The largest part of the Earth's surface lies beneath the sea; events taking place in, on and directly above the oceans have an enormous impact on our lives. Consequently, maritime remote sensing and surveillance are of great importance. Since its discovery during the 1930s, radar has played a central role in these activities. Much of their military development was driven by the circumstances of the Cold War; now this era is past and a different set of imperatives holds sway. Military surveillance does, however, remain a key requirement. Great progress has been made recently on non-military applications, particularly the remote sensing of the environment, of which the sea is the most important component. These newly emerging concerns, whether they are ecological or geopolitical, currently define the requirements imposed on maritime radar systems. Nonetheless, the underlying principles of the systems' operation, and the interpretation of their output, remain the same; the body of knowledge developed in the twentieth century provides us with the tools with which to address the problems facing the radar engineer of the twenty-first century. This book attempts to bring together those aspects of maritime radar relating to scattering from the sea surface, and their exploitation in radar systems. The presentation aims to emphasise the unity and simplicity of the underlying principles and so should facilitate their application in these changing circumstances.

In this chapter, sea clutter is described in terms of its observed characteristics. These observations will provide the foundations upon which the themes of subsequent chapters are developed. Radars operating in a maritime environment inevitably encounter backscattered radar signals from the sea surface, usually referred to as sea clutter. For some applications, such as remote sensing systems, the reception of this backscattered signal may be the main purpose of the radar. Space-borne synthetic aperture radars, with spatial resolutions of a few metres, are used for oceanographic studies, gathering data on waves and currents, sea ice and so on. Scatterometers measure average backscatter over hundreds of square kilometres to measure wind speed and direction over the sea surface.

The observed characteristics of radar sea clutter were described in the previous chapter. This chapter presents some of the empirical models that describe sea clutter characteristics and that can be applied for signal processing design and radar performance predictions. The models are of limited use to the radar designer unless their parameter values can be matched to different environmental conditions, radar characteristics and signal processing schemes. While some headway has been made with direct electromagnetic modelling of the sea surface, nearly all of the models used for practical applications are based on very many measurements of different conditions. The chapter covers models for three main attributes of sea clutter: the normalised radar cross section (NRCS), the amplitude statistics, and the Doppler spectrum. These characteristics were introduced in the previous chapter and their significance will become clear in later chapters of this book.

In this chapter we consider simulation techniques that enable us to study various aspects of radar performance in circumstances where an analytic attack may not be possible or particularly informative. To complement the analytic clutter modelling discussed in Chapter 3, we will develop methods for the numerical simulation of unwanted radar returns. The clutter models in Chapter 3 are exclusively statistical; this prejudice is still evident in our choice of simulation methods. In essence, we address the problem of generating correlated random numbers with prescribed one- and two-point statistics.

In this chapter, we give an informal review of probability theory and other matters that are of relevance to the modelling of clutter and its impact on radar systems. Our selection of topics is rather eclectic and our treatment pragmatic; the principal aim is to remove those obstacles to the evaluation of radar performance presented by any unfamiliarity with the concepts and practice involved in calculating probabilities. Much fuller accounts can be found in standard texts such as in References 1 and 2. By collecting together the relevant definitions and didactic developments, we also ensure a greater continuity, and brevity, in our discussions in other chapters of the book.

In this chapter we develop the models of sea clutter that will provide a basis for our discussion and calculation of radar performance. As we will see in Part IV (Chapters 15-17), a direct approach to the modelling of sea clutter provides many useful insights. Nonetheless, the interactions between the ocean, atmosphere and microwave radiation are far too complicated to be described usefully in strictly deterministic terms. Consequently, the models we develop in this chapter are unashamedly statistical in character.

In this chapter we have considered a class of models that complement and illuminate the compound form of the K process discussed in Chapter 6. These models have in the main grown out of the optics literature, and tend to highlight the role played by individual scatterers, rather than a speckle process modulated by the partially resolved structure of the sea surface. Consequently, they provide a useful vehicle for the discussion of sea spikes and bursts generated by breaking waves; this analysis allows us to draw together the K, Class A and BEMs into a consistent whole. The formulation of the K process in terms of FP and SDEs has also been discussed; the impetus lent to this methodology by its application in financial modelling and other areas may yet result in this formulation being of practical as well as academic interest.

