Volume 9, Issue 10 e2021EA001974
Research Article
Open Access

Sonar Observation of Heat Flux of Diffuse Hydrothermal Flows

Darrell Jackson

Corresponding Author

Darrell Jackson

Applied Physics Laboratory, University of Washington, Seattle, WA, USA

Correspondence to:

D. Jackson,

[email protected]

Contribution: Conceptualization, ​Investigation, Writing - original draft

Search for more papers by this author
Karen Bemis

Karen Bemis

Department of Marine and Coastal Sciences, Rutgers University, New Brunswick, NJ, USA

Contribution: ​Investigation, Project administration

Search for more papers by this author
Guangyu Xu

Guangyu Xu

Applied Physics Laboratory, University of Washington, Seattle, WA, USA

Contribution: ​Investigation

Search for more papers by this author
Anatoliy Ivakin

Anatoliy Ivakin

Applied Physics Laboratory, University of Washington, Seattle, WA, USA

Contribution: ​Investigation

Search for more papers by this author
First published: 21 September 2022

Abstract

Previous work using multibeam sonar to map diffuse hydrothermal flows is extended to estimate the heat output of diffuse flows. In the first step toward inversion, temperature statistics are obtained from sonar data and compared to thermistor data in order to set the value of an empirical constant. Finally, a simple model is used to obtain heat-flux density from sonar-derived temperature statistics. The method is applied to data from the Cabled Observatory Vent Imaging Sonar (COVIS) deployed on the Ocean Observatories Initiative's Regional Cabled Array at the ASHES vent field on Axial Seamount. Inversion results are presented as maps of heat-flux density in MW/m2 and as time series of heat-flux density averaged over COVIS' field of view.

Key Points

  • Multibeam sonar is used to observe heat flux of diffuse hydrothermal flows

  • The inversion method estimates vertically averaged temperature fluctuation

  • Validation and calibration of method by direct measurement is needed

Plain Language Summary

Hydrothermal activity in the deep ocean is an important component of the processes that cool the Earth. This activity is of two types: venting of hot, turbid water in “black smokers” and diffuse venting of warm, mostly clear water. It has proven difficult to measure the heat output of diffuse venting, as the usual temperature-measuring devices only provide data at a few points, and further, do not give the velocity of the flow as needed to determine the amount of heat being delivered to the ocean. The Cabled Observatory Vent Imaging Sonar (COVIS) has been used to measure heat flowing from an area of 500 square meters in the caldera of Axial Seamount. COVIS was installed in July 2018 on the Ocean Observatories Initiative Regional Cabled Array, which allows shore-based investigators immediate access to COVIS data. It is hoped that COVIS will continue to monitor heat flow long enough to see the effects of an eruption of Axial volcano.

1 Introduction

This article extends the work reported in Xu et al. (2021) with the goal of quantifying the heat output from a portion of the ASHES Vent Field on Axial Seamount. The earlier article gives references to the literature as well as a detailed description of the data provided by the Cabled Observatory Vent Imaging Sonar (COVIS). Diffuse discharges of hydrothermal fluids take many forms and may originate from the sides of sulfide mounds, isolated cracks, lava tubes, and areas of permeable seafloor (Barreyre et al., 2012; K. Bemis et al., 2012, Figure 1). The heat output of diffuse-flow discharge may equal or exceed that of focused flow from associated smokers (Barreyre et al., 2012; Rona & Trivett, 1992; Schultz et al., 1992; Veirs et al., 2006). This article will make frequent mention of both heat flux and heat-flux density. Heat flux has the dimensions of power and can conveniently be measured in MW (megawatts). Heat flux density is the measure of power per unit area. Its integral over an area is the heat flux emitted within that area. This usage follows a convention of physics, where, for example, the integral of current density over a surface is the current passing though that surface.

Details are in the caption following the image

Sketch of thermistor array used for ground-truth measurements.

In Schultz et al. (1992) an electromagnetic-based flowmeter and several thermistors were used on the sulfide mound called Peanut in the southern half of Main Endeavour Field (MEF) to simultaneously measure temperature and vertical velocity in hydrothermal diffuse discharge for about 2 months, estimating a diffuse heat-flux density of 2.91 ± 0.23 MW/m2. Trivett and Williams (1994) used thermistor moorings and tripod-mounted current meters to estimate heat flux from diffuse-flow regions on the Southern Juan de Fuca Ridge, obtaining a net flux of 125 ± 75 MW for the most active region. Their instruments were placed at a distance from the diffuse-flow regions of interest and relied on currents to convect the warm water toward their apparatus. Veirs et al. (2006), using data from an Remotely-Operated Vehicle (ROV), a CTD, and current meters, estimated the along-axis heat flux at MEF to be 8–42 MW. They concluded that the MEF heat flux was about equally divided between focused and diffuse flows. More recently, acoustic estimates of areal extent were combined with limited temperature and flow rate measurements to estimate heat flux by diffuse discharge at North Tower of the sulfide mound Grotto in MEF as 33–380 MW or a heat flux density of 0.33–3.8 MW/m2 over the 100 m2 summit of the North Tower (Rona et al., 2015).

Observations of hydrothermal flows for the northern part of ASHES were reported in previous studies. For example, Rona and Trivett (1992) conducted a thermal survey over a 100 m by 100 m area enclosing the depression region using an ROV carrying a 1-m long vertical thermistor array. They estimated the heat flux from both focused and diffuse flow sources to be 2.4–6.4 and 15–75 MW, respectively, from in situ temperature measurements, which suggested diffuse flow venting was the dominant hydrothermal heat source at ASHES. In this same area, Pruis and Johnson (2004) measured the flow rate and temperature of the discharge from a diffuse-flow source using a fluid sampler cemented to the seafloor. They estimated a volume flux of 48 m3/yr and a heat flux density of 260 W/m2 for the 1 m2 area sealed by the sampler. Most recently, Mittelstaedt et al. (2016) measured heat flux of diffuse flow venting from a narrow fracture near the southern end of the depression region using the Diffuse Effluent Measurement System, which is a camera system augmented with thermistors that can measure the flow rate and temperature of diffuse-flow effluents. They obtained an estimated heat-flux density of 0.07–0.51 MW/m2, which was subsequently extrapolated over all the cracks detected in a photomosaic survey to get a total heat flux from venting fractures of 0.1–4 MW.

