© 2003 by ICES/CIEM International Council for the Exploration of the Sea/Conseil International pour l'Exploration de la Mer
The absorption of sound in seawater in relation to the estimation of deep-water fish biomass
NIWA, Kilbirnie Wellington, New Zealand
*Correspondence to I. J. Doonan, NIWA, PO Box 14-901, Kilbirnie, Wellington, New Zealand; tel: +64 4 386 0300. e-mail: i.doonan{at}niwa.cri.nz.
It is well known that the acoustic estimation of fish abundance requires an estimate of the absorption of sound in seawater. Any errors in this factor will lead to errors in abundance estimates that increase with range. Concerns over the accuracy of the widely used relationship of Francois and Garrison (Journal of the Acoustical Society of America, 72: 896–907; Journal of the Acoustical Society of America, 72: 1879–1890) led to a new analysis of the data they used and a reconsideration of the relationship as a whole.
A major component of sound absorption by seawater is the relaxation frequency of MgSO4, for which there are at least five equations in the literature. We review these and our analysis uses the relaxation equation that best fits the sound-absorption data using modern regression methods. We propose a new equation for the range of frequencies used in fish abundance (10–120 kHz) based on this analysis. For a typical New Zealand deep-water fishery carried out at 38 kHz the new equation gives a biomass 17% lower than that yielded by Francois and Garrison. Whilst we believe that the new equation gives more accurate results for the range of frequencies used in fish abundance, there remains much uncertainty which will only be resolved by the collection of new in situ sound-absorption data in this frequency range.
Keywords: absorption of sound in seawater, Chatham Rise, deep-water fish survey
Received 21 January 2002; accepted 31 July 2002.
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Introduction |
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It is well known that sound is absorbed by seawater such that sound intensity decreases exponentially with distance (Clay and Medwin, 1977), and this effect must be taken into account when using underwater-acoustic methods to estimate fish abundance (Burczynski, 1979).
The absorption of sound by seawater was first investigated in the 1940s (Liebermann, 1948) and by the early 1980s two relationships had emerged: one based on laboratory measurements of artificial seawater (Fisher and Simmons, 1977; hereafter referred to as FiS) and the other on measurements in situ (Francois and Garrison, 1982a, b; hereafter referred to as FrG). The two relationships differ by small amounts at low frequencies but by larger amounts at high frequencies. A simplified version of FrG has been proposed by Ainslie and McColm (1988).
Most fisheries-acoustics work uses frequencies in the range 12–200 kHz, the "standard" frequency being 38 kHz, and most is carried out in relatively shallow water where return paths to fish targets are less than 300 m or so. Fisheries workers have generally used the most recent absorption equation available and adopted that of FrG when it appeared (Foote et al., 1987). For the frequencies and ranges used in most surveys, the estimated absorption for a typical target differs little from the alternative FiS estimate based on the artificial seawater relationship (Do and Coombs, 1989).
