A new approach for data assimilation, which is based on the adjoint method, but allows the computer code for the adjoint to be constructed directly from the model computer code, is described. This technique is straightforward and reduces the chance of introducing errors in the construction of the adjoint code. Implementation of the technique is illustrated by applying it to a simple predator-prey model in a model fitting mode. A series of identical twin numerical experiments are used to show that this data assimilation approach can successfully recover model parameters as well as initial conditions. However, the ease with which these values are recovered is dependent on the form of the model equations as well as on the type and amount of data that are available. Additional numerical experiments show that sufficient coefficient and parameter recoveries are possible even when the assimilated data contain significant random noise. Thus, for biological systems that can be described by ecosystem models, the adjoint method represents a powerful approach for estimating values for little-known biological parameters, such as initial conditions, growth rates, and mortality rates.

A general circulation model and North Atlantic climatological data of temperature, salinity, wind stress, evaporation minus precipitation, and air-sea heat fluxes are used to examine the possibility of solving inverse problems using a full-scale numerical GCM and real oceanographic data, combined through an optimization approach.In this study several solutions for the model inputs and the structure of the cost function as a function of the model inputs are examined to demonstrate two of the main difficulties confronting such large-scale nonlinear inverse problems (about 30 000 unknowns and a similar number of constraints for the problem examined here). The first is the possible existence of local minima of the cost function, which prevents convergence of the optimization to the global minimum representing the desired optimal solution for the model inputs. The second difficulty, which seems the dominant one for many of the problems examined in this part as well as in Part I, is the ill conditioning of the inverse problem. Simple model equations are used to analyze the conditioning of the optimization problem and to analyze the role of both dissipation and waves in the model dynamics in conditioning the problem. The analysis suggests what might be an improved formulation of the cost function resulting in better conditioning of th e problem.The relation between the optimization approach and the robust diagnostic method of Sarmiento and Bryan is explicitly demonstrated, and the solution obtained by combining the two methods is used to examine the performance of the GCM used here for the North Atlantic Ocean.

This paper deals with the solution of inverse problems involving complex numerical models of the oceanic general circulation and large datasets. The goal of these inverse problems is to find values for model inputs consistent with a steady circulation and, at the same time, consistent with the available data. They are formulated as optimization problems. seeking values for the model's inputs that minimize a cost function measuring departures from steady state and from data. The two main objectives of this work are 1) to examine the feasibility of solving inverse problems involving a realistic numerical model of the oceanic general circulation and 2) to understand how the optimization uses various data to calculate the desired model parameters.The model considered here is similar to the primitive equation model of Bryan and of Cox, the principal difference being that here the horizontal momentum balance is essentially geostrophic. The model's inputs calculated by the optimization consist of surface fluxes of heat, water, and momentum, as well as the eddy-mixing parameters. In addition, optimal estimates for the hydrography are obtained by requiring the hydrography to be consistent with both other types of data and the model's dynamics.In the examples presented here, the data have been generated by the model from known inputs; in some cases, simulated noise has been added. The cost function is a sum of terms quadratic in the differences between the data and their model counterparts and terms quadratic in the temperature and salinity time rate of change as evaluated using the model equations. The different inverse problems considered differ in the choice of the model inputs calculated by the optimization and in the data used in the cost function. Optimal values of the model's inputs are computed using a conjugate-gradient minimization algorithm, with the gradient computed using the so-called adjoint method.In examples without added noise, solutions for the model inputs were found efficiently and accurately. This was not the case when simulated data with randomly generated noise were used. Amplification of noise was especially felt in regions of deep-water formation due to the strong vertical mixing in these regions. Away from deep-water formation regions, the performance of the optimization with noisy data was still not satisfactory, possibly due to bad conditioning of the problem. The conditioning of the optimization and the difficulties due to the noise amplification are further discussed in Part II of this work using real oceanographic data for the North Atlantic Ocean.

A formalism is presented for fitting dynamic forecast models to asynoptic data. Because of the importance of wind stress forcing in oceanic models and of the inadequacies of wind stress observations, the formalism allows an oceanic model to be fit to both Oceanographic and meteorological data. Within the context of this formalism the important question of whether an asynoptic data set contains sufficient information to determine the model state completely and unambiguously is discussed. Because the information travels along wave characteristics, it is clear that for the data to be sufficient to determine the model state, they must be distributed so that every feature of the flow is seen at some time or another. Such widespread coverage of the oceans requires a data collection system that relies heavily on satellites. The formalism is illustrated using a highly truncated model of the wind‐driven equatorial ocean and computational examples demonstrate how surface elevation and wind stress observations might be used to recover the model state.

A sophisticated analysis technique is applied to a subset of pitch-roll buoy data collected by the research vessels Gilliss and Quadra during the GARP Tropical Atlantic Experiment (GATE) in September 1974. The procedure enables the examination of directional properties of the wave field at a level of detail not previously along 44°N latitude. BY comparing properties of the observed spectra with the predictions of a simple schematic model of the storm, we conclude that swell reaching the GATE area was emitted during the first half of the storm's lifetime; swell subsequently radiated from the storm was heavily attenuated, either by sheltering of the site by the Cape Verde Islands or because of radically lower emission levels from the storm itself. This work illustrates the power, as well as the limitations of the pitch-roll buoy when used in conjunction with a fully effective analysis technique.

Airborne radar observations of the Southern Bight of the North Sea have revealed sea-surface patterns that reflect a constructional sea-floor topography of tide-built sand waves and sand ridges. The patterns result from local horizontal differences in the amplitude of water surface waves with wavelengths less than 1 cm. Such modulations result from surface convergences and divergences as the water flows across structures on the ocean floor.

A useful parametrical model for the vertical structure of the pressure field induced by wind blowing over a field of surface gravity waves is proposed. The model is a linear expansion in a set of empirical orthogonal functions, derived from a set of 110 complex pressure profiles computed according to the theory of Miles (1957), and provides a compact, quantitative description of those profiles. The model has been used as an element in the analysis of a body of experimental data on wave-induced atmospheric pressure fluctuations obtained by Snyderet al.(1980).

The weak interaction of a field of gravity waves with an irregular bottom topography has been analyzed theoretically by Hasselmann. His theory provides the time rate of change of the wave spectrum as a linear functional of the spectrum of the bottom irregularity with coupling coefficients determined by the eigenfunctions of the linear flat bottom problem. To evaluate this effect on surface waves, the coupling coefficient has been derived for this special case. The result indicates that the effect may be sufficiently large to account for swell decay observed during the Joint North Sea Wave Project in 1969.