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Abstract
In this paper, results are presented of an investigation to study the tropical intraseasonal oscillation (ISO) and its impact on the extended-range forecast in the NMC operational model during Phase II (14 December 1986–31 March 1987) of the Dynamical Extended Range Forecast (DERF). Based on principal component analysis of the velocity potential and streamfunction, evidence was found of tropical–extratropical interaction associated with the ISO. The NMC model possesses significant forecast skills for the principal streamfunction and velocity potential modes up to the first ten days. Results of the error growth analysis suggest that the principal modes of velocity potential have large initial errors comparable to the model random errors. By comparison, the initial errors in the streamfunction are much smaller. The error growth for both tropical and extratropical modes are found to be significantly suppressed during periods of strong ISO relative to periods of weak ISO. During strong ISO, a reduction of forecast error, implying a recovery of forecast skill, is also found at about three weeks into the forecast. This is presumably due to the oscillatory nature of the ISO.
The overall results suggest that the forecast skill of ISO and of low-frequency extratropical modes may be considerably enhanced during a special period of DERF when the ISO was strong. The increase in extratropical forecast skill is likely due to (a) the model's ability to better capture ISO signals in the tropics and (b) the increased coupling between the tropics and extratropics during periods of strong ISO. However, because of the limited data, the applicability of the present results to a more general case needs to be ascertained by further studies.
Abstract
In this paper, results are presented of an investigation to study the tropical intraseasonal oscillation (ISO) and its impact on the extended-range forecast in the NMC operational model during Phase II (14 December 1986–31 March 1987) of the Dynamical Extended Range Forecast (DERF). Based on principal component analysis of the velocity potential and streamfunction, evidence was found of tropical–extratropical interaction associated with the ISO. The NMC model possesses significant forecast skills for the principal streamfunction and velocity potential modes up to the first ten days. Results of the error growth analysis suggest that the principal modes of velocity potential have large initial errors comparable to the model random errors. By comparison, the initial errors in the streamfunction are much smaller. The error growth for both tropical and extratropical modes are found to be significantly suppressed during periods of strong ISO relative to periods of weak ISO. During strong ISO, a reduction of forecast error, implying a recovery of forecast skill, is also found at about three weeks into the forecast. This is presumably due to the oscillatory nature of the ISO.
The overall results suggest that the forecast skill of ISO and of low-frequency extratropical modes may be considerably enhanced during a special period of DERF when the ISO was strong. The increase in extratropical forecast skill is likely due to (a) the model's ability to better capture ISO signals in the tropics and (b) the increased coupling between the tropics and extratropics during periods of strong ISO. However, because of the limited data, the applicability of the present results to a more general case needs to be ascertained by further studies.
Abstract
Multidecadal and longer changes to the Atlantic interhemispheric sea surface temperature gradient (AITG) in phase 5 of the Coupled Model Intercomparison Project (CMIP5) historical simulations are investigated. Observations show a secular trend to this gradient over most of the twentieth century, with the southern lobe warming faster relative to its northern counterpart. A previous study of phase 3 of the CMIP (CMIP3) suggests that this trend is partially forced by anthropogenic sulfate aerosols. This analysis collectively confirms the partially forced trend for the CMIP5 and by anthropogenic aerosols. Like the CMIP3, the CMIP5 also simulates a reversal in the AITG trend in the late 1970s, which was attributed to a leveling off of the anthropogenic aerosol influence and increased influence of greenhouse gases in the late twentieth century. Two (of 25) CMIP5 models, however, systematically simulate a twentieth-century trend opposite to observed, leading to some uncertainty regarding the forced nature of the AITG trend. The observed AITG also exhibits a pronounced multidecadal modulation on top of the trend, associated with the Atlantic multidecadal oscillation (AMO). Motivated by a recent suggestion that the AMO is a forced response to aerosols, the causes of this multidecadal behavior were also examined. A few of the CMIP5 models analyzed do produce multidecadal AITG variations that are correlated to the observed AMO-like variation, but only one, the Hadley Centre Global Environmental Model, version 2 (HadGEM2), systematically simulates AMO-like behavior with both the requisite amplitude and phase. The CMIP5 simulations thus point to a robust aerosol influence on the historical AITG trend but not to the AMO-like multidecadal behavior.
