Adjoint formulation of the radiative transfer equation

Problems in radiative transfer theory can be solved by various solution methods. In the previous chapter we have discussed a number of these which we will classify as the forward or the regular methods. In addition to applying the forward solutions, it is also possible to use the so-called adjoint solution techniques which offer the decisive advantage that for certain types of transfer problems the numerical effort can be drastically reduced.

In this section we will formulate the adjoint technique. It will be necessary to introduce a new terminology involving expressions such as the radiative effect and the atmospheric response due to the presence of energy sources. By means of an important but simple example we will demonstrate the numerical advantage that the adjoint technique offers in comparison to the forward formulation. In Section 5.2 we will introduce the perturbation technique and show how to apply it to the forward as well as to the adjoint formulation.

The adjoint method originated as a purely mathematical tool for the solution of linear operator equations. Discussions on this subject can be found in textbooks on principles of applied mathematics such as Courant and Hilbert (1953), Friedman (1956) and Keener (1988).

Introduction

In this chapter we will deal with the application of radiative transfer theory to atmospheric remote sensing. Remote sensing means that measurements are performed at a large distance from the physical object or medium under consideration with the purpose of retrieving its physical properties. In many cases the carrier of the physical information is electromagnetic waves. Nevertheless it is possible to observe the atmosphere, the soil, or the ocean by means of sound waves. Sodar (sound detection and ranging) and sonar (sound navigation and ranging) are techniques that employ acoustic waves.

Two basic methods known as active and passive remote sensing can be applied. ‘Active’ means that the source of the waves is man-made; for example, a laser transmitter can be used to emit light pulses which propagate through the medium under consideration. The laser light is scattered by air molecules, or it is scattered and partly absorbed by aerosol and cloud particles. The scattered laser light is then collected by a detector telescope. The amount and the amplitude of the detected pulses can then be used as a measure for transmission losses. This particular technique is called lidar (light detection and ranging). Another widely employed active technique is radar (radiation detection and ranging) where antennas emitting microwave radiation are being used.

Specification of the primitive equation model

The primitive equations, commonly understood as the Navier–Stokes equations in a rotating reference frame, with the vertical momentum equation replaced by the hydrostatic relation, form the basis of the most commonly used models of the ocean on regional to global scales. The flow is generally presumed to be incompressible but stratified; density variations are neglected save in the buoyancy terms. This is known as a shallow Boussinesq approximation. It is much more suitable for the oceans than for the atmosphere.

As in a truly incompressible fluid, there are no exchanges of energy between the flow and the internal energy of any fluid parcel, and a cubic meter of bottom water traveling at 1ms–1 is assumed to have the same momentum as a cubic meter of surface water traveling at the same speed. This has the effect of filtering sound waves, which would otherwise render practical large-scale models impractical.

It is sometimes useful to discard the hydrostatic approximation. This is necessary for the study of the details of deep convection in high latitudes and in some coastal ocean modeling applications. Marshall and co-workers (Marshall et al., 1997a, b) have suggested that nonhydrostatic models may be practical for simulation of the ocean on large scales, given the speed of modern computers. This would have the advantage of a single model for the global circulation, including formation of deep water.

The primitive equations by their nature present a number of obstacles to practical computation. For the most part, practical models differ in the fashion in which they cope with these problems.

Introduction

In recent years interest in the coastal ocean has increased throughout the world scientific community. Knowledge of the coastal oceans is directly relevant to issues of resource management and security, among others, and is therefore of broad societal interest, since a large proportion of the world's population lives near coastlines. In a purely scientific context, new instruments such as surface velocity mapping radars have been developed, and advances in computers and computing techniques have made detailed models of the coastal ocean practical.

The essential physical mechanisms that determine the most interesting aspects of coastal flow differ from season to season and from location to location. Coastal flows are affected by tides. Nonlinear effects can include rectification, so periodic tidal forcing of the coastal ocean can lead to significant residual steady flows. Interaction of the barotropic tide with topography can result in significant baroclinic motion. Interaction with motions on longer timescales can be nontrivial, and simply averaging over tidal periods or implementing other strategies for filtering out the relatively high frequency tidal motions may not be sufficient to deal with tidal interactions. River outflow, with its associated buoyancy fluxes, can be important.

Coastal upwelling, with its ecological implications, is an important feature of coastal circulation in many areas, as are the coastal jet and the ubiquitous coastally trapped waves, which propagate with the coast on their right as you face in the direction of propagation in the northern hemisphere (see Exercises 6.2 and 6.3).

Surface and bottom boundary layers are of the order of tens of meters thick, and often contain many of the phenomena of interest, beyond simple vertical transfer of momentum and heat.

This text grew out of notes for a course taught to graduate students in physical oceanography at the College of Oceanic and Atmospheric Sciences at Oregon State University. The students are typically in their second year of graduate school, having passed introductory courses in theoretical physical oceanography. This is the background assumed for the course. Most of the students at this point have seen some numerical analysis.

The course, and hence this text, is intended for all students of physical oceanography. Major emphasis is on those features that distinguish models of the ocean from other models in computational fluid mechanics. The intent is to examine ocean models critically, and determine what they do well and what they do poorly. We will ask when we can be confident that the model reflects nature, and when we can say that it is likely that we are looking at a feature of the model itself.

