1. Introduction

An inverted echo sounder (IES) is an ocean-bottom instrument that measures the time for a 10-kHz sound pulse to travel round-trip to the ocean surface and back (Watts and Rossby 1977; Chaplin and Watts 1984). In use since the mid-1970s, these instruments provide up to two-year-long hourly time series of acoustic travel time τ. The τ measurements can be used to estimate the depth of isothermal surfaces in the main thermocline, the geopotential height anomaly between two pressure levels, or other dynamic and descriptive quantities (Rossby 1969; Watts and Johns 1982; He et al. 1997). Historical hydrography can be used to determine the empirical relationship between these quantities for a given region. Coincident measurements from a conductivity–temperature–depth (CTD) probe or an expendable bathythermograph (XBT) have been used to calibrate these relationships from τ (usually just determining an additive constant); effectively, this calibration is required to determine the precise depth of each instrument.

Although some IESs in earlier experiments have been equipped with pressure gauges (PIES), the τ and bottom pressure P records were calibrated and used independently. In particular, Watts and Kontoyiannis (1990) used the pressure measurements to test the drift and accuracy of the pressure sensors, and Shay et al. (1995) and Watts et al. (1995) used the pressure measurements to study deep geostrophic flows. This paper presents a new method of using P measurements to calibrate the measured acoustic travel times τ. The pressure measurement provides the instrument depth with improved accuracy. The new calibration, therefore, provides improved accuracy for the dynamic variables that can be estimated from the τ measurement, as will be shown.

2. Simulating an IES using historical hydrographic data

Examples of the functional relationships between acoustic travel time and various standard dynamic variables may be found in Watts and Rossby (1977) (Sargasso Sea), Watts and Johns (1982) (Gulf Stream), Chiswell et al. (1986) (eastern equatorial Pacific), Hallock (1987) (Gulf Stream and Sargasso Sea), Trivers and Wimbush (1994) (North Atlantic), James and Wimbush (1995) (North Pacific and Kuroshio Current), Garzoli and Bianchi (1987) (Malvinas and Brazil Currents), Garzoli and Gordon (1996) (Benguela Current), and Chiswell (1994) (Hawaii).

These papers have used a number of different methods to represent the vertical integral of acoustic travel time. The common goal among them is to simulate (from historical hydrographic data) an IES that measures temporal variations while moored at a fixed (x, y, z) point [where z represents absolute height, not depth measured below the sea surface, whose height itself varies with (x, y, t)]. Common to all of these simulations is the assumption that temporal variations at one site due to mesoscale eddy variability may be simulated from the combined (x, y, t) variations among a set of hydrographic profiles for the region. The need then is, on the one hand, to select from a space–time region data that are limited enough to exclude variability that would occur only “away” from the desired (x, y) site and, on the other hand, to select enough data to include and represent the full range of variability that can occur at the site (Hallock 1987; Trivers and Wimbush 1990; James and Wimbush 1995).

The choice of integration variable (z or P), and in particular the specification of integration limits, also differs among the aforementioned authors. This paper will not review the different approaches. The common goal is to represent the round-trip vertical integral of acoustic travel time to a fixed height z in the ocean. This is represented mathematically as

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where ρ, g, c, and P are the density, gravity, sound speed, and pressure, respectively. The choice to integrate between constant pressure limits is motivated by the following argument. First, bear in mind that the ocean surface height η may change with steric height and with atmospheric pressure change and ocean barotropic pressure change, but hydrography can determine only the steric height changes. An alternative choice for the τ integral would be to integrate the hydrographic profiles with height z between the surface η and a fixed distance below the sea surface. However, because of these substantial variations in η, this is not as good a representation of a fixed height. A typical range of variation in the absolute height is approximately 1 m. On the other hand, it has been observed in the deep ocean that the bottom pressures and the surface atmospheric pressures vary independently, each by about 0.3 db (Qian and Watts 1992), which corresponds to approximately 0.3 m of hydrostatic height changes. Hence, the τ representation in Eq. (1) is preferable because it more nearly represents a constant absolute height than an integration between z limits.

Another subtle detail is important in the practical application of Eq. (1) to a regional dataset. The gravitational acceleration g depends upon latitude. Hence, for the same profile of salinity and temperature occurring at different latitudes, the (P, z) relation would differ (by more than 1 db per 5° of latitude at 3500 db). The approach taken here is to select a target latitude, ϕ_t, and stretch or shrink the P axis from that that it has at the observed latitude, ϕ_o, to match that of the target latitude. This is done by using the algorithm of Fofonoff and Millard (1983) to convert the pressures from the hydrographic cast into depths, where g = g_o(ϕ_o, z) is calculated using the observed latitude ϕ_o. The algorithm of Fofonoff and Millard is then inverted, and the depths are converted back into pressures, P_t. For this second calculation, however, g = g_t(ϕ_t, z) is determined using the target latitude ϕ_t. This procedure is repeated for all of the hydrographic casts to be used from the region. In this manner, the latitudinal dependence of the pressure range of integration is removed, leaving the integration range as close to constant as possible, and the simulation of the IES is at a fixed latitude. For example, if P_sim = 2000 db, we denote

