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## Abstract

In this paper, persistence characteristics of the Northern Hemisphere (NH) extratropical circulation have been studied. A simple method, based on the speed with which the atmosphere moves in the phase space (PSS, measured by 2-day lag distances), was adopted to partition the circulation data series into 5-day or longer quasi-stationary periods (QSP) and alternating transition periods (TP). The method is based on the assumptions that large-scale circulation regimes often develop abruptly and that during their development transient activity is either unchanged or enhanced. The partitioning results reveal that a whole cycle of QSP and TP on the hemisphere has an average duration of 20 days with considerable amount of variability. The average length of a QSP-TP cycle is not sensitive to changes of a relatively wide range in the PSS limit value employed in the method. In this range of limit values, the average length of the cycle changes less than 17%, while the ratio between the length of QSP and TP increases dramatically from 0.73 to 4.42.

The partitioning results are statistically very similar for three complementary sectors of the hemisphere. However, we found very little synchronicity in the changes in the three sectors. The correlation between changes in any of the three sectors and on the whole hemisphere is at a much higher level, around 0.75. Although the length of the cycle on height values at individual grid points is in the range we can expect from a red-noise process, this cycle length is considerably (20%) shorter than that in a larger region or on the whole hemisphere. This is an indication that the persistence characteristics of larger-scale circulation, due to spatial interactions, show more persistence than, and cannot be well modeled by, a simple autoregressive process.

Statistical tests indicate that the hemispheric QSPs are largely temporally uncorrelated and cannot be results of a random partitioning method. All these results suggest that the basic assumption about the regime-like behavior of the atmosphere is at least partially true: large-scale regime changes are indeed accompanied with higher speed of changes in the circulation phase space.

Further evidence is presented that the circulation patterns in the phase space are distributed as a multivariate normal distribution in a phase-average sense (i.e., as a function of distance from the mean). Hence, the characteristic distance between neighboring circulation patterns is smaller close to the climate mean than farther away from it. As a consequence that had to be considered in this study, the day-to-day changes in the circulation (an inverse measure of persistence) are also smaller close to the climate mean. It is also argued that phase-average multinormality is the primary characteristic of the distribution of circulation patterns in the phase space and any secondary characteristic (local density maximum) should be searched for and interpreted in this context.

## Abstract

In this paper, persistence characteristics of the Northern Hemisphere (NH) extratropical circulation have been studied. A simple method, based on the speed with which the atmosphere moves in the phase space (PSS, measured by 2-day lag distances), was adopted to partition the circulation data series into 5-day or longer quasi-stationary periods (QSP) and alternating transition periods (TP). The method is based on the assumptions that large-scale circulation regimes often develop abruptly and that during their development transient activity is either unchanged or enhanced. The partitioning results reveal that a whole cycle of QSP and TP on the hemisphere has an average duration of 20 days with considerable amount of variability. The average length of a QSP-TP cycle is not sensitive to changes of a relatively wide range in the PSS limit value employed in the method. In this range of limit values, the average length of the cycle changes less than 17%, while the ratio between the length of QSP and TP increases dramatically from 0.73 to 4.42.

The partitioning results are statistically very similar for three complementary sectors of the hemisphere. However, we found very little synchronicity in the changes in the three sectors. The correlation between changes in any of the three sectors and on the whole hemisphere is at a much higher level, around 0.75. Although the length of the cycle on height values at individual grid points is in the range we can expect from a red-noise process, this cycle length is considerably (20%) shorter than that in a larger region or on the whole hemisphere. This is an indication that the persistence characteristics of larger-scale circulation, due to spatial interactions, show more persistence than, and cannot be well modeled by, a simple autoregressive process.

Statistical tests indicate that the hemispheric QSPs are largely temporally uncorrelated and cannot be results of a random partitioning method. All these results suggest that the basic assumption about the regime-like behavior of the atmosphere is at least partially true: large-scale regime changes are indeed accompanied with higher speed of changes in the circulation phase space.

Further evidence is presented that the circulation patterns in the phase space are distributed as a multivariate normal distribution in a phase-average sense (i.e., as a function of distance from the mean). Hence, the characteristic distance between neighboring circulation patterns is smaller close to the climate mean than farther away from it. As a consequence that had to be considered in this study, the day-to-day changes in the circulation (an inverse measure of persistence) are also smaller close to the climate mean. It is also argued that phase-average multinormality is the primary characteristic of the distribution of circulation patterns in the phase space and any secondary characteristic (local density maximum) should be searched for and interpreted in this context.

