## 1. Introduction

Ocean climate observations (e.g., temperature, salinity, nutrients, flow speeds, and directions) are typically sparse in space and time across the globe. Long-term moored observing stations and/or repeat observational transects tend to be costly and are not easily resourced. This is not only true for the deep ocean but also in nearshore regions across the continental shelf. Satellite-observing instruments such as the Advanced Very High Resolution Radiometer (AVHRR) mounted on orbiting satellites provide high-resolution repeat measurements of sea surface temperature (SST) along track. Unfortunately, these instruments lack a temporal sample rate high enough to consistently capture high-resolution SST variability on daily to weekly time scales, for example, variability resulting from mesoscale eddies or synoptic weather forcing. Fixed measurement stations and moorings can provide location-specific observations with high sample rates but, unless there are many such instruments as part of a large network, lack the spatial coverage or homogeneity in both space and time. Global ocean models and reanalysis data can also be valuable tools and products to help synthesize our understanding but are typically insufficient on their own to capture the physical processes that govern SST variability on the continental shelf, since they tend to be conditioned to represent the large-scale ocean circulation.

One method for synthesizing irregularly spaced observations is objective mapping, which takes account of relevant space and time signal and noise scales. For example, Holbrook and Bindoff (2000) generated a dataset of uniformly gridded upper-ocean temperatures using an objective mapping technique that takes advantage of the strong vertical correlations in ocean temperature data to minimize the cost. Ridgway et al. (2002) interpolated irregularly spaced ocean temperature and salinity observations onto a regular grid using a loess filter that takes into account data at neighboring vertical levels as well as the influence of bathymetry and land barriers.

Here, however, rather than interpolating existing data, we wish to improve global ocean model estimates of the marine climate across the shelf. Downscaling is an approach often used to perform such a task. It provides the link between the large-scale state of some variable, for which there is typically coarsely gridded information, and the smaller-scale state of that or another variable, for which there is no accurate scale-relevant information (von Storch et al. 1993; Benestad et al. 2008). For example, global climate models provide reasonably accurate predictions of global surface air temperature at coarse spatial resolutions (e.g., 1°–2° in latitude and longitude) but cannot reliably predict smaller-scale variations due to the presence of unresolved features such as clouds or steep orography. There are two approaches to downscaling: dynamical downscaling and statistical downscaling (e.g., Murphy 1999).

Dynamical downscaling nests one or more fine-resolution general circulation model(s) within the domain of a coarse-resolution circulation model. Dynamical downscaling allows for important physical processes to be resolved at high resolution in the part of the domain of interest, while the remainder of the domain is represented using the coarse-resolution large-scale model. Dynamical downscaling is commonly applied to global climate model output (e.g., Giorgi 1990; Christensen et al. 1998; Murphy 1999; Corney et al. 2010), but it has also been used in an ocean-only context using nested grids (e.g., Oey and Chen 1992; Guo et al. 2003; Oliver 2011). Alternatively, a heterogeneous grid spacing can allow for specific regions to be resolved more finely than others, within the same grid [e.g., the Australian marine region (Oke et al. 2008), or the tropical Pacific Ocean (Zhang et al. 2010)]. An extension of dynamical downscaling is telescoping, in which multiple nested models are used to focus in on a particular region of interest (e.g., Christensen et al. 1998; Guo et al. 2003; Trapp et al. 2007). For example, Trapp et al. (2007) used global 2.5° × 2.5°–resolution National Center for Atmospheric Research (NCAR) atmospheric reanalysis data as the coarse-scale information and nested a higher-resolution (~55 km) regional climate model within a domain covering the continental United States. A convection-permitting model, with a horizontal resolution of less than 4 km, was further nested within the regional climate model to allow for convective storm behavior to be resolved over small-scale local domains. The most significant downside to dynamical downscaling is that the required circulation models are costly to develop, implement, and validate.

