related to pertinent physical quantities, the backscattered signal needs to be calibrated by means of extra sensors that measure independently the quantity one wants to relate to echo intensity, such as sediment concentration and grain-size distribution or plankton activity. This is done by means of water sampling (Wall et al. 2006), sediment traps, or nets (Flagg and Smith 1989). Optical backscatter sensors (OBS) can also be used, because their calibration can easily be performed in the laboratory using the above-mentioned water samples (Kim and Voulgaris 2003). The calibration of profiling acoustic devices in the laboratory is more complex, because a large volume of fluid has do be ensonified. It can be performed successfully for short-range acoustic backscatter devices (Betteridge et al. 2008) but not for long-range ADCP to our knowledge.

We found several datasets for which comparison of the time-averaged echo intensity profiles show poor agreement with the theoretical predictions given by Deines (1999), particularly in the tail of the profile when the backscattered intensity is less than one order of magnitude above the noise level. However, any quantitative study of backscattering strength variability is based on this comparison. A simple modification of Deines’ equation that better fits with the data is the purpose of this letter.

## 2. Method

*I*from a signal initially emitted with an acoustic intensity

*I*

_{0}, transmitted along the acoustic beam over a distance

*R*, and scattered back to the source as Here the second term on the right-hand side is spherical spreading and exponential attenuation between the transducer and the scatterer (

*α*is the attenuation coefficient of sound in the media). The third term is backscattering by reflectors proportional to their volume backscattering strength

*s*, which is a function of local concentration, cross section of the scatterer, and frequency of the acoustic signal; thus, it is a function of time and space. Note the

_{υ}*γR*

^{2}term to account for the spreading of the ensonified volume, with

*γ*being a geometrical factor specific to the beam-pattern function and proportional to the transmit pulse length. The last term on the RHS is spherical spreading and exponential attenuation during the return path of the sound from the scatterer to the transducer.

*I*

_{dB}= 10 log(

*I*/

*I*

_{ref}), with

*I*

_{ref}being an arbitrary reference intensity level] and introducing a constant

*A*that incorporates

*I*

_{0}and the time- and distance-independent coefficients, (1) (in decibels) becomes where

*A*is independent of time and space. Using this equation, relative backscattering strength (in decibels, to a distance

*R*

_{0}) can be computed as

This relation is used to calibrate ADCP data with simultaneous water samples or OBS measurements that can provide an estimate of *s _{υ}*(

*R*

_{0}) at a given time

*t*

_{0}. When the temporal and spatial variations of

*s*are due to concentration changes only, the relative concentrations of scatterers can be computed the same way. However, we would like to emphasize that, although (2) is true as for acoustic intensity, systematic errors occur when this expression is applied directly to the measured ADCP backscattered intensity.

_{υ}*k*is used to convert RSSI counts to decibels (RDI 2001). The ADCP records both the backscattered signal and the ambient noise of the water in its receiving band. Thus, in the presence of noise, we have (in decibels) where

_{c}*E*is the RSSI,

*I*

_{noise}is the acoustic intensity of the noise, and

*I*

_{ref}is a reference intensity imposed by the hardware. The precise determination of

*k*is crucial if any quantitative measurement is to be done, because fluctuations of 20% are observed around the commonly used value of

_{c}*k*= 0.45 (Deines 1999) and also between the individual transducers of a single ADCP.

_{c}*E*

_{noise}= (10/

*k*) log(

_{c}*I*

_{noise}/

*I*

_{ref}) as the noise level in counts, we have (in decibels)

*x*+

*y*) ≠ log

*x*+ log

*y*, this expression obviously differs from the statement

*k*(

_{c}*E*−

*E*

_{noise}) =

*I*

_{dB}. It is surprising that in the work of Deines (1999) this problem, although briefly mentioned in one equation of the appendix, is not included in the final expression for backscatter strength. If we literally reproduce here from Deines [(1999), Eq. (A-5)] the expression where

*S*is the backscattered signal power,

*N*is the noise power, and

*E*=

_{r}*E*

_{noise}, and correct it for an unfortunate typing error as follows, we have a rigorous introduction of

*N*in the signal-to-noise ratio because the power recorded by the ADCP is indeed

*S*+

*N*. However, because the sonar equation in (A-1) proposed by Deines (1999) gives the theoretical value of the ratio

*S*/

*N*, we see that, to identify

*S*/

*N*with (

*S*+

*N*)/

*N*, the assumption

*S*+

*N*≃

*S*was made by the author without being mentioned. This is not obvious, because the author suggests that

*E*=

_{r}*E*

_{noise}can be estimated at the tail (large

*R*) of the profile (where thus

*S*≃

*N*). We will show in the following how the above error modifies the estimation of absolute backscattering strength as well as relative concentration estimates, when

*S*/

*N*≤ 10, that is, for

*k*(

_{c}*E*−

*E*

_{noise}) < 10.

