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- Author or Editor: Mark A. Askelson x

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## Abstract

The response function is a commonly used measure of analysis scheme properties. Its use in the interpretation of analyses of real-valued data, however, is unnecessarily complicated by the structure of the standard form of the Fourier transform. Specifically, interpretation using this form of the Fourier transform requires knowledge of the relationship between Fourier transform values that are symmetric about the origin. Here, these relationships are used to simplify the application of the response function to the interpretation of analysis scheme properties.

In doing so, Fourier transforms are used because they can be applied to studying effects that both data sampling *and* weight functions have upon analyses. A complication arises, however, in the treatment of constant and sinusoidal input since they do not have Fourier transforms in the traditional sense. To handle these highly useful forms, distribution theory is used to generalize Fourier transform theory. This extension enables Fourier transform theory to handle both functions that have Fourier transforms in the traditional sense and functions that can be represented using Fourier series.

The key step in simplifying the use of the response function is the expression of the inverse Fourier transform in a magnitude and phase form, which involves folding the integration domain onto itself so that integration is performed over only half of the domain. Once this is accomplished, interpretation of the response function is in terms of amplitude and phase modulations, which indicate how amplitudes and phases of input waves are affected by an analysis scheme. This interpretation is quite elegant since its formulation in terms of properties of input waves results in a one-to-one input-to-output wave interpretation of analysis scheme effects.

## Abstract

The response function is a commonly used measure of analysis scheme properties. Its use in the interpretation of analyses of real-valued data, however, is unnecessarily complicated by the structure of the standard form of the Fourier transform. Specifically, interpretation using this form of the Fourier transform requires knowledge of the relationship between Fourier transform values that are symmetric about the origin. Here, these relationships are used to simplify the application of the response function to the interpretation of analysis scheme properties.

In doing so, Fourier transforms are used because they can be applied to studying effects that both data sampling *and* weight functions have upon analyses. A complication arises, however, in the treatment of constant and sinusoidal input since they do not have Fourier transforms in the traditional sense. To handle these highly useful forms, distribution theory is used to generalize Fourier transform theory. This extension enables Fourier transform theory to handle both functions that have Fourier transforms in the traditional sense and functions that can be represented using Fourier series.

The key step in simplifying the use of the response function is the expression of the inverse Fourier transform in a magnitude and phase form, which involves folding the integration domain onto itself so that integration is performed over only half of the domain. Once this is accomplished, interpretation of the response function is in terms of amplitude and phase modulations, which indicate how amplitudes and phases of input waves are affected by an analysis scheme. This interpretation is quite elegant since its formulation in terms of properties of input waves results in a one-to-one input-to-output wave interpretation of analysis scheme effects.

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## Abstract

Despite great strides in understanding the tornadic near-storm environment (NSE), at times it remains difficult to determine why some storms produce significant tornadoes, while others produce none, given similar pretornadic radar reflectivity and velocity signatures. Previous studies have shown that this is likely related to the potential buoyancy (*θ _{ep}
*) of the rear-flank downdraft (RFD) air. Unfortunately, to date there are few ways to operationally anticipate possible RFD thermodynamic character. Based upon previous research indicating that capping inversions may restrict much of the low-level RFD air to come from within the boundary layer, this study considers the relation of Δ

*θ*(vertical change in

_{ep}*θ*within the boundary layer below the cap) to tornadogenesis potential. This is because when a cap exists above a boundary layer and the descent of lower-

_{ep}*θ*air from aloft to the surface is potentially limited, then minimal Δ

_{ep}*θ*may indicate more RFD air that has greater potential buoyancy. The Rapid Update Cycle (RUC) soundings used in this study and several observed soundings taken in the vicinity of violent tornadoes suggest that boundary layer Δ

_{ep}*θ*shows promise as an additional means of discriminating between tornadic and nontornadic NSEs.

