1. Introduction

Investigating the effects of ducts on EM wave propagation can be accurately conducted using simulations that utilize the parabolic equation to solve Maxwell’s equations, provided the refractive conditions are known (Sirkova 2012). However, estimating the atmospheric properties that influence refractivity has its own limitations and as such, several approaches to modeling the environment have been developed, each with its own strengths and weaknesses. One approach is to use similarity functions [e.g., Monin–Obukhov (MO) similarity theory] and bulk environmental measurements (Garratt 1992; Rowland et al. 1996; Fairall et al. 1996, 2003). Similarity theory–based models, such as the Navy Atmospheric Vertical Surface Layer Model (NAVSLAM; Frederickson 2017) or the Coupled Ocean–Atmosphere Response Experiment (COARE) algorithm (Fairall et al. 1996, 2003), are restricted to the surface layer, have limited representation of physical processes due to model assumptions, and have nonunique solutions in stable conditions (Wang and Bras 2010). Another approach is to model the atmosphere using mesoscale numerical weather prediction (NWP) models like the Weather Research and Forecasting (WRF) Model (Michalakes et al. 2001), the Coupled Ocean–Atmosphere Mesoscale Prediction System (COAMPS; Hodur 1997),¹ or other similar models. However, because of their limited vertical resolution, particularly near the ocean surface, NWP model forecast variables are often blended with MO theory–based surface layer model variables, to increase resolution at the lowest altitudes.

Inversion methods represent another approach to estimating the refractive environment directly. Inversion methods use a measured EM signal, either clutter, or received power using separated transmitters and receivers (point-to-point radar measurements) (e.g., Rogers et al. 2000; Karimian et al. 2011; Penton and Hackett 2018; Pozderac et al. 2018; Pastore et al. 2022). Limitations of inversion methods include the adequacy of the parametric refractive model applied, nonunique solutions, and the various sources of uncertainty associated with the radar measurements such as the amount and location of data (Matsko and Hackett 2019). Rather than treat each of the approaches for characterizing the environment independently, one can seek to integrate them to estimate the environment more accurately or more quickly. A more rapid estimation method can allow for real-time characterization of the environment (as it evolves in time).

Karimian et al. (2013) merged an inversion method based on radar clutter observations with ensemble NWP model forecasts and found that use of the combined method yielded more accurate humidity and air–sea temperature difference estimates than using either method in isolation. This NWP–inversion hybrid method focused on evaporation ducts. Zhao and Huang (2014) also suggest that using NWP refractivity forecasts as a regularization term in an adjoint method inversion approach can improve accuracies and avoid convergence onto incorrect environments that predict the correct propagation, although the authors did not demonstrate this idea with NWP explicitly. Gordon et al. (2015) also discuss a “refractivity data fusion” approach that incorporates NWP forecasts into a refractivity from clutter inversion approach.

Here, we extend the ideas presented in these studies by incorporating NWP forecasts into an inversion method and evaluate whether the incorporation of this information reduces the duration or increases the accuracy of inversions for surface ducting conditions. Analogous to Gordon et al. (2015), we map ensemble NWP refractivity profiles onto the diagnostic parameters that the inversion is solving. Similar to ideas presented in Zhao and Huang (2014) and following the inversion method outlined in Penton and Hackett (2018), we evaluate a parametric model to describe surface ducting conditions and utilize genetic algorithms (GA) to solve the inversion problem by initiating and constraining the GA using parameters estimated from NWP ensemble forecasts. This incorporation of NWP forecasts into the GA is aimed at reducing the probability of converging onto an incorrect refractive environment, which can occur due to limitations of radio frequency data sampling (Matsko and Hackett 2019) and/or the influence of the sea state (Penton and Hackett 2018).

We test this fusion of NWP and the inversion approach in three numerical experiments performed with 13 surface ducting cases measured during the Wallops Island field experiment (Haack et al. 2010; Zhao et al. 2016), where one experiment does not use NWP and the other two introduce the NWP into the inversion approach in two separate ways. One way includes using the NWP ensembles to set the initial population used by the GA, while the other constrains the search space of environmental parameters estimated by the GA using NWP ensembles and also uses them for the initial population. These experiments allow investigation into the degree to which the NWP ensembles are able to aid the inversion process. We find that NWP ensembles can both reduce the duration and improve the accuracy of an inversion approach by restraining the GA search space, but not when used solely to set the GA’s initial population.

2. Data

a. Wallops Island field experiment

This study incorporates atmospheric data taken from an instrumented helicopter between 28 April and 4 May 2000 from the shoreline of Wallops Island, Virginia, to around 60 km offshore during the Wallops Island field experiment (Babin and Rowland 1992; Babin 1995, 1996; Haack et al. 2010). This coastal environment is favorable to a complex variety of spatially and temporally evolving refractivity structures including surface ducting scenarios. Furthermore, these data have been used to verify NWP data (Haack et al. 2010) and to assess the performance of NWP ensembles (Zhao et al. 2016).

