The sine qua non of any inverse problem is an accurate treatment of the forward problem: to construct an acoustic arrival pattern given “the ocean.” If the forward problem cannot be solved, then the measured data cannot be inverted to reconstruct the ocean.

When we entered this field in 1978, it was under the impression that the forward problem had been solved and that our efforts could be directed toward the oceanographically interesting inverse problem. As it turned out, the forward problem had not been solved; oceanographic and acoustic fields had never been simultaneously measured with accuracy adequate for a critical test. When such measurements were finally made in the late 1980s as part of tomographic work, significant discrepancies were found between the measured and computed sound fields. The most dramatic discrepancy was traced to the equations of state for sea water; some of the commonly used sound-speed equations were shown to be in error.

That era of trouble with the forward problem came to an end with the so-called SLICE89 experiment. The forward problem is now sufficiently well understood to permit oceanographically useful applications of inverse methods. Some discrepancies remain; they appear to be associated with internal waves and other range-dependent phenomena (see chapter 4).

There is a hierarchy of techniques available to predict the propagation of acoustic pulses in the ocean, given the sound-speed field. The geometric-optics approximation (i.e., ray theory) is adequate to predict travel times in the majority of situations encountered in ocean acoustic tomography.

The fundamental experimental requirement of ocean acoustic tomography is to make precise measurements of acoustic propagation in the ocean. Any of a variety of characteristics of acoustic propagation potentially can provide useful information about the ocean through which the sound has traveled, provided that the forward problem is thoroughly understood. The task of the measurement system is to provide estimates of whatever parameters are desired, with useful precision, together with estimates of the errors of the measurements.

In this chapter we focus on the use of broadband acoustic signals to measure the impulse response of the ocean with sufficient resolution (in time and/or vertical angle) to separate individual ray arrivals. The precision with which travel times and other parameters of the individual ray arrivals can be estimated is limited by the ambient acoustic noise in the ocean. In addition, and more important in most cases, small-scale ocean variability causes fluctuations in ray amplitudes, travel times, phases, and arrival angles. Travel time is the most robust observable and is the one we emphasize in this book. The inversion of travel-time data to obtain information on the ocean sound-speed and current fields was outlined in chapter 1. Because the expected magnitude of the travel-time perturbations is O(100 ms), travel times need to be measured with a precision of a few milliseconds, corresponding to a few parts per million over 1 Mm range.

The problem of ocean acoustic tomography is to infer from precise measurements of travel time, or of other properties of acoustic propagation, the state of the ocean traversed by the sound field. The tomographic method was introduced by Munk and Wunsch (1979) in direct response to the demonstration in the 1970s that about 99% of the kinetic energy of the ocean circulation is associated with features that are only about 100 km in diameter, called the mesoscale. Measuring and understanding the behaviors of both the mesoscale and the larger-scale features associated with the general circulation present a formidable sampling task. Not only are the flow elements very compact spatially, but also they have long time scales (order 100 days). To produce statistically significant measurements of the fluid behavior, even in an area as compact as 1 Mm × 1 Mm (1 megameter = 1000 km), about 1% of an ocean basin, requires several full-time vessels or several hundred fixed moorings. One is accordingly led to the technology of sound propagation to measure the properties of the fluid between moorings.

Ocean acoustic tomography takes advantage of the facts that (i) travel time and other measurable acoustic parameters are functions of temperature, water velocity, and other parameters of oceanographic interest and can be interpreted to provide information about the intervening ocean using inverse methods, and (ii) the ocean is nearly transparent to low-frequency sound, so that signals can be transmitted over distances of many thousands of kilometers.

Over drinks in the Cosmos Club in 1979, Athelstan Spilhaus, who had perfected the bathythermograph for measuring temperature profiles to predict the ranges at which submarines could be detected acoustically, held forth that it should be done the other way around: the measured sonar transmission should serve to determine the ocean temperature field. Unknown to Spilhaus, we were in Washington to persuade the Office of Naval Research and the National Science Foundation to fund an experiment to do just that.

In seismology, the inversion of travel times to map the interior of the Earth has been the time-honored procedure, since the Earth is not readily accessible to direct intrusive measurements. In medicine, intrusive methods are viewed with some reluctance (at least on the part of the patient), and this has led to the development of computed tomographic inverse methods using X-rays. In contrast, the oceans are accessible to direct intrusive measurements; the limits are set by the availability of costly platforms for adequate sampling. Unlike the seismological and medical applications, ocean time variability is an essential component, and the requirements for sampling in space and time are severe. With only a few research vessels plying the world's oceans, it is not surprising that the first century of oceanography had a strong climatological flavor.

It came as a great shock in the 1960s that the oceans, like the atmosphere, had an active weather at all depths. The storms within the sea are called eddies.