With the exception of Trivett and Williams (1994), previous work either provides time series at specific points or large-scale snapshots of heat flux. The uncertainty in these measurements is great, with the ratio of upper to lower bounds being as large as 5 or 10. These facts motivate the present attempt to use multibeam sonar to estimate heat flux across substantial areas over long times.

COVIS was installed in July 2018 on the Ocean Observatories Initiative (OOI) Regional Cabled Array at the ASHES vent field on Axial Seamount. The resulting data have been used to obtain time series of diffuse-flow activity, primarily through estimation of the area of diffuse flow regions (Xu et al., 2021). The present effort seeks to extend this work and that of Jackson et al. (2017), to provide a method for sonar observation of the heat output of diffuse flows. In Section 2, the thermistor data used as ground-truth are described, and in Section 3, sonar data are discussed, and the first stage of inversion used to estimate temperature statistics is developed. Section 4 compares inverted and directly measured temperature statistics, and the final stage of inversion for heat-flux density is developed in Section 5. Inversion results for heat flux are given in Section 6, and Section 7 offers conclusions and a suggestion for future work.

2 Thermistor Data

The thermistor array pictured in Figure 1 was set on the seafloor by ROV Jason at the two sites shown in Figure 2. The thermistors are RBRsolo T, with response time of 0.1 s (https://docs.rbr-global.com/support/sensors/temperature/time-response-spectral), consistent with the sonar ping-to-ping spacing of 0.25 s. At sites 1H and 4C several stations were occupied on 6 June 2019, with nominal 25-cm spacing. Seven stations were occupied at 1H and 11 at 4C. Figure 3 shows the stations for site 4C. A close-up image of Site 1H can be found in Xu et al. (2021). The sources of warm water at the two sites are diverse, with small, nearly point sources evident in video observation, wider sources due to dispersal by tube worm bushes, and even possible sheet-like sources emanating from cracks (Xu et al., 2021). Close video inspection at some sites showed a few focused flows emitting slightly turbid water. These sources resembled miniature black smokers.

Details are in the caption following the image

Sites in ASHES vent field where thermistor arrays were deployed in 2018 and 2019. Sites 1H and 4C are marked by green squares. Cabled Observatory Vent Imaging Sonar's (COVIS's) location is marked by a black dot. The smaller bathymetric feature to the NW of COVIS is the Mushroom black smoker and the larger one to the east of COVIS is Inferno. The color scale gives the height in meters above the base of COVIS.

Details are in the caption following the image

Locations of stations for the thermistor array (visible on right) at site 4C. The array was placed at locations 1, 3, 6–8, 10–14, and 17. The other numbered locations were occupied by a high-temperature thermistor probe. Tube-worm bushes are evident as well as localized anhydrite and biofilm deposits. Both are indicators of diffuse hydrothermal flow.

Time series having lengths of 2–6 min were obtained at each station, with sampling frequencies of 4 Hz on two thermistors in the array and 16 Hz on the remaining eight thermistors. In processing, the faster sampling rate was reduced to 4 Hz by decimation. Figure 4 shows the mean and standard deviation of temperature profiles at the 7 stations occupied at 1H and the 11 stations occupied at 4C. The “temperature anomaly” is defined as the difference with ambient temperature and is substantial only up to heights about 50 cm above the seafloor. This maximum height may depend upon ambient current, but no co-located current measurements are available. Measurements of the vertical component of current would be welcome, as the product of temperature anomaly and vertical current is proportional to the vertical component of heat-flux density (Rona & Trivett, 1992). Some stations show peak temperatures occurring above the bottom which indicates that these stations were offset from sources producing small plumes of warm water, with ambient cross-currents moving the small plumes toward the thermistor array. A 3D single-point velocity meter deployed at the International District hydrothermal vent field on the southeastern side of Axial's caldera, approximately 30 m distant from ASHES, has provided times series of transverse current components at a height of approximately 1 m. The measured current components oscillate with typical amplitudes of about 1 cm/s with excursions approaching and sometimes exceeding 2 cm/s. The effect of current on diffuse-flow activity at ASHES is discussed in (Xu et al., 2021). It is of particular importance that the standard deviation of temperature is very nearly equal to the mean anomaly. This behavior was noted at all stations having substantial anomaly, and as will be seen, is an important element in our inversion algorithm. Near-equality of the standard deviation and mean is not observed in laboratory measurements. Papantoniou and List (1989) give probability density functions for tracer concentration from which standard deviation/mean ratios of about 0.7 can be determined. In more recent work, Wang and Law (2002) measured ratios between 0.4 and 0.45 near the centerlines of buoyant plumes. Our thermistor measurements are different from these laboratory observations in that temperature fluctuations are partially due to the horizontal advection of plume fluids. For example, a thermistor can frequently shift from being inside to outside of a wafting plume, resulting in a larger standard deviation than if the thermistor were at a fixed location relative to the plume's centerline. Finally, it will be noted that there is large variability in the magnitude of the temperature anomaly from station to station, and this spatial intermittency must be accounted for in the inversion algorithm.

Details are in the caption following the image

Vertical profiles of mean and standard deviation of temperature at diffuse-flow sites (a–g) 1H and (h–r) 4C. The heights of individual thermistors are marked by circles on the curves for average temperature.

Although no model is available that would fit the flows from the variegated sources at sites 1H and 4C, for example, it can be shown that the temperature profiles of Figure 4 are not at odds with buoyant plume theory for an axisymmetric source (Morton et al., 1956). Figure 5 shows the temperature profile that would result from a circular source of diameter 1 cm, and initial temperature anomaly 8°C, with entrainment coefficient 0.083 and temperature-anomaly radius/velocity radius ratio 1.19, discharging into an unstratified water column. It is assumed that there is no cross-current, and the profile is for a vertical line centered on the source. While only a few of the profiles in Figure 4 exhibit the sharp peak at zero height seen in the model curve, the lack of this behavior for other profiles can be explained as a deliberate experimental bias: the thermistor array was not placed directly over the hottest sources in order to avoid damage to the lowest thermistors. Allowing for this and for cross-currents to move plumes toward the thermistors not on the seafloor, the model profile is suggestive of the observed behavior.

