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## Abstract

For use in a model on the semi-staggered E (in the Arakawa notation) grid, a number of conserving schemes for the horizontal advection are developed and analyzed. For the rotation terms of the momentum advection, the second-order enstrophy and energy-conserving scheme of Janjić (1977) is generalized to conserve energy in case of divergent flow. A family of analogs of the Arakawa (1966) fourth-order scheme is obtained following a transformation of its component Jacobians. For the kinetic energy advection terms, a fourth- (or approximately fourth) order scheme is developed which maintains the total kinetic energy and, in addition, makes no contribution to the change in the finite-difference vorticity. For the resulting both second- and fourth-order momentum advection scheme, a modification is pointed out which avoids the non-cancellation of terms considered recently by Hollingsworth and Källberg (1979), and shown to lead to a linear instability of a zonally uniform inertia-gravity wave. Finally, a second- order as well as a fourth-order (or approximately so) advection scheme for temperature (and moisture) advection is given, preserving the total energy (and moisture) inside the integration region.

## Abstract

For use in a model on the semi-staggered E (in the Arakawa notation) grid, a number of conserving schemes for the horizontal advection are developed and analyzed. For the rotation terms of the momentum advection, the second-order enstrophy and energy-conserving scheme of Janjić (1977) is generalized to conserve energy in case of divergent flow. A family of analogs of the Arakawa (1966) fourth-order scheme is obtained following a transformation of its component Jacobians. For the kinetic energy advection terms, a fourth- (or approximately fourth) order scheme is developed which maintains the total kinetic energy and, in addition, makes no contribution to the change in the finite-difference vorticity. For the resulting both second- and fourth-order momentum advection scheme, a modification is pointed out which avoids the non-cancellation of terms considered recently by Hollingsworth and Källberg (1979), and shown to lead to a linear instability of a zonally uniform inertia-gravity wave. Finally, a second- order as well as a fourth-order (or approximately so) advection scheme for temperature (and moisture) advection is given, preserving the total energy (and moisture) inside the integration region.

## Abstract

A numerical experiment is performed with the purpose of investigating the behavior of the trajectories of a very large number of constant-volume particles. Practical significance is given to this problem by the possibility of using superpressured, constant-volume balloons for routine upper air observations.

The computational scheme of the experiment is described with emphasis on some aspects of trajectory computations. Thirty-day diagnostic trajectories are computed for two levels using the total velocity components, and for one level using only the nondivergent ones, and the resulting spacing of the trajectory points is discussed. Theory of the distances to the nearest among a large number of points is developed and applied for the statistical description of the results. Histograms of distances from the constant-volume particles as well as from random points in space to the nearest neighboring constant-volume particle are computed, and compared with the frequency function of those distances for the case of a random distribution of particles. It is shown that the nondivergent part of the atmospheric motions gives rise to a random distribution of initially regularly spaced particles. Departures from the random distribution are therefore produced by the divergent part of the atmospheric motions. In the experiment they resulted in the increase in distances from random points in space to the nearest constant-volume particle of about 12 and 4 per cent, at the levels corresponding to 800 and 300 mb, respectively. The computational region was approximately equal to the area north of 15N; a somewhat larger effect of the divergent part of the wind should be expected in the case of the global constant-volume trajectories.

## Abstract

A numerical experiment is performed with the purpose of investigating the behavior of the trajectories of a very large number of constant-volume particles. Practical significance is given to this problem by the possibility of using superpressured, constant-volume balloons for routine upper air observations.

The computational scheme of the experiment is described with emphasis on some aspects of trajectory computations. Thirty-day diagnostic trajectories are computed for two levels using the total velocity components, and for one level using only the nondivergent ones, and the resulting spacing of the trajectory points is discussed. Theory of the distances to the nearest among a large number of points is developed and applied for the statistical description of the results. Histograms of distances from the constant-volume particles as well as from random points in space to the nearest neighboring constant-volume particle are computed, and compared with the frequency function of those distances for the case of a random distribution of particles. It is shown that the nondivergent part of the atmospheric motions gives rise to a random distribution of initially regularly spaced particles. Departures from the random distribution are therefore produced by the divergent part of the atmospheric motions. In the experiment they resulted in the increase in distances from random points in space to the nearest constant-volume particle of about 12 and 4 per cent, at the levels corresponding to 800 and 300 mb, respectively. The computational region was approximately equal to the area north of 15N; a somewhat larger effect of the divergent part of the wind should be expected in the case of the global constant-volume trajectories.

