Characterizing Range-Dependent Variations of the Evaporation Duct: A Meteorological Perspective

Evaporation ducts (EDs) are an observed electromagnetic phenomenon caused by rapid decreases of humidity with altitude. Sensing technologies that operate at X-band frequencies (8–12 GHz) exhibit the extension of radar signals beyond the radar horizon and holes in coverage at high altitudes during evaporative ducting conditions. In addition, the evolution of the ED over range has been reported to cause further adverse effects on these technologies in some circumstances. However, in terms of the predictability of these effects, more research is still needed. This research explores ED variations over range and characterizes them based-on two numerical datasets from different latitudes, different hemispheres, and during different seasons as well as correlating their range distributions to distributions of other atmospheric variables. It is found, for both datasets, that most often a linear function accurately represents the variations of ED characteristics over ~60 km in range, but higher order functional distributions of ED characteristics with range do occur albeit less frequently. Furthermore, the range distributions of mean wind speed and mean specific humidity within the evaporation layer are strongly correlated to the range distributions of duct height; while range distributions of duct shape are mainly related to the specific humidity gradient in the evaporation layer. The midlatitude dataset exhibits more complex range distributions during frontal events, while the equatorial dataset shows the most complex range distributions near sunrise.


I. INTRODUCTION
E VAPORATION ducts (EDs) are an atmospheric phe- nomenon driven by rapid decreases in humidity with height that cause X-band electromagnetic (EM) waves to bend their trajectory toward the earth's surface leading to increases in maximum detection ranges as well as positioning errors in radar systems.Because EDs are nearly permanent worldwide Fig. 1.Primary characteristics (in red) of an M-profile in evaporation ducting conditions that impact X-band radar wave propagation within the lowest 100 m of the atmosphere.
features in marine and coastal environments [1], the study of their impacts on radar signals is integral.
The vertical profiles of the index of refraction (n), atmospheric refractivity (N ), or modified refractivity (M) are commonly used to explore radar wave propagation in evaporative ducting conditions.Atmospheric refractivity (N ) is described by [2] where T is the temperature (K), p is the pressure (mb), and e is the partial water vapor pressure (mb).Modified refractivity is considered, in lieu of atmospheric refractivity because it makes evaporative ducting conditions easy to detect as it accounts for earth's curvature where R e is the radius of the earth (m) and z is the altitude (m).Fig. 1 shows an example of a typical M-profile in evaporation ducting conditions and a few of its primary characteristics known to cause changes in radar wave propagation in the lowest 100 m of the atmosphere [3].These primary characteristics are the duct height, which is the critical point within a modified refractivity profile (i.e., where (∂ M/∂z) = 0), and the duct shape previously quantified in literature by the altitudinal-mean second derivative of modified refractivity within the evaporation layer ((∂ 2 M/∂z 2 ); [4]).The evaporation layer extends from the surface (z = 0 m) to twice the duct height (z = 2z d ).
Many studies assume lateral homogeneity of the ED, where a single profile (e.g., Fig. 1) is assumed to remain consistent over ranges up to 60 km.However, the lateral homogeneity assumption breaks down frequently in coastal zones [5], [6] or in intense weather events, such as tropical cyclones [7].
Furthermore, studies have shown nonnegligible impacts on radar wave propagation in horizontally heterogeneous environments.Bean and Cahoon [8] first reported that rays emitted at low elevation angles are sensitive to extreme horizontal variations of the atmosphere near the surface.Goldhirsh and Dockery [5] further found that radar propagation differences between homogeneous and heterogeneous environments were large at ranges greater than 30 km from the radar, suggesting that one needs to account for heterogeneous environments to accurately predict long-range propagation.Other studies have reached similar conclusions [6], [9], [10], [11], [12], [13].However, these studies mainly focus on heterogeneous variations of refractivity and their impacts on propagation but do not investigate the likelihood of different heterogeneous distributions of refractivity and how they connect to prevailing meteorological conditions.
The horizontal distribution of ED height has been most commonly examined.Brooks et al. [9] revealed that ED heights typically vary by just a few meters over ranges of up to 200 km.Brooks [11] showed that drastic decreases in ED height of 5 m over less than 10 km can occur around a coastal headland, which is subject to large horizontal changes in wind speed, depending on the wind direction.Alappattu et al. [14] found that ED heights typically increased from a height of 7 m near California to 15 m near Hawaii.Recently, Ulate et al. [13] have reported that the ED height increases offshore of Duck, North Carolina during autumn.Despite the fact that studies have shown that propagation can be impacted by heterogeneity of duct characteristics, especially the duct height, the sparsity of studies examining what leads to and when heterogeneous conditions occur represents a knowledge gap, along with simple ways to model such variations when present.Thus, this research focuses on investigating the probabilities of occurrence and meteorological causes of certain horizontal duct height distributions.Such trends could be seasonally dependent at certain latitudes or modified during certain synoptic-scale events, such as weather fronts.Last, to the best of our knowledge, horizontal variations of ED shape, independent of shape changes due to duct height variations, have yet to be studied.
This research aims to address some of these knowledge gaps by exploring the interplay of atmospheric variables well known to affect EDs [wind speed, humidity, temperature, air-sea temperature difference (ASTD), humidity gradient, wind shear, gradient Richardson number, and sea level pressure (SLP)] but has not been carefully quantified [1], [7], [14], [15], [16], [17], [18], [19], [20], [21], with ED characteristics (duct height and shape) using two verified numerical weather prediction (NWP) model forecasts in coastal environments, where heterogeneous environments have been observed.These NWP forecasts were generated as part of large field campaigns, located in different hemispheres in different seasons and at different latitudes and have been verified against direct meteorological measurements [13], [20].The NWP forecasts must be used in this study rather than direct measurements because they contain sufficient range resolution to carry out the analyses; obtaining numerous concurrent meteorological measurements over a large area at high vertical resolution is typically not logistically and/or economically feasible and the authors are unaware of any such direct measurement sets.Thus, these NWP forecasts that were extensively compared to direct meteorological measurements at times and locations when direct measurements were made enable a higher level of confidence that these NWP forecasts accurately represent measured conditions.Typical distributions of duct height and duct shape over range are classified and probabilities of occurrence are investigated.These distribution types are correlated to the aforementioned range distributions of atmospheric variables to determine linear relationships, while principal component analysis (PCA) is used to decipher multivariate relationships.