Each helicopter dataset consists of vertical profiles of three different atmospheric variables: pressure, relative humidity, and temperature, measured simultaneously over ∼100 s at near-0.75-m vertical intervals within the lowest 150 m of the atmosphere (Babin 1995). Each helicopter dataset is used to calculate the vertical profile of atmospheric refractivity and modified refractivity via Eqs. (1) and (2).

Of all helicopter datasets measured during the Wallops Island field experiment, 13 exhibit modified refractivity profiles (M profiles) with clear surface ducting scenarios. These 13 profiles are used for analyses in this study and are referred to as separate surface ducting “cases.” Figure 1 shows M profiles during these surface ducting cases, and Table 1 indicates the start time of each M profile estimated from each of these helicopter datasets. The vertical distribution of each M profile is different, each containing between 48 and 86 points up to 110 m. No measurements of surface variables, and hence surface-modified refractivity, were made by the helicopter.

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Thirteen cases of M profiles that contain surface ducts in the lower 100 m of altitude calculated from helicopter datasets during the Wallops Island field experiment. Since helicopter datasets do not include surface refractivity, surface points are from extrapolation of nonlinear least squares fits to the parametric model explained at the beginning of section 3.

Citation: Journal of Applied Meteorology and Climatology 62, 9; 10.1175/JAMC-D-22-0127.1

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Table 1.

Details of M profiles used in this study, including: helicopter measurement start time, the initialization time of the COAMPS ensemble forecasts, the number of hours after initialization for which each NWP ensemble forecast is valid (forecast hour), and the distance (km) of the COAMPS grid location from each case’s helicopter profile starting location.

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b. COAMPS numerical weather prediction ensembles

COAMPS is used to generate ensembles of NWP forecasts used in this study, and are the same as those used in Zhao et al. (2016). Each ensemble contains 32 members that were generated by COAMPS using lateral boundary conditions and initial conditions estimated from the Navy Operational Global Atmospheric Prediction System (NOGAPS; Hogan and Rosmond 1991; Peng et al. 2004). Global perturbations seeding the NOGAPS ensembles are generated through the ensemble transform (ET) technique (McLay et al. 2008, 2010). Further information about mesoscale perturbations included in COAMPS’s initial conditions to obtain the ensemble forecasts can be found in Zhao et al. (2016); only a brief description is elaborated on below.

COAMPS is a high-resolution, nonhydrostatic mesoscale model based upon the Navier–Stokes equations that describe temporal changes in atmospheric variables: temperature, water vapor, wind components, turbulent kinetic energy, and five moisture species including clouds, rain, ice, snow, and graupel (Hodur 1997). To close the equations, some physical processes are parameterized, such as vertical turbulent mixing, surface energy exchange, and cloud microphysics. COAMPS has been used extensively since 1995 to provide operational forecasts to the U.S. Naval Fleet and in a research capacity by using historical case studies to validate and advance the modeling and forecasting skill (Haack and Burk 2001; Thompson and Haack 2011; Kulessa et al. 2017; Alappattu et al. 2022; Xu et al. 2022). The Wallops model setup comprised three nested domains with each inner grid having 3 times the horizontal resolution of the parent grid. All nests have 71 levels distributed vertically between 4 and 40 000 m, and 6 levels reside in the lowest 111 m, the lowest level being 4 m above the surface. In this study, we utilized only the highest-resolution third nest covering an area 484 km by 485 km, whose domain is represented in Fig. 2 and has an average horizontal grid spacing of 4 km. For the ensemble predictions, 32 COAMPS model runs were initialized 48 h prior to the period of interest using 32 sets of NOGAPS fields that had been seeded with observational data and the ensemble perturbations. At each subsequent 12-h forecast interval, current observational data were assimilated into forecasts to initialize the next 12-h forecast cycle, incrementally updating the fields and allowing mesoscale detail and vertical structure to develop and carry through the model during the experiment period.

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Region covered by the third-level nested grid from COAMPS utilized in this study. Locations of helicopter measurements within the domain are illustrated by the colored stars (see the legend).

Citation: Journal of Applied Meteorology and Climatology 62, 9; 10.1175/JAMC-D-22-0127.1

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COAMPS modified refractivity ensemble members that are closest to the nearest hour and location of each helicopter dataset (Table 1) are used in this study. The distance between the COAMPS grid location and the helicopter’s starting location is computed using the Haversine distance formula. Figure 3 shows an example comparison between the helicopter data and the associated COAMPS ensemble members’ refractivity vertical profiles for case 2. In Fig. 3, it is evident that the ensemble covered the range of M in the measured data, but no single forecast accurately reflects the measured refractivity, similar to the results published in Zhao et al. (2016). Notice that similar to the helicopter data, the ensemble data do not contain surface estimates of modified refractivity.

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An example of a COAMPS ensemble forecast and the corresponding helicopter refractivity profile for case 2 (see the legend).