_{ep}## Abstract

Despite great strides in understanding the tornadic near-storm environment (NSE), at times it remains difficult to determine why some storms produce significant tornadoes, while others produce none, given similar pretornadic radar reflectivity and velocity signatures. Previous studies have shown that this is likely related to the potential buoyancy (*θ _{ep}
*) of the rear-flank downdraft (RFD) air. Unfortunately, to date there are few ways to operationally anticipate possible RFD thermodynamic character. Based upon previous research indicating that capping inversions may restrict much of the low-level RFD air to come from within the boundary layer, this study considers the relation of Δ

*θ*(vertical change in

_{ep}*θ*within the boundary layer below the cap) to tornadogenesis potential. This is because when a cap exists above a boundary layer and the descent of lower-

_{ep}*θ*air from aloft to the surface is potentially limited, then minimal Δ

_{ep}*θ*may indicate more RFD air that has greater potential buoyancy. The Rapid Update Cycle (RUC) soundings used in this study and several observed soundings taken in the vicinity of violent tornadoes suggest that boundary layer Δ

_{ep}*θ*shows promise as an additional means of discriminating between tornadic and nontornadic NSEs.

_{ep}^{ }

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## Abstract

Various combinations of smoothing parameters within a two-pass Barnes objective analysis scheme are applied to analytic observations obtained by regular and irregular sampling of a one-dimensional sinusoidal analytic wave to obtain gridded fields. Each of these various combinations of smoothing parameters would produce equivalent analyses if the observations were continuous and infinite (unbounded). The authors demonstrate that owing to the discreteness of the analytic observations, the actual analyses resulting from these various combinations of smoothing parameters are different. When derivatives are computed and as stations become more irregularly distributed, these differences increase. An awareness of these potentially significant analysis differences should prompt the analyst to consider carefully the choice of smoothing parameters when applying an objective analysis scheme to real observations.

## Abstract

Various combinations of smoothing parameters within a two-pass Barnes objective analysis scheme are applied to analytic observations obtained by regular and irregular sampling of a one-dimensional sinusoidal analytic wave to obtain gridded fields. Each of these various combinations of smoothing parameters would produce equivalent analyses if the observations were continuous and infinite (unbounded). The authors demonstrate that owing to the discreteness of the analytic observations, the actual analyses resulting from these various combinations of smoothing parameters are different. When derivatives are computed and as stations become more irregularly distributed, these differences increase. An awareness of these potentially significant analysis differences should prompt the analyst to consider carefully the choice of smoothing parameters when applying an objective analysis scheme to real observations.

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## Abstract

Two accepted postulates for applications of ground-based weather radars are that Earth’s surface is a perfect sphere and that all the rays launched at low-elevation angles have the same constant small curvature. To accommodate a straight vertically launched ray, we amend the second postulate by making the ray curvature dependent on the cosine of the launch angle. A standard atmospheric stratification determines the ray-curvature value at zero launch angle. Granted this amended postulate, we develop exact formulas for ray height, ground range, and ray slope angle as functions of slant range and launch angle on the real Earth. Standard practice assumes a hypothetical equivalent magnified earth, for which the rays become straight while ray height above radar level remains virtually the same function of the radar coordinates. The real-Earth and equivalent-earth formulas for height agree to within 1 m. Our ultimate goal is to place a virtual Doppler radar within a numerical or analytical model of a supercell and compute virtual signatures of simulated storms for development and testing of new warning algorithms. Since supercell models have a flat lower boundary, we must first compute the ray curvature that preserves the height function as the earth curvature tends to zero. Using an approximate height formula, we find that keeping planetary curvature minus the ray curvature at zero launch angle constant preserves ray height to within 5 m. For standard refraction the resulting ray curvature is negative, indicating that rays bend concavely upward relative to a flat earth.