## Abstract

The problem of the forced adjustment of the wind field to the height field is experimentally studied with the Mintz-Arakawa two-level atmospheric general circulation model.

In all but one of the experiments, the height field was assumed to be perfectly observed at 6-hr intervals, over a time period of one day or less, and from this height data the vector wind field was computed by forced dynamical adjustment. In one experiment, the temperature alone was prescribed. The winds computed in these experiments were compared with the “control” winds of the general circulation simulation.

The best agreement between the computed and the control winds was obtained when the time-differencing scheme in the governing finite-difference equations of motion had a large rate of damping of high-frequency motions. This damping rate also determined the optimum fraction and frequency of restoration of the height (or temperature) fields. With strong damping, total restoration every time step gave the most rapid rate of wind error reduction and the smallest asymptotic limit of the wind error.

The information content of the height field and its time derivatives was analysed. The first time derivative of the height field was of much greater importance than the next higher time derivatives. In middle latitudes, where the time variation of the height field was large, the first time derivative reduced the computed wind error to about half of the error when using no time derivative. When the information is limited to 24 hr or less, the total height field information (surface pressure as well as temperature) produced a much smaller wind error than temperature information alone.

With the first time derivative of the height field, the asymptotic limit of the computed wind error was about 1–1.5 m sec^{−1} in middle latitudes and about 2.5 m sec^{−1} in the tropics.

## Abstract

The problem of the forced adjustment of the wind field to the height field is experimentally studied with the Mintz-Arakawa two-level atmospheric general circulation model.

In all but one of the experiments, the height field was assumed to be perfectly observed at 6-hr intervals, over a time period of one day or less, and from this height data the vector wind field was computed by forced dynamical adjustment. In one experiment, the temperature alone was prescribed. The winds computed in these experiments were compared with the “control” winds of the general circulation simulation.

The best agreement between the computed and the control winds was obtained when the time-differencing scheme in the governing finite-difference equations of motion had a large rate of damping of high-frequency motions. This damping rate also determined the optimum fraction and frequency of restoration of the height (or temperature) fields. With strong damping, total restoration every time step gave the most rapid rate of wind error reduction and the smallest asymptotic limit of the wind error.

The information content of the height field and its time derivatives was analysed. The first time derivative of the height field was of much greater importance than the next higher time derivatives. In middle latitudes, where the time variation of the height field was large, the first time derivative reduced the computed wind error to about half of the error when using no time derivative. When the information is limited to 24 hr or less, the total height field information (surface pressure as well as temperature) produced a much smaller wind error than temperature information alone.

With the first time derivative of the height field, the asymptotic limit of the computed wind error was about 1–1.5 m sec^{−1} in middle latitudes and about 2.5 m sec^{−1} in the tropics.

Since 9 June 1993, the eta coordinate regional model has been run twice daily at the National Centers for Environmental Prediction (NCEP, previously the National Meteorological Center) as the NCEP's “early” operational model. Its performance is regularly monitored in a variety of ways, with particular attention given to precipitation forecasts. Throughout this period, the eta model has demonstrated significantly increased accuracy in forecasting daily precipitation amounts compared to NCEP's Nested Grid Model (NGM). The model has shown a smaller but equally consistent advantage in skill against that of NCEP's global spectral model.

Precipitation scores of these three operational models for the 6-month period March–August 1995 are presented. This interval is chosen because the 6-month-long periods September–February and March–August have been used in previous model comparisons and because an upgraded version of the eta model, run at 48-km resolution, was also regularly executed twice daily during the March–August 1995 period. It is thus included and highlighted in the present comparison. The 48-km eta carries cloud water as a prognostic variable and is coupled to a 12-h eta-based intermittent data assimilation system. It replaced the 80-km eta as the NCEP's early operational model on 12 October 1995.

Compared to the then-operational 80-km eta, the 48-km eta has demonstrated substantially increased skill at all eight precipitation categories for which verifications are made. The increase in skill was greatest for the most intense precipitation, at the threshold of 2 in. (24 h)^{−1}. A 24-48-h forecast of accumulated precipitation, resulting from Hurricane Allison as it was crossing the extreme southeastern United States, is shown as an example of a successful forecast of intense precipitation by the 48-km model.