Citation: Journal of Applied Meteorology and Climatology 62, 9; 10.1175/JAMC-D-22-0127.1

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3. Methods

Three numerical experiments are performed with the aforementioned 13 cases measured during the Wallops Island field experiment and coinciding NWP ensembles (Table 1). Helicopter refractivity datasets are fit to a parametric refractivity model, whose fit is scrutinized to ensure the parametric model is an accurate representation of measured surface ducting conditions. Fit helicopter profiles are used in a radar propagation simulation, and propagation predictions are used as simulated propagation measurements for an inversion method to determine refractivity, which is evaluated by comparing it with the refractivity used to generate the propagation predictions. Simultaneous NWP ensembles (see Table 1) are also fit to the parametric model, whose fits are examined for their accurate representation of the ensembles. Parameters from these ensemble fits are further used to initialize the inversion method. More details about the inversion method, the parametric refractivity model, and the propagation simulation are outlined in section 3a, while the numerical experiments are explained in section 3b.

a. Inversion method

The inversion method consists of a parametric refractivity model (PRM), a propagation simulation, the variable terrain radio parabolic equation (VTRPE) model (Ryan 1991), and the GA, where the last is the optimization engine for the inversion. The inversion method is driven by synthetic radar data generated from the VTRPE simulation using the fit helicopter profiles. The GA iteratively optimizes parameters of the PRM to best match the synthetic radar data.

The PRM is a modified version of the layered models implemented by Gerstoft et al. (2003) and Saeger et al. (2015):

$M(z)=\hspace{0.17em}{M}_0+\hspace{0.17em}\{\begin{array}{c}\frac{\mathrm{\Delta }M}{z_d}z,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}z\le z_d\\ m_{1}z-\hspace{0.17em}(m_1{z}_{\mathrm{d}}-\hspace{0.17em}\mathrm{\Delta }M),\hspace{0.17em}z>\hspace{0.17em}{z}_d\end{array},$(3)

where M₀ is surface modified refractivity, ΔM is M deficit (i.e., difference between the minimum refractivity and the surface refractivity), m₁ is refractivity slope above the duct, referred to as the mixed layer slope, and z_d is duct height. Refractivity parameters included in Eq. (3) (M₀, ΔM, m₁, and z_d) are estimated for helicopter profiles using an iterative nonlinear least squares regression technique, referred to as a PRM-fit, using initial estimates determined by visual inspection of the helicopter-based refractivity profiles. Table 2 illustrates resulting parameters estimated from nonlinear regressions for each case and refractivity root-mean-square errors (RMSEs) calculated over all altitudes. RMSEs are calculated by $R_M=\hspace{0.17em}{\left[{{\displaystyle \sum }}_{i=1}^{n_z}(M_i-{\widehat{M}}_i{)}^2/n_z\right]}^{1/2}$, where M_i is modified refractivity computed from helicopter measurements at one altitude, $\widehat{M_i}$ is modeled modified refractivity from Eq. (3) at the same altitude, and n_z is the number of altitudes. RMSEs illustrated in Table 2 are calculated between PRM-fit profiles $\widehat{M_i}$ and each helicopter M profile (M_i), where both profiles ($\widehat{M_i}$ and M_i) are down sampled to the same vertical resolution as the COAMPS data to allow direct comparison (i.e., n_z = 7) with the fit COAMPS ensembles (discussed in section 3b). The low RMSEs suggest that the four-parameter PRM provides a good fit to the measured surface ducts.

Table 2.

PRM parameters estimated from fits to measured helicopter data and coinciding RMSEs between the PRM and helicopter data down sampled to the same levels (n_z = 7) as COAMPS for each case.

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Radar wave propagation is simulated via VTRPE, which computes EM wave propagation in complicated environments including surface ducting scenarios using the parabolic equation (PE) approximation to predict electromagnetic fields (Sirkova 2012). VTRPE simulates the propagation factor:

$P_f=20\log \left|\frac{\mathbf{E}}{{\mathbf{E}}_0}\right|,$(4)

where E is the electric field of the modeled environment, and E₀ is the electric field in free space; P_f is used to compute propagation loss P_L:

$P_L=20\log 2k_{0}r-P_f,$(5)

where k₀ is the wavenumber of the electromagnetic wave, and r is slant range (Ryan 1991). For this study, VTRPE uses transmitter and domain properties shown in Table 3 and assumes a flat water surface, and refractivity is horizontally homogeneous. Although recent studies have shown that refractivity can be horizontally heterogeneous in coastal zones (Ulate et al. 2019) and could have nonnegligible effects on radar wave propagation (Goldhirsh and Dockery 1998; Greenway 2020; Xu et al. 2022), this work assumes homogeneity because it is a commonly made assumption and the focus of this work is on fusing NWP ensembles into the inversions (rather than on ducting heterogeneity effects). Figure 4 shows an example of a VTRPE synthetic P_L pattern based on the fit helicopter M profile for case 6.

Table 3.

Unvaried parameters used in propagation simulations associated with the transmitter, sea surface dielectric properties, and numerical domain.

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Propagation loss pattern simulated by VTRPE using refractivity from the PRM-fit helicopter M profile for case 6.