## Abstract

Most circulation studies use the root-mean-square difference (RMSD) or correlation (COR) (or both) as a toot for comparing different (observed or forecast) circulation patterns. However, there are some other measures introduced into the literature (e.g., mean absolute error, rms vector error, S1 score) and one might easily construct other measures of circulation similarity. The question of the appropriate choice among possible similarity measures rarely arises. In this paper an objective intercomparison of nine different similarity measures (also called distance functions) is presented. The similarity measures were evaluated through the 7OO-hPa hemispheric analog forecasts obtained by them. In the indirect evaluation, the analogs to the base cases found by each individual distance function were checked whether they were identical with the best analogs (selected by all nine functions) to the circulation pattern that actually followed. The number of coincidences is an indication of the quality of the similarity measures and is found, both for daily and pentad data, to be largest for a dynamically oriented distance function that measure the difference between the gradient of height of two maps. For daily data, RMSD also appears to be significantly better than COR. However, in a direct assessment, where analog forecasts by each distance function were compared to the analysis fields using one of the distance functions to measure the difference, practically no performance differences were found among the functions that performed differently in the indirect evaluation.

It should be noted that the results of both intercompaiison methods are, in a strict sense, valid only for forecast situations. For other purposes, other distance functions might be more appropriate. However, there are some indications that the similarity measure that performed best in the forecast experiments (difference in the gradient of height) remains superior in other applications, too.

## Abstract

Most circulation studies use the root-mean-square difference (RMSD) or correlation (COR) (or both) as a toot for comparing different (observed or forecast) circulation patterns. However, there are some other measures introduced into the literature (e.g., mean absolute error, rms vector error, S1 score) and one might easily construct other measures of circulation similarity. The question of the appropriate choice among possible similarity measures rarely arises. In this paper an objective intercomparison of nine different similarity measures (also called distance functions) is presented. The similarity measures were evaluated through the 7OO-hPa hemispheric analog forecasts obtained by them. In the indirect evaluation, the analogs to the base cases found by each individual distance function were checked whether they were identical with the best analogs (selected by all nine functions) to the circulation pattern that actually followed. The number of coincidences is an indication of the quality of the similarity measures and is found, both for daily and pentad data, to be largest for a dynamically oriented distance function that measure the difference between the gradient of height of two maps. For daily data, RMSD also appears to be significantly better than COR. However, in a direct assessment, where analog forecasts by each distance function were compared to the analysis fields using one of the distance functions to measure the difference, practically no performance differences were found among the functions that performed differently in the indirect evaluation.

It should be noted that the results of both intercompaiison methods are, in a strict sense, valid only for forecast situations. For other purposes, other distance functions might be more appropriate. However, there are some indications that the similarity measure that performed best in the forecast experiments (difference in the gradient of height) remains superior in other applications, too.

## Abstract

An empirical predictability study is presented based on 7OO-hPa Northern Hemispheric circulation analogs. A linear relationship between the initial root-mean-square difference of analog pairs and the time taken for the error to reach a certain limit value is used to extrapolate the predictability with initial errors considerably smaller than those in the present database. The relationship, first used in predictability experiments with the NMC numerical weather prediction (NWP) model, conforms to the experimental data in that the effor growth depends not only on the magnitude of the error but also, to a lesser extent, on the initial error.

Despite the fact that earlier error growth studies did not reflect this dependence on the initial error predictability results with two state-of-the-art numerical models using different analysis methods, and those derived here by the linear relationship mentioned above from circulation analogs are gratifyingly similar. These estimates indicate that given the present observational error (about 12 m rms) and spatial resolution of the data, in the NH winter, the atmosphere seems to have 17-18 days of predictability before the initial difference reaches 95% of the saturation Revel (random error). In present models, the forecast error reaches the same 95% level at around ten and a half days. Since the climate mean as a forecast has considerably less error than a random forecast, from a forecaster's point of view it is more appropriate to use the climate mor as a reference level (71% of the saturation level). With the same conditions as above and using this alternative error reference level, the atmosphere might have a predictability of nine days, while the two models considered currently exhaust predictability at close to six days, leaving considerable room for improvement. Note that these atmospheric predictability estimates were obtained without considering a possible enhancement of predictability due to interactions with the slowly changing ocean and other geospheres. Hence, these estimates can be considered as lower limits to atmospheric predictability.