II. DATA
The Coupled Ocean/Atmosphere Mesoscale Prediction System (COAMPS; 1 [22]) is used to generate weather forecasts to explore heterogeneous variabilities in EDs.COAMPS 1 is a high-resolution, nonhydrostatic mesoscale model based upon the Navier-Stokes equations that describe temporal changes in atmospheric variables, such as pressure ( p), temperature (T ), and humidity (q), which can be used to estimate modified refractivity (M), as discussed in Section I [22].COAMPS 1 has been used to provide operational forecasts for the U.S. Naval Fleet as well as used in historical case studies to validate and advance forecast skill [20], [23], [24], [25], [26], [27].
More specifically, the simulated vertical profiles of modified refractivity, wind speed, specific humidity, and air temperature along with bulk sea-surface temperatures (SSTs) along crossshore transects predicted by COAMPS 1 , verified for two separate field campaigns, are utilized for analysis.The two field campaigns include the Coupled Air-Sea Processes and EM Ducting Research-East field campaign (CASPER-East; [28]) and the Tropical Air Propagation Study (TAPS; [20]).The numerical data from these two field campaigns are chosen exclusively because they have been reported to accurately represent directly measured atmospheric conditions and show heterogeneous conditions with hourly sampling and approximately 2-km range resolution over large horizontal scales (∼60 km).While the conditions forecast may be somewhat limited, this limitation is mitigated by using forecasts from separate field campaigns that occur in different hemispheres at different latitudes, and during different seasons, offering some insight into whether phenomenon examined herein are strongly location or seasonally dependent.Due to the verified nature of the NWP data, the study results may be limited in what stability conditions are included but 1 Registered trademark.are still valuable due to their known accuracy.Other in situ meteorological datasets measured during both field campaigns, but not used extensively here, are measurements from repeated winched tethered balloons, research vessels (R/V), and on piers.See [20] and [28] for further information on CASPER-East and TAPS field campaigns, respectively.
COAMPS 1 forecasts are two-way coupled with NCOM and use initial boundary conditions estimated from the Navy Global Environmental Model (NAVGEM; [29]) for the atmosphere and the Global Hybrid Coordinate Ocean Model (HYCOM; [30]) for the ocean.Forecasts are hourly using a 12-h update cycle during both field campaigns.To enable the resolution of EDs near the ocean surface, COAMPS 1 is blended with a surface layer model-the Navy Atmospheric Vertical Surface Layer Model (NAVSLaM; [31]).The blending process, similar to that discussed by [4], [27], and [32], results in cross-shore transects of meteorological vertical profiles for both field campaigns examined in this study.The blended method uses COAMPS 1 specific humidity, atmospheric pressure ( p), and wind speed near the surface along with bulk SSTs from the Navy Coupled Ocean Model (NCOM) for NAVSLaM estimates of vertical profiles of specific humidity, wind speed, air temperature, and modified refractivity.These blended forecasts are referred to as COAMPS 1 hereafter for simplicity.
CASPER-East transects are forecast from Duck Pier in North Carolina to ∼62-km offshore with ∼2-km horizontal spacing and decimeter vertical spacing and are estimated hourly between October 12 and November 6, 2015.Crossshore transects from TAPS occur from a jetty offshore of Ingham, Queensland to ∼86-km offshore with ∼1.5-km horizontal spacing and decimeter vertical spacing hourly between November 24 and December 5, 2013.Both transects are shown in Fig. 2. TAPS forecasts are linearly interpolated over range at each altitude to the same horizontal spacing as CASPER-East forecasts for the ease of analysis.If a forecast contains an elevated duct within the first 100 m of altitude at any range along a transect, then the entire transect is removed to focus exclusively on EDs.This filtering results in a total of 460 cross-shore transects (each with 31 vertical profiles for a total of 14 260 M-profiles) for CASPER-East and 216 crossshore transects (31 vertical profiles per transect for a total of 6696 M-profiles) for TAPS.The examples of heterogenous modified refractivity profiles used in this study from both the CASPER-East and (interpolated) TAPS datasets are illustrated in Fig. 3.

