## 1. Introduction

We use a recently developed measure-theoretic method that requires minimal assumptions, can be used to identify optimal measurement locations, and can be applied to any computational model to estimate parameters. The focus is on coastal ocean modeling during a hurricane with specific attention on critical parameters affecting storm surge as a way to introduce to the community this advanced mathematical approach. We lay the groundwork for future research, which could apply this methodology to atmospheric parameters in a coupled atmospheric–ocean hurricane model. This future application could use both water surface elevation and meteorological data to improve estimates of parameters critical to accurate hurricane forecasting such as the central pressure, radius of maximum wind, maximum wind speed, forward speed, and the hurricane track. Furthermore, this methodology is general enough that it is broadly applicable to any physics-based predictive model assuming that the quantities of interest map varies smoothly with respect to small perturbations to the model parameters.

### a. Storm surge and Manning’s n

In recent years a succession of hurricanes has caused significant damage to communities along the U.S. coastline, with the majority of damage and loss of life due to storm surge (i.e., inundation caused as hurricanes make landfall). During a hurricane, the balance between momentum gain from wind stress and momentum loss from bottom stress is one of the primary controlling factors of storm surge (Schubert et al. 2008). Water levels and currents during a storm surge event are usually modeled by the shallow-water equations (SWEs), with bottom friction described using a standard parameterization, such as the Gauckler–Manning–Strickler formula. In this formula, the variation of bottom friction due to varying roughness is described through the Manning’s *n* parameter. Correctly characterizing bottom friction was shown to be especially important in hindcasting Hurricane Ike (Zheng et al. 2013; Martyr et al. 2013; Kerr et al. 2013; Kennedy et al. 2011). It is therefore critical to determine the Manning’s *n* values in order to ensure accurate and precise predictions of storm surge.

### b. Uncertainty in Manning’s n

Manning’s *n* is currently used to model momentum loss due to bottom stress in a variety of coastal applications, but determination of a proper value of Manning’s *n* in any model is fraught with uncertainties. On the modeling side, Manning’s *n* as developed in the Gauckler–Manning–Strickler formula only relates average square cross-sectional velocity, hydraulic radius, and the channel bed slope for open-channel flow in a fully turbulent fairly regular unvegetated domain (Gioia and Bombardelli 2001; Yen 1992b,a, 2002; Leopold et al. 1964). Manning’s *n* can be empirically calculated for environmental flows that fit within the original scope of the Gauckler–Manning–Strickler formula by sampling the diameters of roughness elements or from field or laboratory measurements (Myers et al. 1999; Barnes 1967; Bathurst 1985; Clifford et al. 1992). However, the Gauckler–Manning–Strickler formula is commonly used beyond this original scope to model flow resistance for various environmental flows (Forzieri et al. 2011; Schubert et al. 2008). For environmental flows, Arcement and Schneider (1989), along with Chow (2008) and Barnes (1967), have provided guidelines for selecting Manning’s *n* that extend the original domain of application to flood plains, vegetated water ways, and channels with irregular boundaries. There are multiple subgrid processes that cause momentum loss, two of which are 1) the bottom friction due to bed forms and bed material (sand, silt, mud, etc.) and 2) the form drag due to flexible vegetation. Manning’s formula only represents bottom friction, but is used as a convenient model to capture form drag caused by vegetation (Lane 2014; Wamsley et al. 2010; Kouwen and Unny 1973). Regardless, complex models require a representation of momentum loss due to subgrid processes such as bottom friction, form drag due to vegetation, bed forms, and various porous-media-like structures that are present in coastal estuaries (Dietrich et al. 2010, 2011a; Mignot et al. 2006).

It is possible to determine the Manning’s coefficient through hydraulic calibration for a specific geographic location (Orlandini 2002). Unfortunately, using this process to develop a Manning’s *n* field for a highly resolved finite element (FE) mesh over a large, complex physical domain, such as those often used in storm surge modeling, would be extremely cost prohibitive. Alternatively, a Manning’s *n* value can be developed for specific types of land cover. The Coastal Change Analysis Program (C-CAP) Regional Land Cover database (NOAA) and the National Land Cover Database (NLCD, USGS) document the geographical distribution of different types of land cover using a land cover classification scheme. We use these datasets to determine the spatial distribution of land cover classifications (Agnew 2012; Bunya et al. 2010; Schumann et al. 2007; Medeiros et al. 2012; Wamsley et al. 2010; Tao and Kouwen 1989).

Because of the modeling error and uncertainty inherent in empirical observations discussed above, the actual values of Manning’s *n* are subject to a great deal of uncertainty. This uncertainty must be quantified, and reduced wherever possible, in order to improve inferences drawn from storm surge predictions.