Comparing the predictability estimates gained from twin model experiments to those from observational data is a special, complex method of model verification. Keeping in mind the uncertainties in the observational studies, one can ascertain that the models produce quite similar error growth characteristics to those of the real atmosphere. Hence, the NWP models are quite good on the tirne and spatial scales for which they were designed. However, there are some indications that they probably could not be reliably used to answer the theoretical questions regarding the gain in predictability with very small initial errors or with very high spatial resolution. Some kind of dynamic-empirical approach based on the interactions between different scales of motion is required to enhance current knowledge on these topics.

## Abstract

An empirical predictability study is presented based on 7OO-hPa Northern Hemispheric circulation analogs. A linear relationship between the initial root-mean-square difference of analog pairs and the time taken for the error to reach a certain limit value is used to extrapolate the predictability with initial errors considerably smaller than those in the present database. The relationship, first used in predictability experiments with the NMC numerical weather prediction (NWP) model, conforms to the experimental data in that the effor growth depends not only on the magnitude of the error but also, to a lesser extent, on the initial error.

Despite the fact that earlier error growth studies did not reflect this dependence on the initial error predictability results with two state-of-the-art numerical models using different analysis methods, and those derived here by the linear relationship mentioned above from circulation analogs are gratifyingly similar. These estimates indicate that given the present observational error (about 12 m rms) and spatial resolution of the data, in the NH winter, the atmosphere seems to have 17-18 days of predictability before the initial difference reaches 95% of the saturation Revel (random error). In present models, the forecast error reaches the same 95% level at around ten and a half days. Since the climate mean as a forecast has considerably less error than a random forecast, from a forecaster's point of view it is more appropriate to use the climate mor as a reference level (71% of the saturation level). With the same conditions as above and using this alternative error reference level, the atmosphere might have a predictability of nine days, while the two models considered currently exhaust predictability at close to six days, leaving considerable room for improvement. Note that these atmospheric predictability estimates were obtained without considering a possible enhancement of predictability due to interactions with the slowly changing ocean and other geospheres. Hence, these estimates can be considered as lower limits to atmospheric predictability.

Comparing the predictability estimates gained from twin model experiments to those from observational data is a special, complex method of model verification. Keeping in mind the uncertainties in the observational studies, one can ascertain that the models produce quite similar error growth characteristics to those of the real atmosphere. Hence, the NWP models are quite good on the tirne and spatial scales for which they were designed. However, there are some indications that they probably could not be reliably used to answer the theoretical questions regarding the gain in predictability with very small initial errors or with very high spatial resolution. Some kind of dynamic-empirical approach based on the interactions between different scales of motion is required to enhance current knowledge on these topics.

## Abstract

This study addresses two questions: 1) whether there are local density maxima and minima in the Northern Hemisphere extratropical wintertime circulation phase space and 2) if so, what the preferred circulation types are. All investigations are based on the null hypothesis that the statistical distribution of circulation patterns in the phase space is a multinormal distribution. If this is a good approximation (as it was shown in a phase-average sense in earlier studies) and the assumed independent variables have equal variance, then the theoretical distribution of circulation patterns can be uniquely described by the (climatological) mean, a single standard deviation, and the number of independent variables (dimension). Having the estimates of all these variables, the local density of the actual circulation data can be easily compared to the theoretical expectation of a multinormal distribution. With randomly generated multinormal samples the significance of such discrepancies can also be tested.

The results at the 1% significance level show that out of the 273 circulation maps investigated in the phase space there are 28(15) that have a higher (lower) local density than that expected from a multinormal distribution. Moreover, the high number of local discrepancies is a statistically clear indication that the circulation data sample cannot come from a symmetric, fully multinormal distribution (global significance). The positive deviation from normal density properties in certain areas of the phase space (preferred maps) is offset by opposite sign deviations in other areas (unpreferred maps), ensuring multinormality only in a phase-average sense. This is clear evidence for the existence of multiple flow regimes in the hemispheric circulation.

As to the second question, the preferred and unpreferred circulation maps were found to cluster around 6 and 5 distinct area of the phase space, respectively. The average of the preferred or unpreferred circulation maps for each cluster was interpreted as an estimate of local density maximum or minimum areas in the phase space. Large changes in the database and the statistical methods made little change in the estimates (especially for local maxima).