Statistical downscaling is an alternative approach that takes the observed statistical relationship between the large-scale variability (e.g., large-scale general circulation features) and local variability (e.g., station measurements)—a relationship that depends on processes not resolved by coarse climate models—and uses it to “downscale” climate model output and provide predictions of local variability at some point location. One of the simplest approaches is a multivariate linear regression model in which the local variability is regressed onto a set of predictors taken from the large-scale dataset (e.g., Hellstrom et al. 2001; Huth 2002). More complex methods include the use of principal components or canonical correlations analysis (e.g., von Storch and Zwiers 2002; Huth 2002), stochastic weather generators (e.g., Wilks 1999; Palutikof et al. 2002; Qian et al. 2002), and neural networks (e.g., Zorita and von Storch 1999), among others. Statistical downscaling has primarily been used to downscale climate for the terrestrial environment (e.g., Wilby et al. 1998) and has not been used as extensively for the marine environment; existing downscaling studies for oceanic variables have primarily focused on ocean wave climate (e.g., Wang et al. 2010). In general, statistical downscaling is more computationally efficient than dynamical downscaling, but it lacks the dynamical consistency between variables provided by the latter approach.

Our focus here, however, is not strictly about statistical downscaling per se, but rather about what we would call “statistical boundary scaling.” Given the nature of the oceanographic climate problem, the approach presented in this paper does not scale coarse predictions to a finer resolution over the same domain. Instead, we use accurate and robust ocean data from the larger-scale geographic domain—specifically, deeper ocean points proximal to our shelf region of interest—to provide estimates in the domain where the ocean model performs poorly (here, the coastal ocean) based on the observed relationship between those two domains. Unlike statistical downscaling, the statistical boundary-scaling approach presented in this paper does not relate the local finescale variability to the local coarse-scale variability; rather, it relates the local coastal ocean variability to the nonlocal open-ocean variability. We are careful here to detail our approach thoroughly, and contend that this approach is likely to have utility for application more generally to coastal scaling problems outside of the Australian context, which we use here as a case study example of its potential.

Australia has over 35 000 km of coastline and nearly 85% of the population lives within 50 km of the sea (ABS 2001). These marine-related and coastal climate issues are particularly important for Australia, given their proximity to people, its biodiversity and resources (ecosystem services), and amenity. The Ocean Forecasting Australia Model (OFAM) provides the highest-resolution model information available for short-term (7 days) ocean forecasts at an approximate 10-km scale for the ocean environment surrounding Australia. Further, the Bluelink Reanalysis (BRAN) product provides ocean-reanalyzed daily estimates of ocean variability over the 1992–2008 satellite-observing period using the data-assimilative eddy-resolving OFAM, used primarily for ocean and climate diagnostic studies. BRAN reproduces well the large-scale circulation around Australia. For example, the volume transport of the Indonesian Throughflow, the Leeuwin Current, the South Australian Current, and the eddy-rich East Australian Current (EAC) are all well captured (Oke et al. 2008; Schiller et al. 2008). However, since OFAM is designed for the primary purpose of modeling features of the open-ocean circulation rather than the bounding coastal and shelf regions, the BRAN output in space and time tends to evaluate poorly against available across-shelf observations in contrast to the open ocean, where BRAN performs very well. For example, nearshore daily SSTs have been shown to be poorly represented by BRAN, especially along the coast of southeast Australia (Hobday et al. 2011).

To summarize our approach tested here in the Australian context, separate model components are developed for the mean SST, the SST seasonal cycle, and the residual SST variability across southeast and southern Australian marine waters (domain shown in Fig. 1). The model parameters for each component are determined from the observed relationships between AVHRR SSTs on the continental shelf and offshore. For example, the residual component is simply a linear regression of nonseasonal shelf SST variability onto nonseasonal offshore SST variability. The model components are then informed by estimates of offshore SST from BRAN to provide daily estimates of inshore SST on a

This paper is organized as follows. The data sources and a preliminary statistical analysis are presented in section 2. The statistical model is described in section 3. In section 4, the model SST estimates are initially evaluated at a point location in Bass Strait and then extended to the entire continental shelf in three regions: (i) Tasmania and Victoria, (ii) the Great Australian Bight, and (iii) New South Wales and southern Queensland. Model refinements, based on a Lagrangian connectivity analysis utilizing the mean circulation information, are presented in section 5. Conclusions and further work are discussed in section 6.