Some of the most profitable fisheries off New Zealand are for deep-water species such as orange roughy (Hoplostethus atlanticus Collett) and smooth oreos (Pseudocyttus maculates Gilchrist), which live in depths of 800–1300 m. The main deep-water fisheries are on the Chatham Rise to the east of New Zealand around latitude 44°S. Acoustic techniques at a frequency of 38 kHz are now the main source of biomass information for these fish. This frequency offers a reasonable compromise of signal-to-noise ratio, range, transducer size, beam angle etc. However, there are problems with the usage of the appropriate absorption relationship. Although towed bodies are ordinarily used for these surveys, return-travel distances may still be up to 1800 m and absorption losses at 38 kHz over these distances are significant. Foote (1981) showed that the effects on estimates of fish density of errors in the absorption coefficient used could be significant and in our study, for example, the difference between FrG and FiS (about 1.7 dB km–1) results in a difference in the biomass estimate of 77% for smooth oreos. Although, in principle, in situ measurements are preferable to those made in the laboratory, the FrG data contain few measurements of absorption at the depths, temperatures, and salinities typical of the Chatham Rise and other New Zealand waters, and no measurements at 38 kHz. As a consequence, we continued to use FiS when FrG was published because it was based on standard seawater and better represented our range of frequencies (Do and Coombs, 1989). However, other, later, orange-roughy workers adopted FrG (Elliott and Kloser, 1993; Kloser et al., 1996; Boyer and Hampton, 2001). Given the impact of the differences in the two relationships on our survey results, this is a matter that needs resolution. Ideally, new in situ measurements at 38 kHz and the other frequencies commonly used for fish surveys should be made on the deep-water fishing grounds. Unfortunately, such measurements are both costly and difficult and so, as an interim measure, in this paper we present a new analysis of the data used by FrG focusing on frequencies of interest for fisheries work and deep-water hydrographic conditions, employing statistical techniques developed since the earlier analyses. In formulating an absorption, temperature, salinity, and depth relationship we have given particular consideration to the best relaxation equation to use and consider this in some detail.
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Factors affecting sound absorption
As pressure waves propagate through water, the transfer of thermal energy associated with chemical relaxation processes drives the absorption of sound by seawater salts. When the acoustic wavelength corresponds to certain ranges of the relaxation frequency, alternate compression and rarefaction of parcels of water cause dissociation of the magnesium and boron salts. The exact physical mechanisms are still a matter of debate. The relaxation frequency for magnesium salts depends on temperature and also slightly on salinity. The relationship between relaxation frequency and temperature has been determined from experimental data several times (Table 1). However, the results differ by up to 15 kHz in the 0–15°C range (Figure 1). This relationship is critical in comparing attenuation values collected at different frequencies.
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The magnitude of the absorption coefficient in seawater is a function of three components: the absorption of sound by pure water, by boric acid, and by the salts of magnesium. For frequencies >10 and <100 kHz, most of the absorption is due to magnesium salts, ranging from 96% at 10 kHz to 87% at 100 kHz. Below 10 kHz the boric acid term is important and above 1000 kHz absorption by pure water is significant (FiS). The frequencies of interest for acoustic surveys of fish (12–200 kHz) are within the range of absorptions dominated by magnesium salts.
The chemical speciation of magnesium in seawater is dominated by free Mg2+ ions (87–89%). The remaining fraction is paired with SO42– (9–12%) (Grasshoff et al., 1983). Earlier reports suggested that there were differences in the magnesium/chlorinity ratio in the world's oceans, but these differences are now attributed to analytical uncertainties rather than to natural variability (Grasshoff et al., 1983). There is no evidence that oceanic magnesium/chlorinity ratio varies with depth or time (Grasshoff et al., 1983). The sulphate/chlorinity ratio is also constant at all depths of the open ocean (Grasshoff et al., 1983). A compilation of the results for magnesium and sulphate concentrations in the oceans can be found in Culkin and Cox (1966), Morris and Riley (1966), Carpenter and Manella (1973), Millero (1974), and Wilson (1975). Chlorinity is linearly related to salinity, so that the absorption coefficient for sound at frequencies between 10 and 1000 kHz is affected by salinity over the depth ranges relevant to deep-water fisheries.
Summary of the current predictive equations
Both FrG and FiS equations date from before the 1982 adjustment to salinity units from parts per thousand (ppt) to the Practical Salinity Scale (dimensionless units). Consequently, our re-analyses of the relaxation frequency, which is a function of temperature and salinity, is also in ppt units, as is our final equation for sound absorption, which is a function of relaxation frequency and pressure. Quantitative differences between these units are very small and conversion between them is not generally attempted, so we make no conversion here.
