## 1. Introduction

[2] Nonstandard atmospheric refractivity structures significantly affect the performance of shipboard radar systems in the detection of low-altitude targets. The refractivity structure associated with the marine atmospheric surface layer often provides a thin (a few meters to a few tens of meters) and leaky “evaporation duct” that results in up to a factor of 2 extension in detection ranges over what would be expected with a standard atmosphere. The refractivity profile of the surface layer is often characterized by a single parameter, the evaporation duct height δ. The term “surface-based duct” is usually associated with anomalous refractivity structures that are formed by inversion layers or internal boundary layers (i.e., the result of the offshore flow of a relatively hot and dry planetary boundary layer [*Gossard and Strauch*, 1983]). Surface-based ducts are associated with clutter rings and instances where shorelines can be observed using shipboard radars from 200 km or more. The refractivity profiles associated with surface-based ducts are complex in both the vertical and horizontal dimensions and are difficult to characterize in just a few parameters.

[3] Beginning in roughly 1980, the ability to model these effects led designers to incorporate consideration of them in system analysis [*Hitney et al.*, 1985]. By the mid-1990s, systems were being demonstrated for near-real-time shipboard capability for characterization of atmospheric refractivity and its associated impact on radars [*Rowland et al.*, 1996]. At that time, the obvious choice for characterizing refractivity was to use the bulk method [*Liu et al.*, 1979] for characterization of the atmospheric surface layer. Radiosondes, rocketsondes [*Rowland et al.*, 1996], and the output of numerical weather prediction models [e.g., *Haack and Burk*, 2001] have been viewed as the means for estimating refractivity structures associated with surface-based ducts. In the absence of the refractivity characterization provided by measurements or models, the default (normally resulting in pessimistic predictions of radar range) is to make the assumption of a standard atmosphere.

[4] In the late 1990s the authors and colleagues began looking at the feasibility of inferring low-altitude atmospheric refractivity from radar sea clutter. *Rogers et al.* [2000] demonstrated an algorithm that generated a maximum likelihood (ML) estimate of the evaporation duct height, using radar sea clutter observations at 3.0 GHz, for the case where evaporation ducting is the dominant mechanism of propagation and there is a sufficient clutter-to-noise ratio. To a significant degree this problem has a unique solution at this frequency.

[5] The generation of ML estimators or the maximum a posteriori (MAP) of parameters describing the refractive environment for the case where surface-based ducting is the dominant mechanism of propagation has been addressed by several authors [*Krolik and Tabrikian*, 1998; *Vasudevan and Krolik*, 2001; *Gerstoft et al.*, 2003a, 2003b]. When surface-based ducting is the dominant mechanism of propagation, and both model error and noise are present, the solutions are often nonunique. A weakness of many formulations of ML or MAP estimators, however, is that they contain no information on the certainty of one point vis-à-vis the certainty of others. For cases where the probability in the a posteriori distribution is not concentrated about the MAP or ML estimate (*Gerstoft et al.* [2003a] illustrates this behavior), the ML or MAP solutions may not be the proper answer to the problem; rather, the full solution is the a posteriori probability distribution. *Gerstoft et al.* [2004] explicitly deals with the generation of the a posteriori distributions based on Gaussian assumption of the likelihood function. In that paper, the mismatch between observed and modeled clutter was assumed to be Gaussian (using an ad hoc estimate of the associated covariance), and the result was mapped into the space of the information usage (propagation loss in that instance) so as to provide a measure of the cost of ambiguity and noise.

[6] This paper continues with the theme of *Gerstoft et al.* [2004] of using the Bayesian approach to the generation of a posteriori distribution, but it differs in its implementation. Furthermore, it extends the analysis to examine how well the a posteriori distributions of a scalar usage variable contain the true values of the usage variable.

### 1.1. Bayesian Framework

[7] An array of observed radar sea clutter observations **d**^{o} is used to generate an estimated “usage” quantity *u*, such as a signal threshold level. We have no direct mapping from **d**^{o} to *u* but have “forward” models *f*(·) and *g*(·) that can map the *i*th modeled range and height-dependent refractivity realization **m**^{i} into modeled deterministic clutter (**d**_{i}) and usage value (*u*_{i}), respectively. The true refractivity (**m**^{true}) and usage value (*u*^{true}) are unknown (see Table 1).