This is not a mathematics text as such, but it has a high mathematical content. Numerical analysis is introduced as needed. The reader may wish to consult supplementary references for basic numerical analysis of partial differential equations such as Sod (1985) (many typos, but reasonably current on fundamentals) or Richtmyer and Morton (1967), which is useful and commonly cited, though outdated. The reader might also find Isaacson and Keller (1966) or Allen et al. (1988) useful as general references. We will take examples from the two-volume work by Fletcher (1991) and the monograph by Leveque (1992). Durran (1999) contains useful and closely related material, from a slightly different viewpoint.

Numerical modeling has become an essential component of most research in physical oceanography. Once the domain of specialists, interpretation of model results has become part of the routine scientific practice. The highly schematic models of the 1970s and 1980s with their highly idealized geometry, coarse resolution and crudely parameterized small-scale processes could give only rough qualitative insights into applicability of theory. With the computing power available to nearly every scientist today, numerical models are expected to compare in detail with observation.

Field experiments are now designed with the intention of providing models with initial conditions, boundary conditions and verification data. Non-specialists, who, even five years ago, would not have given much thought to the implications of modeling results, now routinely ask themselves, “What do the models tell us?” Increasingly, this is the way oceanography is done.

Ocean modeling is closely related in method and spirit to atmospheric modeling, but atmospheric modeling was developed earlier, driven by the need for operational weather prediction. The basic methodology for weather forecasting was first set out by Richardson in a remarkable book (Richardson, 1965), first published in 1922. In that book, all of the steps for constructing a numerical weather forecast were set out in detail. A sample calculation was performed for two points in Europe, but the forecast turned out to be markedly different from the state of the atmosphere observed at the forecast time.

Introduction

Some of the most notable early successes in ocean modeling came in the 1980s in models of the tropical oceans, with most being applied to the tropical Pacific Ocean due to its role in interannual climate variability. The success of these models, many of which involved coarsely resolved simple linear dynamics, was due to a variety of factors. The standard by which success of these and many other models were judged was by comparison to pressure anomaly data in the form of sea level height anomalies or dynamic height increments. Sea level data carry the El Nino-Southern Oscillation (ENSO) signal, which is the phenomenon of greatest interest in the study of the tropical Pacific. Relatively rapid progress in models of the tropical ocean, as opposed to the mid-latitude or polar oceans is due, at least in part, to the agenda in tropical oceanography, which is focused on climate variability.

The success of the simple models stems from the fact that to some extent they contain the basic phenomenology of the interaction of largescale low-frequency atmospheric variability with the large-scale lowfrequency behavior of the fluid. When the early studies were performed, the only wind data sets available were coarsely resolved in space and time, and the only available ocean data sets with long enough time series to capture interannual climate variability consisted of depthintegrated pressure anomalies, i.e., data from tide gauges and expendable bathythermograph (XBT) data from volunteer observing ships (VOS); this latter data source is usually processed as dynamic height.

Introduction

Typically, the term “risk” is defined as either the product or a composite of (a) the probability or likelihood that any accident or limit state leading to severe consequences, such as human injuries, environmental damage, and loss of property or financial expenditure, occurs; and (b) the resulting consequences. In the design and operation of ship-shaped offshore units, as in many other types of structures, there are a number of hazards that must be dealt with in the process of risk assessment. Wherever there are potential hazards, a risk always exists.

To minimize the risk, one may either attempt to reduce the likelihood of occurrence of the undesirable events or hazards concerned, or contain, reduce, or mitigate the consequences, or both. In the lifecycle of a ship-shaped offshore installation, assessing managing, and controlling the risk is required so that it remains under a tolerable level. The risk management and control should, in fact, be an ongoing process throughout the lifecycle of an installation – that is, involving feasibility study, concept, or front-end design, detailed design, operation, and decommissioning. The different stages of the lifecycle will offer different opportunities for risk management and control, as may be expected.

Substantial efforts, such as the SAFEDOR project (http://www.safedor.org), are being directed by the maritime industry toward the application of the risk-assessment techniques together with risk-evaluation criteria to offshore design, operation, and human and environmental safety (e.g., Skjong et al. 2005).

Introduction

While in service, most structural systems such as ships, offshore structures, bridges, industrial plants, land-based structures, and other infrastructure will be subject to age-related deterioration that can potentially cause significant issues in terms of safety, health, the environment, and financial expenditures. Indeed, such age-related deterioration has reportedly been involved in many of the known failures of ships and offshore structures, including total losses. Although the loss of a total system typically causes great concern, repair and maintenance of damaged structures is also very costly to society, in general, and important to the economic viability of the enterprises involved, in particular. It is thus of great importance to develop advanced technologies, which can allow for the proper management and control of such age-related deterioration.

One of the most important factors in the safety and integrity of ship-shaped offshore units at sea is that they are affected by corrosion. Corrosion for such structures becomes a problem particularly where surfaces are unprotected, and this can be an issue for internal surfaces in cargo tanks where conditions of high humidity, and often inspection and access difficulties, also exist.

Periodic surveys can detect corrosion problems and help estimate the remaining thickness of structural components prior to their needing replacement. For high-quality asset management and for the development of optimal maintenance programs, refined strategies for corrosion management and control, including corrosion corrective or preventive measures, are required.