In this chapter we have seen how the relatively simple theoretical ideas underlying the K distribution model of sea clutter that were discussed in Chapters 6 and 7 can be extended, both in terms of their physical content and of the mathematical techniques used in their analysis. The simple isotropic random walk model of scattering that formed the foundation of the modelling of strong scattering observed in backscattered clutter returns can be relaxed by imposing either a bias or offset to the random walk; in the absence of number fluctuation effects, both these models yield the familiar Rice distribution model, which in turn has formed the basis for more elementary discussions of weak scattering. The impacts of negative binomial number fluctuations on the offset and biased random walk models are different; in the former case the homodyned K distribution model (which also underpins the radar performance modelling in Chapter 12) emerges and has proved to be a useful model for weak scattering. This model has recently found application in the analysis of medical ultrasound imagery. The biased random walk model preserves the infinite divisibility characteristic of the K distribution (this is not the case for the HK model) and yields an analytically tractable PDF; these pleasing formal attributes are rendered rather superfluous by the model's inability to produce a sensible very weak scattering limiting behaviour. This seemingly parlous state of affairs is redeemed, however, by the rather unexpected applicability of the GK model to the analysis of the performance of an interferometric SAR system.

In this chapter, we discuss the special functions associated with the K distribution. These functions are gamma function, K distribution PDF, Bessel function, Laguerre polynomial and Hermite polynomial.

In this chapter, we have established a well-defined framework, provided by the likelihood ratio concept and the statistical models developed in Chapters 3 and 6, within which useful detection schemes can be developed in a systematic way. The closely related problem of parameter estimation is also considered: maximum likelihood techniques derived from Bayes' theorem prove to be quite tractable for simple Gaussian clutter models, and can be incorporated into generalised likelihood ratio-based detection methods. Some relatively simple examples of the application of these principles have been discussed in detail, to the point where contact is made with the small target detection strategies discussed in Chapters 12 and 13. The principles demonstrated here can then be applied to other detection scenarios. The estimation of parameters characterising non-Gaussian clutter is more problematic: the maximum likelihood derived equations are now much less easy to solve. Nonetheless, useful estimation methods have been derived that can be applied to gamma, Weibull and K-distributed data. The compound representation of clutter developed in Chapters 3 and 6 plays a central, but rather subtle, role in this work. Small target detection procedures are applied to localised samples of data, to which the relatively simple and tractable Gaussian derived methods can be applied. Consequently, we should expect the methods described in Section 10.7 to be relatively effective even in spiky, non-Gaussian clutter; their performance can then be improved incrementally and relatively straightforwardly, when prior knowledge of the gamma distribution of local power can then be brought to bear, as is described in Section 10.10.

In the previous chapter, we considered how we might best detect small, localised targets in a background of sea clutter. The identification of the likelihood ratio as an optimum discriminant, and of its more practically useful approximations, provided us with a unifying framework for this discussion. However, small target returns are not the only features of interest in maritime radar imagery. Large-scale correlated structures arising from surface currents, ship wakes, the presence of surfactants and other sources can frequently be discerned and are a valuable source of information in many circumstances. In this chapter, we will discuss how such ocean surface features might best be enhanced and detected. Once again the likelihood ratio concept is a very useful guiding principle, which leads us to methods that enable us to both enhance these features and exploit our prior knowledge of their structure to detect them more effectively. So, paradoxically, a discussion of the processing of images, which are frequently interpreted and assessed in qualitative terms, will involve us in a fair amount of detailed formal analysis. Much of this will be based on the multivariate Gaussian distribution; we commence this chapter with a review of its pertinent properties, which are discussed in more detail in Chapter 5.

We consider just one system in detail; an airborne, scanning sea surface surveillance radar, which operates at I-Band (9-10 GHz), is non-coherent, uses frequency agility and has pulse-to-pulse and scan-to-scan integration. The problem is that of detecting small, point sea-surface targets. Having worked through and understood this example, the reader should be well placed to carry out similar analyses of other types of radar. A few pointers towards how this may be done are given towards the end of the chapter, where results are presented of performance with a logarithmic detector; a comparison of predicted performance using the K, Weibull and lognormal distributions; and performance of Pulsed Doppler radar processing in K distributed clutter.