III. METHODS
The purpose of this study is to examine typical distributions of heterogeneous EDs in coastal regions and generally explore the relationships between horizontal variations of ED characteristics and the horizontal variations of other atmospheric variables.These explorations shed light on which atmospheric parameters may be driving changes in the ED over cross-shore transects.
Each M-profile within each transect (described in Section II) is fit using a weighted iterative nonlinear least squares regression to an ED parametric refractivity model described by [33] where z is the altitude, M 0 is the surface modified refractivity, c 0 is the potential refractivity gradient-a parameter shown to be directly related to duct curvature [19], m 1 is the modified refractivity slope of the mixed layer, z d is the ED height, M 1 is a parameter ensuring continuity between the two layers (and is defined by the other parameters), z 0 is the aerodynamic roughness factor, and z L is the altitude of the top of the evaporation layer, where z L ≡ 2z d .z 0 = 0.0015 m, which is the average value for oceanic conditions [34], and remaining parameters (M 0 , c 0 , m 1 , and z d ) are estimated in the regression.A regression technique is used to estimate all parameters for two reasons.First, the use of a parametric model helps to enable extension of the results presented herein to modeling heterogeneity in propagation simulations in a simplified manner.This simple parametric model can be used to model heterogeneous conditions if there are known distribution types with range for the c 0 and z d parameters.
Second, while z d , M 0 , and m 1 can be directly estimated from numerical data, c 0 cannot, and therefore to be consistent and avoid any discontinuities in the profiles, all parameters are fit except z 0 .The duct height in combination with the c 0 parameter sets the curvature ((∂ 2 M/∂z 2 )) of the M-profile within the evaporation layer [19].The aerodynamic roughness (z 0 ) can also influence the curvature of the M-profile below the duct height, but because c 0 can make similar adjustments, it is usually sufficient to fix this value [35].Furthermore, the inclusion of z 0 in the regression technique more often than not created less-accurate representations of the M-profile [larger root-mean-square errors (RMSEs)].Initial guesses for each parameter are estimated using the surface modified refractivity (M 0 ) of the COAMPS 1 M-profile, the commonly assumed neutral stability value for c 0 (0.125 M-units•m −1 ) [35], [36], [37], [38], [39], the altitude of the minimum modified refractivity for z d , and the slope of modified refractivity (( M/ z)) between the highest altitude (z = 100 m) and z L (where z d defining z L uses the z d initial guess).The weights of the regression technique are distributed, such that data at altitudes ±1.5 m closest to the initial guess of z d are weighted 25 times more than that of other altitudes.Because the duct height influences propagation most significantly [3], [4] greater emphasis is placed on its accurate prediction in the regression.A histogram of RMSEs calculated between each COAMPS 1 M-profile and its corresponding parametric model fit for all transects is illustrated in Fig. 4.This weighting scheme and regression technique results in all fits having an RMSE of less than 1 Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.M-unit, making the parametric model an ideal representation of the modified refractivity vertical profiles.These regressions, performed for vertical refractivity profiles at each range and for each forecast time, result in a single range distribution of these fit ED characteristics (M 0 , c 0 , m 1 , and z d ) for every forecast during both field campaigns.Range distributions of duct height (z d ) and duct shape (via the c 0 parameter) for each forecast are used to categorize the various range-dependent variations of the ED for both datasets.These distributions are further utilized to compute Pearson's correlation coefficient with various atmospheric properties as well as PCA to characterize which atmospheric properties may be driving range variations of the ED.
Distributions of z d and c 0 over range are categorized into one of the six categories for each forecast: linear, quadratic, cubic, fourth-degree, oscillatory, or step.Categorizations are determined by fitting z d or c 0 over range for each forecast to a polynomial function using linear least squares regression.
Range distributions of z d are categorized using an ordered operation of linear least squares regression polynomial fits.Polynomials from linear up to fourth degree are fit to range distributions of z d for each forecast until a polynomial fit produces an RMSE of less than 0.60 m and contains an absolute bias at all ranges that is less than 2 m.These thresholds followed guidelines found in [40], which states that range variations of z d as small as 1 m over range cause differences in propagation loss of at least 10 dB at ranges far from the transmitter (>45 km).The order of polynomial fits applied over range to each forecast is from lowest degree (linear) to highest degree (fourth degree) to avoid unneeded model complexity when multiple models meet the accuracy requirements.The lowest degree that fulfills the above conditions (RMSE < 0.6 m and max bias < 2 m) is the classification of the z d distribution for that forecast.If none of these polynomials fulfill the criteria, then a seventhdegree polynomial is fit to the distribution of z d over range.