Reasons for the advantage of the eta model over its predecessor, the NGM, are reviewed. The work in progress is outlined.

Since 9 June 1993, the eta coordinate regional model has been run twice daily at the National Centers for Environmental Prediction (NCEP, previously the National Meteorological Center) as the NCEP's “early” operational model. Its performance is regularly monitored in a variety of ways, with particular attention given to precipitation forecasts. Throughout this period, the eta model has demonstrated significantly increased accuracy in forecasting daily precipitation amounts compared to NCEP's Nested Grid Model (NGM). The model has shown a smaller but equally consistent advantage in skill against that of NCEP's global spectral model.

Precipitation scores of these three operational models for the 6-month period March–August 1995 are presented. This interval is chosen because the 6-month-long periods September–February and March–August have been used in previous model comparisons and because an upgraded version of the eta model, run at 48-km resolution, was also regularly executed twice daily during the March–August 1995 period. It is thus included and highlighted in the present comparison. The 48-km eta carries cloud water as a prognostic variable and is coupled to a 12-h eta-based intermittent data assimilation system. It replaced the 80-km eta as the NCEP's early operational model on 12 October 1995.

Compared to the then-operational 80-km eta, the 48-km eta has demonstrated substantially increased skill at all eight precipitation categories for which verifications are made. The increase in skill was greatest for the most intense precipitation, at the threshold of 2 in. (24 h)^{−1}. A 24-48-h forecast of accumulated precipitation, resulting from Hurricane Allison as it was crossing the extreme southeastern United States, is shown as an example of a successful forecast of intense precipitation by the 48-km model.

Reasons for the advantage of the eta model over its predecessor, the NGM, are reviewed. The work in progress is outlined.

## Abstract

An earlier attempt to estimate the effect of hail suppression by silver iodide seeding in eastern parts of Yugoslavia, based on hail-frequency data at stations having professional observers, is extended here. Hail-frequency data only are considered, rather than the hail- and the ice pellet-frequency data taken together. The period of the data is extended from 37 to 40 years. A statistical analysis of the probability of the observed result being obtained by chance is made, based on the permutation test; a sensitivity test of the possible observer-subjectivity effect is done; and several tests of and corrections for any climate and observing practices change are also made.

The ratio of the average hail frequency during the seeding activities in the area of the station and the average frequency before these activities shows a reduction in the hail frequency by about 25%. A synthetic histogram of the frequency ratios resulting from 10 000 random permutations (station by station) of the observed frequency data gave the probability of this observed frequency reduction being obtained by chance, if in fact no positive effect of seeding or climate change existed, of about 2 in 10 000.

A sensitivity test of the observer-subjectivity effect was made by removing from the available sample of 23 stations the station showing the greatest reduction in hail frequency. This decreased the apparent effectiveness from about 25% to about 23%, and the probability of the positive result became 4 in 10 000.

Tests as well as corrections for the effects of possible climate fluctuations and/or a change in hail-observing practices were performed by using the two neighboring regions of Vojvodina and Bosnia and Herzegovina, which had no hail suppression programs as the control area. The effectiveness calculations as well as the permutation tests were than repeated using “corrected” data. These various corrections reduced the effectiveness of the seeding activities from about 25% to between 22% and 15% and increased the probability of the positive result being obtained by chance to between about 6 and 141 in 10 000. Thus, it appears unlikely that the seeding activities have no positive effect whatsoever; and the reduction in hail frequency seems to be of the order of 15%–20%.

## Abstract

An earlier attempt to estimate the effect of hail suppression by silver iodide seeding in eastern parts of Yugoslavia, based on hail-frequency data at stations having professional observers, is extended here. Hail-frequency data only are considered, rather than the hail- and the ice pellet-frequency data taken together. The period of the data is extended from 37 to 40 years. A statistical analysis of the probability of the observed result being obtained by chance is made, based on the permutation test; a sensitivity test of the possible observer-subjectivity effect is done; and several tests of and corrections for any climate and observing practices change are also made.

The ratio of the average hail frequency during the seeding activities in the area of the station and the average frequency before these activities shows a reduction in the hail frequency by about 25%. A synthetic histogram of the frequency ratios resulting from 10 000 random permutations (station by station) of the observed frequency data gave the probability of this observed frequency reduction being obtained by chance, if in fact no positive effect of seeding or climate change existed, of about 2 in 10 000.