Citation: Journal of Applied Meteorology and Climatology 62, 9; 10.1175/JAMC-D-22-0127.1

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The optimization method, which utilizes GA, follows closely the methods outlined in Penton and Hackett (2018), where sets of refractive parameters (i.e., ΔM, m₁, and z_d) are used to build M profiles, and each M profile is used in the radar propagation simulation (VTRPE). P_L produced from the simulation (i.e., Fig. 4) is compared with a “known” P_L domain using a fitness function. The GA minimizes the fitness function reaching a set of refractive parameters (generating a refractive environment) that accurately portrays the known P_L domain. However, the current study differs from Penton and Hackett (2018) in that the inversion is being applied for a surface ducting scenario; hence, the PRM used is different but utilizes the same number of parameters. The current study also alters the initial population of the GA to incorporate information from NWP as well as limits the search space of the refractive parameters estimated by the GA based on the NWP data (these aspects are discussed more thoroughly in the next subsection). Similar to Penton and Hackett (2018), the inversion does not estimate the surface modified refractivity (M₀); it is assumed to be known and set to the surface value estimated by the nonlinear fits of each helicopter profile (see Table 2).

A weighted mean-square error (MSE) function is the fitness/objective function used in the GA (Penton and Hackett 2018):

$\widehat{\text{Γ}}=\hspace{0.17em}\frac{1}{2}{\text{Γ}}_{\text{LA}}+\hspace{0.17em}\frac{1}{4}{\text{Γ}}_{\text{LR}}+\hspace{0.17em}\frac{1}{4}{\text{Γ}}_W,$(6)

where Γ_LA represents P_L MSE for a low-altitude region between 0 and 60 m in altitude over all ranges, Γ_LR represents P_L MSE for a long-range region covering all altitudes over ranges 30–60 km, and Γ_W represents P_L MSE over the whole domain. Note that $\widehat{\text{Γ}}$ represents a fitness score in dB² and will be referred to as such. Weights for the various regions in the fitness function [Eq. (6)] are chosen based on sensitivity analysis results from Lentini and Hackett (2015). They find propagation loss is most sensitive to duct height z_d at low altitudes and to the vertical gradient below the duct height (i.e., ΔM) at long ranges. The low-altitude region has the highest weight because it is well established that z_d has significant effects on radar wave propagation. Table 4 includes the parameters for the GA setup; for further information regarding these settings, see Penton and Hackett (2018).

Table 4.

GA implementation parameters.

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b. Numerical experiments

This study includes 39 inverse solutions (13 cases; three experiments), which are computed in three numerical experiments. Two of the experiments incorporate NWP ensemble forecasts into the inversion process, and one experiment does not utilize any NWP data. In the first experiment, referred to as the Random experiment, inversions are carried out with a random initial GA population determined using a pseudorandom number drawn from a standard normal distribution that is used to select initial populates from within the following parameter ranges: 1–90 m for z_d, 0.001–1 M units per meter for m₁, and 0.001–45 M units for ΔM. All parameter estimates from the GA for this experiment are within this liberal but constrained search space, which is inclusive of typically observed ranges for these parameters (Sirkova 2015) and is comfortably inclusive of all parameter values from the helicopter-measured data and the NWP ensembles. The second experiment, referred to as the Ensemble experiment, fits ensemble numerical weather predictions corresponding with each helicopter case (Table 2) to the PRM, whose parameters are used as the initial population for the GA optimization. Specifically, each ensemble member acts as one member of the GA initial population. Unlike the random initial population (from the Random experiment), where it is possible that nonphysical combinations of parameters could be randomly selected, using an initial population based on the NWP ensembles guarantees all initial populates are physically realistic. The Ensemble experiment utilizes the same parameter search spaces as the Random experiment. For the third experiment, called the Spread experiment, the initial population is the same as the Ensemble experiment, but the parameter search spaces for the GA are determined from the range of the estimated ensemble members’ parameters for each case. These search spaces are shown in Table 5.

Table 5.

The constrained GA search space for each parameter based on the fit NWP ensemble data. The constrained search spaces are used in the Spread experiment.