True | Modeled | |
---|---|---|

Data domain | d^{o} | d_{i} = f(m_{i}) |

Environmental domain | m^{true} | m_{i} |

Usage domain | u^{true} | u_{i} = g(m_{i}) |

[8] Each **m**_{i} is associated with one **d**_{i} and one *u*_{i} to form a three-tuple {**m**_{i}, **d**_{i}, *u*_{i}} = {**m**, **d**, *u*}_{i}. The a priori probability of **m**_{i} and {**m**, **d**, *u*}_{i} are the same, that is, *P*(**m**_{i}) = *P*({**m**, **d**, *u*}_{i}) = *P*_{i}. Bayes rule applied to this problem is

where *P*(**m**_{i}∣**d**^{o}) is the a posteriori probability (refining our estimate of **m** by adding the information in **d**^{o}), and *p*(**d**^{o}∣**m**_{i}) is the conditional probability density of **d** given **m**_{i}. The forward model **f**(·) is embedded in *p*(**d**^{o}∣**m**_{i}). The small *p* denotes probability density as opposed to a probability (**d**^{o} is generally a continuous quantity). The index of the **m** corresponding to the maximum a posteriori estimator is

The posterior distribution of *u* is found via the empiric or sample method [*Dudewicz and Mishra*, 1988], which can be realized as

where

#### 1.1.1. Common Assumptions

[9] The Bayesian framework does not impose a model for *p*(**d**^{o}∣**m**_{i}). However, it is often modeled as a multivariate Gaussian process [see *Gerstoft et al.*, 2003a]

with the covariance matrix **C** of the form

There are at least two reasons why (5) and (6) may not be physically consistent with the nature of this problem:

[10] 1. Contamination of **d**^{o} due to the horizontal variability of the sea clutter radar cross section is expected to be a major component of the mismatch **f**(**d**_{i}) − **d**^{o}. That would be expected to be a colored process (e.g., a Markov process); thus **C** would not have the form of a scaled identity matrix.

[11] 2. The effect of nuisance parameters (i.e., what is not modeled in **m** but can be present in **m**^{true}) on **d**^{o} includes the behavior of horizontal displacements of clutter features relative to where the features (e.g., clutter rings) were the nuisance parameters not present. The evidence of this is illustrated in work by *Gerstoft et al.* [2003a, Figures 8 and 9]. Displacements are not accounted for in equation (5).

#### 1.1.2. Attributes of Implementation of Bayes Rule

[12] We present an inverse method that differs from what is described in section 1.1.1. We use random functions to characterize the mismatch between **f**(**m**) and **d**^{o}. Then we incorporate threshold likelihood function. In the context of the preceding discussions, the inversion algorithm has the following attributes.

[13] The probability distribution of **m** is represented by **m**_{i} for *i* = 1, 2, ⊂ … *M* samples. The samples are equally likely, that is,

Each sample is mapped into one deterministic realization of clutter **d**_{i}^{det} = **f**(**m**_{i}) and into the scalar usage variable *u*_{i}. Random functions Ψ_{j} (·) for *j* = 1, 2, … *N* map the **d**_{i}^{det} into random replicas **d**_{i,j} = Ψ_{j}(**d**_{i}^{det}). That leads to *M* × *N* equally likely tuples of the form {**m**_{i,j}, **d**_{i}^{det}_{,}* _{j}*,

**d**

_{i,j},

*u*

_{i,j}} = {

**m**,

**d**

^{det},

**d**,

*u*}

_{i,j}, where

**m**

_{i,j}=

**m**

_{i}for

*j*= 1, 2, …

*N*, and likewise for

**d**

_{i}

^{det}and

*u*

_{i}.

[14] We implement a thresholded likelihood by defining a sample region *S* around the observation (**d**^{o}) with experimental probability

Note that *P*(**d**^{o}∣**m**_{i,j}) = 1 for **d**_{i,j} ∈ *S* because there is only one **d**^{o} in *S*. On the other hand, letting be the number of **d**_{i,j} in *S* leads to

hence Bayes rule is satisfied for this case where the **m**_{i,j} have equal a priori likelihood. The probability of a given **m**_{i} is found by summing over *j*:

Finally, the distribution function of *u* is found as in (3) and (4).

### 1.2. Inversion Calibration

[15] The behavior of Ψ and *S* are determined by their nature and parameters used to control them. These ultimately control the shape of the a posteriori distribution. The desired behavior is that the ground truth *u*^{true} falls between the 0th and 20th percentile levels of a posteriori distribution of *u* 20% of the time, between the 20th to 80th percentile levels 60% of the time, etc. A trial-and-error process was used to adjust the distribution for the simulation cases. Ideally, a maximum likelihood procedure is where the parameters controlling the inversion are varied so as to maximize the a posteriori probability of *u*^{true}.

### 1.3. Usage Scalar *u*(**m**)

[16] In the case of noise-limited detection, one factor determining whether or not a target can be kept in track is the signal-to-noise (S/N) ratio of the target at points along the track. A robust tracking algorithm will tolerate some dropouts (hits where S/N drops below some threshold value) but will drop a track if a series of drop-outs exceeds some maximum “coasting” distance. An approximate answer as to whether a drop-track event will occur (when all other factors such as the target's radar cross section are taken into account) is the percentile level of the target's S/N dropouts. In this paper, we use a non-system-specific statistic of the propagation loss as our usage variable *u*. Here *u*(**m**) is the 20th percentile level of the two-way propagation loss (−2*L*) for a given environment **m** at a height of 5 m over the range interval of 10–60 km. An example of *u* showing how it relates to propagation loss on the described track is shown in Figure 1.