A sensitivity test of the observer-subjectivity effect was made by removing from the available sample of 23 stations the station showing the greatest reduction in hail frequency. This decreased the apparent effectiveness from about 25% to about 23%, and the probability of the positive result became 4 in 10 000.

Tests as well as corrections for the effects of possible climate fluctuations and/or a change in hail-observing practices were performed by using the two neighboring regions of Vojvodina and Bosnia and Herzegovina, which had no hail suppression programs as the control area. The effectiveness calculations as well as the permutation tests were than repeated using “corrected” data. These various corrections reduced the effectiveness of the seeding activities from about 25% to between 22% and 15% and increased the probability of the positive result being obtained by chance to between about 6 and 141 in 10 000. Thus, it appears unlikely that the seeding activities have no positive effect whatsoever; and the reduction in hail frequency seems to be of the order of 15%–20%.

## Abstract

A Lagrangean-type numerical forecasting method is developed in which the computational (grid) points are advected by the wind and the necessary space derivatives (in the pressure gradient terms, for example) are computed using the values of the variables at all the computation points that at the particular moment are within a prescribed distance of the point for which the computation is done. In this way, the forecasting problem reduces to solving the ordinary differential equations of motion and thermodynamics for each computation point, instead of solving the partial differential equations in the Eulerian or classical Lagrangean way. The method has some advantages over the conventional Eulerian scheme: simplicity (there are no advection terms), lack of computational dispersion in the advection terms and therefore better simulation of atmospheric advection and deformation effects, very little inconvenience due to the spherical shape of the earth, and the possibility for a variable space resolution if desired. On the other hand, some artificial smoothing may be necessary, and it may be difficult (or impossible) to conserve the global integrals of certain quantities.

A more detailed discussion of the differencing scheme used for the time integration is included in a separate section, This is the scheme obtained by linear extrapolation of computed time derivatives to a time value of *t*
_{0} + aΔ*t* where *t*
_{0} is the value of time at the beginning of the considered time step Δ*t* and where *a* is a parameter that can be used to control the properties of the scheme. When choosing a value of *a* between ½ and 1, a scheme is obtained that damps the high-frequency motions, in a similar way as the Matsuno scheme, but needs somewhat less computer time and, with the same damping intensity, has a higher accuracy for low-frequency meteorologically significant motions.

Using the described method, a 4-day experimental forecast has been made, starting with a stationary Haurwitz-Neamtan solution, for a primitive equation, global, and homogeneous model. The final geopotential height map showed no visible phase errors and only a modest accumulation of truncation errors and effects of numerical smoothing mechanisms. Two shorter experiments have also been made to analyze the effects of space resolution and damping in the process of time differencing. It is felt that the experimental results strongly encourage further testing and investigation of the proposed method.

## Abstract

A Lagrangean-type numerical forecasting method is developed in which the computational (grid) points are advected by the wind and the necessary space derivatives (in the pressure gradient terms, for example) are computed using the values of the variables at all the computation points that at the particular moment are within a prescribed distance of the point for which the computation is done. In this way, the forecasting problem reduces to solving the ordinary differential equations of motion and thermodynamics for each computation point, instead of solving the partial differential equations in the Eulerian or classical Lagrangean way. The method has some advantages over the conventional Eulerian scheme: simplicity (there are no advection terms), lack of computational dispersion in the advection terms and therefore better simulation of atmospheric advection and deformation effects, very little inconvenience due to the spherical shape of the earth, and the possibility for a variable space resolution if desired. On the other hand, some artificial smoothing may be necessary, and it may be difficult (or impossible) to conserve the global integrals of certain quantities.

A more detailed discussion of the differencing scheme used for the time integration is included in a separate section, This is the scheme obtained by linear extrapolation of computed time derivatives to a time value of *t*
_{0} + aΔ*t* where *t*
_{0} is the value of time at the beginning of the considered time step Δ*t* and where *a* is a parameter that can be used to control the properties of the scheme. When choosing a value of *a* between ½ and 1, a scheme is obtained that damps the high-frequency motions, in a similar way as the Matsuno scheme, but needs somewhat less computer time and, with the same damping intensity, has a higher accuracy for low-frequency meteorologically significant motions.