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GA initial population ensemble parameter sets, used in both Ensemble and Spread experiments, are estimated with nonlinear least squares regression, like the helicopter profiles, except that the surface refractivity M₀ is not estimated in the regression and a weighting scheme is also applied. Coarse vertical resolution of the ensemble predictions in the lowest 100 m makes accurately estimating the parameters of the PRM more difficult than the helicopter data, particularly when estimating the surface refractivity M₀. Fewer data points means lower accuracy using nonlinear least squares fitting to estimate M₀ (i.e., the same method applied to helicopter profiles). Applying this technique to the NWP data results in converting many surface ducts to nonsurface ducts, so other means of estimating M₀ are explored. Using extrapolation to estimate the surface refractivity (M_0x) can result in profile changes that create nonsurface ducting conditions and largely depends on the slope between the lowest two points in the NWP (Fig. 3). Alternatively, surface refractivity can be calculated via Eq. (2) using the ensemble member’s estimated sea surface temperature (SST) and assuming a relative humidity of 98% (M_0S) (Buck 1981). However, similar to using M_0x, using M_0S as a prediction of surface refractivity can transform M profiles to nonsurface ducts. It is found that an average of both surface refractivity estimations [$\overline{M_0}=(M_0{}_x+\hspace{0.17em}{M}_0{}_S)/2$] created more consistent surface ducting scenarios (with respect to the rest of the profile) than either surface refractivity method in isolation. This average surface refractivity is used for each ensemble member. M profiles using $\overline{M_0}$ are subsequently shifted at all altitudes by the difference between $\overline{M_0}$ and the M₀ estimated from the helicopter profile closest in time and space to each respective ensemble profile (Table 2). This process ensures that the gradients of refractivity with altitude remain consistent with that predicted by the ensemble but allow the NWP and helicopter data to have the same surface refractivity (M₀), reducing the number of parameters the inversion has to estimate. The shifting is deemed acceptable because it is the vertical of the gradient of refractivity that affects propagation (Gerstoft et al. 2003; Zhao and Huang 2014). These shifted profiles are used in a weighted nonlinear least squares PRM-fit technique to estimate the remaining parameters (ΔM, m₁, and z_d). The weights are distributed such that the altitude closest to the initial guess of z_d is weighted double relative to other altitudes because of the strong influence of duct height on propagation (e.g., Lentini and Hackett 2015; Pastore et al. 2021). To evaluate the goodness of fit of the PRM to the ensemble forecasts, a histogram of RMSEs calculated between these shifted ensemble M profiles (M_i) and their respective PRM-fit profiles of each ensemble member ($\widehat{M_i}$) over all COAMPS altitudes (n_z = 7) is shown in Fig. 5, where each bin width is 1 M unit. Higher RMSEs (relative to the PRM-fit helicopter data shown in Table 2) reflect that the PRM fits the NWP ensembles less well than it does the helicopter data. These larger RMSEs are reflective of the fact that the least squares regression is based on far fewer altitudes for the NWP ensembles than the helicopter data; essentially, the refractivity slope within the duct is represented more poorly by the PRM for the NWP forecasts than for the helicopter data due to the low number of points below the duct height.

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Histogram of RMSE between each ensemble member profile and associated PRM-fit profile for all cases and all levels (n_z = 7).

Citation: Journal of Applied Meteorology and Climatology 62, 9; 10.1175/JAMC-D-22-0127.1

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Although using $\overline{M_0}$ to calculate the surface value produces ducting scenarios more frequently than using either method previously specified above in isolation, most cases (8/13) still have some ensemble members that do not predict surface ducting scenarios. Ensemble members that do not contain surface ducts are removed from initial GA populations because they are not suitable to be fit to the PRM [Eq. (3)]. Randomly selected parameters are placed in lieu of these ensemble members to ensure that all inversions use the same-size initial population regardless of member removal. At most, 7 (of 32) ensemble members were replaced.

Inversion solutions produced by the GA, which estimate the PRM parameters, z_d, m₁, and ΔM, are evaluated against the PRM-fit helicopter data using a variety of metrics. Percent errors are calculated between each parameter estimated by inverse solutions (z_d, m₁, and ΔM) and the corresponding parameters from fits of helicopter profiles (Table 2). RMSE is calculated using the M profiles of the inverse solutions [i.e., inverse parameters used in Eq. (3)] and the M profiles generated using estimated parameters of the helicopter profiles (Table 2) substituted into Eq. (3). Propagation loss (P_L) using the inversion-based M profile is compared with P_L patterns produced using the PRM-fit helicopter profiles via a weighted MSE [Eq. (6)]. Last, the duration of inverse solutions is evaluated by the number of generations required to reach a fitness score of less than 10 dB², which is well within the accepted frame of error for propagation studies [e.g., Goldhirsh and Dockery (1998) suggest 25 dB²]. All metrics are evaluated for each inversion solution in the three experiments to evaluate whether the addition of NWP data to the inversion process reduces the duration or improves the accuracy of the inversion.

4. Results and discussion

M profiles resulting from the three inversion experiments, along with modified refractivity based on helicopter measurements, are illustrated in Fig. 6. This figure highlights the wide variety of surface ducts considered in this study, including those with weak M deficits (Figs. 6a,c), strong M deficits (Fig. 6m), low duct heights (Fig. 6b), high duct heights (Figs. 6a,m), and a variety of mixed layer slopes. M profiles from inversion results suggest the refractivity above the duct height is the least accurate part of the modified refractivity profile predicted by the inversions. Furthermore, although there are several cases where both the Random and Ensemble experiments result in accurate M profiles, there are also many cases where these inversion experiments result in inaccurate M profiles. On the contrary, the Spread experiment frequently reproduced the M profile accurately, with the exception of case 10 (Fig. 6j).

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The M profiles from three inversion experiments (see the legend) with coinciding helicopter measurements of modified refractivity for each surface ducting case examined in this study.