*E*(RSSI counts) along

*R*in an ADCP beam as (in decibels) This equation can be used for estimates of the absolute backscatter strength, provided the constant

*A*is determined using the set of formulas proposed by Deines (1999). It can also be used to estimate relative backscatter strength or relative backscatter concentration for which the determination of

*A*is not needed.

## 3. Observations

*R*. Doing so, 〈log(

*s*)〉

_{V}*can be incorporated in the constant*

_{t}*A*, and the time-averaged RSSI curves can be fitted with (9):

For comparison, we reproduce (10) inferred from Deines (1999). In Fig. 1a, it is shown how (9) and (10) diverge as *E* approaches *E*_{noise} to within 10 dB. In the case *E* > *E*_{noise} + 10, the uncorrected (10) can be used, however, with a 0.5-dB confidence interval. This is consistent with an observation in Zedel (2001) of the lowest significant sound levels resolved by an ADCP.

*α*is estimated using Schulkin and Marsh equations as proposed by Urick (1975), and correction for the attenuation by the scatterers themselves can be neglected in first approximation, because we are not working with frequencies larger than 10

^{6}Hz (Gartner 2004), except for one dataset using a 1200-kHz ADCP (limit case). The term

*E*

_{noise}is the tail value of the minimum RSSI profile over the total record. Because the RSSI counts are integers and we take the minimum of this value over long time series, we allow ourselves to add an adjustable constant

*δE*

_{noise}to

*E*

_{noise}that ranges between 0 and 1 to compensate for this discretization. For an ADCP with beams slanted at an angle

*θ*to the vertical, range

*R*is computed from where

*R*

_{1}is the distance to the center of the first bin,

*n*is the vertical bin number, and

*D*is the size of a depth cell. All of these variables are provided in the RDI ADCP fixed leader data, and we checked that the first bin distance is computed using

*R*

_{1}=

*B*+ (

*L*+

*D*+ La)/2, where

*B*is the blanking distance,

*L*is the transmit pulse length, and La is the transmit lag. This expression differs from Deines [(1999), Eq. (3)], where the transmit lag distance is not included.

Scale factor *k _{c}* and constant

*A*are adjusted to obtain the best fit of (9) with the data, with

*k*being constrained to the range [0.35, 0.55]. However, it can be shown that, when following Deines (1999), no value of

_{c}*A*and

*k*can be found that gives a correct fit along the profile. Using the correct (9), deviation of the time-averaged profiles from (9) can now be interpreted as spatial (depth) variations of the time-averaged backscattering strength

_{c}*s*

_{υ}.

The different datasets from which Figs. 1–f are obtained are presented in Table 1. An inset plot with the RSSI distance–time series also is visible in Fig. 1 for each dataset. We have the following five datasets:

- Astronomy with a Neutrino Telescope and Abyss Environmental Research (ANTARES): This ADCP is part of the ANTARES neutrino telescope that directly sends information to the coast using the submarine network developed for cosmic particle detection. It is downward looking in the deep Mediterranean Sea waters at a depth of 2300 m. The RSSI values are very weak and do not show strong backscattering variability (J. A. Aguilar 2010, manuscript submitted to
*Geophys. Res. Lett.*). - Processes above Continental Slopes (PROCS): The ADCP mooring was part of an extensive multidisciplinary program to study the effects of internal wave–induced mixing on sediment transport and nutrient redistribution on the continental slope of the Faeroe–Shetland Channel. The upward-looking ADCP was mounted in a bottom frame to which a thermistor string was attached. The echo data ranged between 0.9 and 8.15 m above the bottom and hence are affected by sediment resuspension (Hosegood and van Haren 2004).
- Long-Term Ocean Climate Observations (LOCO): On this long-term mooring (530 days in total), the ADCP was mounted in the top buoy, downward-looking from a depth of 1400 m to 2000 m in the central part of the Canary Basin. Total water depth is 5270 m. Most of the backscattering is due to living organisms that move up and down with diurnal periodicity (van Haren 2007b).
- Integrated North Sea Program (INP): This is an ADCP upward looking from the seafloor (45 m) to the surface in the shallow North Sea. Backscatterers are a combination of sediments, air bubbles, and zooplankton (van Haren et al. 1999).
- Deep Ocean Current (DOC): The ADCP was attached to a bottom lander on the eastern slope of the Great Meteor Seamount to monitor internal wave breaking and fronts. Echo data range from 4 to 80 m above bottom and reveal zooplankton migration at the solar diurnal constituent (S1) modulated by the internal tide (M2; van Haren 2009).