Using the described method, a 4-day experimental forecast has been made, starting with a stationary Haurwitz-Neamtan solution, for a primitive equation, global, and homogeneous model. The final geopotential height map showed no visible phase errors and only a modest accumulation of truncation errors and effects of numerical smoothing mechanisms. Two shorter experiments have also been made to analyze the effects of space resolution and damping in the process of time differencing. It is felt that the experimental results strongly encourage further testing and investigation of the proposed method.

## Abstract

The benefits of assimilation of precipitation data had been demonstrated in diabetic initialization and nudging-type experiments some years ago. In four-dimensional variational (4DVAR) data assimilation, however, the precipitation data have not yet been used. To correctly assimilate the precipitation data by the 4DVAR technique, the problems related to the first-order discontinuities in the “full-physics” forecast model should be solved first. To address this problem in the full-physics regional NMC eta forecast model, a modified, more continuous version of the Beta-Miller cumulus convection scheme is defined and examined as a possible solution.

The 4DVAR data assimilation experiments ate performed using the conventional data (in this case, analyses of *T*, *p _{s}*,

*u*,

*v*, and

*q*) and the precipitation data (the analysis of 24-h accumulated precipitation). The full-physics NMC eta model and the adjoint model with convective processes are used in the experiments. The control variable of the minimization problem is defined to include the initial conditions and model's systematic error parameter. An extreme synoptic situation from June 1993, with strong effects of precipitation over the United States is chosen for the experiments. The results of the 4DVAR experiments show convergence of the minimization process within 10 iterations and an improvement of the precipitation forecast, during and after the data assimilation period, when using the modified cumulus convection scheme and the precipitation data. In particular, the 4DVAR method outperforms the optimal interpolation method by improving the precipitation forecast.

## Abstract

The benefits of assimilation of precipitation data had been demonstrated in diabetic initialization and nudging-type experiments some years ago. In four-dimensional variational (4DVAR) data assimilation, however, the precipitation data have not yet been used. To correctly assimilate the precipitation data by the 4DVAR technique, the problems related to the first-order discontinuities in the “full-physics” forecast model should be solved first. To address this problem in the full-physics regional NMC eta forecast model, a modified, more continuous version of the Beta-Miller cumulus convection scheme is defined and examined as a possible solution.

The 4DVAR data assimilation experiments ate performed using the conventional data (in this case, analyses of *T*, *p _{s}*,

*u*,

*v*, and

*q*) and the precipitation data (the analysis of 24-h accumulated precipitation). The full-physics NMC eta model and the adjoint model with convective processes are used in the experiments. The control variable of the minimization problem is defined to include the initial conditions and model's systematic error parameter. An extreme synoptic situation from June 1993, with strong effects of precipitation over the United States is chosen for the experiments. The results of the 4DVAR experiments show convergence of the minimization process within 10 iterations and an improvement of the precipitation forecast, during and after the data assimilation period, when using the modified cumulus convection scheme and the precipitation data. In particular, the 4DVAR method outperforms the optimal interpolation method by improving the precipitation forecast.

## Abstract

Bias-adjusted threat and equitable threat scores were designed to account for the effects of placement errors in assessing the performance of under- or overbiased forecasts. These bias-adjusted performance measures exhibit bias sensitivity. The critical performance ratio (CPR) is the minimum fraction of added forecasts that are correct for a performance measure to indicate improvement if bias is increased. In the opposite case, the CPR is the maximum fraction of removed forecasts that are correct for a performance measure to indicate improvement if bias is decreased. The CPR is derived here for the bias-adjusted threat and equitable threat scores to quantify bias sensitivity relative to several other measures of performance including conventional threat and equitable threat scores. The CPR for a bias-adjusted equitable threat score may indicate the likelihood of preserving or increasing the conventional equitable threat score if forecasts are bias corrected based on past performance.

## Abstract

Bias-adjusted threat and equitable threat scores were designed to account for the effects of placement errors in assessing the performance of under- or overbiased forecasts. These bias-adjusted performance measures exhibit bias sensitivity. The critical performance ratio (CPR) is the minimum fraction of added forecasts that are correct for a performance measure to indicate improvement if bias is increased. In the opposite case, the CPR is the maximum fraction of removed forecasts that are correct for a performance measure to indicate improvement if bias is decreased. The CPR is derived here for the bias-adjusted threat and equitable threat scores to quantify bias sensitivity relative to several other measures of performance including conventional threat and equitable threat scores. The CPR for a bias-adjusted equitable threat score may indicate the likelihood of preserving or increasing the conventional equitable threat score if forecasts are bias corrected based on past performance.