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with ρ′ and c′ designating profiles as a function of P_t. (Note that deeper bottom pressures than 2000 db are simulated later; this calibration method is applicable to the full depth of any IES.)¹

For the purpose of providing an example to compare the new calibration method using the measurement of bottom pressure to the traditional method for calibrating an IES, this paper will focus on the relationship between the geopotential height anomaly, integrated between 100 and 4000 db and τ₂₀₀₀. The geopotential height anomaly is determined from the same hydrography via

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where δ represents the specific volume anomaly as a function of P = P_t. Figure 1 shows ΔΦ¹⁰⁰₄₀₀₀ plotted against τ₂₀₀₀, calculated from Eqs. (2) and (1) using about 130 hydrographic casts from the Newfoundland Basin near 42°N. The relationship between these quantities could be represented by a polynomial; it is adequate for our purpose of demonstrating the improved accuracy of the pressure method of calibration to approximate this relationship as linear (shown in Fig. 1 as a solid line):

ΔΦ¹⁰⁰₄₀₀₀Aτ₂₀₀₀B,(3)

where A ≈ −270 m² s⁻³ and B ≈ 737 m² s⁻². If an IES in the Newfoundland Basin region was moored at a depth of precisely 2000 db, the slope and intercept values derived from this simulation could be directly applied to the time series of τ₂₀₀₀ measurements to obtain a time series of ΔΦ¹⁰⁰₄₀₀₀. Unfortunately, the depth of the IES in steep topography might not be known by bathymetric survey to an accuracy of better than ±30 m, which would result in a travel-time bias of 40+ ms. Since the entire signal of a major current like the North Atlantic Current or the Gulf Stream is 40–50 ms, such a bias must be avoided by using an alternative calibration method, such as the following.

3. Calibration of the IES

The traditional method of calibrating the IES assumes that the slope A in Eq. (3) is not dependent on the pressure at which τ is simulated, so long as P_sim is far below the main pycnocline. Rather, the depth dependence of τ is solely absorbed by changing the intercept B to B′. Under this assumption, all that is required to determine ΔΦ¹⁰⁰₄₀₀₀ is to determine the appropriate B′ for each IES site.

The final calibrated values of ΔΦ¹⁰⁰₄₀₀₀ using this method are subject to two kinds of errors: biases due to errors in the determination of B′ and random scatter. The two sources of random scatter are 1) the errors in the τ_meas record after 40-h low-pass filtering [(rms scatter of hourly measurements)/degrees of freedom = 1 ms/40 = 0.15 ms that corresponds to ε_a = 0.045 m² s⁻²], which propagate through Eq. (3) to be errors in ΔΦ¹⁰⁰₄₀₀₀; and 2) the scatter of the fit of Eq. (3) (see Fig. 1, ε_b = 0.42 m² s⁻²). Errors in the determination of B′ result in biases for the time series of ΔΦ¹⁰⁰₄₀₀₀. There are three sources of error in the determination of B′: 1) the error in the hourly τ_meas [1 ms (Chaplin and Watts 1984) that corresponds to ε_c = 0.27 m² s⁻²] at the time of the coincident CTD, which affects the determination of B′ via Eq. (4); 2) the scatter introduced because τ_meas in Eq. (4) represents an integration over the full water column, while the integration of ΔΦ¹⁰⁰₄₀₀₀ from the coincident CTD measurements is affected only by variability in the range of integration between 100 and 4000 db (ε_d = 0.14 m² s⁻² based on the observed variability in density structure below 4000 db for this region); and 3) due to the spatial offset between the CTD and the IES sites (0.27 m² s⁻² per kilometer distance between the CTD and IES sites based on the maximum observed change in geopotential height anomaly across the front). The spatial offset error introduces a random scatter due to the deflections of isotherms by internal waves during the several hours involved in the CTD measurement as well as a bias due to the ambient horizontal gradient of the main thermocline depth. Note that when multiple CTDs are taken at a site, ε_c, ε_d, and ε_e are reduced by a factor of N, where N is the number of CTD casts. An estimate of the total rms error in this traditional method, denoted ε_tm, is given by

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when the concurrent CTDs were taken 2.5 km from the IES site, and two CTDs were taken at the site during the record.

4. Calibrating a PIES

The inclusion of pressure sensors on the PIES provides an alternative calibration method. ParoScientific, Inc., the manufacturer of the pressure sensors used in the PIES, states in its technical brochures an absolute pressure accuracy of 0.01% of full scale, or 0.5 db, in up to 5000-m depth. The accuracy of the pressure sensors has also been tested by predicting the bottom pressure of the PIES using the measured travel times and a sound-speed profile based on coincident full-water-column CTDs. For 11 PIES in three different experiments the resulting mean offset between the predicted pressure and the measured pressure was about 1 db, about half of which may be attributable to errors in the sound-speed algorithm rather than the pressure sensor (Meinen and Watts 1997). Thus, the pressure sensor provides the accurate depth information required to calibrate an IES τ record. Calibration for an individual PIES using this new method consists only of using the historical hydrography to simulate IESs, where P_sim = P_ies in Eq. (1), and then the A and B in Eq. (3) will apply directly to the measured travel times (rather than a 2000-db simulation). Also note that this calibration requires that we account for three well-known constant offsets intrinsic to the measurements used in this study. A discussion of these offsets is given in appendix A.