The advantages and innovations of the above analysis were the following. 1) The phase space was studied in its full dimensionality. 2) Based on the appropriate null hypothesis (multinormality), the results were presented with a clear determination of statistical significance. 3) Due to the statistically significant results of the local density analysis, the possibility of a physical interpretation of clustering results (preferred circulation types) is guaranteed. A relationship between local maximum points and various boundary conditions is suspected. 4) Lacunar areas or unpreferred types that are theoretically as interesting as the preferred ones have been identified for the first time in the circulation phase space.

## Abstract

This study addresses two questions: 1) whether there are local density maxima and minima in the Northern Hemisphere extratropical wintertime circulation phase space and 2) if so, what the preferred circulation types are. All investigations are based on the null hypothesis that the statistical distribution of circulation patterns in the phase space is a multinormal distribution. If this is a good approximation (as it was shown in a phase-average sense in earlier studies) and the assumed independent variables have equal variance, then the theoretical distribution of circulation patterns can be uniquely described by the (climatological) mean, a single standard deviation, and the number of independent variables (dimension). Having the estimates of all these variables, the local density of the actual circulation data can be easily compared to the theoretical expectation of a multinormal distribution. With randomly generated multinormal samples the significance of such discrepancies can also be tested.

The results at the 1% significance level show that out of the 273 circulation maps investigated in the phase space there are 28(15) that have a higher (lower) local density than that expected from a multinormal distribution. Moreover, the high number of local discrepancies is a statistically clear indication that the circulation data sample cannot come from a symmetric, fully multinormal distribution (global significance). The positive deviation from normal density properties in certain areas of the phase space (preferred maps) is offset by opposite sign deviations in other areas (unpreferred maps), ensuring multinormality only in a phase-average sense. This is clear evidence for the existence of multiple flow regimes in the hemispheric circulation.

As to the second question, the preferred and unpreferred circulation maps were found to cluster around 6 and 5 distinct area of the phase space, respectively. The average of the preferred or unpreferred circulation maps for each cluster was interpreted as an estimate of local density maximum or minimum areas in the phase space. Large changes in the database and the statistical methods made little change in the estimates (especially for local maxima).

The advantages and innovations of the above analysis were the following. 1) The phase space was studied in its full dimensionality. 2) Based on the appropriate null hypothesis (multinormality), the results were presented with a clear determination of statistical significance. 3) Due to the statistically significant results of the local density analysis, the possibility of a physical interpretation of clustering results (preferred circulation types) is guaranteed. A relationship between local maximum points and various boundary conditions is suspected. 4) Lacunar areas or unpreferred types that are theoretically as interesting as the preferred ones have been identified for the first time in the circulation phase space.

## Abstract

In this paper daily wintertime extratropical Northern Hemisphere (NH) circulation analogs are studied. First the analog forecasts are compared to various common benchmark methods such as random or persistence forecasts and the climate mean as a forecast. In line with earlier work, it is concluded that beyond a few days lead time the analogs offer no advantage over any of these benchmark methods. The same is true for derivative analogs (where only the time derivative of the analogs is used and added to the base case), although they perform considerably better than the traditional analogs on the first time step(s).

Even though the circulation analogs have no extended-range forecast capability, they nevertheless offer a convenient way of studying the gross structure of the phase space of circulation patterns. A thorough study of the root-mean-square distances (rmsd) between the best circulation analogs, considered as an indicator for the relative frequency in the phase space, has been performed. It was shown that in a phase-average sense, when the density characteristics are considered only as a function of distance from the climate mean, the distribution of circulation patterns is statistically indistinguishable from a multinormal distribution.

This simple but previously unobserved fact has a series of consequences, some of which are presented here. First, since the density of the distribution of circulation patterns is increasing with decreasing distance from the climate mean, the best analog to a particular circulation pattern is more likely to be closer to the climate mean than the base case. A second observation is that the persistence of the flow increases with decreasing distance from the climate mean. A double stratification of the analogs according to their initial difference and the persistence of the flow showed no enhanced predictability in persistent cases; the forecast error depends only on the initial error of the analogs. This is an indication that the higher numerical forecast skill in persistent cases (reported in earlier studies) may be related to the fact that those cases are relatively close to the climate mean, where even random forecasts have smaller rms error. A third point is that analog predictability does not depend on the initial flow's distance from the climate mean either.