## 2. A basic quality assessment of BRAN SSTs

Daily fields of reanalyzed SSTs were obtained from BRAN, version 2.1, for the period 14 October 1992–30 December 2006 and from BRAN, version 2.2, for the period 29 November 2006–13 May 2008. The two datasets were merged by a linearly weighted average over the 30 days of overlap between them, and the combined dataset is referred to hereafter simply as BRAN. Daily BRAN SST fields are denoted *m* × *n* matrix, where *m* and *n* are the number of grid points in the latitudinal and longitudinal directions, respectively, and *t* is a time index.

Daily fields of observed SSTs were obtained from AVHRR for the period 14 October –13 May 2008 and are defined on a 4-km-resolution grid. Only the nearest AVHRR grid point to each BRAN location is retained, and these fields are denoted _{t} (also an *m* × *n* matrix). We would like to note that while the AVHRR SST observations have been assimilated into BRAN, there is still a large degree of independence between BRAN and AVHRR over the continental shelf because (i) only SST data in water depth greater than 200 m are assimilated (P. Oke 2013, personal communication), and (ii) only 1 day out of every 7 days of SST data is assimilated and therefore, to the extent that we can ignore serial correlation in time, 6 out of every 7 days of SST data are independent (Oke and Schiller 2007).

_{t}and

*T*

_{ijt}at each location (

*i*,

*j*) can be written aswhere

_{t}. Separate quality functions—

^{S}, and

*x*| denotes the absolute value of

*x*,

*ρ*(

*x*

_{t},

*y*

_{t}) denotes the temporal correlation (e.g., Priestley 1981) between

*x*

_{t}and

*y*

_{t}, and

*T*

_{0}is a reference temperature (taken to be the freezing point of water, i.e., 0°C). These equations are invariant under a linear transformation of variables (e.g., a change of units). In each case, a quality of one is perfect correspondence between that component of

_{t}; values less than one indicate a discrepancy between

_{t}.

In general, the quality of the mean, seasonal cycle, and residual variability of ^{S} is low in the same regions, except coastal New South Wales; and

## 3. Statistical model

*i*

_{X},

*j*

_{X}) and an inshore location (

*i*

_{Y},

*j*

_{Y}):andwith offshore and inshore SST from BRAN,

*X*

_{t}and

*Y*

_{t}(i.e., the submodels are trained on observed SST). Then a predictor time series

The choices of the offshore and inshore locations—(*i*_{X}, *j*_{X}) and (*i*_{Y}, *j*_{Y}), respectively—are in general free. The choice of the inshore location is given by the location at which a prediction of SST is desired. The choice of the offshore location (i.e., the predictor location) is also free, although in section 3d we outline a relatively simple but systematic method of optimally choosing the offshore predictor location, based on jointly maximizing the fundamental relationships between *X*_{t} and *Y*_{t}.

### a. Mean

*X*

_{t}and

*Y*

_{t}are scalars and are related linearly bywhich leads to a simple expression for the parameter

*Y*

_{t}is

### b. Seasonal cycle

*X*

_{t}and

*Y*

_{t}can be written as a sum of

*K*harmonics of the annual cycle:andrespectively, where

*A*

_{k},

*φ*

_{k}, and

*ω*

_{k}are the amplitude, phase, and frequency of the

*k*th harmonic, respectively. These amplitudes and phases are estimated from

*X*

_{t}and

*Y*

_{t}by harmonic regression (see appendix A for more details).