For a deployment involving multiple PIES, the method described above has the weakness that it would require determining a set of coefficients A and B for each individual PIES since no two instruments would be at exactly the same pressure. If these PIES were being calibrated into not only ΔΦ¹⁰⁰₄₀₀₀ but also heat content and potential energy anomaly, for example, this would be cumbersome. It might also lead to inconsistencies between different IESs due to small errors in fitting the coefficients, especially for variables for which it is necessary to fit a nonlinear function.

A better approach is for the travel-time measurements from different sites to be projected onto a common (deep) pressure level, P_com, using τ_com = A × τ_meas + B (note script A, B differ from A, B). This approach requires only a single set of coefficients for other dynamic variables to be estimated from the τ_com measurements. Using historical hydrography to simulate IESs at various pressures, it can be shown that τ_sim1 at any pressure level, P₁, which is significantly below the thermocline, is linearly related to τ_sim2 at any other deep pressure level, P₂. Figure 2 shows a number of examples. The slopes are very nearly but not exactly 1. By studying a number of these relations it has been determined that the slope and intercept of the linear relationships between τ_sim at different pressures are simple functions of pressure themselves, A(P) and B(P). Appendix B presents the details of the conversion of the measured τ_meas at P_ies into τ_com on a common pressure level, P_com, which we take to be 2000 db in this study. This allows the use of a single set of A and B coefficients in Eq. (3) derived from a simulation of IESs at P_com.

5. Comparison of the two calibration methods

From August 1993 until June 1995, four PIES were deployed in a line across the North Atlantic Current near 42°N (Tracey et al. 1996). During the period of deployment, one to three full-water-column CTDs were taken at each PIES site. Each CTD was used to determine a B′ intercept [via Eq. (4)] for determining ΔΦ¹⁰⁰₄₀₀₀ from τ_meas, and the τ_meas records were calibrated in the traditional manner described in section 3. These same τ_meas records were also calibrated using the pressure method described above, and the results of the two calibration methods were compared.

Figure 3 shows the mean of the 22-month time series of ΔΦ¹⁰⁰₄₀₀₀ from each of the four PIES calibrated using the pressure method (circles) and using the traditional method (crosses). The one standard deviation errors are shown for each method. Note that these standard deviations represent only the sources of error that are independent between the two methods. Both of the methods for obtaining calibrated ΔΦ¹⁰⁰₄₀₀₀ are impacted in the same manner by the error due to the individual measurements of the τ_meas time series and by the error due to the scatter in Eq. (3). Thus, these two error sources are not included in the error bars shown in Fig. 3. The remaining sources of error are all biases, so the error bars would be the same for an individual day as they are for the 22-month averages shown in Fig. 3.

The error bars indicate that the ΔΦ¹⁰⁰₄₀₀₀ time series calibrated using the two different methods are not statistically different. The pressure-calibrated travel times agree with those calibrated by the traditional method to within about 0.47 m² s⁻², which is about 4% of the total North Atlantic Current signal in this region.

6. Summary

The addition of a pressure sensor to the IES has permitted the development of a new method of calibrating a time series of measured acoustic travel times into other dynamic variables without the need for coincident XBTs or CTDs at the IES site during the deployment. Instead, the method relies on the combination of historical hydrography from the region and the measurement of the pressure sensor.

One advantage of calibrating the PIES using the pressure method is that there is a smaller error inherent in the calibration. During the North Atlantic Current experiment the pressure method had a standard deviation in ΔΦ¹⁰⁰₄₀₀₀ of 0.52 m² s⁻², while the traditional method had a standard deviation of 0.65 m² s⁻². The majority (0.42 m² s⁻²) of these errors are due to the rms scatter of the linear relationship between τ and ΔΦ¹⁰⁰₄₀₀₀, which is smaller by as much as a factor of 2 in other ocean regions with tighter temperature–salinity relationships (such as the Gulf Stream). The scatter about the fitted curve could be reduced by using a curved functional relationship between ΔΦ¹⁰⁰₄₀₀₀ and τ; however, the improvement would apply to both calibration methods equally. Hence, for the simple purpose of demonstrating the improved accuracy of the pressure method of calibration, a linear fit is adequate.

A second advantage to the pressure method is the elimination of the need for coincident CTDs or XBTs, which reduces the cost and logistical efforts during deployment and recovery. Of course, the bottom pressure record is valuable in its own right to measure the barotropic pressure field (particularly if leveled by combination with deep current measurements), as shown in Shay et al. (1995), Howden (1996), and Lindstrom et al. (1997).