## Abstract

Two accepted postulates for applications of ground-based weather radars are that Earth’s surface is a perfect sphere and that all the rays launched at low-elevation angles have the same constant small curvature. To accommodate a straight vertically launched ray, we amend the second postulate by making the ray curvature dependent on the cosine of the launch angle. A standard atmospheric stratification determines the ray-curvature value at zero launch angle. Granted this amended postulate, we develop exact formulas for ray height, ground range, and ray slope angle as functions of slant range and launch angle on the real Earth. Standard practice assumes a hypothetical equivalent magnified earth, for which the rays become straight while ray height above radar level remains virtually the same function of the radar coordinates. The real-Earth and equivalent-earth formulas for height agree to within 1 m. Our ultimate goal is to place a virtual Doppler radar within a numerical or analytical model of a supercell and compute virtual signatures of simulated storms for development and testing of new warning algorithms. Since supercell models have a flat lower boundary, we must first compute the ray curvature that preserves the height function as the earth curvature tends to zero. Using an approximate height formula, we find that keeping planetary curvature minus the ray curvature at zero launch angle constant preserves ray height to within 5 m. For standard refraction the resulting ray curvature is negative, indicating that rays bend concavely upward relative to a flat earth.

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## Abstract

Distance-dependent weighted averaging (DDWA) is a process that is fundamental to most of the objective analysis schemes that are used in meteorology. Despite its ubiquity, aspects of its effects are still poorly understood. This is especially true for the most typical situation of observations that are discrete, bounded, and irregularly distributed.

To facilitate understanding of the effects of DDWA schemes, a framework that enables the determination of response functions for arbitrary weight functions and data distributions is developed. An essential element of this approach is the equivalent analysis, which is a hypothetical analysis that is produced by using, throughout the analysis domain, the same weight function and data distribution that apply at the point where the response function is desired. This artifice enables the derivation of the response function by way of the convolution theorem. Although this approach requires a bit more effort than an alternative one, the reward is additional insight into the impacts of DDWA analyses.

An important insight gained through this approach is the exact nature of the DDWA response function. For DDWA schemes the response function is the complex conjugate of the normalized Fourier transform of the effective weight function. In facilitating this result, this approach affords a better understanding of which elements (weight functions, data distributions, normalization factors, etc.) affect response functions and how they interact to do so.

Tests of the response function for continuous, bounded data and discrete, irregularly distributed data verify the validity of the response functions obtained herein. They also reinforce previous findings regarding the dependence of response functions on analysis location and the impacts of data boundaries and irregular data spacing.

Interpretation of the response function in terms of amplitude and phase modulations is illustrated using examples. Inclusion of phase shift information is important in the evaluation of DDWA schemes when they are applied to situations that may produce significant phase shifts. These situations include those where data boundaries influence the analysis value and where data are irregularly distributed. By illustrating the attendant movement, or shift, of data, phase shift information also provides an elegant interpretation of extrapolation.

## Abstract

Distance-dependent weighted averaging (DDWA) is a process that is fundamental to most of the objective analysis schemes that are used in meteorology. Despite its ubiquity, aspects of its effects are still poorly understood. This is especially true for the most typical situation of observations that are discrete, bounded, and irregularly distributed.

To facilitate understanding of the effects of DDWA schemes, a framework that enables the determination of response functions for arbitrary weight functions and data distributions is developed. An essential element of this approach is the equivalent analysis, which is a hypothetical analysis that is produced by using, throughout the analysis domain, the same weight function and data distribution that apply at the point where the response function is desired. This artifice enables the derivation of the response function by way of the convolution theorem. Although this approach requires a bit more effort than an alternative one, the reward is additional insight into the impacts of DDWA analyses.

An important insight gained through this approach is the exact nature of the DDWA response function. For DDWA schemes the response function is the complex conjugate of the normalized Fourier transform of the effective weight function. In facilitating this result, this approach affords a better understanding of which elements (weight functions, data distributions, normalization factors, etc.) affect response functions and how they interact to do so.

Tests of the response function for continuous, bounded data and discrete, irregularly distributed data verify the validity of the response functions obtained herein. They also reinforce previous findings regarding the dependence of response functions on analysis location and the impacts of data boundaries and irregular data spacing.

Interpretation of the response function in terms of amplitude and phase modulations is illustrated using examples. Inclusion of phase shift information is important in the evaluation of DDWA schemes when they are applied to situations that may produce significant phase shifts. These situations include those where data boundaries influence the analysis value and where data are irregularly distributed. By illustrating the attendant movement, or shift, of data, phase shift information also provides an elegant interpretation of extrapolation.