Following the notation of FiS, the absorption coefficient, 
, at a salinity of 35 and a pH of 8.0, is given by:
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| (1) |
1 kHz and the third term gives absorption by pure water. To compare the FiS and FrG equations we focus on the second term describing absorption by MgSO4. FiS give the following predictive equations for A2 (s m–1), P2 (dimensionless), and f2 (Hz): |
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To use this at other salinities with frequencies over 10 kHz, the term A2 is multiplied by (S/35). To convert to dB km–1, is multiplied by 8686.
FrG use the same notation as FiS but their units differ as they calculate in units of dB km–1. They give the following alternative predictive equations for A2 (dB km–1 kHz–1), P2 (dimensionless), and f2 (kHz):
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The equations were fitted in a stepwise manner using different parts of the data set to estimate different sets of parameters, typically using linear regression. In particular, Glotov's relaxation equation (Glotov, 1964) was used.
Analytical approach
Our new analysis uses computer-intensive methods on non-linear regressions that are formulated directly on the equations for predicting sound absorption. This approach gives a relatively simple structure to the analysis, avoiding the necessity for linear approximations and allowing all appropriate data to be included. Error structures are modelled explicitly and the statistical framework provides a natural way to evaluate the adequacy of adding extra parameters to the model. This ensures that spurious parameters have not been included. We check that the residuals fit the assumed distribution and we estimate the precision of the estimated parameters by bootstrapping the data (Efron and Tibshirani, 1986). The latter is used to estimate the precision of the predicted sound absorption and thus can be used to calculate errors from this source in biomass estimates.
We check the effects of the assumptions used in the analysis by simulating data with the same patterns of salinity (S), temperature (T), and depth (D) as the real data. We estimate the parameters from these simulated data using deliberately false assumptions to assess the robustness of the results to the assumptions used. For this we assumed a proportional salinity effect and used a fixed relaxation frequency, neither of which may be true. In addition, we use simulation to test the effect of unbalanced data design on parameter estimates. If bias is present, it will vary with frequency and we focus on 38 kHz since this is the one we mainly use.
In situ data
All the data used in our analysis came from FrG or the papers they cite. FrG collated all previously published data available at the time (1982). The full data covered frequencies from 7.1 to 467 kHz (Figure 2). No new in situ data for frequencies between 12 and 200 kHz have been published since.
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The frequency of most data points was either 60 or 75.8 kHz. Only 15 points had frequencies within 10 kHz of our primary frequency of 38 kHz and there were none actually at 38 kHz. The only field measurements of sound absorption in oceanic waters in the whole Pacific around 38 kHz were those of Bezdek (1973). These were made in 1971 and 1972, 519 km southwest of San Diego, CA, and included measurements at 30 and 45 kHz, respectively. Other measurements in this frequency range (30–41 kHz) were from the Arctic (Greene, 1966), Dabob Bay in Puget Sound, Washington State, USA (FrG, 1982a), and the North Atlantic (Schulkin and Marsh, 1962).
FrG tabulated 
35 for most of their measurements (35 is absorption normalized by wavelength, adjusted to a salinity of 35). Variability in 35 was high for all frequencies (Figure 2) (coefficient of variation=22.6%, standard error = 0.113 x 10–5 Np m–1, n=89).
The S, T, D combinations in the FrG data were unbalanced for a regression analysis. A balanced design would contain the same number of points in each S, T, D combination and thus there would be no correlations between these variables. Given the nature of the marine environment, it is probably impossible to achieve data balance and certainly there was a correlation between salinity and depth of 0.5 and consequently salinity and pressure effects are confounded in the regression.
For our analysis we selected points in the frequency range 10–120 kHz from the overall data set. Bezdek (1973) gave 29 points, Schulkin and Marsh (1962) 6 points, Green (1966) 9 points, and FrG (Applied Physics Laboratory (APL), University of Washington, 1953–1980) 30 points. Measurement errors were missing from Schulkin and Marsh (1962) and were assigned 2.6 dB, which is the 75% quartile of the measurement errors estimated using all other data.