If the RMSE for the seventh-degree polynomial fit is <1 m, the forecast is categorized as oscillatory; otherwise, it is categorized as a step function.These categorizing criteria (i.e., RMSE < 0.6 m and max bias < 2 m) are chosen because visual inspection of fits and range distribution patterns showed most distinct variations between each category with these criteria.The authors found that stricter conditions would lead to too many seemingly unnecessary higher order categorizations, while more lenient conditions would lead to too many linear categorizations that were poor-fits visually.Fig. 5 shows the examples of each z d range-distribution category.
Range distributions of c 0 are categorized similar to z d distributions but instead utilize criteria of an RMSE < 0.03 M-units•m −1 and an absolute bias at every range that is less than 0.12 M-units•m −1 .These criteria were chosen by cursory findings that suggested variations of c 0 , which were as small as 0.03 M-units•m −1 over range, could cause nonnegligible effects on X-band radar wave propagation.These thresholds should be further verified, but they were found to best represent the different types of range distributions visually as well.A c 0 distribution is categorized as oscillatory if a seventh-order polynomial fit produces an RMSE < 0.03 M-units•m −1 ; otherwise, it is categorized as a step function.Fig. 6 shows the examples of each c 0 range-distribution category.
Potential atmospheric causes of the different categories of z d and c 0 range distributions are investigated using range distributions of atmospheric variables.Here, we focus on atmospheric variables previously suggested to impact EDs but not carefully quantified in [15], [16], [17], [18], [19], [20], and [21].Note that although evaporation has been carefully quantified in previous literature, it is omitted from this study because it is directly related to humidity, humidity gradient, wind speed, and wind shear; thus, inclusion of evaporation as a separate metric seemed duplicative.Atmospheric variables include SLP, mean gradient Richardson number (Ri), mean temperature (T ), ASTD, mean specific humidity (q), specific humidity gradient ((dq/dz)), mean wind speed (U ), wind shear ((dU/dz)), and bulk SST.
Ri is calculated where g is the gravitational acceleration (9.81 m•s −2 ), θ V is the virtual potential temperature, T v is the average virtual temperature over all altitudes between the surface (z = 0 m), and z = z ref , where z ref is a reference altitude.u and v are the wind components toward the east and north, respectively.Ri is calculated at each altitude and range, the altitudinal average between the surface and z = z ref at each range is used to calculate Ri transects for this study.The mean temperature, specific humidity, and wind speed are calculated by taking the mean of each variable over all altitudes from the surface to z = z ref .
ASTD is calculated by subtracting the bulk SST from the air temperature at z = z ref .
The gradients of specific humidity and wind speed (i.e., wind shear) are  For each hourly forecast, the forecast range distributions of each atmospheric variable previously described (i.e., SLP, T , Ri, ASTD, q, (dq/dz), U , (dU/dz), and SST) are correlated over range to both ED parameters (z d and c 0 ) range distributions estimated from the nonlinear regressions.Thus, for each forecast from both CASPER-East and TAPS, correlation coefficients are calculated for all Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.Multivariate relationships are also investigated using PCA.In this study, PCA utilizes a 10-D dataset that includes range distributions for one hourly forecast of each atmospheric parameter (i.e., SLP, Ri, T , ASTD, etc.) and a range distribution of either z d or c 0 as the 10th-dimension during the same hourly forecast.Thus, PCA will be applied to each forecast twice, once using a transect of z d , and once using a transect of c 0 .A line is fit through each 10-D space referred to as the first principal component, which generally describes the mean variation within the dataset.Further components are derived by creating lines of best fit orthonormal to the previous component.By projecting each of the ten variables (i.e., SLP, Ri, z d , etc.) onto each component, one can glean potential relationships between parameters, providing insight as to which variables vary together.This projection of a variable onto a component is commonly referred to as a loading.Atmospheric variable loadings onto the component where the highest loading of either z d or c 0 occurs are determined for all forecasts.PCA is ideal for this analysis because it enables correlated variables to be recast into "new variables" (components) that are completely independent of each other.
Variables that load onto the same component indicate that they are related, and given the large number of variables that are intercorrelated here, PCA provides a way of handling that interdependence while still providing insights into factors affecting the range distributions of z d and c 0 .Variables that are intercorrelated will load onto the same component.Last, the occurrence frequency of categorically sorted z d and c 0 range distributions and their variation over the diurnal cycle during both field campaigns allows discussion about common heterogeneous distributions, how often they occur, and when each distribution type might be expected.Results from linear correlations and PCA loadings offer fresh insights about which atmospheric conditions lead to heterogeneous evaporation ducting scenarios.Comparisons between both field campaigns can provide insight into which atmospheric variables, if any, can be used to parameterize heterogeneous EDs in a variety of locations and seasons.