Citation: Journal of Applied Meteorology and Climatology 62, 9; 10.1175/JAMC-D-22-0127.1

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Investigation of case 10’s Spread experiment results (Fig. 6j) revealed that the NWP ensemble did not encompass the refractivity parameters of the helicopter data; further, it was revealed that helicopter-based duct heights were also not encompassed within the NWP ensembles for cases 1 and 3 (Table 2). The parameter search spaces that do not include the helicopter-measured parameters could be due to the nonlinear least squares fitting of NWP ensembles to the PRM [Eq. (3)], or by the estimates of refractivity produced by the NWP ensemble. For the former, multiple methods of nonlinear regression were considered before settling on the method previously described in section 3b, which most often produced fits having RMSEs of 0–4 M units (Fig. 5), which is relatively low. Furthermore, ensembles of cases 1 and 3 produced RMSEs between 0 and 1 M units, while RMSEs for case 10 were between 0 and 5 M units, with a vast majority lying between 2 and 3 M units (still within the overall average). The relatively good performance of fits during these outlying cases leads the authors to think limitations of the NWP ensembles are the more likely reason for the search space inadequacies (Ulate et al. 2019; Zhao et al. 2016; Haack et al. 2010). Haack et al. (2010) state that a common difficulty of COAMPS during the Wallops Island field experiment was accurate prediction of the timing of ducting conditions. Since this timing issue may be the cause of the inaccurate PRM parameter estimates for cases 1, 3, and 10, the authors repeat the Spread experiment for these cases using multiple ensemble forecasts over an extended forecast window to set the search spaces for the inversions. Table 6 illustrates the number of forecast hours needed for the search space to encompass the helicopter data, which varies for each case. Table 6 also shows the resulting parameter search spaces for each experiment. Haack et al. (2010) report that COAMPS predictions tend to have a lag in the formation of surface ducts, and consistent with this result, most of the extended forecasts required to achieve a search space inclusive of the helicopter-based parameters are for forecast hours after the helicopter measurement time (Table 6). Figure 7 demonstrates that extended forecast hours for the Spread experiment improved the estimation of modified refractivity by the inversion process in cases 1 and 3 and especially case 10. As such, all future results and discussions will utilize these extended forecast windows in lieu of the single forecast for the Spread experiment in cases 1, 3, and 10 for the purposes of clarity and conciseness. It should also be noted that the range of each parameter’s search spaces (Table 6) is increased when including extended forecast windows during the Spread experiment; for case 10, the duct height spread (3.3–95 m) is quite similar to the search space set by the GA for the Random and Ensemble experiments (1–90 m).

Table 6.

Extended ensemble forecast hours used to estimate parameter search spaces for select Spread experiment cases. Note that boldface forecast hours are those used in the original Spread experiment. Boldface forecast hours do not reproduce parameter search spaces that contain the parameters estimated by the helicopter data (Table 2) but are forecasts that are closest in time to the helicopter measurements.

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The M profiles estimated by Spread experiment inversions for cases (a) 1, (b) 3, and (c) 10, where green lines show results using the closest time ensemble forecast to set the parameter search space of the inversion (i.e., the same results as shown in Figs. 6a,c,j; see Table 5), and blue lines show inversion results using an extended forecast window to set the parameter search spaces of the inversion (see Table 6).

Citation: Journal of Applied Meteorology and Climatology 62, 9; 10.1175/JAMC-D-22-0127.1

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RMSEs are calculated between each inversion-estimated modified refractivity profile ($\widehat{M_i}$) and each parametric model fit of the helicopter profile (M_i) for each case over all altitudes measured by the helicopter (n_z = 46–88; case dependent). Figure 8 illustrates these RMSEs. Please note that these RMSEs are not calculated at the same altitudes as those illustrated in Fig. 5 because these RMSEs are being used to assess the accuracy of the inversion solutions, rather than evaluate the PRM’s ability to represent vertical profiles of NWP ensembles or helicopter profiles as done in section 3; thus, these RMSEs should not be compared with those in Fig. 5 or Table 2. In general, most of the modified refractivity profiles are estimated to an error of less than 5 M units, with an obvious exception from the Random experiment in case 1. Furthermore, a few cases (3, 4, and 6) show higher RMSEs estimated by the Ensemble experiment than any other experiment; case 13 illustrates the highest RMSEs over the three experiments, and case 10 is the lone case whose RMSE is considerably higher during the Spread experiment (relative to the other experiments). On average, the Spread experiment inversions more accurately estimate the modified refractivity profile than either the Random or Ensemble experiments. However, all RMSEs are typically less than 3 M units, indicating modified refractivity was estimated accurately on average for all experiments. Although the generally low RMSEs of the Random experiment (∼1 M unit) suggest that using NWP ensembles may be unnecessary to accurately predict modified refractivity via inversion techniques, RMSEs are computed over the entire profile; previous work has shown that correct propagation predictions rely most heavily on accurate duct heights and M deficits (Lentini and Hackett 2015; Pastore et al. 2021), which are found to be less accurately estimated in the Random and Ensemble experiments, as discussed in the next paragraph.

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RMSEs between M profiles estimated by each inversion solution and the M profiles calculated from fits of the helicopter data to the PRM [Eq. (3)].