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## Abstract

Idealized simulations using the Weather Research and Forecasting Model (WRF) were performed to examine the role of capping inversions on the near-surface thermodynamic structure of outflow from simulated supercells. Two simulations were performed: one with the traditional noncapped Weisman and Klemp (WK) analytic sounding and the second with a WK sounding modified to contain a capping inversion. Both sounding environments favor splitting storms and a right-moving supercell by 90 min into the simulation. These two supercell simulations evolve in a qualitatively similar fashion, with both storms exhibiting large, quasi-steady updrafts, hook-shaped appendages in the precipitation mixing ratio field, and prominent localized downdrafts.

Results show that the supercell simulated in the capped environment has a surface cold pool with larger values of pseudoequivalent potential temperature (*θ*
_{ep}) than the cold pool of the supercell produced in the noncapped simulation. Parcels in the surface cold pool of the supercell produced in the capped sounding simulation have a lower origin height than those in the surface cold pool of the supercell produced in the noncapped simulation for all times. Although *θ*
_{ep} values in the surface cold pool are primarily associated with the origin height of downdraft parcels and the environmental *θ*
_{ep} at that level, it is shown that nonconservation of *θ*
_{ep} primarily associated with hydrometeor melting can decrease *θ*
_{ep} values of downdraft parcels as they descend by several degrees.

## Abstract

Idealized simulations using the Weather Research and Forecasting Model (WRF) were performed to examine the role of capping inversions on the near-surface thermodynamic structure of outflow from simulated supercells. Two simulations were performed: one with the traditional noncapped Weisman and Klemp (WK) analytic sounding and the second with a WK sounding modified to contain a capping inversion. Both sounding environments favor splitting storms and a right-moving supercell by 90 min into the simulation. These two supercell simulations evolve in a qualitatively similar fashion, with both storms exhibiting large, quasi-steady updrafts, hook-shaped appendages in the precipitation mixing ratio field, and prominent localized downdrafts.

Results show that the supercell simulated in the capped environment has a surface cold pool with larger values of pseudoequivalent potential temperature (*θ*
_{ep}) than the cold pool of the supercell produced in the noncapped simulation. Parcels in the surface cold pool of the supercell produced in the capped sounding simulation have a lower origin height than those in the surface cold pool of the supercell produced in the noncapped simulation for all times. Although *θ*
_{ep} values in the surface cold pool are primarily associated with the origin height of downdraft parcels and the environmental *θ*
_{ep} at that level, it is shown that nonconservation of *θ*
_{ep} primarily associated with hydrometeor melting can decrease *θ*
_{ep} values of downdraft parcels as they descend by several degrees.

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## Abstract

Spatial objective analysis is routinely performed in several applications that utilize radar data. Because of their relative simplicity and computational efficiency, one-pass distance-dependent weighted-average (DDWA) schemes that utilize either the Cressman or the Barnes filter are often used in these applications. The DDWA schemes that have traditionally been used do not, however, directly account for two fundamental characteristics of radar data. These are 1) the spacing of radar data depends on direction and 2) radar data density systematically decreases with increasing range.

A DDWA scheme based on an adaptation of the Barnes filter is proposed. This scheme, termed the adaptive Barnes (A-B) scheme, explicitly takes into account radar data properties 1 and 2 above. Both theoretical and experimental investigations indicate that two attributes of the A-B scheme, direction-splitting and automatic adaptation to data density, may facilitate the preservation of the maximum amount of meaningful information possible within the confines of one-pass DDWA schemes.

It is shown that in the idealized situation of infinite, continuous data and for an analysis in rectangular-Cartesian coordinates, a direction-splitting scheme does not induce phase shifts if the weight function is even in each direction. Moreover, for radar data that are infinite, collected at regular radial, azimuthal, and elevational increments, and collocated with analysis points, the direction-splitting design of the A-B filter removes gradients in the analysis weights. This is a beneficial attribute when considering the treatment of gradient information of rectangular Cartesian data by an analysis system because then postanalysis gradients equal the analysis of gradients. The direction-splitting design of the A-B filter is unable, however, to circumvent the impact of the varying physical distances between adjacent measurements that are inherent to the spherical coordinate system of ground-based weather radars. Because of this, even with the direction-splitting design of the A-B filter postanalysis gradients do not equal the analysis of gradients.