Details are in the caption following the image

Model for vertical profile using buoyant plume theory. A source temperature anomaly of 8°C is assumed with orifice diameter 1 cm and ambient temperature 2.4°C.

The thermistors are autonomous, with independent sampling, and the drift of the clocks on each unit was sufficient to cause lack of synchronization on the order of a few seconds. As the sonar samples at a rate of 4 Hz, this lack of synchronization is of concern. To assess the magnitude of clock drift, cross-correlations between different thermistors were examined, but no peaks were evident over lags up to 10 s. This leads to the conclusion that the random parts of the time series are essentially independent between thermistors. If the random thermal structure were “frozen” and advected vertically, a strong correlation would be expected at lags determined by the vertical component of velocity and clock offsets due to drift. This lack of correlation is additional evidence for the influence of cross currents as seen in Xu et al. (2021). As the signature of upward advection is not visible in the thermistor data, clock drift is of no concern, and statistics appropriate for comparison with inversions of sonar data are estimated as described below.

In the development of the sonar inversion method described in the next section, the statistic of interest is the “structure function” for vertically averaged temperature anomaly. This particular structure function was defined by Jackson et al. (2017), but as the discussion is minimal, its definition will be given in detail here. The structure function of interest is
urn:x-wiley:23335084:media:ess21284:ess21284-math-0001(1)
where the brackets denote a formal average over an (unattainable) infinite ensemble of measurements at horizontal coordinates (x and y). The change in path-averaged temperature is the difference between measurements at times t and t − τ. Note that it is assumed that the average is independent of horizontal coordinates. In other words, it is assumed that while the flow may have complicated dependence on both horizontal and vertical coordinates, its average over members of the ensemble and over x and y is perfectly stratified. The quantity being averaged is the square of
urn:x-wiley:23335084:media:ess21284:ess21284-math-0002(2)
which is the vertical average of the difference in temperature, T, occurring between times t and t − τ for measurements made at horizontal coordinates (x and y). The average is over the height of the sonar, H, and the variable τ will be called the “lag.”

These formal definitions are motivated by the requirements of the inversion algorithm to be defined later and must be adapted to our thermistor data with all its limitations. First, the sonar height H, is 4.2 m, greater than the height of the thermistor array. Fortunately, the temperature anomaly is essentially zero for heights greater than about 0.5 m, so there is no appreciable error in replacing H in the upper limit of the integral Equation 2 with the 2-m height of the thermistor array. The average in Equation 1 is over an infinite ensemble of measurements and horizontal positions, and must be replaced by estimates based on a finite data set. The average over positions will be replaced by an average over stations, and the average over measurements will be replaced by an average over the time interval over which measurements were made at each station. Although this estimation method seems reasonable, it may suffer from undersampling. The sonar inversions to be discussed later are averages over 1 day, which is approximately two tidal cycles. While this averaging interval may not be large enough to smooth out all tidal influence, longer averaging periods would degrade the time resolution of possible events. The thermistor data are averaged over 6 min, so tidal effects are not averaged over to any degree. We cannot estimate the error involved, but error in path-averaged temperature will result in a proportionate error in heat flux. It must be realized that the bar for heat-flux measurement accuracy is set very low, with previous estimates having cited error as large as factors of five to ten.

The integral in Equation 2 will be approximated by a discrete sum over the difference in temperatures measured at times tj and tj − τi by individual thermistors in the array, and the structure function (1) needed for comparison with sonar inversions is estimated as the average over time samples of the squared change in temperature difference.

An issue arises from the thermal inertia of the thermistors. Even though their response time of 0.1 s is shorter than the sonar sampling interval of 0.25 s, the inversion method to be discussed later is partially dependent on short lags in the structure function, and these are affected by the thermal inertia of the thermistors. To compensate for this, the temperature time series are corrected by dividing their Fourier transforms by a transfer function that is itself a Fourier transform of the response of the thermistor to a unit impulse of temperature. According to the manufacturer, this response can be approximated by an exponential with e-folding time T = 0.1 s (https://docs.rbr-global.com/support/sensors/temperature/time-response-spectral). The corresponding transfer function is 1/(urn:x-wiley:23335084:media:ess21284:ess21284-math-0003). The effect of this correction on the estimated structure function is illustrated in Figure 6.

Details are in the caption following the image

Structure function for path-averaged temperature with and without correction for thermal inertia of thermistors. This is an average over the seven stations occupied at 1H.

3 Sonar Data

3.1 Sonar System

COVIS is a high-frequency (200 and 400 kHz) sonar system, based on the Reson 7,125 multibeam sonar. The source and receiver transducers are mounted on a tripod 4.2 m high on a tri-axial rotator which provides freedom to set pitch, roll, and yaw. COVIS provides near real-time images and monitoring of hydrothermal flow (K. G. Bemis et al., 2015). A description of COVIS is given in Jackson et al. (2017). As in the earlier acoustic mapping efforts of Rona et al. (1997), COVIS exploits the expected constancy of ping returns from the seafloor to detect the phase changes caused by travel time variations due to the variable water temperature between COVIS and the seafloor (Rona & Jones, 2009). As there is time jitter between the instant of transmission and beginning of digitization, the transmitted signal that leaks into data channels via cross talk is used to align the phase of successive echo time series. In the typical mode of operation for diffuse-flow study, COVIS transmits a burst of 39 pings (sometimes more) at a rate of 4 Hz, with a burst transmitted every hour or half-hour. The transmitted pulse has an approximately rectangular envelope of width 0.3 ms and a nominal source level of 200 dB re 1 μPa@1m.