IV. RESULTS AND DISCUSSION
Pie charts showing the frequency of occurrence of each range-distribution category are shown in Fig. 7. From this figure, it is clearly evident that linear z d and c 0 range distributions occur most often for both datasets (>60% and >80% of the time, respectively).These results suggest that further work investigating heterogeneous effects on EM propagation should first focus on effects caused by linear distributions of both z d and c 0 .It is also clear that z d range distributions are more frequently higher order than c 0 distributions.Although generally similar, TAPS forecasts illustrate fewer z d or c 0 steplike range distributions compared to CASPER-East.These differences between CASPER and TAPS may be due to differences between the diurnal cycles of both locations or the presence of different mesoscale atmospheric phenomena.Following these hypotheses, the next subsections explore the frequency of occurrence between range distributions with respect to the diurnal cycle (see Section IV-A), as well as linear (see Section IV-B) and multivariate (see Section IV-B) correlations of z d and c 0 range distributions to range distributions of atmospheric variables previously mentioned.

A. Diurnal Cycle
The patterns of both z d and c 0 range-distribution categories with the diurnal cycle are explored in Fig. 8. Fig. 8(a) shows that linear distributions over range may be more common throughout the hottest part of the day (i.e., between 12:00 and 18:00 EDT) as evidenced by the brighter blue colors during this time period for CASPER-East.Furthermore, higher order range distributions of z d (cubic and above) occur least frequently during this hottest part of the day, where linear distributions dominate.TAPS results show that linear distributions of z d peak a little later in the day comparatively and are infrequent near sunrise and early morning [see Fig. 8(b)].The different timing of the peak could be associated with the lower latitude of the TAPS data causing longer daytime hours.Linear range distributions of c 0 tend to follow the same trends as z d for each respective field campaign but in a manner that is slightly less coherent than the z d trends.Trends for the higher order distributions of c 0 are inconsistent, especially for CASPER-East, and are based on only a few cases, but typically occur during nighttime for CASPER-East and during sunrise to late morning for TAPS.These differences might suggest temporary dominance of more localized weather patterns or land-sea interactions that are associated with the higher order distributions, which is why they are also less common.
In general, linear variations of z d are relatively similar with slight offset in timing between the two field campaigns, suggesting that the diurnal cycle may play a part in the commonly occurring linear variation with range.This connection might be expected as the influence of the land-sea boundary diminishes moving seaward along the cross-shore transect.During the daytime, differential heating between the land and sea can drive the development of internal boundary layers near the coast influencing duct height distributions cross shore.