Ringing in the response function of a one-dimensional Barnes filter is illustrated. The negative impact of data windows on the main lobe of the response function is found to decrease rapidly as the window is widened relative to the weight function. Unless an analysis point is near a data boundary, in which case both ringing and phase shifting will adversely affect the analysis, window effects are unlikely to be significant in applications of the A-B filter to radar data.

The A-B filter has potential drawbacks, the most significant of which is misinterpretations owing to the use of the A-B filter without comprehension of its direction- and range-dependent response function. Despite its drawbacks, the A-B filter has the potential to improve analyses owing to the aforementioned attributes and thus to aid research efforts in areas such as multiple-Doppler wind analyses, pseudo-dual-Doppler analyses, and retrieval studies.

## Abstract

Spatial objective analysis is routinely performed in several applications that utilize radar data. Because of their relative simplicity and computational efficiency, one-pass distance-dependent weighted-average (DDWA) schemes that utilize either the Cressman or the Barnes filter are often used in these applications. The DDWA schemes that have traditionally been used do not, however, directly account for two fundamental characteristics of radar data. These are 1) the spacing of radar data depends on direction and 2) radar data density systematically decreases with increasing range.

A DDWA scheme based on an adaptation of the Barnes filter is proposed. This scheme, termed the adaptive Barnes (A-B) scheme, explicitly takes into account radar data properties 1 and 2 above. Both theoretical and experimental investigations indicate that two attributes of the A-B scheme, direction-splitting and automatic adaptation to data density, may facilitate the preservation of the maximum amount of meaningful information possible within the confines of one-pass DDWA schemes.

It is shown that in the idealized situation of infinite, continuous data and for an analysis in rectangular-Cartesian coordinates, a direction-splitting scheme does not induce phase shifts if the weight function is even in each direction. Moreover, for radar data that are infinite, collected at regular radial, azimuthal, and elevational increments, and collocated with analysis points, the direction-splitting design of the A-B filter removes gradients in the analysis weights. This is a beneficial attribute when considering the treatment of gradient information of rectangular Cartesian data by an analysis system because then postanalysis gradients equal the analysis of gradients. The direction-splitting design of the A-B filter is unable, however, to circumvent the impact of the varying physical distances between adjacent measurements that are inherent to the spherical coordinate system of ground-based weather radars. Because of this, even with the direction-splitting design of the A-B filter postanalysis gradients do not equal the analysis of gradients.

Ringing in the response function of a one-dimensional Barnes filter is illustrated. The negative impact of data windows on the main lobe of the response function is found to decrease rapidly as the window is widened relative to the weight function. Unless an analysis point is near a data boundary, in which case both ringing and phase shifting will adversely affect the analysis, window effects are unlikely to be significant in applications of the A-B filter to radar data.

The A-B filter has potential drawbacks, the most significant of which is misinterpretations owing to the use of the A-B filter without comprehension of its direction- and range-dependent response function. Despite its drawbacks, the A-B filter has the potential to improve analyses owing to the aforementioned attributes and thus to aid research efforts in areas such as multiple-Doppler wind analyses, pseudo-dual-Doppler analyses, and retrieval studies.

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## Abstract

Owing to their ease of use, “simplified” propagation models, like the equivalent Earth model, are commonly employed to determine radar data locations. With the assumption that electromagnetic rays follow paths of constant curvature, which is a fundamental assumption in the equivalent Earth model, propagation equations that do not depend upon the spatial transformation that is utilized in the equivalent Earth model are derived. This set of equations provides the true constant curvature solution and is less complicated, conceptually, as it does not depend upon a spatial transformation. Moreover, with the assumption of constant curvature, the relations derived herein arise naturally from ray tracing relations. Tests show that this new set of equations is more accurate than the equivalent Earth equations for a “typical” propagation environment in which the index of refraction *n* decreases linearly at the rate *dn*/*dh* = −1/4*a*, where *h* is height above ground and *a* is Earth’s radius. Moreover, this new set of equations performs better than the equivalent Earth equations for an exponential reference atmosphere, which provides a very accurate representation of the average atmospheric *n* structure in the United States. However, with this *n* profile the equations derived herein, the equivalent Earth equations, and the relation associated with a flat Earth constant curvature model produce relatively large height errors at low elevations and large ranges. Taylor series approximations of the new equations are examined. While a second-order Taylor series approximation for height performs well under “typical” propagation conditions, a convenient Taylor series approximation for great circle distance was not obtained.