Regression analysis
The original formulation for the MgSO4 term of sound absorption Equation (1) is, ignoring constants, of the form S(1+T)(1+P+P2)g(f), where g(f) is given by f2f2/(f22+f2). f2 is the relaxation frequency for the MgSO4 effect, S salinity, T temperature, and P is pressure. An alternative form for the (1 + P + P2) term is exp(P) proposed by Ainslie and McColm (1998) who found that it fitted well in their approximation of FrG. We used the exponential form because it reduces the potential number of parameters needed.
We estimated the A2 and P2 coefficients in the MgSO4 term of Equation (1) using observed from published in situ data. The A2 and P2 terms were estimated for each relaxation equation in Table 1 and the one with the best fit chosen as the final predictor.
We followed the assumption of FiS that the attenuation from different components of seawater is additive, and we ignored the boric acid and magnesium carbonate components. The pure water component has a small effect, so we used previously published values for Aw and Pw (FrG):
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Thus, y=observed–AwPwf2 was regressed against functions of T, S, and D. The units used here are (notation in brackets): °C for temperature (T), ppt for salinity (S), dB km–1 for attenuation (), kHz for frequency (f), and metres for depth (D, an alias for pressure).
For the ith experiment the regression is:
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| (2) |
There are three contributions to the error term:
- Measurement error,
m (normally distributed with zero mean and standard deviation).
- Within-experiment drift (normally distributed with zero mean and standard deviation).
- Formulation error (ignoring the boric acid contribution, this includes errors in the assumptions and errors in the parameter values of the water component).
The last two sources of error were combined into w. m and w were assumed to be independent. Measurement errors were reported with the observed by FrG so our analysis only needs to estimate w.
Within-experiment drift is caused by deviations in the assumed spreading law and changes in the effective gain of the acoustic systems between measurements at different ranges. Values for within-experiment drift come from the adjustment by FrG to make data vary with frequency according to relaxation theory, 
, of which there were three values: –0.56 dB km–1 (APL, Kane Basin, 1979), 0.64 dB km–1 (APL, Chukchi Sea, 1974), and 1.3 dB km–1 (APL, Chukchi Sea, 1974). These three values give an approximate estimate of the lower bound for w of 1.1. Preliminary analyses showed that for the data used here, w was much greater when temperatures were less than zero. Therefore, two w values were estimated: one for temperatures less than zero (w1) and another for temperatures greater than or equal to zero (w2).
Because errors were assumed to be normally distributed and uncorrelated, parameters were estimated by maximising the log-likelihood, ignoring constants, given by:
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w,i2 was (a) either w2 or w22 depending on T, or (b) 2 if there was no temperature dependence on errors. The maximum number of parameters is 6: a0, a1, a2, p1, w, and w2 (see Equation (2)). These parameters were added one by one, and kept if the log-likelihood increased by 1.8 (likelihood ratio test). The starting parameter set was to estimate a0 and one w value with a1, a2, and p1 set to zero. The two parameters in the relaxation equations could also be estimated by our regression analysis. However, we have used the published equations instead and evaluated these in the regression. We took this approach because some, particularly the more recent equation of Qiu and Han (1987) (QiH), are based on more data covering wider temperature ranges than are represented in the FrG data and the former are not available to us to include in our analysis.
Estimating precision in predictions
We estimated precision specifically with respect to our main deep-water fishery on the Chatham Rise using three independent conductivity–temperature–depth (CTD) profiles measured during an acoustic survey on the south Chatham Rise in 1997. These profiles can be characterised by a (Practical Salinity Scale) salinity of 34.5 with five samples that have temperatures of 8.3, 7.3, 6.6, 5.3, and 4.3°C at depths of 250, 450, 650, 850, and 1000 m, respectively. The absorption coefficient was calculated from these profiles and averaged to give 
. The coefficient of variation (c.v.), of the estimated parameters and the predicted 
was calculated using parametric bootstrapping (Efron and Tibshirani, 1986). For this, pseudodata were generated by drawing random numbers through the estimated error structure and using the estimated parameters at the same configuration of frequency, S, T, and D to estimate the parameters. The pseudodata were used to re-estimate the parameters and by repeating this procedure, the distribution of the estimated parameters found. To keep the absorption values positive, the absolute value was used for any negative simulated values (i.e. a reflected normal distribution).