Citation: Journal of Applied Meteorology and Climatology 62, 9; 10.1175/JAMC-D-22-0127.1

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Figure 9 illustrates percent errors between the inversion parameter $\widehat{p}$ (i.e., z_d, m₁, and ΔM) and the same parameters estimated from nonlinear least squares regression fits of the helicopter profiles (p; Table 2) to the PRM [Eq. (3)] for each of the three experiments:

${\sigma }_p=\frac{\left|p-\widehat{p}\right|}{p}\times 100\%.$

Overall, the parameter with the lowest percent error for the inversion solutions was duct height z_d. This result is expected since the inversion optimizes the parameters using $\widehat{\text{Γ}}$ [Eq. (6)] and it is commonly reported that duct height is a primary driver of changes in propagation loss at microwave frequencies (Skolnik 1990; Lentini and Hackett 2015; Pastore et al. 2021). The least accurately estimated parameter throughout all cases is the mixed layer slope, where in a few cases (1, 3, and 6), mixed layer errors are over 100%. This result is also expected since the slope of the mixed layer has the least effect on propagation in comparison with other parameters (Lentini and Hackett 2015). In general, the Spread experiment inversions more accurately predicted all parameters. Surprisingly, the Ensemble experiment m₁ and ΔM percent errors have the most variability over the cases of all three experiments. These results suggest that using NWP to produce the initial population for the inversion process yielded limited benefits (and potentially detracted from the accuracy), but when NWP is used to constrain the search space, it can aid the inversion’s accuracy at estimating refractive parameters during surface ducting conditions. Notably, the greatest accuracy improvements occurred for the M deficit (ΔM) and mixed layer slope (m₁), while the duct height percent errors for all experiments were quite small. Furthermore, Fig. 9 shows the parameter values for each case (black dashed lines and left y axis), where it is evident that there is only a weak relationship between the percent errors of the parameters and the “true” values.

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Percent errors σ_p between each experiment’s inversion-estimated refractivity parameter and that based on the helicopter data, for each case. The corresponding parameter estimated from the helicopter data for each case is shown by the black dashed line referenced to the right y axis.

Citation: Journal of Applied Meteorology and Climatology 62, 9; 10.1175/JAMC-D-22-0127.1

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Figure 10 shows the fitness scores [Eq. (6); P_L MSE] for each case for all experiments. For 27 of the 39 cases examined, the inversions accurately predict P_L, evidenced as bars below the P_L fitness limit (Table 4); however, there are inversion results that did not reach the fitness limit (12/39). Of these cases, 5, 10, and 12 show higher P_L MSEs for the Spread experiment, while cases 1, 4, 8, and 13 P_L MSEs are highest during the Random experiment, and case 3 illustrates higher P_L MSEs during the Ensemble experiment. For cases where the Spread experiment showed larger P_L MSEs than the Random and Ensemble experiments, the difference in P_L MSE between the experiments was essentially negligible. Furthermore, these minor differences are mainly driven by positioning of the multipath nulls, for which a small change in position can impact the MSE statistic (Penton and Hackett 2018). On the other hand, cases with high P_L MSE in Random experiment cases (1, 4, 8, and 13) are significantly higher than that of the Spread experiment, suggesting that using NWP can improve the accuracy of inverse estimates of refractive parameters that influence P_L. Case 13 had the highest P_L MSE for all experiments, potentially due to the high duct height, which is higher than half of the altitudinal domain (∼60 m) and/or due to having the strongest M deficit (∼30 M units) of all cases. In fact, none of the inversion results for any of the experiments were able to predict case 13’s P_L below the GA’s fitness threshold within 60 generations (Table 4; 0.2 dB²; Penton and Hackett 2018). Overall, the results from the Spread experiment show the most accurate estimates of P_L; notably, all estimations of P_L throughout all cases and experiments are less than the commonly referenced “baseline error” of 5 dB (MSE of ∼25 dB²) mentioned by Goldhirsh and Dockery (1998).

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Weighted P_L MSE [Eq. (6)] between P_L based on inversion solutions and P_L predictions based on the fit helicopter data. The P_L fitness limit (Table 4) is shown by the black dashed line.

Citation: Journal of Applied Meteorology and Climatology 62, 9; 10.1175/JAMC-D-22-0127.1

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The results thus far have demonstrated that, on average, inversions were more accurate with the use of NWP to constrict the inversion search space; in addition, using NWP could also have advantages by reducing the duration required for the inversion to converge on an accurate solution. Obtaining solutions in less time could be integral for applications targeting approximately real-time environmental sensing. As such, we compare the duration required for the inversion technique to converge on an accurate solution using the number of generations required to estimate PL to an MSE threshold of less than 10 dB², as illustrated in Fig. 11. We use the number of generations as the duration metric because computational times vary considerably based on computer hardware; thus, the use of generations is a machine-independent metric of duration.

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Number of generations for each inversion to reach a P_L MSE [Eq. (6)] of less than 10 dB² along with associated statistics over all cases shown in the embedded text boxes.