3.2 Sonar Data Processing

COVIS uses digital beamforming in the horizontal with a Hamming window to provide azimuthal resolution of 1° at 200 kHz. For each of the 128 beams there is a complex (baseband) time series sampled at a rate of 34 kHz with corresponding time interval between samples of 29.4 μs. It should be noted that there are three different sampling times in the data set. First, data are acquired at a rate of one burst per hour (or half hour). These sampling times can be regarded as “true” times, that is, clock times, for example, UTC. In the processing employed here, averages will be taken over 24-hr time intervals. We believe this averaging is crucial to the method, bringing the average temperature field into play and avoiding the problem of modeling details of the complex flows observed at the thermistor sites. As for the second sampling interval, each sonar burst consists of 39 or more pings with time spacing Δτ = 0.25 s between pings. Finally, as mentioned above, each ping is sampled at a time spacing of 29.4 μs.

Correlation between pings in the same burst is central to our method. The ping-to-ping correlation is formed using two (complex) echo time series urn:x-wiley:23335084:media:ess21284:ess21284-math-0004 and urn:x-wiley:23335084:media:ess21284:ess21284-math-0005 from the same burst and the same sonar beam, transmitted at times different by the lag τ = NΔτ. Here, N ranges from 0 to 38 for a 39-ping burst. The time arguments t are measured from the center point of each transmission and are sampled at the interval 29.4 μs mentioned previously. The ping-to-ping correlation estimator is
urn:x-wiley:23335084:media:ess21284:ess21284-math-0006(3)
where
urn:x-wiley:23335084:media:ess21284:ess21284-math-0007(4)
and with urn:x-wiley:23335084:media:ess21284:ess21284-math-0008 and urn:x-wiley:23335084:media:ess21284:ess21284-math-0009 being given by similar integrals with the squared magnitude of each echo time series. If the two echo time series are identical, urn:x-wiley:23335084:media:ess21284:ess21284-math-0010 has a value of unity, and if the two are different, urn:x-wiley:23335084:media:ess21284:ess21284-math-0011 will have a magnitude smaller than unity. The signal product in Equation 4 is weighted by a Hamming window function urn:x-wiley:23335084:media:ess21284:ess21284-math-0012 whose position time, urn:x-wiley:23335084:media:ess21284:ess21284-math-0013, determines path length (“slant range”), urn:x-wiley:23335084:media:ess21284:ess21284-math-0014, and whose width (1.5 ms) is set by a trade-off between range resolution and the desire to reduce statistical fluctuations. The time urn:x-wiley:23335084:media:ess21284:ess21284-math-0015 is the “round-trip time” required for the acoustic wave to reach a part of the seafloor situated at a distance r from the sonar and then return to the sonar as an echo. The integral is performed as a sum over samples in the time domain. The estimate provided by Equation 3 is averaged over all the data available for a 24-hr period. This rather long period was chosen in order to average over tidal effects, which will be discussed in Section 6. COVIS was programmed to obtain diffuse-flow data either every one-half hour or every hour. Except for occasional data loss, 24 or 48 such estimates are averaged. For each day, averaged ping-to-ping correlation is computed for 38 lags with spacing 0.25 s, for all 128 sonar beams, and for ranges out to 30 m with an interval of 0.56 m between placements of the Hamming window in range. These results, originally in a polar coordinate system, are interpolated onto a 2D rectangular grid with 0.5-m spacing in both dimensions. In this gridding process, the bathymetry as measured by COVIS (Xu et al., 2021) is taken into account in converting the coordinates (azimuthal angle and slant range to seafloor) to x- and y-coordinates. To understand this process, note that slant range and beam angle are not sufficient to determine transverse (x and y) coordinates except in the case of a flat, level seafloor. To take an extreme example, if a portion of the seafloor is a vertical cliff that faces the sonar, x and y will be equal to the slant range, r, multiplied by the sine and cosine of the beam angle. If the seafloor is flat and level, the slant range must be replaced by the “horizontal range” urn:x-wiley:23335084:media:ess21284:ess21284-math-0016, where H is the height of the sonar, before multiplication by the trigonometric functions.

3.3 Structure Function

As a first step toward inversion, a relation between ping-to-ping correlation and the temperature structure function will be sought. Changes in sound speed cause changes in the sonar echo from the seafloor quantified by the ping-to-ping correlation. There is an approximate linear relationship between temperature change and sound-speed (urn:x-wiley:23335084:media:ess21284:ess21284-math-0017) change (Jackson et al., 2017). For this work, urn:x-wiley:23335084:media:ess21284:ess21284-math-0018 ms−1K−1 is used, appropriate for nominal temperature 2.4°C, salinity 34.5 psu, and depth 1,440 m. Salinity fluctuations have a negligible effect, as a salinity change of about 15 psu would be required to match the sound-speed change caused by a (plausible) 5° temperature change. Current fluctuations have no short-term effect on the ping-ping correlation, as the time shift over the outgoing acoustic path is equal and opposite to the time shift over the return path. Thus, the time shift of the echo signal due to this mechanism is zero owing to cancellation over the round-trip acoustic path. Of course, currents do transport and alter the turbulent temperature structure, contributing to ping-to-ping change in the sonar signals.

It is not feasible to obtain a closed-form model for the expected value of the estimator urn:x-wiley:23335084:media:ess21284:ess21284-math-0019, as Equation 3 is an irrational functional of the two time series. In previous work (Jackson et al., 2017), we employed a formal approximation, but the lack of any measure of the bias that might result from this approximation motivates a new approach. In this approach, Monte Carlo simulations (described later in this article) are used to generate synthetic sonar signals using a point-scatterer model. This type of model has been employed extensively, with origins in Faure (1964) and Ol'shevskii (1967). Our application uses the simplest possible form of point-scatterer model, in contrast to recent, for example, more sophisticated versions (Brown et al., 2017).