Estimating bias in predictions
To investigate the prediction bias of arising from misspecification of fixed parameters, simulated data were generated at the same configuration of S, T, and D values, and using the estimated parameter values as though they were the true values, but with different fixed parameters. The parameters were re-estimated using the simulated data and the same values for the fixed parameters as used in the main analysis. This process mimics the analysis when an assumption used in it was misspecified. Two cases were considered: misspecifications in the relaxation equations and a misspecified form of the salinity term in Equation (2). The relaxation equations' parameters were changed so that the relaxation frequency was different by 5 kHz. This was done in two ways: by shifting the curve up and down, and by changing the slope such that the frequency was the same at 10°C but was 5 kHz out at 0°C. For the salinity term, the exponential form, S1.3, was used to generate data, but parameters estimated using the form, S1.0. All the authors cited here assumed that the relationship between absorption and salinity is linear. When we estimated the exponent for S we obtained 1.3, which was not statistically significantly different from 1.
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Regression analysis
The best regression fit in our analysis was obtained with the QiH 2 relaxation equation (Table 2), and the standard error for predicting
was 0.09 dB. The parameter values for the best fit are given in Table 3. The FrG relaxation equation gave the worst fit in our analysis (Table 2). The range over all five relaxation equations for the mean absorption coefficient () was 1.1 dB.
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For all relaxation equations, plotting the quantiles of the standardised residuals against the quantiles of the standard normal distribution gave an approximately straight line, validating the assumption that errors were more or less normally distributed. Plots of residuals against temperature, salinity, and depth showed no trends (Figure 3). However, for frequencies below 28 kHz, residuals were such that nearly all had predicted values that were too low (Figure 3) and these had a median bias that ranged from 0.35 to 0.75 dB, depending on the relaxation equations used. For the QiH 2 equation, this bias was 0.47 dB.
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The bias in predictions of
from misspecified fixed parameters was low. When f2 was misspecified by increments of 5 kHz, the bias was 0.3 dB. Misspecifying the salinity exponent also produced a small negative bias in of about 0.3 dB km–1. The errors in biomass estimates resulting from these biases are small by comparison with other errors. |
Discussion |
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From our analysis it seems that the difference between FiS and FrG cannot be attributed to a simple pattern in the relationship of temperature, depth, and frequency. In the hydrographic conditions of the south Chatham Rise, our proposed equations (using the QiH 2 relaxation equation) generally lie between the curves for FiS and FrG over a frequency range of 10–100 kHz (Figure 4). It appears that with respect to sound absorption, 38 kHz was an unfortunate choice for the south Chatham Rise surveys because this frequency is close to a local maximum in the difference in sound absorption between the FiS and FrG equations.
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For a frequency of 38 kHz in south Chatham Rise conditions or over a range of temperatures at a salinity of 34.5, the effects of our equation on estimating biomass with range again lie between that for the FrG and FiS equations (Figure 5). At 0°C, there are minimal differences in the effect between our equation and FrG's, but not at the temperatures encountered on the south Chatham Rise for which the differences are near maximal. The consequences on the estimated abundance of recruited smooth oreos on the south Chatham Rise from using our new formula for absorption (24 000 t) is a 17% reduction over that using the FrG equation, and a 45% increase over that using FiS.
