Goals and examples of sequence analysis

Sequences of data, either in space or in time, appear all the time in ocean research. You may have a time series of measurements at a location (e.g. sediment trap data, or ocean surface temperature), a series of stations along a hydrographic section, or isotope measurements on a long sediment core. For the sake of simplicity (initially) we shall discuss only regularly sampled data; that is, samples taken at identical intervals in space or time. The analysis becomes more difficult and more complicated when we discuss irregularly spaced samples, but the principles are similar and best understood in terms of the simplest case first. What do we hope to achieve in the analysis of data sequences? There are as many reasons (or perhaps more) as there are data sequences. The next subsections outline briefly some of the major conceptual motivations.

Searching or testing for structure or periodicities

Within a single data set you might be looking or testing for changes in a system due to periodic forcing, for example the effect of seasonal changes on biological production, or the effect of lunar tides on shell-fish contamination. This may be extended to spatial regularity as well, in that you may be looking for evidence of large-scale Kelvin waves (rapidly propagating variations of the thermocline depth) on dissolved nutrients near convergence zones in the ocean.

Until now we have concentrated on what may be loosely termed “data analysis methods”. In some respects, this is a form of modeling in that we are attempting to interpret our data within the context of some intrinsic model of how our data should behave, whether it be assuming the data follow an underlying probability distribution, vary as a function of some other variables, or exhibit some periodic behavior as a function of time. We hope you are beginning to see that all of these methods share common mathematical and algorithmic roots, and we want you to realize that many of these tools will come in handy as we now embark on a more model-intensive course.

Before doing so, we want to outline the basic aspects of model design, implementation, and analysis. Selecting the most accurate and efficient algorithms and developing robust and usable MATLAB code is important, but most of your intellectual energies should be directed at the design and analysis steps. Moreover, although correct design is vital to any successful modeling effort, developing the tools to efficiently analyze model output is just as important. It is critical to assessing the mechanics of how a model is performing as well as ultimately understanding the underlying system dynamics and how well a model compares to observations.

Basic principles

Constructing numerical models of marine systems usually involves setting up a series of partial differential equations, specifying boundary conditions and then “running the model”. Your purpose may be to establish the value of parameters (e.g. rates of reaction or the magnitude of some property), estimate fluxes, or make some prediction about the future state of the system. Although you can sometimes choose a physical problem that is simple enough to be modeled with analytic solutions (an example would be Munk's 1966 “Abyssal recipes” model; Chapter 13), more often than not you will encounter situations where the processes or the geometry of the system are too complex to allow analytic solutions.

Don't get us wrong; analytic solutions are nice. They can often provide you with a nice conceptual, intuitive feel for how the system responds, especially in an asymptotic sense. However, for realistic geometries, you will find that the few analytical solutions provided in many books are infinite series solutions. Be very, very careful when dealing with those series solutions. Pay particular attention to the assumptions made in deriving the solutions, to the conditions under which they ought to be applied, and especially to convergence issues.

Onward to the next dimension

Although one-dimensional models provide useful insight into basic biogeochemical processes, we are forced to admit that the world is made of more than one spatial dimension. The addition of an extra dimension to a model often does more than “fill space”, but rather imbues the model with behavior that is qualitatively different from its lower-dimensional analogue. The opportunity presented by the extra dimension is that more interesting, and perhaps more “realistic” phenomena may be modeled. This opportunity brings with it challenges, however, that are not just computational in nature. The choices of model geometry, circulation scheme, and boundary conditions become more complicated. Seemingly innocuous choices can have subtle or profound effects on how your model behaves. Moreover, matching model results to observations often requires decisions about whether features result from intrinsic processes of interest, or are mere artifacts of the choices made in model configuration.

For instructional purposes, we'll stick to a genre called gyre models which, as you might guess, are characterized by a quasi-circular flow on a plane. Such models have utility in the subtropics – at least that's where we'll be dwelling here – but can be used in many other parts of the ocean.

We treat one-dimensional models of sedimentary systems separately in this book because of the added complication that they contain two phases – solid material and pore waters – that not only can interact biogeochemically, exchanging chemicals, but also can move in relation to one another. In fact, with a reference system fixed at the sediment–water interface, the solid phase is actually moving owing to a combination of sedimentation (addition of material at the interface) and compaction. If you're thinking that this makes the construction of models a little more complicated, those are exactly our sediments!

We will be talking about the process of diagenesis, i.e. the sum total of all processes that bring about changes to sediments after they have been deposited on the seafloor. This includes everything from bioturbation through chemical transformation to compaction and pore water extrusion. The general topic of diagenesis extends to even longer timescale processes that include metamorphism and weathering of sedimentary rocks after uplift, but we will focus on Early Diagenesis (Berner, 1980), which encompasses changes that occur at or near the sedimentary surface or in the upper portion of the sedimentary column.

Why scientific visualization?

Throughout this book we have used a number of MATLAB's graphical capabilities as tools to monitor the progression of our mathematical and numerical travails, to demonstrate some characteristic of our results, or to reveal underlying relationships in data. Our emphasis now will be on the basic process of scientific visualization and providing you with some advice on how to effectively use (and not abuse) the many tools available to you. You may think that scientific visualization is an easy and natural thing to do, especially given the relatively powerful and reasonably intuitive tools built into MATLAB and other “point and click” packages so readily available. However, in the many years that we have been attending conferences, reading journals, and perusing text-books, we have encountered some ghastly instances of computer graphics abuse (or more to the point, abuse of the poor viewer/reader). This is a shame, because invariably the presenter has worked hard, often under difficult circumstances, to acquire scientific data, execute a model, or discover an erstwhile hidden relationship … only to fail to communicate the final result effectively. After all, isn't communication the final end-product of all our scientific endeavors?

We could also regale you with the awe-inspiring size of today's huge data sets, but rest assured that tomorrow's will be even more impressive.

Rationale

Our main objective in studying one-dimensional, open-ocean advection–diffusion models is pedagogical. The fact that they have relatively simple analytical solutions makes them a useful starting point for studying ocean models. In fact, you may find yourself turning to these more idealized representations as a tool for building intuition about the behavior of more complex models. That is, you might build a “model of your model” to explore what is happening within it. Perhaps more important to the student of modeling, they represent an elegant example of how we can use spatial distributions to illuminate underlying physical and biogeochemical dynamics.

They're not really considered “state-of-the-art”, having been extensively exploited by geochemists starting in the 1950s and used by others many decades before that. In truth, there are few parts of the ocean that can be regarded as truly satisfying the assumptions and requirements of this class of model. Even then it is highly debatable how generalizable the parameters derived from such models really are to the rest of the world. However, it is instructive to think of the abyssal ocean in terms of simple one-dimensional balances because it helps build intuition about open-ocean processes. Certainly it is an interesting historical stage in the evolution of geochemical ocean modeling, and has much to offer as a learning tool for understanding the process of ocean modeling.

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.