Citation: Journal of Applied Meteorology and Climatology 62, 9; 10.1175/JAMC-D-22-0127.1

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Figure 11 shows that, most conclusively, the Spread experiment had the shortest inversion duration a majority of the time (9/13 cases), suggesting that the use of NWP data to constrain the search space decreases the duration required to estimate an accurate PL environment. However, in 4/13 inversion cases (6, 7, 10, and 12), the Random experiment had the shortest duration of all experiments. This result is not surprising since the random initial population could serendipitously be closer to the actual solution than the Spread experiment on occasion, which would allow the inversion to evolve to a correct answer quickly by random chance via elitism. Conversely, the ensemble experiment performs slower than one (or both) of the other experiments for all cases. This result suggests that using NWP ensembles to set the initial population did not reduce the duration of the inversions. However, the Ensemble experiment likely performed worse than the Random experiment in terms of duration because of the lower spread of the initial population in comparison with the Random experiment initial population. Although the initial population used by the Ensemble experiment was presumably closer to the “true” answer and physically realistic, mutation and crossover to the optimal solution are more hindered by using NWP ensembles as the initial population due to its relative uniformity. These results align with those found by Zhao and Huang (2014), who found that too much regularization can hinder the accuracy of an inversion in reproducing accurate environments.

This uniformity of initial populations based on NWP ensembles likely occurs when NOGAPS initialization fields and the ensemble perturbations result in similar NWP forecasts for all ensemble members. Larger ensemble spreads could suggest larger uncertainty in the NWP forecasts, while smaller spreads could suggest less uncertainty in the NWP forecasts. In other words, a forecast that has a higher uncertainty may actually be more beneficial to the duration of the inversion since its initial population has a larger parameter spread. However, the occurrence of these uniform ensembles should be further investigated to determine the validity of this hypothesis.

The fusion of NWP ensemble data with a GA-based inversion approach clearly provided encouraging results, which in many cases benefited either the accuracy or duration (or both) of the inversion result; however, the necessity to use the extended ensemble forecasts during the Spread experiment in a few cases brings to light a few points of discussion. First, all Spread experiment cases, which required extended forecast hours to establish valid parameter search spaces for the inversion, were at least 5 h from the model initialization time. Second, parameters estimated using the helicopter profiles for cases 1 and 3, which required extended ensemble forecasts, both illustrate the smallest M deficits of all cases. These latter observations may suggest that utilizing multiple ensemble forecasts is necessary to aid inversion processes when M deficits are small, but a larger number of cases with small M deficits is needed to test this hypothesis. Last, case 10 required an extended time frame of many ensemble forecasts (Table 6) to produce parameter search spaces that contained the fit helicopter parameters (Table 2). Upon further inspection, the need for many ensemble forecasts stems from the fact that consecutive hourly ensemble forecasts were relatively uniform in terms of the (fit) parameter search spaces. Thus, to incorporate the helicopter fit parameters, many additional forecasts were needed. This fact makes it difficult for the authors to recommend a set number of ensemble forecasts that can be sufficiently used to ensure parameter search spaces contain the fit helicopter parameters, because it depends on the steadiness of the modeled environment. Further research should investigate more cases such as those in 1, 3, and 10 to determine how many ensemble forecasts are typically needed for the search spaces to be inclusive of fit helicopter parameters. Thus, although using ensemble NWP to aid refractivity inversions is seemingly beneficial to estimating surface ducting conditions, it is important to use these data only when NWP ensembles have a large spread and do not produce extremely similar environments; otherwise, a GA-based inversion process may not benefit from this additional information (and it may in fact hinder the inversion process).

5. Summary and conclusions

This study uses numerical weather prediction ensembles to aid a GA-based inversion technique, which uses synthetic propagation loss data based on measured refractivity profiles taken during surface ducting conditions off the coast of Wallops Island, Virginia. Refractivity profiles are modeled using a simple two-layer parametric refractivity model, which accurately describes modified refractivity profiles during surface ducting conditions, and is used to simplify the number of variables estimated by the inversion process. Numerical weather prediction ensembles are used to both initialize and constrain the search space of the inversion technique. These results are compared with a control inversion set that has no influence from NWP. Limited to no benefit is found by using each NWP ensemble member as an initial population member for the genetic algorithm inversion technique; however, using them to constrict the parameter search space of the inversion technique benefited either accuracy or duration (or both) for the majority of inversion results. Furthermore, when the ensemble spread is small, the “true” solution has a high chance of falling outside of the ensemble search space. If this happens, using these ensembles will have a negative impact on the inversion result. In fact, in a few cases, extended forecast hours had to be used to ensure that the ensembles contained the true solution. This forecast extension would be difficult to predict a priori, and further research on the likelihood of ensembles not containing an observed condition is needed. Thus, overall, the authors recommend initializing GA-based inversions using NWP only when NWP ensembles have a large spread, or during times when NWP forecasts are expected to contain high uncertainty. Admittingly with a few limitations, this study reveals that NWP ensembles can often be used to enhance inversion accuracy and reduce inversion duration, offering a promising step forward in the fusion of NWP with refractivity inversion problems.

Data availability statement.

Access to the helicopter and NWP ensemble modified refractivity and their associated optimized refractivity parameters, inverse solutions, and associated propagation predictions are available at the Mendeley data repository (https://data.mendeley.com/datasets/2sv737k56h/draft?a=c409cffc-02fb-4bb2-94ff-f7e20a2ba859).