## Abstract

It is suggested that there are two major problems with the “standard” methods of reducing pressure to sea level based on the surface temperature or the lowest-layer(s) temperature of a numerical model. The first is that using air temperatures above elevated terrain for reducing pressure to sea level is in conflict with the presumed objective of the reduction. The authors take this to be the derivation of a pressure field appropriate to sea level that to the extent possible maintains the shape of the constant-elevation isobars and reflects the changes in the horizontal of the magnitudes of horizontal pressure gradients, as these exist at the ground surface. The other problem is that evidence is emerging showing that with the increasing realism in the representation of mountains in numerical models the performance of the standard reduction methods is about to deteriorate to the point of becoming unacceptable.

Fortunately, as proposed earlier by the first author, an alternative exists that is both simple and consistent with the objective of the reduction as presumed above. It is to replace the downward extrapolation of temperature by the *horizontal* interpolation of (virtual) temperature where the temperatures are given *at the sides* of mountains.

Performance of the “horizontal” reduction method is here compared against the so-called Shuell method, which is a conventional part of the U.S. National Meteorological Center's postprocessing packages. This is done by examining the sea level pressure centers of initial conditions and forecasts, at 12-h intervals, of the National Meteorological Center's eta model, as obtained via the Shuell and horizontal reduction methods. The comparison is done for a sample of late summer initial conditions and forecasts verifying at 16 consecutive 0000 and 1200 UTC initial times. Note that the Shuell reduction method was specifically designed to improve upon a standard lapse rate reduction to sea level during the warm season.

In terms of the agreement with the analyst-assessed values, the two methods showed an overall comparable performance. The horizontal reduction method performed much better for Mexican heat lows, while the Shuell method was clearly superior in reproducing the analyzed values at high centers over the United States and Canadian highlands. The horizontal reduction method performed somewhat better in depicting the values at the centers of lows over the United States and Canadian mountainous region of the study. As its main benefit, the horizontal reduction method eliminated formidable noise and artifact problems of the Shuell reduction method without resorting to smoothing devices.

## Abstract

It is suggested that there are two major problems with the “standard” methods of reducing pressure to sea level based on the surface temperature or the lowest-layer(s) temperature of a numerical model. The first is that using air temperatures above elevated terrain for reducing pressure to sea level is in conflict with the presumed objective of the reduction. The authors take this to be the derivation of a pressure field appropriate to sea level that to the extent possible maintains the shape of the constant-elevation isobars and reflects the changes in the horizontal of the magnitudes of horizontal pressure gradients, as these exist at the ground surface. The other problem is that evidence is emerging showing that with the increasing realism in the representation of mountains in numerical models the performance of the standard reduction methods is about to deteriorate to the point of becoming unacceptable.

Fortunately, as proposed earlier by the first author, an alternative exists that is both simple and consistent with the objective of the reduction as presumed above. It is to replace the downward extrapolation of temperature by the *horizontal* interpolation of (virtual) temperature where the temperatures are given *at the sides* of mountains.

Performance of the “horizontal” reduction method is here compared against the so-called Shuell method, which is a conventional part of the U.S. National Meteorological Center's postprocessing packages. This is done by examining the sea level pressure centers of initial conditions and forecasts, at 12-h intervals, of the National Meteorological Center's eta model, as obtained via the Shuell and horizontal reduction methods. The comparison is done for a sample of late summer initial conditions and forecasts verifying at 16 consecutive 0000 and 1200 UTC initial times. Note that the Shuell reduction method was specifically designed to improve upon a standard lapse rate reduction to sea level during the warm season.

In terms of the agreement with the analyst-assessed values, the two methods showed an overall comparable performance. The horizontal reduction method performed much better for Mexican heat lows, while the Shuell method was clearly superior in reproducing the analyzed values at high centers over the United States and Canadian highlands. The horizontal reduction method performed somewhat better in depicting the values at the centers of lows over the United States and Canadian mountainous region of the study. As its main benefit, the horizontal reduction method eliminated formidable noise and artifact problems of the Shuell reduction method without resorting to smoothing devices.