B. Atmospheric Conditions
Fig. 9 shows the forecast-averaged Pearson's correlation coefficients calculated between range distributions of many atmospheric variables and range distributions of either z d or c 0 to further investigate the physical properties associated with horizontal variations of ED characteristics.Dominant trends across both field campaigns reveal strong relationships both between mean wind speed or mean specific humidity and z d along range.Mean temperature also varies strongly with duct height along range in TAPS, but this relationship is not as influential or consistent in CASPER-East.For both field campaigns, specific humidity gradient is most strongly (inversely) correlated to c 0 along range; although mean wind speed, mean specific humidity, and mean temperature also show relatively strong significant correlations, with these relationships in TAPS being more consistent than those in CASPER-East.Range correlations of other atmospheric variables to z d or c 0 differ based on field campaign or rangedistribution type or contain large fractions of insignificant results [see Fig. 9(c) and (f)].
The direct relationship with wind speed indicates that as the average wind speed beneath the duct increases/decreases offshore, z d typically increases/decreases and alternatively, the indirect relationship with mean specific humidity, indicates that as mean specific humidity increases/decreases below the duct height offshore, duct height decreases/increases.These results support statements from [1], [7], and [18], whom also noted similar relationships between the variation of z d with wind speed or humidity.Physically, these results imply that more mechanical mixing near the surface is associated with increasing duct heights while more humid air at low altitudes can be associated with lower duct heights.Thus, for example, an increasing wind offshore may indicate a linear duct height trend in range with higher duct heights offshore.
Another clear finding from Fig. 9 is that c 0 distributions are inversely correlated to distributions of humidity gradient.Also, large variance of the correlation coefficients occurs for most variables examined for CASPER-East, and in TAPS, few significant results are found for c 0 step distributions.Nevertheless, the inverse correlation with humidity gradient suggests that as the humidity gradient decreases (becomes more negative) or increases (approaches zero) the duct curvature, c 0 , increases or decreases, respectively, supporting results reported by [19], which found a strong inverse correlation between the humidity gradient and duct curvature/shape.
Although the above correlations are evident in both campaigns, some strong relationships appear during only one.TAPS forecasts indicate a strong correlation between z d and mean temperature along range, but such a strong consistent correlation is not observed in the CASPER-East forecasts.Furthermore, during TAPS, mean wind speed and mean temperature are inversely correlated with c 0 more consistently; and while these correlations exist for CASPER-East, they are more variable and weaker.
The lack of a consistent relationship between z d or c 0 with atmospheric stability metrics (i.e., ASTD and Ri) is somewhat surprising but may be reflective of the relatively restricted stability regimes within each field campaign.During TAPS, the atmosphere was almost always unstable due to the low latitude (warm surface water temperatures) and the season (late spring/early summer), whereas more complex stability regimes are found along cross-shore transects during CASPER-East due to SST influences from the Gulf Stream and the fall season.Frequently, CASPER-East regimes include environments that switch from stable to unstable along a transect.The possibility that a lack of data is influencing these results is partly reflected in the larger percentages of insignificant results for these atmospheric variables.ASTD does not correlate well to range distributions of c 0 for all distribution types during CASPER-East but shows moderate to strong inverse correlations during TAPS for higher order polynomial distribution types.Furthermore, it was found that during all linear distributions of c 0 , Ri < 0 at all ranges.This observation suggests, for these data, that linear distributions of c 0 along a transect tend to occur when the environment is unstable across the entire transect and conversely that higher order distributions of c 0 are likely if the stability regime over range changes from unstable to free convective or unstable to stable.
Although based on limited statistics, for CASPER-East, frontal systems may explain the somewhat consistent stronger correlations between mean wind shear or mean temperature and z d or c 0 for higher order range-distribution types.In contrast, the absence of higher order distributions of z d correlating to wind shear during TAPS, as well as the low occurrence of step functions in general, further suggests frontogenesis as the cause of some of the higher order distributions for CASPER-East since Queensland's latitude makes it insusceptible to frontogenesis.Also, all CASPER-East step distributions of c 0 occur during periods where mean wind speeds > 8 ms −1 , and when frontal effects from a cold front or a warm front were reported to occur during CASPER-East [28].Further work should investigate correlations between air temperature and wind shear variations with z d or c 0 distributions during atmospheric phenomena, such as thunderstorms or synoptic systems to further evaluate the hypotheses above.
The distributions of z d and c 0 over range are related to complex nonlinear processes occurring at and above the air-sea interface, as such, although some consistent linear correlations were found, a multivariate approach provides insight into coupled effects.The PCA method outlined in Section III is used to ascertain potential multivariate relationships between variables, where the results are illustrated in Fig. 10.Recall, PCA analysis is performed for a set of atmospheric variable transects and either z d or c 0 transects for each forecast within a range-distribution category.The component on which either z d or c 0 loads most strongly is identified and other variables that load onto that same component are identified and quantified by their loading.These loadings are averaged over all forecasts within a distribution category and are shown as the bars in Fig. 10.Importantly, the atmospheric properties are not all independent.For example, the wind shear and humidity gradient are related to time-averaged wind speed, temperature, and humidity.Thus, when considering these results, it is important to remember that some variables could be loading together because the variables are intercorrelated.
In general, z d tends to on-average load most heavily with both SST and ASTD rather than wind speed and specific humidity found previously.However, it is important to note that although the SST and ASTD loadings are large, they are also highly variable noted by the large error bars.These large error bars may be related to the reported inaccuracies of the NAVSLaM model under stable atmospheric conditions [41] and/or the high sensitivity of duct height to SST in transitional stability regimes.In contrast, mean temperature and mean wind speed show small but consistent loadings illustrated by the very small error bars.Thus, these findings do not necessarily contradict those found in the linear correlation analysis.Instead, it illuminates the expected relationship of z d with atmospheric stability that was seemingly absent in the linear correlation analysis.The high PCA loadings of SST and ASTD reflect sentiments made by [15], which states that the ASTD has a characterizing effect on the relationship of z d with its environment.Furthermore, SST is a primary variable used to calculate evaporation rates, which have been reported to be directly related to the ED height [16], [17], [21].These results further suggest that important nonlinear relationships between stability (associated with SST and ASTD) and z d exist.
In contrast, the PCA results for c 0 show obvious consistent results with those from the direct linear correlations that the strongest relationship across most range-distribution categories is with the specific humidity gradient, especially for TAPS.Relative to z d , the number of variables with high loading onto the same component as c 0 is much larger-potentially suggesting a more complex relationship between duct shape and atmospheric conditions than that between those conditions and duct height.The (c 0 ) loadings of mean wind speed, mean temperature, and wind shear being slightly higher for CASPER-East than TAPS, especially for higher order distribution types, further support the notion of fronts playing a role in the higher order distribution types for CASPER-East.