The phase-average multinormality of the wintertime extratropical NH circulation phase space discussed in this study does not rule out the possibility of a fine structure with several local maxima (multiple equilibria) embedded in the overall gross structure of approximate normality. Indeed, a refined methodology revealing the existence of such “dense” areas will be reported in a later paper.

## Abstract

In this paper daily wintertime extratropical Northern Hemisphere (NH) circulation analogs are studied. First the analog forecasts are compared to various common benchmark methods such as random or persistence forecasts and the climate mean as a forecast. In line with earlier work, it is concluded that beyond a few days lead time the analogs offer no advantage over any of these benchmark methods. The same is true for derivative analogs (where only the time derivative of the analogs is used and added to the base case), although they perform considerably better than the traditional analogs on the first time step(s).

Even though the circulation analogs have no extended-range forecast capability, they nevertheless offer a convenient way of studying the gross structure of the phase space of circulation patterns. A thorough study of the root-mean-square distances (rmsd) between the best circulation analogs, considered as an indicator for the relative frequency in the phase space, has been performed. It was shown that in a phase-average sense, when the density characteristics are considered only as a function of distance from the climate mean, the distribution of circulation patterns is statistically indistinguishable from a multinormal distribution.

This simple but previously unobserved fact has a series of consequences, some of which are presented here. First, since the density of the distribution of circulation patterns is increasing with decreasing distance from the climate mean, the best analog to a particular circulation pattern is more likely to be closer to the climate mean than the base case. A second observation is that the persistence of the flow increases with decreasing distance from the climate mean. A double stratification of the analogs according to their initial difference and the persistence of the flow showed no enhanced predictability in persistent cases; the forecast error depends only on the initial error of the analogs. This is an indication that the higher numerical forecast skill in persistent cases (reported in earlier studies) may be related to the fact that those cases are relatively close to the climate mean, where even random forecasts have smaller rms error. A third point is that analog predictability does not depend on the initial flow's distance from the climate mean either.

The phase-average multinormality of the wintertime extratropical NH circulation phase space discussed in this study does not rule out the possibility of a fine structure with several local maxima (multiple equilibria) embedded in the overall gross structure of approximate normality. Indeed, a refined methodology revealing the existence of such “dense” areas will be reported in a later paper.

## Abstract

An analog selection method relying an the coincidence of main features (large ridge lines) in the Northern Hemisphere is presented and used for making 30-day weather forecasts for Hungary. Numerous analog model trials were tested, with the aid of the advance selection of the “best circulation analogs” of the Atlantic-European forecast region, for every target month of the 27-yr calibration period and the 5.5 yr test period. The best predictor types are a one pentad (i.e., 5-day) predictor period with spatial smoothing (allowing slight longitudinal shifts between pressure patterns), and a 2 pentad predictor period with time averaging (with a weighting factor of 0.4 on data from outside the forecast region in both cases). A subset of each group of analogs with similar circulations during the forecast period was identified. Using the subset 1eads to further significant increases

in skill.

Monthly weather forecast for temperature (5-day subperiods) and precipitation quantity (10-day subperiods) in any of three climatologically equal probable categories were given. Different statistics, which were slightly but significantly better than chance expectation and persistence, were employed to area the skill of the forecast. By means of the previously chosen best circulation analogs, the potential monthly analog predictability based on our dataset and methods were also determined. Accordingly, the operable forecasting method realizes 30%–60% of potential predictability. Using lengthened data series for selecting analogs, the improvement in both analog predictability and actual forecasting skills was investigated. Extrapolating the experimental data for the future by comparing it with a logistic curve, an estimate was obtained of increased forecast skill from the present 38%–39% to 42% within 15 yr.

## Abstract

An analog selection method relying an the coincidence of main features (large ridge lines) in the Northern Hemisphere is presented and used for making 30-day weather forecasts for Hungary. Numerous analog model trials were tested, with the aid of the advance selection of the “best circulation analogs” of the Atlantic-European forecast region, for every target month of the 27-yr calibration period and the 5.5 yr test period. The best predictor types are a one pentad (i.e., 5-day) predictor period with spatial smoothing (allowing slight longitudinal shifts between pressure patterns), and a 2 pentad predictor period with time averaging (with a weighting factor of 0.4 on data from outside the forecast region in both cases). A subset of each group of analogs with similar circulations during the forecast period was identified. Using the subset 1eads to further significant increases

in skill.