*γ*

_{k}and Δ

_{k}:Given a predictor variable

*Y*

_{t}can be predicted by the model

### c. Residual

*β*is the regression coefficient and

*ε*

_{t}is an error term representing effects not included in the model. However, we note that the predictors are imperfect observations of SST and are subject to some measurement error

*η*

_{t}is a noise term representing measurement error. This is known as an “error in variables” regression model and, as an estimator of the regression coefficient, the ordinary least squares estimate

*λ*is known as the “reliability ratio.” The reliability ratio accounts for the attenuation of the regression coefficient due to measurement error and is given by

*η*

_{t}, respectively. We take a value of

*σ*

_{η}= 0.25°C for the measurement error on the SST observations (e.g., Oke et al. 2008; Schiller et al. 2008).

*Y*

_{t}can be predicted by the simple modeland the total inshore SST can be estimated as

### d. Choice of the predictor location

It is possible to use different predictor locations for each of the individual submodels outlined above. This is desirable, as the mean, seasonal cycle, and residual SST at the same inshore location may be best estimated separately by offshore SST predictors at different locations. For example, the best location from which to predict the mean may not be the best from which to predict the residual variability. There may be good physical reasons why this might be observed, for example, different physical processes operate on different time scales and may lead to differing spatial relationships. We show later that mean circulation (connectivity) can be considered one such physical process.

*i*,

*j*). The quality matrices were defined for the mean, the seasonal cycle, and the residual components [see Eqs. (2)–(4)]. Second, the strength matrix

*i*

_{Y},

*j*

_{Y}) and each location (

*i*,

*j*). The strength matrices for the mean, seasonal cycle, and residual components are defined in a similar way as the quality matrices:andwhere, as above,

*T*

_{0}is a reference temperature (0°C). The strength matrices are equal to 1 at (

*i*

_{Y},

*j*

_{Y}) and decrease away from that location. Finally, the localization matrix

*i*

_{Y},

*j*

_{Y}) with a decay scale

*σ*,

## 4. Application of the statistical model in southeastern Australia

The statistical model was used to estimate SSTs on the continental shelf in southeastern Australia. First, the model was tested at a point location in Bass Strait (section 4a). Then, the method was applied to the entire southeast Australian continental shelf including Victoria and Tasmania, the Great Australian Bight, New South Wales, and southern Queensland (section 4b).

### a. Estimating SST at a point location in Bass Strait

Time series of observed SST (*Y*_{t}) and BRAN SST (*Y*_{t}. The mean SST is underpredicted by 0.4°C (*ρ* = 0.972 and root-mean-square error (RMSE) of 0.503°C, middle panel of Fig. 3]. The residual variability also compares poorly with the observations (*ρ* = 0.488) with an RMSE of 1.01°C (bottom panel of Fig. 3).

The linear statistical model described in section 3 was used to predict SST at 40°S, 147°E. The locations of *T*_{ijt} at each location (*i*, *j*) given by Eqs. (2)–(4). The

The strength matrices are represented by a comparison between *Y*_{t} and *T*_{ijt} across the domain given by Eqs. (19)–(21). The

The inverse of the product of these three fields gives the cost matrix [see Eq. (23) and Figs. 4c,f,i], the minimum of which corresponds to the location of

The linear statistical model provides a substantially more accurate estimate of SST in Bass Strait (*ρ* = 0.998, a slight improvement of 0.026) and the RMSE has reduced by about two-thirds to 0.157°C. The residual variability is much better correlated (*ρ* = 0.692, an improvement of 0.204) and the RMSE has reduced by about half to 0.535°C. Time series of

### b. Estimating SST across the continental shelf

The approach described in the previous section (section 4a) was applied across the entire continental shelf. The statistical model was systematically applied to each location with water depths less than 200 m off the coasts of Victoria and Tasmania [section 4b(1)], the Great Australian Bight [section 4b(2)], and New South Wales and southern Queensland [section 4b(3)]. The cost function was masked for all locations in water depth less than 200 m to exclude those locations as potential predictors. The combined field of all