V. CONCLUSION
This research examined range-dependent distributions of the ED, characterized using two parameters known to greatly affect radar wave propagation in the marine atmospheric surface layer: the duct height and the duct curvature, using verified NWP forecasts from two separate field campaigns.It is found that a majority of range-dependent distributions of duct height and duct curvature are linearly distributed along coastal cross-shore transects.Linear duct height range-dependent distributions most commonly occurred during the warmest portions of the day and were strongly related to variations in mean wind speed (direct) and mean specific humidity (indirect) within the duct.Range-dependent distributions of duct curvature are associated most closely to horizontal changes in the humidity gradient within the evaporation layer (z < 2z d ).
It is found that relationships between higher order rangedistribution patterns of duct height and duct curvature with mean temperature, wind shear, and mean wind speed differ between CASPER-East and TAPS.This comparison between relationships suggests that fronts may play a role in these sparsely sampled events for CASPER-East, and their timing coincides with the arrival of reported fronts during CASPER-East.In contrast, for TAPS, relationships of duct height with wind shear are weaker or nonexistent, and therefore, mean temperature range variations dominated as a secondary influence on z d .TAPS occurred at a latitude where frontogenesis is unlikely.
Although these results are informative, it should be emphasized that the data used in this study are from blended NWP models due to current limitations in measurement technology (and/or dataset availability/existence). NWP models incorporate assumptions and simplifications that can cause the (prediction) forecasts to deviate from observations [42].Furthermore, it is emphasized that this study examines a limited range of atmospheric conditions in favor of using NWP forecasts that have been verified against measured data; however, this limitation was somewhat mitigated by using verified NWP datasets from two different field campaigns that differed in latitude and season.Research efforts to obtain vertical profiles of direct atmospheric measurements simultaneously along horizontal transects with high vertical resolution over long horizontal scales in coastal regions should continue to be a goal to enable validation of the trends observed based on these NWP forecasts.