The techniques that have been described and analysed in this chapter can be exploited in various ways to develop practical CFAR detection systems. If a CA CFAR system is to be used to estimate the local mean level, choices must be made among the many possible configurations, such as the length (M) of the cell-averagers, the gap (G) surrounding the cell under test and the various schemes described in Section 13.2.3. It may be appropriate to have these parameters or choice of scheme selectable by the radar operator or automatically adapting, according to the prevailing conditions. The method for setting the threshold multiplier must also be determined. The CA CFAR suggests that the surrounding clutter may be used directly to estimate the distribution shape and hence the threshold multiplier, a. The various methods for estimating the clutter shape parameter in Section 13.3 could be assessed for this purpose. As discussed in Section 13.1, the clutter conditions may vary very widely with range and azimuth, and a problem is likely to be encountered in achieving a sufficient number of independent clutter samples within an area of constant statistics. One approach may be to gather statistics over a number of scans. Other estimates of clutter statistics, such as the U estimator described in Section 13.3, may also be used. A closed-loop system (Section 13.4.2) that directly estimates the PFA may also be a practical solution, although there are similar considerations on the number of samples required to give a satisfactory estimate, especially if a low value of PFA is required.

One of the ultimate aims of modelling sea clutter is to inform the development of improved radar systems that meet the operational needs of their users. However, the translation from a user's requirements to the entry into service of an equipment that meets these needs in an acceptable way is a very complex process. From the viewpoint of a radar designer, an important part of the process is the methodology for specifying and measuring radar detection performance. This methodology is the subject of this chapter, with emphasis being placed on the issues relating to maritime radars and the detection of targets in sea clutter.

In Chapter 2, we considered the phenomenology of sea clutter and its impact on the operation of microwave radar systems. To exploit this knowledge and improve radar performance, we need to understand the underlying physical mechanisms responsible for these clutter properties, so that they can be modelled realistically. At first sight this might seem, in principle at least, to be a simple matter, particularly if use is made of a computer. The fundamental physics of the generation and transmission of microwaves, their interaction with the ocean surface and scattering to the radar receiver is well understood. The ocean and atmosphere have been subject to intensive study for at least a century; the underlying laws of fluid motion have been known for much longer. Can clutter modelling be any more than a matter of assembling these constituent parts and turning the handle?

The previous chapter considered scattering at high grazing angles where the Physical Optics approximation is accurate. At lower grazing angles this approximation breaks down and it is necessary to adopt a different approach. In Section 16.2, electromagnetic (EM) scattering theory is extended to the composite model for scattering from an imperfectly conducting dielectric rough surface. In contrast to Physical Optics, the EM polarisation sensitivity of the composite model is able to explain the difference between vertical (V) and horizontal (H) polarisations at medium grazing angles. However, at low grazing angles (LGAs) there are many characteristics, as described in Chapter 2, which are not consistent with the composite model. Section 16.3 therefore considers extending LGA scattering theory beyond the composite model. The principal LGA clutter phenomena of multipath interference and scattering from breaking waves (and their manifestation in radar `sea spikes') may be understood from scattering off corrugated (one-dimensional, 1D) surfaces. Thus, it is here that we show how for corrugated surfaces vector EM scattering simplifies to scalar scattering. We introduce numerical methods for calculating the backscatter from rough surfaces at LGA and compare the accuracy of various techniques for idealised surfaces. In Section 16.4, the radar cross-sections (RCS) of spikes and the effect of multipath are calculated for realistic surfaces. These are put together in Section 16.5 with a model for the occurrence of breaking waves to derive trends for the average RCS of sea clutter versus grazing angle and sea state. It is shown that the results are well matched with a wide range of experimental LGA data. Finally, in Section 16.6, we return to the remote sensing problem of Section 15.4 and illustrate how the imaging of tidal flow over bottom topography is affected by LGA scattering.

This chapter will be long and perhaps overburdened with technical detail. Possibly as a result of this, it should provide a reasonably thorough introduction to the theory of EM scattering in the LGA regime that is particularly pertinent to the discussion of sea clutter. The reader who wishes to carry out controlled calculations of this kind, and to interpret their output sensibly, has no alternative but to study their theoretical background in some depth. Much of the relevant material is spread throughout the physics and engineering literature and is expressed in a variety of notations. It is hoped that, by focussing our attention on the less general scalar formulation of the problem, we will be able to lead the reader through the finer points of the theory and its implementation, and elucidate details too often omitted in articles in the literature, without incurring an unacceptable overhead of formulaic obfuscation. The authors sincerely hope that they can succeed in achieving this aim, as they remember only too well the difficulties they experienced when encountering this material for the first time.