### c. Quantifying uncertainties in Manning’s n

Model input parameters such as Manning’s *n* are often inferred by solution to an inverse problem using model output data sensitive to the model input parameters of interest. The primary goal of this paper is to demonstrate how a measure-theoretic approach for inverse sensitivity analysis can be used to both *quantify and reduce* uncertainties in the inputs of a storm surge model. Specific focus is given to how this novel mathematical approach can aid in the design of observation networks in order to increase precision in quantified uncertainties. While the approach used here is quite general, we focus on the important Manning’s *n* parameter in the Bay St. Louis area using simulations of Hurricane Gustav (2008) to design spatial configurations of buoys recording maximum water elevation data. This is a compelling case study and also serves to make the mathematical ideas concrete. To simulate data from Hurricane Gustav, we use the Advanced Circulation model for oceanic, coastal, and estuarine waters (ADCIRC). The ADCIRC model solves the SWEs on highly resolved, unstructured FE meshes (Luettich and Westerink 2004, 2010).

There are many types of uncertainty quantification (UQ) methods currently adopted in the scientific community for quantifying uncertainties in the inputs of a model. Below, in order to locate and distinguish the contribution of this work within the vast field of UQ research, we provide a brief literature review of some of the more popular UQ methods and draw clear distinctions to the measure-theoretic approach.

First, to provide a common framework for understanding the differences between UQ methods that formulate and solve inverse problems, we introduce some general notation. Define the quantities of interest (QoI) as the set of observable output data of a physics-based model used to formulate the inverse problem. The QoI define a physics-based map from the input parameter space, which we denote by *set-valued* function. Essentially, the mapping between the input and output is nonunique in that a single datum maps to a set of parameters rather than a single point in parameter space. This is what is often referred to as ill-posedness of the inverse problem. Another complication to the formulation and solution of the inverse problem is that observed data, even from the computational model, are often uncertain due to measurement, modeling, and discretization errors. It is common to describe these uncertainties in terms of probabilities (e.g., using a probability measure

The set-valued inverses of the QoI map impart a significant amount of geometric information on the parameter space that should be respected if we are to maintain confidence in the physical interpretation of solutions to the inverse problem. This is precisely the advantage of using a measure-theoretic framework for the formulation, solution, analysis, and numerical approximation of the inverse problem for scientific inference (Breidt et al. 2011; Butler and Estep 2013; Butler et al. 2012, 2014; T. Butler et al. 2014, unpublished manuscript, hereafter B14). We have previously studied this problem for the hydrodynamic model of an idealized inlet in Butler et al. (2015). The takeaway message is that the solution to the stochastic inverse problem is a nonparametric probability measure that fully exploits the geometric information imparted by the physics-based QoI map. A major contribution of this work is a demonstration that the geometry can be further exploited to aid in optimal experimental design, which here means choosing the optimal spatial configuration of buoys to record maximum water elevations and reduce uncertainties in Manning’s *n*. This is significantly different than other work in this community that has studied the effectiveness of targeted measurements without the use of such geometric analysis to mixed results (e.g., see Hamill et al. 2013).

Approaches for solving the core deterministic inverse problem of mapping a datum to parameters often form the foundation of UQ methods for solving stochastic inverse problems. One common approach for deterministic inversion is to “regularize the map” or use “regularization,” which implies that the map *Q* is fundamentally changed, usually in favor of defining a misfit functional mapping between *Q*, which comes purely from the physics in the model, creates a new mapping between the spaces

Alternate methods for solving stochastic inverse problems include Bayesian approaches (Yang et al. 2007; Hall et al. 2011). A Bayesian approach often formulates the solution on *Q* often appears as an argument of the likelihood function [e.g., using discrepancies between the model-predicted value of *Q* with a statistical map given by the likelihood. As with regularization based approaches, since the map is fundamentally different between the spaces

In the coastal ocean modeling and atmospheric science communities, data assimilation techniques based upon the ensemble Kalman filter are perhaps the most commonly used methods for estimating both parameters and state variables of models (Mayo et al. 2014; Aksoy 2015; Ruiz et al. 2013). The attractiveness of these type of data assimilation methods are both their ease of implementation and ability to provide so-called real-time updates to parameter estimates and state variable forecasts as new data become available. In Ruiz et al. (2013), it was shown that proper estimation of parameters can have positive effects on predictions both in the short term and in medium-range forecasts. However, one of the most difficult steps in using data assimilation techniques is the initialization step. For example, in Mayo et al. (2014), it was shown that incorrect initialization of Manning’s *n* values in Galveston Bay lead to final ensemble estimates of Manning’s *n* values that are incorrect by as much as 63% despite relatively little error in predicting water elevations from tidal forcing. Fundamentally, this is due to the fact that the inverse of the map from Manning’s *n* to water elevation data is set valued. The consequences of such incorrect estimates of Manning’s *n* values for predictions of storm surge during extreme events are uncertain. We demonstrate that it may be possible to exploit the measure-theoretic approach to provide better initialization of an ensemble of model parameters in real-time data assimilation schemes. In fact, since the probability measures are often nonparametric, this approach may in particular enhance initializations of ensembles for particle filtering methods (van Leeuwen 2009). While a full exploration is outside the scope of this study, we provide some results showing how to utilize the solution to the measure-theoretic solution to generate and analyze ensembles of Manning’s *n* fields. Moreover, open source software, data files used in this study, and instructions are provided online for the interested reader to reconstruct the numerical results and experiment with different applications of these results.

