In science, it's not enough to perform a measurement or make an estimate and report a number. It is equally important that you quantify how well you know the number. This is crucial for comparing measurements, testing theories, debunking bad models, and making real-world decisions. In order to do this, you need to know something about basic measurement theory, and how different sources of uncertainty combine to influence your final results. In this chapter we will be going over the fundamental statistical concepts that lie underneath all that we do when analyzing our data and models. We discuss some important issues that lie at the heart of measurement theory (scales, errors, and roundoff) and look at the concept of probability distributions. Related to this is the very delicate question of whether, how, and why some experimental data should be rejected. In short, this chapter summarizes the science of knowing what you know and how well you know it.

Measurement theory

Systems of measurements (scales)

How we measure or characterize things in science depends on our objectives.

So far, we have introduced many of the elements of ocean modeling but in simplified situations with reduced dimensionality (i.e. box models, vertical 1D models, 2D gyre models). Here we pull all of the elements together, introducing the topic of 3D ocean general circulation models (GCMs). As you might expect, the topic is complex, and our GCM tour will be necessarily brief and focused. While you probably won't be able to construct your own GCM, you should at least be able to understand the conversation and perhaps even utilize 3D GCM output. Several good review articles and books have been written on ocean GCMs that the reader can refer to for more details (e.g. Haidvogel and Beckmann, 1999; Griffies et al., 2000; Griffies, 2004). While our emphasis is on marine systems, most of the fundamental concepts about GCMs are applicable to a wide range of environmental fluid systems, from the atmosphere to mantle convection to groundwaters.

Several themes emerge when considering ocean GCMs. First, no matter how fast technology develops, the cutting edge of ocean modeling is always “compute bound”, which is why you won't be able to build a decent GCM using MATLAB. Ocean GCM development is linked to the evolution of supercomputers, and in fact GCMs are commonly used to test new supercomputers.

Scope, background, and purpose

So far in our models the physical characteristics of velocity and diffusivity have been specified or “hard-wired” into the calculations. The next step to consider is allowing them to respond to changing conditions. In this chapter, we will be developing and exploring a class of models aimed at simulating the seasonal behavior of the upper ocean in response to changing atmospheric forcing. We subsequently will extend this model to simulate the response of dissolved gases in the upper ocean. This approach can be more generally applied to other shallow water column properties (including bio-optical modeling, particle dynamics, etc.) with very minor modifications. What we're trying to show you here is not just how to design, build, and extend the model, but more importantly how to figure out what the model is actually doing, and how to compare its performance quantitatively with actual observations.

Generalizations

There are two general types of upper ocean models (although there are hybrids of these two as well). There are the bulk mixed layer models which, as the name suggests, treat the mixed layer as a homogeneous, well-mixed box, within which properties including chemical species, temperature, salinity, and physical momentum are uniformly distributed.

Suppose you're looking for patterns or relationships in your data. For example, you may be trying to quantify the presence and distribution of certain water masses in a hydrographic section, or you may be looking for evidence and patterns of nitrogen fixation or denitrification in some nutrient data. Perhaps you're trying to find the best way to account for interferences from other elements (“matrix effects”) in your ICPMS data. You've gathered your data, maybe obtained from a cleverly designed experiment, or extracted from a hydrographic atlas or a collection of cruise data. The information you require lies within the relationships or correlations between the different properties or variables in your data set. But where (and how) do you look? If instinct leads you to look at the data covariance matrix, then your instinct is right! In this chapter we'll show you some techniques for extracting and analyzing this structure. We will start with some underlying basics that you'll need to understand these techniques, and we'll mention a few relatively intuitive approaches for analyzing data structure.

Most of you are familiar with topographic contour maps. Those squiggly lines represent locations on the map of equal elevation. Many of you have probably seen a similar mode of presentation for scientific data, contour plots with isolines of constant property values (e.g. isotherms and isopycnals). What many of you are probably not familiar with are the mathematics that lie behind the creation of those “maps” and their uses beyond visualization.

Contouring and gridding concepts

This chapter covers the question: “What do you do when your data are not on a regular grid?” This question comes up frequently with ocean field data, which are rarely sampled at exactly equal intervals of space or time. The grid dimensions could be latitude–longitude, like the familiar topographic map, or involve other dimensions such as time, depth, or even property values (e.g. temperature, oxygen, chlorophyll). Mathematical gridding is common in visualization because computers can only draw contour lines if they know where to draw them. Often, a contouring package will first grid your data using a default method, and this may be acceptable. But there is more to it than making pretty pictures.

Welcome to Modeling Methods for Marine Science. The main purpose of this book is to give you, as ocean scientists, a basic set of tools to use for interpreting and analyzing data, for modeling, and for scientific visualization. Skills in these areas are becoming increasingly necessary and useful for a variety of reasons, not the least of which are the burgeoning supply of ocean data, the ready availability and increasing power of computers, and sophisticated software tools. In a world such as this, a spreadsheet program is not enough. We don't expect the reader to have any experience in programming, although you should be comfortable with working with computers and web browsers. Also, we do not require any background in sophisticated mathematics; undergraduate calculus will be enough, with some nodding acquaintance with differential equations. However, much of what we will do will not require expertise in either of these areas. Your most valuable tool will be common sense.

Resources

The activities of modeling, data analysis, and scientific visualization are closely related, both technically and philosophically, so we thought it important to present them as a unified whole. Many of the mathematical techniques and concepts are identical, although often masked by different terminology. You'll be surprised at how frequently the same ideas and tools keep coming up.