Monthly weather forecast for temperature (5-day subperiods) and precipitation quantity (10-day subperiods) in any of three climatologically equal probable categories were given. Different statistics, which were slightly but significantly better than chance expectation and persistence, were employed to area the skill of the forecast. By means of the previously chosen best circulation analogs, the potential monthly analog predictability based on our dataset and methods were also determined. Accordingly, the operable forecasting method realizes 30%–60% of potential predictability. Using lengthened data series for selecting analogs, the improvement in both analog predictability and actual forecasting skills was investigated. Extrapolating the experimental data for the future by comparing it with a logistic curve, an estimate was obtained of increased forecast skill from the present 38%–39% to 42% within 15 yr.

## Abstract

The breeding method has been used to generate perturbations for ensemble forecasting at the National Centers for Environmental Prediction (formerly known as the National Meteorological Center) since December 1992. At that time a single breeding cycle with a pair of bred forecasts was implemented. In March 1994, the ensemble was expanded to seven independent breeding cycles on the Cray C90 supercomputer, and the forecasts were extended to 16 days. This provides 17 independent global forecasts valid for two weeks every day.

For efficient ensemble forecasting, the initial perturbations to the control analysis should adequately sample the space of possible analysis errors. It is shown that the analysis cycle is like a breeding cycle: it acts as a nonlinear perturbation model upon the evolution of the real atmosphere. The perturbation (i.e., the analysis error), carried forward in the first-guess forecasts, is “scaled down” at regular intervals by the use of observations. Because of this, growing errors associated with the evolving state of the atmosphere develop within the analysis cycle and dominate subsequent forecast error growth.

The breeding method simulates the development of growing errors in the analysis cycle. A difference field between two nonlinear forecasts is carried forward (and scaled down at regular intervals) upon the evolving atmospheric analysis fields. By construction, the bred vectors are superpositions of the leading local (time-dependent) Lyapunov vectors (LLVs) of the atmosphere. An important property is that all random perturbations assume the structure of the leading LLVs after a transient period, which for large-scale atmospheric processes is about 3 days. When several independent breeding cycles are performed, the phases and amplitudes of individual (and regional) leading LLVs are random, which ensures quasi-orthogonality among the global bred vectors from independent breeding cycles.

Experimental runs with a 10-member ensemble (five independent breeding cycles) show that the ensemble mean is superior to an optimally smoothed control and to randomly generated ensemble forecasts, and compares favorably with the medium-range double horizontal resolution control. Moreover, a potentially useful relationship between ensemble spread and forecast error is also found both in the spatial and time domain. The improvement in skill of 0.04–0.11 in pattern anomaly correlation for forecasts at and beyond 7 days, together with the potential for estimation of the skill, indicate that this system is a useful operational forecast tool.

The two methods used so far to produce operational ensemble forecasts—that is, breeding and the adjoint (or “optimal perturbations”) technique applied at the European Centre for Medium-Range Weather Forecasts—have several significant differences, but they both attempt to estimate the subspace of fast growing perturbations. The bred vectors provide estimates of fastest sustainable growth and thus represent probable growing analysis errors. The optimal perturbations, on the other hand, estimate vectors with fastest *transient growth* in the future. A practical difference between the two methods for ensemble forecasting is that breeding is simpler and less expensive than the adjoint technique.

## Abstract

The breeding method has been used to generate perturbations for ensemble forecasting at the National Centers for Environmental Prediction (formerly known as the National Meteorological Center) since December 1992. At that time a single breeding cycle with a pair of bred forecasts was implemented. In March 1994, the ensemble was expanded to seven independent breeding cycles on the Cray C90 supercomputer, and the forecasts were extended to 16 days. This provides 17 independent global forecasts valid for two weeks every day.

For efficient ensemble forecasting, the initial perturbations to the control analysis should adequately sample the space of possible analysis errors. It is shown that the analysis cycle is like a breeding cycle: it acts as a nonlinear perturbation model upon the evolution of the real atmosphere. The perturbation (i.e., the analysis error), carried forward in the first-guess forecasts, is “scaled down” at regular intervals by the use of observations. Because of this, growing errors associated with the evolving state of the atmosphere develop within the analysis cycle and dominate subsequent forecast error growth.