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There are three main differences between our analysis and that of FrG: how the data were weighted in the analysis, the data used, and the choice of relaxation equation. The weightings used in our analysis are 1/(
m2+w2), where m is the measurement error, compared to 1/m used by FrG; statistically, it should be (1/m)2, but given that some m values were as low as 0.1 and the maximum was 3, using this form would have tended to disregard data with high m. When we used 1/m as the weighting in our analysis, we obtained similar parameter values to FrG. Using our weighting method generally increased the absorption value. When we added more data into the analysis (i.e. data excluded by FrG, but included in the analysis here) keeping the FrG relaxation equation, the predicted absorption shifted to lower values, i.e. back towards those predicted by FiS. The largest effect was due to the relaxation equation and using the QiH 2 equation moved the sound absorption more towards that given by FiS.
An illustration of the above three effects can be given using salinity of 35 at a depth of 900 m and at a temperature of 5°C. Absorption is 8.5 dB for FiS and 10.1 dB for FrG. Using the FrG relaxation equation, the data set used by FrG and using 1/(m2+w2) to weight the data in our non-linear regression gave 10.4 dB, which reduces to 10.0 dB when the extra data were added, and reduced further to 9.3 dB when the QiH 2 relaxation equation was used (i.e. the current parameters as estimated here).
The summation of these differences in the estimated parameters is shown in Table 3. FrG had a temperature effect that was just statistically higher that our estimate. FiS, on the other hand, had a lower pressure effect, but higher overall level (a0).
Our analysis shows that measuring absorption is not a trivial task. Our estimated w1 of 2.0 dB suggests that there are large biases for in situ measurements of sound absorption at temperatures colder than 0°C. For temperatures between 0 and 8°C (the range for deep-water fish), the bias is lower, but still not insubstantial (estimated w2 = 0.37 dB). The significance of these large estimates is that in situ measurements should be conducted as independent experiments, so that bias can be treated as random error. (A reviewer has pointed out that the errors could be due to a biological source and if this were so it would mean that the repeated experiments would have to be done over a range of biological conditions.) To obtain a mean absorption coefficient accurate to 0.1 dB would require 13 such experiments given w=0.35 and m is 0.1 dB, and assuming no component due to formulation error. For w=2.0, 400 experiments would be required.
For most short-range, fisheries-acoustic survey purposes the choice of equation for estimating sound absorption is perhaps not of great significance. However, it can, as we have shown, have a substantial effect for surveys of deep-water fish. Given the standard error of 0.09 dB for , there are significant statistical differences between the equations and the question arises as to which is the best. Clearly, there is no absolute answer to this; in situ measurements ought to produce the best results but are difficult to make, particularly in a statistically rigorous manner. Laboratory measurements of standard seawater are more accurate but do not necessarily give results that are valid in the open ocean. We believe our analysis offers the best relationship currently available for the 10–120 kHz frequency range. By comparison with the earlier work of FrG, we treat within-experiment drift in a more comprehensive way and consequently weight the data correctly. We use statistical evidence in the data to choose the relaxation equation. Lastly, we use more data so that the temperature range is extended and gaps in the temperature distribution filled. We believe that this places our analysis on a surer footing although real confidence will only come when field and laboratory studies can be reconciled with better measurements.
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The data available are inadequate in many respects and, in particular, they are very sparse near 38 kHz mostly only covering salinities lower than 34. Our new analysis of the FrG data set suggests that the relaxation equation chosen is the critical factor in estimating sound absorption and that the equation used by FrG is probably inaccurate. Based on our analysis of the published in situ magnesium sulphate sound-absorption data, we propose the following new formulation for sound absorption in seawater for frequencies in the range 10–120 kHz (i.e. ignoring the boric acid component) and for temperatures less than 20°C:
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describes the magnesium sulphate absorption and the third term gives absorption by pure water. The boric acid component is ignored as it is minor for frequencies 1 kHz. The predictive equations for A2 and P2 are: |
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For A3 and P3, we use the following predictive equations given by FiS:
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Uncited references |
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Lyman and Fleming, 1940
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Acknowledgements |
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Funding for this research was provided by New Zealand Ministry of Fisheries contract ORH9701. We thank the anonymous reviewers for comments on the manuscript.
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