### d. Outline

We describe the state-of-the-art ADCIRC model in section 2. In section 3, we give a concrete description of the measure-theoretic approach for quantifying uncertainties as well as considerations in designing experiments in the context of storm surge and Manning’s *n*. In section 4, we provide details on the setup of the parameter and data spaces for the Hurricane Gustav case study as well as computational details on the simulation. In section 5, we show numerical results for the solution to the stochastic inverse problem to estimate Manning’s *n*, and we describe how choices of buoy configurations affect the ensembles of Manning’s *n* fields generated from the solution to the inverse problem. Finally, conclusions and directions of possible future research are provided in section 6.

## 2. ADCIRC coastal circulation model and Manning’s *n*

*n*

*h*is the bathymetric depth, and

*ζ*is the free surface above the geoid or datum then the continuity equation iswhere

*Q*. Note that in ADCIRC the continuity equation (2.1) is replaced by the generalized wave continuity equation (GWCE) for superior accuracy properties (Luettich and Westerink 2010).

*g*is gravity,

*f*is the Coriolis parameter,

*η*is the Newtonian equilibrium tidal potential. The terms

*n*is a spatially heterogeneous parameter denoting the Manning’s

*n*roughness coefficient. For

*H*sufficiently small,

*H*is small (i.e., estuaries, tidal flats, shallow channels, etc.) the bottom stress is especially sensitive to Manning’s coefficient.

## 3. Measure-theoretic inversion for Manning’s *n*

*n*

To the authors’ knowledge this is the first time the measure-theoretic approach used in this work for formulating and solving stochastic inverse problems has been presented to the intended audience of this journal. We therefore provide a brief but completely abstract description of the measure-theoretic approach in appendix A, the computational algorithm for computing approximate nonparametric probability measures in appendix B, and some details about the structure of solutions that lead to optimal experimental design in appendix C. The interested reader may use these appendices as well as the references therein to formulate and solve additional measure-theoretic inverse problems in their desired research applications. Here, we summarize the framework and formulation of the stochastic inverse problem from Butler and Estep (2013) and B14 placed in the context of the given application.

### a. Formulation of the stochastic inverse problem

In this application, the ADCIRC model provides a mapping from the Manning’s *n* values to the space of temporally variable free surface elevations and velocities. The mapping is complicated to study for a variety of reasons; for example, the Manning’s *n* values enter into the ADCIRC model nonlinearly through the bottom stress terms [(2.1)–(2.4)] and the values of Manning’s *n* are uncertain for a variety of reasons (see section 1). Moreover, the space of functions defining the free surface elevations and velocities is an infinite dimensional space. Practical considerations demand the determination of a finite set of observations that are useful for quantifying uncertainties in Manning’s *n*. Since the ADCIRC model has been well validated against maximum storm surge data (Bunya et al. 2010; Dietrich et al. 2010; Kennedy et al. 2011; Hope et al. 2013; Dietrich et al. 2011a), we limit the focus in this study to observation stations recording maximum free surface elevations (i.e., maximum storm surge).

Another key consideration in the use of this data is that small perturbations to input parameters do not lead to arbitrarily large variations in such output data. This has been numerically observed in the responses of the ADCIRC model in a previous work demonstrating both continuity and differentiability of maximum water elevation data with respect to Manning’s *n* (Butler et al. 2015). Thus, given a defined set of observation stations and a compact set of physically possible Manning’s *n* values associated with a set of land classification types, the range of observable data can be well approximated by evaluating the ADCIRC model using a finite sampling of Manning’s *n* values.

The use of the ADCIRC model and maximum storm surge data cannot, in general, determine a unique vector of Manning’s *n* values. However, given the smooth response of the storm surge data to Manning’s *n*, the inverse set of Manning’s *n* values associated to a particular output datum is identifiable and can be approximated (see appendix A). This implies that the map from Manning’s *n* to storm surge data defines a bijection between *sets of Manning’s n values* and maximum storm surge data. Conceptually, it is useful to think of this simply as a way to generalize a contour map.

By simple properties of random variables/vectors, if a set of maximum storm surge data is assigned a particular probability, then the set of all Manning’s *n* values that map to this set must also be assigned the same probability. In other words, any description of uncertainty in the maximum storm surge data as a probability measure uniquely defines a probability measure on the generalized contour map of Manning’s *n* values. This is one type of solution to the stochastic inverse problem, but it is not particularly satisfying since we generally would prefer to be able to compute the probabilities of more general sets of Manning’s *n* values and not be forced to work with “contour events” that are complicated to approximate.