The breeding method simulates the development of growing errors in the analysis cycle. A difference field between two nonlinear forecasts is carried forward (and scaled down at regular intervals) upon the evolving atmospheric analysis fields. By construction, the bred vectors are superpositions of the leading local (time-dependent) Lyapunov vectors (LLVs) of the atmosphere. An important property is that all random perturbations assume the structure of the leading LLVs after a transient period, which for large-scale atmospheric processes is about 3 days. When several independent breeding cycles are performed, the phases and amplitudes of individual (and regional) leading LLVs are random, which ensures quasi-orthogonality among the global bred vectors from independent breeding cycles.

Experimental runs with a 10-member ensemble (five independent breeding cycles) show that the ensemble mean is superior to an optimally smoothed control and to randomly generated ensemble forecasts, and compares favorably with the medium-range double horizontal resolution control. Moreover, a potentially useful relationship between ensemble spread and forecast error is also found both in the spatial and time domain. The improvement in skill of 0.04–0.11 in pattern anomaly correlation for forecasts at and beyond 7 days, together with the potential for estimation of the skill, indicate that this system is a useful operational forecast tool.

The two methods used so far to produce operational ensemble forecasts—that is, breeding and the adjoint (or “optimal perturbations”) technique applied at the European Centre for Medium-Range Weather Forecasts—have several significant differences, but they both attempt to estimate the subspace of fast growing perturbations. The bred vectors provide estimates of fastest sustainable growth and thus represent probable growing analysis errors. The optimal perturbations, on the other hand, estimate vectors with fastest *transient growth* in the future. A practical difference between the two methods for ensemble forecasting is that breeding is simpler and less expensive than the adjoint technique.

## Abstract

A new probability distribution, which has some of the advantages of the normal distribution but avoids the constraint of symmetry undesired in many applications, is presented. The distribution, called *binormal*, has three parameters, as the standard deviations on the two sides of the most probable value are different, but it also includes the Gaussian distribution as a special case. A test for symmetry and a method for the estimating the parameters are also shown. The distribution will be employed to express probabilistic temperature forecasts, but its climatological application for modeling certain temperature distributions is also useful and recommended.

## Abstract

A new probability distribution, which has some of the advantages of the normal distribution but avoids the constraint of symmetry undesired in many applications, is presented. The distribution, called *binormal*, has three parameters, as the standard deviations on the two sides of the most probable value are different, but it also includes the Gaussian distribution as a special case. A test for symmetry and a method for the estimating the parameters are also shown. The distribution will be employed to express probabilistic temperature forecasts, but its climatological application for modeling certain temperature distributions is also useful and recommended.

## Abstract

Most existing ensemble forecast verification statistics are influenced by the quality of not only the ensemble generation scheme, but also the forecast model and the analysis scheme. In this study, a new tool called perturbation versus error correlation analysis (PECA) is introduced that lessens the influence of the initial errors that affect the quality of the analysis. PECA evaluates the ensemble perturbations, instead of the forecasts themselves, by measuring their ability to explain forecast error variance. As such, PECA offers a more appropriate tool for the comparison of ensembles generated by using different analysis schemes.

Ensemble perturbations from both the National Centers for Environmental Prediction (NCEP) and the European Centre for Medium-Range Weather Forecasts (ECMWF) were evaluated and found to perform similarly. The error variance explained by either ensemble increases with the number of members and the lead time. The dynamically conditioned NCEP and ECMWF perturbations outperform both randomly chosen perturbations and differences between lagged forecasts [used in the “NMC” (for National Meteorological Center, the former name of NCEP) method for defining forecast error covariance matrices]. Therefore ensemble forecasts potentially could be used to construct flow-dependent short-range forecast error covariance matrices for use in data assimilation schemes.

It is well understood that in a perfectly reliable ensemble the spread of ensemble members around the ensemble mean forecast equals the root-mean-square (rms) error of the mean. Adequate rms spread, however, does not guarantee sufficient variability among the ensemble forecast patterns. A comparison between PECA values and pattern anomaly correlation (PAC) values among the ensemble members reveals that the perturbations in the NCEP ensemble exhibit too much similarity, especially on the smaller scales. Hence a regional orthogonalization of the perturbations may improve ensemble performance.