The acoustic model treats the round-trip path from the sonar to the seafloor and back. This is in contrast to more widely used approaches in which weak scattering theory is, for example, used to model one-way propagation from an acoustic source to a receiver through turbulent water (Di Iorio & Farmer, 1994; Duda & Trivett, 1998). Given the short acoustic path through the near-bottom warm layer, our model simply assumes straight-line propagation and further assumes that changes in propagation are completely characterized by time-of-flight, with no significant amplitude changes. The time-of-flight for a sonar signal to travel to a point on the seafloor and back to the sonar is
urn:x-wiley:23335084:media:ess21284:ess21284-math-0020(5)
for ping 1, with the integral being along the nearly straight path of length urn:x-wiley:23335084:media:ess21284:ess21284-math-0021 shown in Figure 7, which assumes a flat seafloor. As noted earlier, the slant range, r, to a given portion of the seafloor is influenced by bathymetry, but this influence is not relevant here. The point here is that, for a given location on the seafloor having a given slant range, the round-trip travel time will change with variation in sound speed. For a later ping, the position-dependent sound speed will have changed from urn:x-wiley:23335084:media:ess21284:ess21284-math-0022 to urn:x-wiley:23335084:media:ess21284:ess21284-math-0023, and the time-of-flight for ping 2 will be altered to
urn:x-wiley:23335084:media:ess21284:ess21284-math-0024(6)
where
urn:x-wiley:23335084:media:ess21284:ess21284-math-0025(7)
Details are in the caption following the image

Geometry for sonar measurement of a diffuse-flow region. The line-of-sight path length (slant range) r is indicated.

In deriving this expression ray bending is neglected, based on ray-tracing simulations showing that refractive effects are negligible in the circumstances of interest. The parameter urn:x-wiley:23335084:media:ess21284:ess21284-math-0026 is a nominal, position-independent sound speed, and urn:x-wiley:23335084:media:ess21284:ess21284-math-0027 is dependent on position along the path and is assumed to have absolute value much smaller than urn:x-wiley:23335084:media:ess21284:ess21284-math-0028. The model neglects dependence upon coordinates parallel to the seafloor within the sonar resolution cell, thus urn:x-wiley:23335084:media:ess21284:ess21284-math-0029 must be interpreted as an average over these coordinates. This assumption drives the 1-day averaging of sonar data, with the hope stated previously that the average temperature field will approximate, at least over meter horizontal scales, a stratified medium.

Next, consider how the time scaling (6) caused by sound-speed change affects the second echo signal urn:x-wiley:23335084:media:ess21284:ess21284-math-0030 relative to the first signal urn:x-wiley:23335084:media:ess21284:ess21284-math-0031. The sonar signals are in complex or “baseband” format, in which the true, real signals are the real part of the product of the baseband signal with urn:x-wiley:23335084:media:ess21284:ess21284-math-0032, where urn:x-wiley:23335084:media:ess21284:ess21284-math-0033 is the center frequency of the sonar in rad/s. Treating the sonar and seafloor as time-invariant (invariant over the time interval between pings being compared, up to 10 s), one can write
urn:x-wiley:23335084:media:ess21284:ess21284-math-0034(8)
Where the complex function urn:x-wiley:23335084:media:ess21284:ess21284-math-0035 is the “impulse response” of the combined sonar-environment system at the time of the first transmission, urn:x-wiley:23335084:media:ess21284:ess21284-math-0036 is the complex envelope of the transmitted signal. The exponential factor in Equation 8 gives a necessary phase shift, and can be understood by examining the real part of the product of Equation 8 with urn:x-wiley:23335084:media:ess21284:ess21284-math-0037. The impulse response incorporates the complexity of scattering by the seafloor as well as the effects of acoustic transmission and sonar properties. Due to the time scaling shown in Equation 6, the second echo is
urn:x-wiley:23335084:media:ess21284:ess21284-math-0038(9)

For the Monte Carlo calculations, the impulse response is constructed by assuming point scatterers having random, zero-mean, Gaussian-distributed scattering amplitudes and random locations on a flat seafloor. Each scatterer contributes a scaled, delayed copy of the transmitted signal to the final synthesized echo signal. To account for the observed spatial intermittency of the diffuse-flow regions at ASHES, only a fraction of the scatterers at the center of the simulation seafloor patch (referred to later as the “hot spot”) are subject to sound-speed change, which is measured by α defined above. The relevant measure of intermittency will be denoted F, and is the fraction of the 1.5-ms window width over which the sound-speed change is applied. This fraction is adjusted to fit ground-truth thermistor data. The parameter α is changed randomly for each Monte Carlo realization and is drawn from a zero-mean Gaussian distribution. In order that the horizontally averaged, path-averaged sound-speed change have the standard deviation α, the standard deviation of α over the hot spot is assigned the value σα/F.

The Monte Carlo calculations produced synthetic data comprising 400 echo-pair realizations. These were generated with Gaussian-distributed α with 40 different standard deviations at seven different ranges out to 34 m. Thus 7 × 2 × 400 × 40 = 224,000 synthetic signals are generated. The range dependence of correlation was negligible, so correlation was averaged over both range and realizations. The average ping-to-ping correlation was fitted by an algebraic function of the form
urn:x-wiley:23335084:media:ess21284:ess21284-math-0039(10)

The fitted function has urn:x-wiley:23335084:media:ess21284:ess21284-math-0040107, urn:x-wiley:23335084:media:ess21284:ess21284-math-0041 = 0.4535, and urn:x-wiley:23335084:media:ess21284:ess21284-math-0042 = 0.7 and is shown in Figure 8. This curve is for F = 0.14, a value found by fits of sonar inversion of synthetic data to thermistor data as discussed later in this article. The algebraic function is then inverted to give an expression for the standard deviation of α in terms of average correlation. Referring to Equation 10, the decorrelation, urn:x-wiley:23335084:media:ess21284:ess21284-math-0043, is proportional to the square of σα for small σα. As σα grows, urn:x-wiley:23335084:media:ess21284:ess21284-math-0044 approaches a constant, non-zero, asymptote. It is important to realize that the σα on the x axis of Figure 8 represents the spatial average of the normalized, path-averaged sound-speed standard deviation. As such it is smaller than the standard deviation within the hot spot.

Details are in the caption following the image

Average correlation obtained from Monte Carlo simulations as given by Equation 10. The abscissa is the standard deviation of normalized, path-averaged sound-speed change.