Figures 1b–f show for each dataset the minimum, maximum, and geometrically averaged RSSI profiles over a specific time interval of the complete time series. This interval is chosen so as to highlight specific phenomena in the distance–time series visible in the inset. For each dataset, our equation fits the main trend of the different profiles. The only parameter changing among the three profiles for a given dataset is the constant *A* (see Table 2 for the values of the different parameters *A*_{min}, *A*_{mean}, and *A*_{max}). There are, of course, deviations from the theoretical profile, but these are precisely the variation along *R* of the minimum, maximum, or time-averaged backscattering strength that we want to investigate. This demonstrates that finding the best theoretical model is very important to estimate these variations. There is, as a validation of (9), only a very small mismatch between constant backscattering strength theory and observations for the ANTARES data (Fig. 1b), which reflects the strong homogeneity of the deep Mediterranean waters.

## 4. Discussion

We propose a simple modification to the equations proposed by (Deines 1999), in which we subtract the noise level in arithmetic space for a correct use of the sonar equation in logarithmic space. Our equation fits the time-averaged RSSI data from very different datasets up to the tail of each profile, whereas if we followed Deines (1999) we would observe a divergence when the signal goes below 10 times the noise level. For the present datasets, this accounts for half of the ADCP range, which is of course a substantial improvement. The above bias was probably not taken into account (yet being possibly significant) when only the relative backscattering strength profiles are plotted. It results in a systematic overestimate of the backscattering strength at the tail of the profile (see Fig. 2a).

In Figs. 2b–f, we show for the previous datasets the relative backscattering strength *S _{υ}* = 10 log[

*s*(

_{υ}*R*)/

*s*(

_{υ}*R*

_{0})], where

*R*

_{0}is arbitrarily chosen as the first significant bin distance. The choice of

*R*

_{0}only translates the profiles vertically above or below 0. In the inset distance–time series, after subtracting the theoretical profile, different kind of backscatterers can then be observed and their relative strength can be quantified. In Fig. 2b, a very weak backscattering strength variation is thus observed in deep (2300 m) homogeneous Mediterranean waters, which has inertial periodicity. In Fig. 2c, backscattering is dominated by very intense short-term increases resulting from sediment resuspended by upslope moving solitary bores that are also distinguished in the velocity data. In Fig. 2d, there is a strong seasonal plankton migration over a 100-m range. In Fig. 2e, the range of the V-shaped planktonic migration down to 20 m below the sea surface is clearly visible. In Fig. 2f, the initially S1-dominant signal of zooplankton migration is altered when the vertical velocity of the M2 internal tide is in the opposite direction to the animals’ displacement. Our improvement is thus useful to correctly interpret RSSI data in these different conditions and for any quantitative observation based on echo intensity data approaching the noise level by a factor of 10.

## Acknowledgments

The assistance of the crew of R/V *Pelagia* is highly appreciated. We thank Theo Hillebrand for assistance in mooring preparation and handling. We acknowledge the ANTARES collaboration for the use of the ADCP dataset (available online at http://antares.in2p3.fr/). Integrated North Sea Program (INP), Processes above Continental Slopes (PROCS), and Long-Term Ocean Climate Observations (LOCO) were funded in part by the Netherlands Organization for the Advancement of Scientific Research (NWO). Author Gostiaux was funded in part via BSIK.

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Observation details of moored Teledyne RDI ADCPs. Temperature *T* is measured by the ADCP, and here salinity is designated by *S* in parts per thousand and is estimated from corresponding CTD profiles at the depth of the instrument. Start and end dates indicate portions of the time series used.

Acoustic details for the various ADCPs of Table 1. All are RDI four-beam instruments. Approximate transmit frequency is denoted by *F*. Data can be beam dependent and are computed for beam 1. See text for other symbols.