In Chapter 10 we introduced optimization techniques for finding unknown model parameters by minimizing a model–data cost function. Here we extend those concepts to generalized inverse and data assimilation methods. Inverse modeling actually covers a number of related numerical approaches and is applicable to a wide range of oceanographic problems, essentially any system where we want to interrogate data to help better constrain a model. Often we can measure the state of the ocean, say the temperature, density, inorganic carbon, or nutrient distributions, much better than we can determine the governing processes (in this case, circulation, mixing, air–sea fluxes, and biological uptake and release). Inverse techniques allow us to take advantage of the wealth of ocean data. With the rapid growth of satellite and ocean observing system data, inverse modeling will likely continue to grow in popularity.

The chapter starts with an introduction to the concepts behind linear inverse modeling (Section 18.1). Many ocean-related inverse problems involving tracer transport can be written as a set of linear equations, and we therefore focus in some detail on methods for solving under-determined linear systems (Section 18.2). To illustrate some of the basic ideas, we present an example case of computing the horizontal velocity field from geostrophic balance and tracer budgets using ocean hydrographic sections (Section 18.3). We follow with brief introductions to more advanced approaches such as variational data assimilation and Kalman filtering (Section 18.4).

Getting started with MATLAB

At the risk of being revealed as very ordinary indeed, we would like to show you some useful MATLAB and other numerical conjuring tricks, or at least give you some useful tips on how to get the most out of this powerful tool. We start by assuming that you have successfully installed MATLAB on your personal computer, or that you have found out how to access it on a shared system. The version you have access to might differ from the one we are currently using, but unless you are working with a particularly old version, small differences should not matter; the principles should remain the same.

Learning to use MATLAB is a lot like learning to use a bicycle. It seems very difficult at first, but after a while it comes so naturally that you wonder at those who cannot do it too. Like a bicycle, you learn to use it by doing it. There may be books on the theory of balancing on a bicyle or the gyroscopic physics of wheel motion, but all the reading in the world won't keep you upright without trying it and taking your lumps when you fall.

If you are a student of science in the twenty-first century, but are not using computers, then you are probably not doing science. A little harsh, perhaps, and tendentious, undoubtedly. But this bugle-call over-simplification gets to the very heart of the reason that we wrote this book. Over the years we noticed, with increasing alarm, very gifted students entering our graduate program in marine chemistry and geochemistry with very little understanding of the applied mathematics and numerical modeling they would be required to know over the course of their careers. So this book, like many before it, started as a course – in this case, a course in modeling, data analysis, and numerical techniques for geochemistry that we teach every other year in Woods Hole. As the course popularity and web pages grew, we realized our efforts should be set down in a more formal fashion.

We wrote this book first and foremost with the graduate and advanced undergraduate student in mind. In particular, we have aimed the material at the student still in the stages of formulating their Ph.D. or B.Sc. thesis. We feel that the student armed with the knowledge of what will be required of them when they synthesize their data and write their thesis will do a much better job at collecting the data in the first place. Nevertheless, we have found that many students beyond these first years find this book useful as a reference.

Wave breaking represents one of the most interesting and most challenging problems for both fluid mechanics and physical oceanography. It is an intermittent random process, very fast by comparison with other processes in the wave system. The distribution of wave breaking on the water surface is not continuous, but its role in maintaining the energy balance within the continuous wind–wave field is critical.

The challenges thus outlined make understanding of such wave breaking and even an ability to describe its onset very difficult, and as a result knowledge of the physics of the breaking, and even practical parameterisations of the phenonemon have been hindered for decades. Recently, knowledge of the breaking phenomenon has significantly advanced, and this book is an attempt to summarise the facts into a consistent, even if still incomplete, picture of the phenomenon.

If this picture were to be formulated into a few paragraphs of these conclusions, we would like to say the following. The waves break because the water surface reaches some limiting steepness. Apparently, the fluid interface has to have a limit beyond which it will collapse. In the system of nonlinear water-surface waves subject to a variety of external forcings and internal instabilities, there are a number of physical processes which can lead to such a steepness. They are the modulational instability, linear (dispersive and directional) and nonlinear (amplitude-dispersion) focusing, modulation of steepness of shorter waves by longer waves, direct forcing by the wind (if very strong) or by the current, and wave-bottom interactions, among others.

On many occasions earlier in this book, it has been mentioned and emphasised that knowledge of breaking severity is as important as is understanding the physics driving the breaking occurrence. While the latter, however, has received a lot of attention from the wave-research community lately, our information on breaking strength, its variability, environmental dependences and physics remains limited and fragmental.

If the breaking strength is defined as energy loss in a single breaking event (Section 2.7), then the breaking severity coefficient s can be identified in a number of ways, through measurement of the individual breaking wave (2.24), of the group where the breaking occurred (2.32), of spectra of the respective groups before and after the breaking (2.38) and of short waves modulated by the longer wave only (2.42). The magnitude of such a coefficient varies greatly, from s = 10% (Rapp & Melville, 1990, or even less as seen in Figure 6.3 below) up to 99% (2.31) based on the Black Sea estimates (see also Bonmarin, 1989; Babanin et al., 2010a, 2011a).

Such a range of change of course cannot be disregarded or substituted with some mean value in applications that involve the breaking severity. A typical application is the wave-energy dissipation function Sds employed in wave forecast models (2.21), (2.61) and (5.40).