## Abstract

Owing to their ease of use, “simplified” propagation models, like the equivalent Earth model, are commonly employed to determine radar data locations. With the assumption that electromagnetic rays follow paths of constant curvature, which is a fundamental assumption in the equivalent Earth model, propagation equations that do not depend upon the spatial transformation that is utilized in the equivalent Earth model are derived. This set of equations provides the true constant curvature solution and is less complicated, conceptually, as it does not depend upon a spatial transformation. Moreover, with the assumption of constant curvature, the relations derived herein arise naturally from ray tracing relations. Tests show that this new set of equations is more accurate than the equivalent Earth equations for a “typical” propagation environment in which the index of refraction *n* decreases linearly at the rate *dn*/*dh* = −1/4*a*, where *h* is height above ground and *a* is Earth’s radius. Moreover, this new set of equations performs better than the equivalent Earth equations for an exponential reference atmosphere, which provides a very accurate representation of the average atmospheric *n* structure in the United States. However, with this *n* profile the equations derived herein, the equivalent Earth equations, and the relation associated with a flat Earth constant curvature model produce relatively large height errors at low elevations and large ranges. Taylor series approximations of the new equations are examined. While a second-order Taylor series approximation for height performs well under “typical” propagation conditions, a convenient Taylor series approximation for great circle distance was not obtained.

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## Abstract

The human population on Earth has increased by a factor of 4.6 in the last 100 years and has become more centered in urban environments. This expansion and migration pattern has resulted in stresses on the environment. Meteorological applications have helped to understand and mitigate those stresses. This chapter describes several applications that enable the population to interact with the environment in more sustainable ways. The first topic treated is urbanization itself and the types of stresses exerted by population growth and its attendant growth in urban landscapes—buildings and pavement—and how they modify airflow and create a local climate. We describe environmental impacts of these changes and implications for the future. The growing population uses increasing amounts of energy. Traditional sources of energy have taxed the environment, but the increase in renewable energy has used the atmosphere and hydrosphere as its fuel. Utilizing these variable renewable resources requires meteorological information to operate electric systems efficiently and economically while providing reliable power and minimizing environmental impacts. The growing human population also pollutes the environment. Thus, understanding and modeling the transport and dispersion of atmospheric contaminants are important steps toward regulating the pollution and mitigating impacts. This chapter describes how weather information can help to make surface transportation more safe and efficient. It is explained how these applications naturally require transdisciplinary collaboration to address these challenges caused by the expanding population.

## Abstract

The human population on Earth has increased by a factor of 4.6 in the last 100 years and has become more centered in urban environments. This expansion and migration pattern has resulted in stresses on the environment. Meteorological applications have helped to understand and mitigate those stresses. This chapter describes several applications that enable the population to interact with the environment in more sustainable ways. The first topic treated is urbanization itself and the types of stresses exerted by population growth and its attendant growth in urban landscapes—buildings and pavement—and how they modify airflow and create a local climate. We describe environmental impacts of these changes and implications for the future. The growing population uses increasing amounts of energy. Traditional sources of energy have taxed the environment, but the increase in renewable energy has used the atmosphere and hydrosphere as its fuel. Utilizing these variable renewable resources requires meteorological information to operate electric systems efficiently and economically while providing reliable power and minimizing environmental impacts. The growing human population also pollutes the environment. Thus, understanding and modeling the transport and dispersion of atmospheric contaminants are important steps toward regulating the pollution and mitigating impacts. This chapter describes how weather information can help to make surface transportation more safe and efficient. It is explained how these applications naturally require transdisciplinary collaboration to address these challenges caused by the expanding population.