The path-averaged sound-speed change is
urn:x-wiley:23335084:media:ess21284:ess21284-math-0045(11)
It follows that the inversion method allows an estimate of path-averaged temperature change:
urn:x-wiley:23335084:media:ess21284:ess21284-math-0046(12)
Given that path-averaged temperature change, rather than path-averaged temperature itself is estimated, a natural statistical characterization is provided by the structure function (1), here expressed as
urn:x-wiley:23335084:media:ess21284:ess21284-math-0047(13)
where urn:x-wiley:23335084:media:ess21284:ess21284-math-0048 is true time, equivalently, clock time. The variable urn:x-wiley:23335084:media:ess21284:ess21284-math-0049 is the time difference between the two pings being compared. The brackets urn:x-wiley:23335084:media:ess21284:ess21284-math-0050 denote a formal average over a hypothetical infinite ensemble but will be implemented as a sample average over 1 day's data. Note that the structure function is simply the second moment of path-average temperature change occurring after an elapsed time urn:x-wiley:23335084:media:ess21284:ess21284-math-0051, the time lag. As temperature change is proportional to sound-speed change, Equation 13 is equivalent to
urn:x-wiley:23335084:media:ess21284:ess21284-math-0052(14)

This is the expression that is used to estimate the structure function for path-averaged temperature. Recapping, the cross correlation between pings separated in time by τ is computed using Equation 3. The cross correlation is averaged over the 24 or 48 separate transmissions for each day, and the expression illustrated in Figure 8 is used to obtain σα. Finally, Equation 14 is used to obtain the temperature structure function over the range of lags 0 < τ < 10 s.

To facilitate inversion for heat flux density, the following three-parameter fit is made to the structure function for path-averaged temperature:
urn:x-wiley:23335084:media:ess21284:ess21284-math-0053(15)
where urn:x-wiley:23335084:media:ess21284:ess21284-math-0054 is the variance of path-averaged temperature fluctuations, urn:x-wiley:23335084:media:ess21284:ess21284-math-0055 is a time scale, and urn:x-wiley:23335084:media:ess21284:ess21284-math-0056 is an exponent giving the power-law behavior at small lags. Plots of this function are given later in this section. As with any structure function for a stationary random process having finite variance, Equation 15 vanishes at zero lag and approaches twice the variance as lag increases. The covariance for path-averaged temperature fluctuations in terms of the fitting parameters is
urn:x-wiley:23335084:media:ess21284:ess21284-math-0057(16)

Figure 9 gives examples of the parameters urn:x-wiley:23335084:media:ess21284:ess21284-math-0058, urn:x-wiley:23335084:media:ess21284:ess21284-math-0059, and urn:x-wiley:23335084:media:ess21284:ess21284-math-0060 obtained by fitting the acoustically estimated structure function for path-averaged temperature including 5 × 5 grid points centered on Sites 1H and 4C. Figure 10 compares the fit of Equation 15 to sonar-derived structure functions for the central 3 × 3 grid points of Figure 9. It is evident from both figures that Site 4C is more active than Site 1H, with one grid point showing large temperature standard deviation. This is most likely due to a small, focused flow. It is also evident that the chosen 25 grid points, centered on the estimated coordinates of the thermistor measurements may not fully contain the active portions of each site. More will be said about navigational error in the following section.

Details are in the caption following the image

Structure function fit parameters for path-averaged temperature structure function estimated from sonar data. Twenty-five adjacent grid points at Sites 1H (upper row) and 4C (lower row) are shown for 7 July 2019. The x axis gives the distance East of Cabled Observatory Vent Imaging Sonar (COVIS) and the y axis gives the distance North of COVIS.

Details are in the caption following the image

Structure function and three-parameter fit for path-averaged temperature structure function estimated from sonar data. Nine adjacent grid points at Sites (a) 1H and (b) 4C are displayed for 7 July 2019. These points are the 3 × 3 points at the center of the plots of Figure 9.

4 Comparison of Thermistor and Sonar Data

The first step in validation of the inversion method is comparison of the sonar-derived structure function with that obtained from thermistor data. For this, sonar data from 7 July 2019, the day after thermistor data were obtained, were used assuming the fractional area occupied by diffuse venting is F = 0.14. This fraction is an average over the warm layer of thickness of order 0.5 m, rather than a value at the seafloor, where the fraction would presumably be smaller. We are unaware of any measurements that would support our value for F and consider it an empirical parameter peculiar to our inversion method. It was not possible to obtain contemporaneous thermistor and sonar data, and the temporal variability of diffuse-flow activity at this site introduces uncertainty in the comparison. We have not attempted to quantify this uncertainty, but the sonar observations reported in (Xu et al., 2021) indicate that most of the variation is due to current variation, with little long-term variation. This prompts temporal averaging of the ping-to-ping correlation as discussed below. Figure 11 shows the comparison for the two sites of interest, with the thermistor results shown as circles and the sonar results shown as solid curves. The sonar curves are obtained from a 24-hr average of ping-to-ping correlation. Navigational error in placing the thermistor measurement sites with respect to COVIS coordinates could be as large as 2 m. To accommodate this, the mean (solid curves) and sample standard deviation (error bars), averaging over the 25 grid points of Figure 9, are plotted. These standard deviations represent the spatial variability of the inversion which, in agreement with visual observation, is large. This variability translates navigational error into uncertainty that compromises comparison of thermistor and sonar data. To obtain the thermistor statistics, Equation 1 is evaluated for each station, and the statistical error in each of these estimates is neglected, as the number of time samples is very large in comparison to the number of stations. The thermistor estimates (red circles) are averages over all stations with error bars that are the sample standard deviations divided by the square root of the number of stations (7 for 1H and 11 for 4C).

Details are in the caption following the image

Comparison of structure function for vertically integrated temperature estimated from sonar and thermistor data, (a) site 1H and (b) site 4C. The black error bars depict the spatial variability of the inversion over 2.5 × 2.5-m2, while the red error bars represent the standard deviations of the estimates of the structure function using thermistor data.