Obtaining the type of probability measure on the space of Manning’s *n* values we desire requires the adoption of an ansatz. The details of this are rather technical requiring understanding of a disintegration theorem (see appendix A). Suffice it to say that we simply proportion out probabilities onto subsets of the set-valued inverses according to their relative volumes (although other types of ansatz are possible). This particular ansatz further exploits the generalized contour map resulting in geometrically complex nonparametric probability measures solving the inverse problem (e.g., see the marginal probability plots in section 5).

### b. Approximation of the inverse solution

The basic idea is to exploit the smoothness of the maximum storm data with respect to Manning’s *n* values to reduce evaluation of the ADCIRC model to a small number of samples of Manning’s *n* values. Specifically, the collection of samples of Manning’s *n* values implicitly defines a partitioning of the space of all possible Manning’s *n* values into nonoverlapping subsets. Evaluation of the ADCIRC model at each sample of Manning’s *n* values defines a particular maximum storm surge datum. Then, having discretized the probability measure on the space of maximum storm surge data, the smoothness provides the justification for identifying which subsets of Manning’s *n* values approximate which contour events of specified probability. Finally, the ansatz is applied to proportion the probability of a contour event to each subset approximating it. The end result is a probability measure defined on the original space of Manning’s *n* values. The interested reader is referred to appendix B for a more thorough description of the details on the algorithm used in this work.

### c. The condition of the stochastic inverse problem and optimal observations

In Butler et al. (2015), we saw that a quantifiable aspect of the QoI map defined as skewness provides a type of condition number for the stochastic inverse problem. We provide some brief remarks on how this may be used to identify optimal configurations of buoys (i.e., how we may define and solve an optimal experimental design problem). Here, we have chosen to focus on optimal buoy configuration; however, this methodology is applicable to any set of potential observations including sets composed of varying types of observables. For more details, we refer the interested reader to appendix C.

First, to define the perspective in which we consider one configuration of buoys to be more optimal than others, we consider a general application of the solution of the stochastic inverse problem, which is to sample the resulting probability measure in order to generate an ensemble of predictions. The predictions can either be for data at future times or for unobservable data at current (or even past) times (e.g., where there are no experimentally verifiable data due to limitations in the ability to deploy buoys). Thus, for simplicity, we assume the goal of choosing a particular configuration of buoys used to solve the stochastic inverse problem is to subsequently provide ensembles of predictions such that the mean of the ensemble is more accurate and the variance of the ensemble is reduced.

We note that the problem studied in this work is nominally a problem of parameter identification under uncertainty (i.e., there is a true vector of Manning’s *n* values leading to the noisy/uncertain maximum storm surge data). If a particular configuration of buoys leads to more probability concentrated in *smaller sets* containing the true Manning’s *n* value than other configurations of buoys, then we expect that the accuracy of a prediction ensemble mean is improved and variance of the ensemble is reduced. In appendix C, we connect the skewness of a map to this basic idea. We show in the numeric results that this skewness provides a reasonable surrogate for determining which possible configuration of buoy stations is optimal for solving the inverse problem and investigate the effect on predictions. Given this knowledge we can then algorithmically select which observations would be best used for a particular set of parameters. When there are many types of observations to choose from these may simply be added to the space of potential observables from which we choose the subset of observables with the smallest skew as defined in appendix C. An interesting consequence of the measure-theoretic perspective and skewness results is that we often find that “data rich” problems where many QoI are recorded are actually “information poor” in the sense that the information contained in many QoI is redundant and fails to improve the solution to the stochastic inverse problem in meaningful ways. While a full discussion of this is beyond the scope of this work, we demonstrate in the numerical results of section 5 that removing certain components of a “poorly skewed” QoI map may have negligible effect on solutions to the stochastic inverse problem. On the other hand, a careful selection of a small number of optimal (or nearly optimal) buoy stations can provide significant precision implying that the overall cost of collecting useful data may be mitigated by such an analysis.

## 4. Hurricane Gustav case study

In this section we examine parameter estimation of Manning’s *n* using a hindcast of Hurricane Gustav.^{1} We focus on Bay St. Louis, a coastal region in southern Louisiana and Mississippi affected by Hurricane Gustav. Hurricane simulations on meshes fine enough to resolve inundation are computationally expensive. To reduce this cost we employ a recently available subdomain implementation of ADCIRC (Baugh et al. 2013; Simon 2011; Altuntas 2012; Baugh et al. 2015). This also allows us to focus on specific regions of interest. It is important to note that this methodology can be applied to any region of the coast.

### a. Model development and hindcast details

The full-domain ADCIRC model for Hurricane Gustav is based on the model in Dietrich et al. (2011a) using a mesh of the domain containing the bathymetry/topography data shown in Fig. 1 developed over a number of years by various researchers. The corresponding Manning’s *n* field for the full-domain is shown in Fig. 2. We use the same wind field and riverine inflows developed in Dietrich et al. (2011a). After a 20-day tidal and riverine spinup we begin wind forcing at 15-min intervals starting at 0000 UTC 26 August 2008 (approximately 6.5 days before landfall) and ending at 0000 UTC 4 September 2008 (approximately 2.5 days after landfall). However, this model differs from the hindcast in that we do not model waves with Simulating Waves Nearshore (SWAN) (Dietrich et al. 2011b). The full-domain model contains 2 720 591 elements and using P(arallel) ADCIRC (PADCIRC) runs to completion on 288 processors on the Stampede supercomputer at the Texas Advanced Computing Center (TACC 2015) in approximately 11.5 h.