## Abstract

Most existing ensemble forecast verification statistics are influenced by the quality of not only the ensemble generation scheme, but also the forecast model and the analysis scheme. In this study, a new tool called perturbation versus error correlation analysis (PECA) is introduced that lessens the influence of the initial errors that affect the quality of the analysis. PECA evaluates the ensemble perturbations, instead of the forecasts themselves, by measuring their ability to explain forecast error variance. As such, PECA offers a more appropriate tool for the comparison of ensembles generated by using different analysis schemes.

Ensemble perturbations from both the National Centers for Environmental Prediction (NCEP) and the European Centre for Medium-Range Weather Forecasts (ECMWF) were evaluated and found to perform similarly. The error variance explained by either ensemble increases with the number of members and the lead time. The dynamically conditioned NCEP and ECMWF perturbations outperform both randomly chosen perturbations and differences between lagged forecasts [used in the “NMC” (for National Meteorological Center, the former name of NCEP) method for defining forecast error covariance matrices]. Therefore ensemble forecasts potentially could be used to construct flow-dependent short-range forecast error covariance matrices for use in data assimilation schemes.

It is well understood that in a perfectly reliable ensemble the spread of ensemble members around the ensemble mean forecast equals the root-mean-square (rms) error of the mean. Adequate rms spread, however, does not guarantee sufficient variability among the ensemble forecast patterns. A comparison between PECA values and pattern anomaly correlation (PAC) values among the ensemble members reveals that the perturbations in the NCEP ensemble exhibit too much similarity, especially on the smaller scales. Hence a regional orthogonalization of the perturbations may improve ensemble performance.

## Abstract

The success story of numerical weather prediction is often illustrated with the dramatic decrease of errors in tropical cyclone track forecasts over the past decades. In a recent essay, Landsea and Cangialosi, however, note a diminishing trend in the reduction of perceived positional error (PPE; difference between forecast and observed positions) in National Hurricane Center tropical cyclone (TC) forecasts as they contemplate whether “the approaching limit of predictability for tropical cyclone track prediction is near or has already been reached.” In this study we consider a different interpretation of the PPE data. First, we note that PPE is different from true positional error (TPE; difference between forecast and true positions) as it is influenced by the error in the observed position of TCs. PPE is still customarily used as a proxy for TPE since the latter is not directly measurable. As an alternative, TPE is estimated here with an inverse method, using PPE measurements and a theoretically based assumption about the exponential growth of TPE as a function of lead time. Eighty-nine percent variance in the behavior of 36–120-h lead-time 2001–17 seasonally averaged PPE measurements is explained with an error model using just four parameters. Assuming that the level of investments, and the pace of improvements to the observing, modeling, and data assimilation systems continue unabated, the four-parameter error model indicates that the time limit of predictability at the 181 nautical mile error level (n mi; 1 n mi = 1.85 km), reached at day 5 in 2017, may be extended beyond 6 and 8 days in 10 and 30 years’ time, respectively.

## Abstract

The success story of numerical weather prediction is often illustrated with the dramatic decrease of errors in tropical cyclone track forecasts over the past decades. In a recent essay, Landsea and Cangialosi, however, note a diminishing trend in the reduction of perceived positional error (PPE; difference between forecast and observed positions) in National Hurricane Center tropical cyclone (TC) forecasts as they contemplate whether “the approaching limit of predictability for tropical cyclone track prediction is near or has already been reached.” In this study we consider a different interpretation of the PPE data. First, we note that PPE is different from true positional error (TPE; difference between forecast and true positions) as it is influenced by the error in the observed position of TCs. PPE is still customarily used as a proxy for TPE since the latter is not directly measurable. As an alternative, TPE is estimated here with an inverse method, using PPE measurements and a theoretically based assumption about the exponential growth of TPE as a function of lead time. Eighty-nine percent variance in the behavior of 36–120-h lead-time 2001–17 seasonally averaged PPE measurements is explained with an error model using just four parameters. Assuming that the level of investments, and the pace of improvements to the observing, modeling, and data assimilation systems continue unabated, the four-parameter error model indicates that the time limit of predictability at the 181 nautical mile error level (n mi; 1 n mi = 1.85 km), reached at day 5 in 2017, may be extended beyond 6 and 8 days in 10 and 30 years’ time, respectively.