The comparison of sonar and thermistor results is reasonable, given navigational uncertainty and the fact that the thermistor and sonar data are not contemporaneous. There is a mismatch for the shortest lags for both sites. Site 1H had a rather low level of hydrothermal activity compared to site 4C, and it appears that the flattening of the sonar-derived structure function at small lags indicates a “floor” below which good estimates are not possible. This floor is most likely due to random thermal structure in the water column, due to wafting of the focused flows of Mushroom and Inferno. The spatial variability for the sonar inversion at Site 4C is very large, indicating strong spatial intermittency with concomitant sensitivity to navigational error. The mismatch between thermistor and sonar data at small lags is greater than expected due to navigational error and is unexplained. The large downward extent of the error bars representing this uncertainty results from the standard deviation approaching or exceeding the sample mean, and is exaggerated by the use of a logarithmic vertical scale. As noted above, it is assumed that the fractional area occupied by diffuse venting is F = 0.14. As this fraction is increased, the sonar curves move upward, and as it is decreased, they move downward. The estimates of the structure function are rather sensitive to F, with a 7% decrease in F resulting in a decrease in the structure function by about 15%. It is encouraging that a single choice of F gives reasonable agreement between sonar- and thermistor-derived structure functions at two sites having markedly different activity levels. While F must vary to some extent with location at scales larger than the resolution of the inversion, it seems reasonable to assume that the chosen value of F represents a spatial average. This choice of F is the first step in the inversion method. Having a value for F, the relation between ping-to-ping correlation and the structure function is established. With this relation, the sonar data provide structure function estimates. Each estimate, for each point on the 2D grid, is reduced to 3 numbers by fitting the analytic Equation 15. Finally, as described in the following section, two of these three numbers are used in estimating the heat-flux density for each point on the grid.

5 Inversion Algorithm

The heat-flux density, q, in W/m2, is proportional to the product of average vertical velocity, w, and the average of the temperature anomaly Tanom (Rona & Trivett, 1992):
urn:x-wiley:23335084:media:ess21284:ess21284-math-0061(17)
where cvp is the heat capacity per unit volume at constant pressure. This relationship will be applied at depths within less than 1 m of the seafloor for which it can be assumed that currents have not distorted the heat-flux density from its near-seafloor spatial dependence. As noted in Section 2, the mean of the temperature anomaly is approximately equal to the standard deviation of temperature:
urn:x-wiley:23335084:media:ess21284:ess21284-math-0062(18)
This relation is key to the inversion method used here, as the sonar data provide information on fluctuations in the sound speed of the intervening water, but are not responsive to slow changes in sound speed. In contrast to methods that employ acoustic scattering theory (Di Iorio & Farmer, 1994; Duda & Trivett, 1998), a simpler approach as outlined in the previous section is used here with fluctuations in round-trip acoustic travel time (from the sonar to the seafloor and back) determined by the spatial average of sound-speed fluctuations. The sonar inversion gives the standard deviation of path-averaged temperature, which can be expressed as:
urn:x-wiley:23335084:media:ess21284:ess21284-math-0063(19)
where d is the effective thickness of the warm layer, and D is the height of the sonar (4.2 m). The sonar inversion also provides a characteristic time, τ0, that quantifies the time required for the temperature field to undergo significant change. As the path-averaged temperature is dominated by the largest scales of the random temperature field, it seems reasonable to assume that these scales are of order d, and that the vertical velocity is
urn:x-wiley:23335084:media:ess21284:ess21284-math-0064(20)
Combining Equations 17-20, the heat-flux density is
urn:x-wiley:23335084:media:ess21284:ess21284-math-0065(21)

Only two of the three parameters produced by the inversion for the structure function appear in Equation 21. The use of the temperature standard deviation needs no further justification, but the omission of the exponent parameter μ can be justified because τ0 is the lag for which the structure function attains a value equal to 1-1/e of its asymptote, irrespective of the value of μ. For the present work, Equation 21 will be assumed to be an equality, but it is likely that this expression would be modified by multiplication by a constant of order unity as a result of future comparison of sonar-derived heat-flux density with direct measurements.

As evident from inspection of Figure 11, the structure functions for Sites 1H (low activity) and 4C (high activity) are noticeably different. In the three-parameter fit to the 25 structure-function estimates at each site, the temperature standard deviation has a mean that is larger for 4C than 1H by a factor 1.6. While this is not a large factor, is should be recalled that the structure function increases as the square of the standard deviation. Another difference between sites 1H and 4C is apparent in their time constants, with means of 4.9 and 1.15 s, respectively. These numbers can be compared to those obtained from COVIS data at a site on the Main Endeavor Ridge by Jackson et al. (2017), where τ0 varied between 2 and 7 s over a time span of 1 yr. While the diffuse flows of interest in this article are rather complex, they are essentially comprised of miniature buoyant plumes. In laboratory measurements on buoyant jets and plumes by Papanicolaou and List (1987), temporal spectra were obtained for temperature measured at a height 53 times the nozzle diameter. By fitting a model spectrum to the inertial subrange of the two spectra available in Papanicolaou and List (1987) (which are quite similar) and then inverting to obtain a structure function, we obtain a time constant of 0.11 s. This time constant is much smaller than those in our work, highlighting the fact that path averaging of temperature filters out shorter-wavelength, higher-frequency fluctuations. Although the sonar returns employed in our inversions are backscattered echoes, the interaction with intervening water is through forward scattering, for which phase fluctuations are known to be governed by the larger scales of heterogeneity, for example (Ishimaru, 1997, Chapter 17). This could explain the rather long time constants seen in the inversions. A study of the connection between the time constants of temperature and path-averaged temperature could enhance the usefulness of the inversion method.

6 Inversion Results

Figure 12 is an example of the application of Equation 21 to obtain heat-flux density from COVIS data. The two images are formed from a combination of data from the three different azimuthal sectors with ping-to-ping correlation averaged over the period 00:00–23:30 UTC on 7 July 2019 and 11 July 2019. This rather long averaging period is intended to reduce the effect of tidal currents on the heat-flux estimates. In forming Figure 12, data have been deleted at ranges deemed too short or too long to give reliable values for ping-to-ping correlation. Data have also been deleted in regions near the Mushroom (M) and Inferno (I) black smokers. These deletions are necessitated by shadowing of the seafloor, whether by the sulfide structures or the focused plumes, which wander in angle in response to currents.