#### 1) Tasmania and Victoria

The model performs well at estimating SST on the continental shelf around Tasmania and Victoria. The quality matrix for the mean is shown in Fig. 5a. The

#### 2) The GAB

The model also provides substantially improved estimates of SST on the continental shelf around the Great Australian Bight (GAB). The quality matrix for the mean is shown in Fig. 6a. The

#### 3) New South Wales and southern Queensland

Model performance on the continental shelf off New South Wales and Southern Queensland is more mixed than in the previous two regions. The quality matrix for the mean is shown in Fig. 7a. The statistical estimates

## 5. Connectivity with the mean circulation

An analysis of the optimal locations of the offshore predictors based on our previous criteria was also performed. The continental shelf around Tasmania and Victoria was chosen as the case study region. The shelf was divided into five regions (northwest, northeast, Bass Strait, western Tasmania, and eastern Tasmania), and these have been color coded along with the corresponding offshore predictor locations from the model for residual SST variability (see Fig. 8, top panel). Predictors tend to lie either (i) along the shelf break immediately adjacent to the inshore region that they predict or (ii) offshore and predominantly to the west (~84% of all predictors in this region lie to the west of the southern tip of Tasmania). The predictors also tend to lie in regions where the mean surface flow, determined from daily fields of meridional and zonal surface currents from BRAN, is strong and continuous between the shelf and the offshore region (see Fig. 8, top panel, gray arrows). This suggests that connectivity between the shelf and offshore regions may be a useful physically based metric for optimal predictor selection.

**x**(

*t*) = [

*x*

_{1}(

*t*)

*x*

_{2}(

*t*)]

^{T}. Given a location at time

*t*, the location at some time Δ

*t*later can be approximated bywhere

**u**(

**x**) is the mean flow at location

**x**. Similarly, the trajectories backward in time can be approximated byThe mean flow was calculated from BRAN at fixed locations in space across the

**u**at the exact location

**x**(

*t*) was determined by bilinear interpolation. In our algorithm, trajectories stop moving if they reach the coast or the open boundaries. Trajectories forward and backward in time starting from the same initial location are denoted

**x**

_{f}(

*t*) and

**x**

_{b}(

*t*), respectively. To simulate turbulent motion, we add a random kick to the particles at each time step by replacing

**u**(

**x**) with

**u**(

**x**) + Δ

**u**. The random kick velocity Δ

**u**is a Gaussian random number with a mean of zero and a standard deviation given by

*A*

_{h}is a horizontal eddy diffusivity (e.g., Brickman and Smith 2002).

*τ*

_{200,ij}is the total time spent over the continental shelf (water depth less than 200 m) by both

**x**

_{f}(

*t*) and

**x**

_{b}(

*t*) starting from an initial location (

*i*,

*j*) and integrated for a total

*T*. If a trajectory spends the entire time on the shelf, either backward or forward, then

*C*

_{ij}= 1, and if it never reaches the shelf, then

*C*

_{ij}= 0; otherwise, 0 <

*C*

_{ij}< 1. Connectivity around Tasmania and Victoria was calculated for each location in the domain bounded by 35°–48°S and 132°–160°E. An ensemble of 500 forward and backward trajectories (with

*τ*= 300 days, e.g., 150 days forward and 150 days backward; Δ

*t*= 0.5 days;

*A*

_{h}= 4 m

^{2}s

^{−1}) was calculated at each location to yield 500 connectivity matrices [calculated according to Eq. (26)], which were then averaged at each location to give the final

By including the connectivity as an integral part of the model, we have moved from a purely statistical model to a hybrid statistical–connectivity model. The power of such a model is that changes in the connectivity (e.g., under a changing climate) can be incorporated into the model to modify it accordingly. This concept can be generalized to a statistical–dynamical model in which information related to ocean dynamics (in our case, only mean circulation) are explicitly incorporated into the statistical model.