We reduce the cost of forward model evaluations by using Subdomain ADCIRC (Simon 2011; Altuntas 2012; Baugh et al. 2013, 2015). Results are shown for a subdomain extracted from the full-domain mesh. This subdomain was chosen for the proximity to existing observation station data sites and geographic interest (e.g., inundation and complex local land classification distributions). This subdomain focuses on Bay St. Louis in Mississippi, contains 15 001 elements, and is shown in Fig. 3. Subdomain ADCIRC forces the boundary of the subdomain with the conditions from the full-domain: velocity, sea surface height, wet/dry condition (Simon 2011; Altuntas 2012; Baugh et al. 2013, 2015). A full 29-day hurricane simulation of the subdomain runs to completion on Stampede in approximately 5.5 h on two processors (this time includes reconstructing and distributing the Manning’s *n* mesh across the processors). For more details of the steps involved in subdomain modeling in ADCIRC, we refer the interested reader to appendix D.

### b. Defining the parameter domain

We apply domain-specific knowledge inherent in the land cover classification system to reduce the dimension of the parameter space due to spatial variation. Additionally, subdomains can feature as few as 4 dominant land classifications whereas the full domain may feature as many as 23 land classifications thus further reducing the dimension of the parameter space (Dietrich et al. 2011a; Agnew 2012). This *drastically* reduces the dimension of the space of Manning’s *n* values while retaining both a highly spatially variable and high-resolution representation of the Manning’s *n* field.

The parameter space for any subdomain model is determined as in Butler et al. (2015) using a mesoscale representation of a Manning’s *n* field parameterization by land cover classification. The result is a linear mapping from *M* is the total number of nodes in the FE mesh and *n* field using *fixed* values of Manning’s *n* coefficients (Agnew 2012; Butler et al. 2015). We refer the interested reader to Butler et al. (2015) for a detailed description of the construction of the land cover classification meshes.

In this parameterization the uncertain parameters *n* coefficients associated with each land cover classification. We use land cover classification data from C-CAP (NOAA Coastal Services Center 2013) to create a set of land cover classification meshes. We fill in regions not covered by the C-CAP data with existing Manning’s *n* nodal data from the original mesh used for the hindcast. The Manning’s *n* coefficients used for the full-domain model are included in Table 1, the values on the continental shelf in the Gulf of Mexico are set to

Dominant land cover classifications for the subdomain as extracted from the global domain. Nodes that are not specified by the C-CAP data have been included in the open-water category.

We determine the dominant land cover classifications by calculating the total fraction of each land cover classification as the sum of the nodal fractions of the land cover classification normalized by the total number of nodes that are tabulated by percentage in Table 1. We define our uncertain parameters as the Manning’s *n* coefficients for the first four dominant land cover classifications and hold the Manning’s *n* coefficients constant for the remaining land cover classifications. We allow the Manning’s *n* coefficients to vary between 33% and 175% of their original values based on tables available in Furniss et al. (2006). The resulting four-dimensional parameter domain is shown in Table 2 and the corresponding land cover classification meshes are shown in Fig. 4. These define a specific mesoscale representation of the Manning’s *n* field that is parameterized by the dominant land cover classifications in the physical domain. This parameterization removes the consideration of spatial uncertainty in the bottom friction field (as opposed to using a Karhunen–Loève expansion) thus significantly reducing the problem size. In the event such land cover classification data were unavailable, satellite images and airborne lidar could be used to estimate the number of dominant land cover classification (Jung et al. 2014; Song et al. 2002; Antonarakis et al. 2008; Brennan and Webster 2006; Im et al. 2008; Zhou et al. 2009).

Manning’s *n* coefficient ranges for the subdomain.

### c. Defining the data domain

Here, we describe an approach for determining optimal buoy configurations recording the simulated maximum storm surge data that define the QoI map in order to reduce uncertainties in the stochastic inverse problem. As discussed in section 3 and appendix C, an advantage of the measure-theoretic approach is that a relatively small number of carefully chosen buoy stations defining the QoI can provide good precision results in the solution to the stochastic inverse problem. We therefore limit focus to the practical problem of deploying a small number of buoy stations (in this case, at most four) to record maximum storm surge data in Bay St. Louis. The motivation is that since the cone of uncertainty of a hurricane’s trajectory narrows significantly 24–36 h prior to landfall, there is a limited window of time to either deploy new, or reconfigure existing, buoy stations. The numerical results demonstrate the significant impact a carefully chosen configuration of buoys can have on the quantification and reduction of uncertainties.