Fig. 3 .
Fig. 3. Example vertical profiles of modified refractivity at various ranges for a COAMPS 1 -NAVSLaM blended forecast during (a) CASPER-East and (b) TAPS field campaigns.Note that the TAPS forecast shown is linearly interpolated over range to the same horizontal resolution as (a) CASPER-East forecast.

Fig. 4 .
Fig. 4. RMSEs between each NWP M-profile and the M-profile for the corresponding parametric model fit [see (3)] for both (a) CASPER-East and (b) TAPS field campaigns.

Fig. 5 .
Fig. 5. Examples of (a) linear, (b) quadratic, (c) cubic, (d) fourth-degree polynomial, (e) oscillatory, and (f) step-like distributions of z d over range found throughout this study.Corresponding polynomial fits are shown in black on each plot.Note that the black lines in (e) and (f) show the fit of a seventh-degree polynomial to z d over range.

Fig. 6 .
Fig. 6.Examples of (a) linear, (b) quadratic, (c) cubic, (d) fourth degree polynomial, (e) oscillatory, and (f) step-like distributions of c 0 over range found throughout this study.Corresponding polynomial fits are shown in black on each plot.Note that the black lines in (e) and (f) show the fit of a seventh degree polynomial to c 0 over range.

Fig. 7 .
Fig. 7. Pie charts for the frequency of occurrence for each type of ED parameter distribution examined in both (a) and (c) CASPER-East and (b) and (d) TAPS.(a) and (b) show occurrence frequencies for z d distributions, while (c) and (d) show occurrence frequencies for c 0 distributions.

Fig. 8 .
Fig. 8. Frequency of occurrence of (a) and (b) z d and (c) and (d) c 0 range-distribution categories binned by local time during both (a) and (c) CASPER-East and (b) and (d) TAPS field campaigns.Occurrence is represented as a percentage of the number of z d or c 0 distributions occurring at a specific local time divided by the total number of distributions in its range-distribution category.Only percentages between 0% and 15% are shown to improve visibility of variations for the linear category (which is the most common category within TAPS and CASPER-East).Magenta lines illustrate sunrise and sunset.Last, percentages on the x-axis are the percentages of occurrence for each distribution type within each dataset.

Fig. 9 .
Fig. 9. Forecast-averaged correlation coefficients calculated between range-dependent distributions of different atmospheric parameters (see x-axis) and either (a) and (b) z d or (d) and (e) c 0 range distributions during both (a) and (d) CASPER-East and (b) and (e) TAPS field campaigns.Error bars represent two standard deviations of correlation coefficients.(c) and (f) indicate the percentage of correlations which were insignificant in both field campaigns, where dashed bars represent the percentage during the TAPS field campaign.

Fig. 10 .
Fig. 10.Distribution type-averaged magnitudes of principal component loadings for range-distributions of atmospheric data (x-axis) and (a) and (b) z d or (c) and (d) c 0 .Loadings are those associated with the same component on-which z d or c 0 loaded most heavily for (a) and (c) CASPER-East and (b) and (d) TAPS field campaigns.Note that the error bars represent two standard deviations of loadings for each distribution type.