## 6. Conclusions and future work

The approach presented in this paper provides a systematic statistical method for the estimation of across-shelf ocean climate variability (here, the case study of sea surface temperature in an Australian marine context was presented) across a range of time scales. The mean SST, SST seasonal cycle, and residual SST variability on the shelf were predicted using estimates of offshore SST from an ocean reanalysis product (BRAN). Independent linear statistical models were developed to predict each of these components, and the model parameters were informed using the observed (e.g., from AVHRR) relationship between SST offshore and SST on the continental shelf. A method of optimally choosing the offshore predictor using three criteria (data quality, strength of offshore/shelf connection, and proximity) was developed. Finally, the model was refined into a simple hybrid statistical–connectivity model by including passive tracer connectivity, based on the mean surface circulation, as a further criterion for predictor selection. The approach presented in this paper represents a potentially useful method for generating homogeneous daily SST fields on the continental shelf and provides a framework for estimating other climate fields (e.g., sea surface salinity and surface chlorophyll a) at relatively low computational cost.

The model was applied to the continental shelf around southeastern Australia. The statistical model predictions of across-shelf SST are generally much closer to the observations than those available directly from BRAN—especially around Tasmania, Victoria, and the Great Australian Bight. The mean and seasonal cycle are much better predicted. Typical RMSEs for the seasonal cycle were reduced by up to 1°C (60%–80% reduction in Bass Strait and nearly 100% reduction in the Great Australian Bight). The residual variability was also better predicted, except very near to the coast. The RMSEs in central Bass Strait and the upper Great Australian Bight are demonstrably reduced by 0.5°C (50% reduction in Bass Strait and 75% reduction in the Great Australian Bight). Prediction quality for coastal New South Wales and southern Queensland were more mixed, and it is not clear that the statistical model provides better predictions in those regions. Overall, however, our model shows significant promise as a methodological approach for the estimation of across-shelf SST in terms of accuracy, robustness, and potential utility beyond the Australian case study example.

In the simplest statistical model approach, predictor locations were identified by minimizing a cost function depending first on three factors: quality, strength, and proximity. We do not presume that this is an exhaustive list of factors that would influence the choice of predictors. They were selected here based on a combination of intuition and oceanographic understanding and allowed us to develop a relatively simple model. The inclusion of other relevant factors in the analysis, once identified, would be relatively simple as an addition to the model formulation. For example, we have been able to readily expand the definition of the cost function to also include a simple measure of connectivity, creating a hybrid type of statistical–connectivity model.

Connectivity between the shelf and the open ocean is defined in terms of Lagrangian trajectories, given the mean surface circulation. We found that predictors that lie in regions with strong connectivity to the coast provided accurate and spatially smooth results. While a purely statistical approach, such as localized EOF analysis, could also lead to model improvements, our connectivity-based approach has the strength of a hybrid statistical–physical model, in that we are informed about from where the information is coming. However, the connectivity statistics should be improved with a more complete treatment and consideration of the full three-dimensional and time-varying flow field, and not only the mean flow. These connectivity statistics could be calculated by extending our technique appropriately, or they could be obtained, for example, from projects such as the Marine Connectivity Interface (Connie2, http://www.csiro.au/connie2/; Condie et al. 2011). We chose to use the mean circulation as the connectivity metric of relevance in our demonstration of concept study here since analysis of the predictor locations indicated that, to first order, it was the physically based oceanographic feature (i.e., the circulation) that linked SST variability between the shelf and offshore regions. It is clear that the inclusion of this physically based metric provided improved model results and greater spatial robustness of the model predictions.

It may be possible to further refine the model by considering frequency-dependent models of residual variability. The “gappy” nature of the observed SST time series prevents the application of a fully spectral approach, but this could be partly achieved by splitting the residual variability into more than one time series. For example, monthly means of the residual variability could be calculated and this time series would represent the “low frequency” residual variability. The remainder would represent the “high frequency” residual variability. It is possible that different mechanisms are at play on different time scales. Separate models for each could then be constructed, allowing for different predictor locations and regression coefficients. Additionally, other predictors could be included in the model (local winds, climate modes of variability, etc.), as it is likely that different mechanisms may lead to predictability on different time scales. Seasonally dependent regression coefficients may also improve the residual SST prediction.