Details are in the caption following the image

Heat-flux density in MW/m2 as determined using Equation 21, with ping-to-ping correlation averaged over the period 0:00–23:30 UTC (a) 7 July 2019 and (b) 11 July 2019. Data have been deleted in the vicinity of the Mushroom (M) and Inferno (I) smokers. Ground-truth thermistor data were taken at Sites 1H and 4C, noted on the plots.

The two images show appreciable differences, with more activity evident on 7 July than 11 July. These could be artifacts caused by either the Inferno or Mushroom plumes, as interposition of a plume between the sonar and the seafloor might cause inactive regions to appear active, as noted later. The ground-truth sites are marked in Figure 12, and it is apparent that activity at 1H is much lower than at 4C, consistent with thermistor measurements and Figure 11.

A time series of heat output has been formed by integrating the heat-flux density (obtained from daily averages of the ping-to-ping correlation) over area and dividing the result by area to obtain average heat-flux density. The yaw rotator motor failed on 24 November 2019 with the result that data gathering at later times was restricted to Sector 1 which includes the Mushroom and Inferno vents. As a consequence, the time series presented in Figure 13a is restricted to Sector 1 with the deletions noted in connection with Figure 12. In the time before failure of the yaw motor, the sonar azimuthal pointing direction showed occasional changes which resulted in abrupt changes in estimated heat flux. In response, the time series displayed in Figure 13a is further restricted to times after the yaw motor failure, ensuring that the pointing direction was constant. The time series shows no evident trends over the period 24 November 2019 through 27 April 2021.

Details are in the caption following the image

(a) Time series average heat-flux density over Sector 1, (b) Time series for fractional area.

The cause of the time variation seen in Figure 13a is unknown. It may be either due to actual variation in heat output, or it may be an artifact caused by tidal currents. Our efforts to compare the heat-flux density time series with pressure and current time series have been inconclusive, but more effort in this direction is warranted. Sample averages over the heat-flux density shown in the figure yield 261 ± 40 kW m−2. If it is assumed that the variation seen in Figure 13a does not represent actual variation in heat output, then the standard deviation 40 kW m−2 can be taken as the uncertainty in heat-flux estimates from one day's worth of data.

Figure 13b is a time series for “fractional area,” which is defined as follows. First the “active area” is defined as
urn:x-wiley:23335084:media:ess21284:ess21284-math-0066(22)
This definition gives a value that is equal to the total area included in the heat-flux image
urn:x-wiley:23335084:media:ess21284:ess21284-math-0067(23)
if the heat-flux density is constant over the entire area. If the heat flux vanishes everywhere except on patches that occupy an area Ap, where it takes on a single value for all patches, definition (22) yields the intuitive value Aactive = Ap. The fractional area plotted in Figure 13b is Aactive/A. It typically ranges between 0.7 and 0.8, showing that the heat flux density obtained by inversion is not concentrated in a few small patches, but spread significantly. This conflicts with video observations that the diffuse-flow patches are quite localized. We attribute this, at least in part, to the effect of currents and the use of averaging over a 24-hr period for each data point in Figure 13. There is no conflict with the assumed fractional active area of 0.14 assumed in Section 3.3, as this fraction refers to the fine-scale distribution of activity, unresolvable by the inversion method. Figure 14 shows that there is a definite correlation between fractional area and heat-flux density, with fractional area increasing as heat-flux density increases. This may indicate that a part of the inverted heat flux is an artifact, caused by wafting of the focused plumes into the sonar's lines-of-sight, or it may simply indicate that increased activity involves an increased area of the seafloor. Wafting of the focused plumes would cause inactive areas to appear active, increasing fractional area. At present we have no means of correcting for the bias that would result from such an artifact if, in fact, it occurs.
Details are in the caption following the image

Average heat-flux density over Sector 1 versus fractional active area.

Rona and Trivett (1992) made heat-flux measurements at ASHES, and their nearest diffuse-flow region had area 962 m2 and estimated heat flux density 57 kW/m2. Our value averaged over COVIS' 567 m2 field of view (combining all three sectors) is 171 kW/m2 for 7 July 2019. The mean of the time series of Figure 12 is 261 kW/m2 for Sector 1, with area 233 m2 after deleting regions near Mushroom and Inferno. The statistical error in this number is only 2 kW/m2 owing to the large number (443) of samples. The systematic error is unknown but expected to be a number of order unity times this mean. Given the 1:5 ratio of the uncertainty bounds given by Rona and Trivett, their estimate does not help with calibration of our method, although the two measurements are consistent.

7 Conclusions

A method has been developed to estimate the heat output of diffuse hydrothermal flows using multibeam sonar data. Temperature fluctuations modulate seafloor echoes, so that ping-to-ping correlation can be used to estimate properties of the random temperature field from which heat-flux density can be obtained. The method is essentially empirical, and calibration by comparison with directly measured heat flux in a future field exercise would be very useful. Even without calibration, the method provides estimates of heat-flux density up to an unknown factor of order unity. These estimates are consistent with historical data and are given as maps over the sonar field-of-view, with one such map per day. A time series over the period 24 November 2019 through 27 April 2021 shows no significant trends in heat flux.

Acknowledgments

The authors acknowledge the support of the National Science Foundation. Engineering and field support were provided by members of the APL-UW Ocean Engineering Department, and we duly thank the OOI Regional Cabled Array team at the University of Washington and Rutgers University, as well as the ROV Jason team, and the officers and crew of R/V Roger Revelle and R/V Thomas G. Thompson.

    Data Availability Statement

    The COVIS sonar data and the thermistor data presented in this paper are available at http://piweb.ooirsn.uw.edu/covis/. The 3D single-point velocity meter data referred to in Sect. Two of this article are accessible through the OOI data portal (https://ooinet.oceanobservatories.org/).