The configuration of buoys is based upon a preliminary study using a small set of ADCIRC model solves that can be analyzed quickly once the cone of uncertainty narrows. Specifically, in the subdomain of interest, we computed approximate gradient data of proposed QoI using finite forward differences in order to determine the skewness of possible QoI maps. A total of 16 clusters defining a total of 80 samples of Manning’s *n* values shown in Fig. 5 in the parameter space were chosen at random to approximate the gradients of the various proposed QoI.

We note that the numerical experiments herein use the full high-resolution ADCIRC simulation that implies every node in the FE mesh is a potential observation station. Examining every possible combination of maximum water elevation at every point in the FE mesh results in an infeasibly large set of possible observation stations to examine. We therefore interpolate the FE solution onto a set of hypothetical observation stations on a regular grid and then further reduce the problem size by limiting the set of possible observation stations to the set of stations where the maximum storm surge data exhibits sensitivity to the parameters. Specifically, we consider only the stations where the maximum change in maximum water elevation over the set of forward model solves associated with evaluating the samples in the clusters shown in Fig. 5 is greater than 0.10 (m). This results in a set of 194 possible observation stations (see Table 3).

Number of possible data spaces for the subdomain.

We use the Butler–Estep–Tavener (BET)^{2} python package (Graham et al. 2015) to optimally choose QoI to result in a well-conditioned stochastic inverse problem with optimal (approximate) skewness (Walsh 2015). Specifically, we use the average condition numbers of the Jacobians of the possible QoI maps on the set of 16 clusters of parameter samples mentioned above as an approximation to the skewness of a QoI map. For each possible QoI map *Q*, the average condition number is denoted by

QoI map skew numbers for the subdomain.

## 5. Numerical results for measure-theoretic inversion

We define and solve five stochastic inverse problems to quantify uncertainty in Manning’s *n* values using five different QoI maps. In the numerical results below, we use the notation ^{3} centered at *n* field defined by a particular *λ*.

### a. Probability densities

We compare the results of the stochastic inverse problem for three QoI maps using optimal stations

Figures 10 and 11 (and 13) show the resulting approximate marginals of *n* values based on using a particular QoI map, is almost identical in both shape and size for both cases. For example, the size of this set as measured by ^{−3} for the optimal QoI choice and 8.390 × 10^{−3} for the near-optimal QoI choice. Subsequently, the solutions to the stochastic inverse problems associated to these two QoI maps are nearly identical.

Notice that even though we have the most difficulty in identifying

Ratios of wetted nodes for the subdomain. “Total ratio” represents the ratio of wet nodes to the total number of nodes, where the number of wet nodes is weighted by the nodal contributions of each parameter. “Wet ratio” represents the ratio of wet nodes to the total number of wet nodes, where the number of wet nodes is weighted by the nodal contributions of each parameter. Here wet nodes are defined as nodes that are wet at any point during the simulation.

The nonoptimal QoI map stations ^{−2} is about 250% greater than that of *Q* is formed from an optimal or near-optimal set of stations. We can examine the marginals that result if we only use a subset of the nonoptimal stations

*i*th station. For the stations not used in the inversion based on the optimal and near-optimal choices of buoys, the reduction in ranges were on average 89% compared to an uninformed analysis. We tabulate the reduction in ranges for the optimal stations

The reduction in ranges of hindcast simulations for maximum water elevations at various stations based on different solutions for the stochastic inverse problem. The first column lists the station numbers used for solving a stochastic inverse problem where the optimal stations are given by *i* is defined as

### b. Generating ensembles of Manning’s n

We now describe the process of generating an ensemble of samples from the nonparametric probability measures that solve the measure-theoretic inverse problem. Such ensembles can be used as part of an initialization step in data assimilation or other predictor/corrector forecasting schemes. Here, we focus on the description of the process using the available code and the analysis of such ensemble results in terms of the effect of the QoI map on the structure of the resulting ensemble. We refer the interested reader to appendix E for more information on how to obtain the data files and code to generate initial ensemble estimates of Manning’s *n* values for any application in this subdomain of Bay St Louis.

There are multiple ways of generating an ensemble of samples in *o*” as optimal or choice “*n*” for near optimal. We denote the QoI maps being considered as

In Table 7 we report the sample statistics for a single ensemble of 50 and 100 bootstrap samples and in Table 8 we report sample statistics for 100 such ensembles of bootstrap samples. We denote the estimate of the mean and variance for the *i*th ensemble of bootstrap samples by

Sample statistics for a single set of samples generated via bootstrapping. The first pair of rows represent results when the samples are drawn according to the

Sample statistics for 100 sets of samples generated via bootstrapping. The first pair of rows represent results when the samples are drawn according to the

Another point of emphasis is the accuracy of certain point estimates from an ensemble of samples generated from either *k*th sample of the *i*th ensemble of bootstrapped samples to be *i*th ensemble is then

Error sample statistics for a single set of samples generated via bootstrapping. The first pair of rows represent results when the samples are drawn according to the