A possible weakness of the statistical model is that each submodel is informed by a single offshore location. A more complex model version might instead utilize information from multiple locations to inform each inshore SST prediction. There are several ways in which this could be achieved. For example, the residual could be predicted by multiple linear regression, with the offshore predictors selected to maximize the amount of on-shelf variance explained. A more general approach would be to use kernel regression, with the kernel defined in a similar way to the cost matrix. This could then be generalized to the submodels for the mean and seasonal cycle (which are not presented as simple linear regression models).

The AVHRR data were provided by the Group for High Resolution SST (GHRSST) and the U.S. National Oceanographic Data Center. The authors thank Matthew Chamberlain and Simon Wotherspoon for the helpful discussions. The authors acknowledge funding from the Australian Research Council Super Science Fellowship (Grant FS110200029). The authors also acknowledge the two anonymous reviewers for their helpful and insightful comments.

# APPENDIX A

## Estimating the Seasonal Cycle by Harmonic Regression

*x*

_{t}, assume that its seasonal cycle

*K*harmonics of the annual cycle:where

*A*

_{k},

*φ*

_{k}, and

*ω*

_{k}are the amplitude, phase, and frequency of the

*k*th harmonic, respectively. The

*ω*

_{k}are written aswhere

*L*is the length of the year (in units of the sample rate of

*x*

_{t}, for example,

*L*= 12 for monthly data). These frequencies correspond to a period of 1 yr for

*k*= 1, 1/2 yr for

*k*= 2, 1/3 yr for

*k*= 3, etc. Note that Eq. (A1) is equivalent towhere

*φ*

_{k}= arctan(

*b*

_{k}/

*a*

_{k}). We use

*K*= 3 in this study.

*K*of regression coefficients,

**s**

_{t}is a vector of length 2

*K*of sines and cosines, and

*ɛ*

_{t}is an error term representing effects not included in the model. The

*i*th element of

**s**

_{t}is given by:andThe estimated regression coefficients

*a*

_{k}and

*b*

_{k}are given by the (2

*k*− 1)

^{th}and (2

*k*)

^{th}element of

*A*

_{k}and

*φ*

_{k}using the relations given above.

# APPENDIX B

## Cost Function Sensitivity to Relative Weightings of Terms

*w*

_{Q},

*w*

_{S}, and

*w*

_{R}are scalar values constrained to lie between 0 and 1. A value of 1 gives full weighting to that term and a value of 0 gives no weighting to that term. For example, weighting values of (

*w*

_{Q},

*w*

_{S},

*w*

_{R}) = (1, 0.5, 0.2) would give full weighting to the quality, 50% weighting to the strength, and 20% weighting to the localization.

A suite of sensitivity calculations were performed to test the sensitivity of the boundary-scaling model to the relative weighting of the terms in the cost function. For each sensitivity test, the model was fit to the entire continental shelf around the Great Australian Bight, Tasmania, Victoria, New South Wales, and southern Queensland (as in section 4), and the quality values and RMSEs at all grid points were collected together, summarized by their probability distribution (Fig. B1). There were small variations in the mean and spread of the results for different weighting choices, which indicated that the quality appears to be the most important term and localization the least important term (in the sense that these results improve as these terms are weighted higher and lower, respectively). However, considering the overlap in the spread of quality values and RMSEs across all sensitivity runs, including the equally weighted model with (*w*_{Q}, *w*_{S}, *w*_{R}) = (1, 1, 1), we can conclude that the results are not strongly sensitive to changes in the relative weighting of the three terms. Therefore, in the interest of simplicity, we have elected to weight all the terms equally.

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