Climatological sound-speed profiles and predicted acoustic arrival patterns for selected locations worldwide (fig. B.1) are displayed in this appendix. The locations are from a regular grid spaced at 15° increments in latitude and 20° increments in longitude. [Worcester and Ma (in press) provide results for all grid locations exceeding 2000 m depth.] The atlas is organized by location, beginning at 75°N and proceeding southward. At each latitude, results are presented in order of longitude, proceeding westward from the prime meridian. The interpretation of the plots presented here is discussed at length in section 2.16. Each panel is described briefly, proceeding counterclockwise from the bottom left.

The sound-speed profiles (bottom left) were computed from annual-average climatological temperature and salinity data due to Levitus (1982), using the Del Grosso (1974) sound-speed equation. The Levitus climatology is a horizontally smoothed picture of the ocean, so the results do not properly represent the behavior to be expected in frontal regions.

Acoustic normal mode functions 1 and 7 computed for 70 Hz are displayed at bottom center. The amplitude normalization is arbitrary. The group velocity for each mode is given immediately below the mode function.

Time fronts in τ, z-space for a fixed range of 500 km (bottom right) show the arrival structure for a source on the sound-channel axis, when one exceeding 100 m depth exists, or for a source at 100 m depth.

The discussion so far has dealt with a range-independent (RI) sound channel. But the ocean certainly varies horizontally and is always range-dependent (RD); one of the chief goals of tomography is to derive its range-averaged (RA) properties.

The RD treatment will vary depending on whether the scale of the horizontal variations is larger, comparable to, or very much smaller than the ray-loop range (typically 50 km). The term “adiabatic range dependence” is defined to apply to the case of small fractional variation over a ray loop. Variations on a gyre scale can accordingly be treated by the adiabatic approximation, assuming there are no sharp frontal surfaces.

The term “loop resonance” applies to ray travel-time perturbations due to ocean perturbations with horizontal scales equal to the ray-loop scale, or to a fraction of the loop scale. This includes mesoscale activity (which accidentally has a scale comparable to the loop scale) and ranges down to the longer components in the internal wave spectrum. Cornuelle and Howe (1987) have shown that measured travel-time perturbations associated with loop resonance can provide some RD information for even a single source-receiver transmission path.

Internal waves are generally included among the small-scale processes for which the forward problem yields estimates of the variance and other statistical properties of the travel time (Flatté et al., 1979). These estimates are required for inversion of the measured data set. In turn, the measured variances can provide useful information about the small-scale structure (Flatté and Stoughton, 1986).

Introduction

The preceding chapters have demonstrated that a variety of measurable acoustic features, including ray travel time, amplitude, and inclination, mode group velocity, and carrier phase, are integral functions of the oceanic sound-speed field. As discussed in previous chapters, sound-speed is intimately related to the oceanic density field, which is, in turn, a dynamic variable related to the oceanic flow field. Under many circumstances, knowledge of the density field alone is adequate to compute the oceanic flow field to a high degree of approximation. Reciprocal tomographic measurements are direct weighted averages of the flow field in the plane of the source and receiver. Thus, determinations of C and u carry immediate implications for the ocean circulation and must be consistent with known physics.

The forward problem has been presented in detail: Given C (or S) and u, and the characteristics of a sound source, compute the detailed structure of the signal as recorded at a receiver of known characteristics. This problem is labeled “forward” mainly as a reflection of its connection to the classic problem of finding solutions to the wave equation.

The “inverse” problem demands calculation of the ocean properties, C and/or u, given the measured properties of the arriving signal. At this stage, the problem becomes a matter of intense oceanographic interest.

Oceanographers are mostly familiar with point value data (e.g., a current meter reading or a thermometer measurement). In contrast, tomographic data are weighted integrals through the oceanic field.

Let us examine the budget of mass, energy, angular momentum and carbon. The presence of the latter in this list is explained by two reasons. First, the carbon budget is interesting in itself. Second, it is closely related to the atmospheric CO2 problem, without solving which it is impossible to estimate the role of anthropogenic factors in climate changes.

In summarizing the factual data we will follow Peixóto and Oort (1984) for the most part.

Initial information

There are at present about 2000 weather stations for atmospheric sounding where horizontal wind velocity, temperature, humidity and isobaric surface height for 11 levels in the atmosphere are measured twice a day. The amount of information from these stations is enormous. Even if we exclude all monthly series with data gaps exceeding ten days, then during the decade covering May 1963 to April 1973 we obtained 10 × 365 × 2 × 100 × 11 × 5 = 4 × 10 information units. Here the first number specifies the number of years, the second specifies the number of days in a year, the third specifies the number of daily series of measurements, the fourth specifies the number of stations providing qualitative information, the fifth specifies the number of measurement levels in the atmosphere and the sixth specifies the number of measured parameters. The amount of information is thus sufficient to detect large-scale peculiarities in the atmospheric climate.