Error sample statistics for 100 sets of samples generated via bootstrapping. The first pair of rows represent results when the samples are drawn according to the

*i*th ensemble of bootstrap samples aswhere

*i*th ensemble of bootstrap samples. The number of unique samples associated with the

*i*th ensemble of bootstrap samples is then

*i*th ensemble. Since

Sample statistics for a single set of samples generated via bootstrapping. The first pair of rows represents results when the samples are drawn according to the

Sample statistics for 100 sets of samples generated via bootstrapping. The first pair of rows represent results when the samples are drawn according to the

## 6. Conclusions

We formulated and numerically solved a stochastic inverse problem involving spatially heterogeneous Manning’s *n* fields parameterized by land cover classification data using maximum water elevation data obtained from a hindcast of Hurricane Gustav using the ADCIRC model. The novel measure-theoretic framework and computational algorithm used was based on the authors’ previous work (Butler et al. 2015; Breidt et al. 2011; Butler and Estep 2013; Butler et al. 2012, 2014; Graham 2015). This previous work explored the condition of the inverse problem defined in terms of the skewness of contour events. However, this previous work did not address the practical computation of skewness numbers nor use those skewness numbers to inform experimental design through the choice of QoI maps based on the underlying geometric structure of the generalized contours to obtain probability measures with small support (relative to using other QoI maps). We have done both in this work and provided a tool that automates this process. We have extended the definition of the inverse problem for the Manning’s *n* field from the relatively simple idealized inlet example to a complex hurricane hindcast example with a higher-dimensional data space. We employed a recently available subdomain implementation of ADCIRC (Baugh et al. 2013; Simon 2011; Altuntas 2012; Baugh et al. 2015) to reduce simulation time and focus on specific areas of interest rather than the much larger domain required for hurricane simulations.

We wish to emphasize that the methodology for parameter estimation presented herein is model agnostic, meaning that it widely applicable to many physics based models. Generally, the steps required to apply this methodology are as follows: 1) define the parameter space, often this involves domain or expert knowledge and a discretization of the parameter; 2) determine the optimal set of QoI for estimating this particular set of parameters, this often involves estimating gradient or Jacobian information through a preliminary study or sensitivity study; 3) sample the parameter space and run the computational model for these parameters; 4) use the resulting parameter data to solve the stochastic inverse problem using BET (Graham et al. 2015); and finally 5) use these results to generate ensembles from the resulting probability measures to enrich future predictions or calculate probabilities of events. If the parameter space is high dimensional, either spatially or in terms of types of parameters, it might be useful to develop a surrogate or reduced-order model for your system or reduce the dimensionality of your parameter space.

We have also demonstrated that the solution of the stochastic inverse problem can be used to generate an ensemble of samples in the parameter space whose accuracy and precision is dependent on the skewness of the QoI map. A map with less skewness generally results in more accuracy and precision. Moreover, when using bootstrap sampling, we showed that a map with less skewness may result in smaller ensemble sizes defined by the unique subset of samples. This has the potential to lower the computational complexity in a data assimilation scheme when parameters are initialized by such ensemble generation, and is the topic of future work.

## Acknowledgments

L. Graham’s work is supported by Department of Energy Office of Science (DE-SC0009324 and DE-SC0010678) and the National Science Foundation Graduate Research Fellowship (DGE-1110007). T. Butler’s work is supported in part by the National Science Foundation (DMS-1228206). C. Dawson’s work is supported by the National Science Foundation (DMS-1228243). C. Dawson would also like to acknowledge the support of the Gulf of Mexico Research Initiative Consortium for Advanced Research on Transport of Hydrocarbons in the Environment (CARTHE). T. Butler and C. Dawson recognize the support of the U.S. Department of Energy Office of Science, Office of Advanced Scientific Computing Research, Applied Mathematics program under Awards DE-SC0009286 and DE-SC0009279 as part of the DiaMonD Multifaceted Mathematics Integrated Capability Center. S. Walsh’s work is supported by the National Science Foundation (DMS-1228206). J. J. Westerink’s work is supported by the National Science Foundation (DMS-1228212) and supported by the Henry J. Massman and the Joseph and Nona Ahearn endowments at the University of Notre Dame. We thank the developers of Subdomain ADCIRC, J. Baugh, A. Altuntas, T. Dyer, and J. Simon at North Carolina State University, for access to their code without which this work would not have been possible. Any opinion, findings, and conclusions or recommendations expressed in this material are those of the authors(s) and do not necessarily reflect the views of the National Science Foundation. The author(s) also acknowledges the Texas Advanced Computing Center (TACC 2015) at The University of Texas at Austin for providing HPC resources that have contributed to the research results reported within this work.

## APPENDIX A

### Measure-Theoretic Inversion Outline

The following is a brief summary of the abstract framework for the formulation and solution of stochastic inverse problems using measure theory. For more details, especially as they pertain to the existence and uniqueness of solutions, and error analysis of approximate solutions, we refer the interested reader to Breidt et al. (2011), Butler and Estep (2013), and Butler et al. (2012, 2014).

Let *Y* (often called the solution to the model) to model input parameters *λ*. Solutions to the model, *Y*, are implicitly functions of the parameters *λ*. However, no system is completely observable and we are often interested in a small number of directly observable QoI defined as functionals of the state. Since the state variables are implicitly a function of parameters, so are the QoI, and we make this dependence explicit in the notation below. We denote the vector-valued QoI map *d* can be any positive integer, but we are particularly interested in the mathematically challenging (and common) case, where

We now describe the minimal assumption required for formulating and applying the measure-theoretic approach in practice. We assume that the QoI map *Q* is piecewise smooth (i.e., that it is differentiable almost everywhere). The notion of differentiability implies some assumption of metrics or norms on the input and output spaces so that distances between points can be computed. With this minimal assumption, *σ*-algebras

Defining probability measures *consistent* with the data (i.e.,

Here, we give some basic terminology useful in describing the structure of solutions to the inverse problem. Assuming for simplicity, that the Jacobian of *Q* has full rank, the implicit function theorem guarantees the existence of locally smooth *Q*. When *Q* a *generalized contour*, which defines an equivalence relation on

We denote the measure space of equivalence classes imposed by *σ*-algebra and measure on *not* require additional assumptions. It is possible to explicitly represent *d*-dimensional manifold in *transverse parameterization*, see Fig. A1.

We emphasize that it is not necessary to explicitly construct the transverse parameterization or generalized contours in order to solve the inverse problem. However, we find it useful to describe the solution of the inverse problem in terms of these manifolds.

Note that a single point in *Q* defines a bijection between *contour event* in *σ*-algebra of contour events on *contour events* (Butler et al. 2014).

However, the goal is to determine a solution on the parameter space

**Theorem A0.1.**

*(Disintegration of probability theorem).*Assume that

To apply this result, we must obtain a family of probability measures on each measurable generalized contour space *Q* in the determination of the volumes

## APPENDIX B

### Algorithm for Approximating Inverse Measure

*SIAM J. Sci. Comput*., hereafter B16)where

**Algorithm 1:** A Sampled Based Approximation of the Inverse Density (B14; B16)

Within the for-loop of Algorithm 1, we use a discrete version of the ansatz to set the probability *not* require explicit construction of the Voronoi tesselation (B16).

## APPENDIX C

### Condition of the Measure-Theoretic Inverse Problem

We refer the interested reader to Butler et al. (2015) for an in-depth description of the condition of the inverse problem. This is somewhat similar to the condition number for a matrix, which we use to provide a more familiar mathematical context. Suppose *x* for a particular approximation method. Similarly, the accuracy of the numerical solution of the stochastic inverse problem depends on a “skewness” property of the Jacobian of *Q*, which can be used to inform the number of samples required in Algorithm 1 to compute accurate approximations of inverse sets

When

*inverse events*in

If the QoI map used for the stochastic inverse problem has large skewness, the solution of the stochastic inverse *both* the stochastic inverse problem and related prediction problems. Algorithms for estimating the skew and selecting optimal sets of quantities of interest are included in Graham et al. (2015).

## APPENDIX D

### Overview of Subdomain ADCIRC

The steps involved in subdomain modeling in ADCIRC are as follows:

- Create full-domain ADCIRC model.
- Create subdomain ADCIRC model:
- Extract input files from full-domain ADCIRC model.
- Create subdomain specific files.
- Link meteorological forcing files in the subdomain directory.

- Generate full-domain control file to control output of subdomain boundary forcing data.
- Run ADCIRC on the full domain.
- Extract subdomain boundary forcing data from the full domain.
- Run ADCIRC on the subdomain.

*n*values for nonboundary nodes ensuring a linear “buffer zone” between the full domain and subdomain Manning’s

*n*fields. We developed the Python package PolyADCIRC (Graham 2016) to efficiently conduct parameter sweeps of

*n*field, and other required input files for a particular ADCIRC simulation, PolyADCIRC runs batches of PADCIRC simulations and saves the output after each batch.

## APPENDIX E

### Scripts and Data

Please contact the corresponding author for data files and scripts used to generate initial ensemble estimates of Manning’s *n* values for any application in this subdomain of Bay St. Louis, which are available upon request through www.designsafe-ci.org. These files were procuded using ADCIRC v50 (adcirc.org), Subdomain ADCRIC (www4.ncsu.edu/~jwb/subdomain), PolyADCIRC (dx.doi.org/10.5281/zenodo.45083), and BET v1.0-dev (github.com/lcgraham/BET/commit/f4ac1bc0dc95eff2cd536f6148ff896c5994c90a).

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^{1}

An initial presentation of the stochastic inverse problem was included in Graham (2015).

^{2}

BET is a python-based package for solving stochastic inverse and forward problems within a measure-theoretic framework.

^{3}

We choose the lengths of the sides to be 10% the length of