Abstract
Multidecadal and longer changes to the Atlantic interhemispheric sea surface temperature gradient (AITG) in phase 5 of the Coupled Model Intercomparison Project (CMIP5) historical simulations are investigated. Observations show a secular trend to this gradient over most of the twentieth century, with the southern lobe warming faster relative to its northern counterpart. A previous study of phase 3 of the CMIP (CMIP3) suggests that this trend is partially forced by anthropogenic sulfate aerosols. This analysis collectively confirms the partially forced trend for the CMIP5 and by anthropogenic aerosols. Like the CMIP3, the CMIP5 also simulates a reversal in the AITG trend in the late 1970s, which was attributed to a leveling off of the anthropogenic aerosol influence and increased influence of greenhouse gases in the late twentieth century. Two (of 25) CMIP5 models, however, systematically simulate a twentieth-century trend opposite to observed, leading to some uncertainty regarding the forced nature of the AITG trend. The observed AITG also exhibits a pronounced multidecadal modulation on top of the trend, associated with the Atlantic multidecadal oscillation (AMO). Motivated by a recent suggestion that the AMO is a forced response to aerosols, the causes of this multidecadal behavior were also examined. A few of the CMIP5 models analyzed do produce multidecadal AITG variations that are correlated to the observed AMO-like variation, but only one, the Hadley Centre Global Environmental Model, version 2 (HadGEM2), systematically simulates AMO-like behavior with both the requisite amplitude and phase. The CMIP5 simulations thus point to a robust aerosol influence on the historical AITG trend but not to the AMO-like multidecadal behavior.
Abstract
A diagnostic model is proposed to use digitized satellite cloud brightness data to estimate objectively the large-scale flow patterns over data-void tropical regions. The model utilizes a linear barotropic vorticity equation with two primary assumptions: 1) that the area-averaged cloud brightness is positively correlated with large-scale divergence in the tropical upper troposphere; and 2) that the large-scale tropical flow is quasi-barotropic and quasi-non-divergent. It is designed to be used at any upper tropospheric level where divergence is important in determining the vorticity field. Three types of information are required: 1) boundary conditions determined from surrounding wind reports, 2) a mean zonal flow determined from climatology, and 3) an equivalent divergence forcing function constructed empirically from the brightness data.
The model is tested daily over a western North Pacific region for July-August 1971. Results for an 8-day representative period are presented and discussed. In general for 25% of the days tested, the model produces a flow field which accurately resembles the major features of the streamfunction field analyzed by the National Meteorological Center. In another 30% of the days it provides some valuable information about the flow patterns which would be difficult to obtain from boundary information alone. Experiments are also performed for two days in which the brightness data are enhanced by time-interpolated satellite infrared data. The resultant flow fields bear better resemblance to the NMC analysis. It is thus suggested that improved results may be expected when infrared and other types of advanced satellite data are available.
Abstract
A diagnostic model is proposed to use digitized satellite cloud brightness data to estimate objectively the large-scale flow patterns over data-void tropical regions. The model utilizes a linear barotropic vorticity equation with two primary assumptions: 1) that the area-averaged cloud brightness is positively correlated with large-scale divergence in the tropical upper troposphere; and 2) that the large-scale tropical flow is quasi-barotropic and quasi-non-divergent. It is designed to be used at any upper tropospheric level where divergence is important in determining the vorticity field. Three types of information are required: 1) boundary conditions determined from surrounding wind reports, 2) a mean zonal flow determined from climatology, and 3) an equivalent divergence forcing function constructed empirically from the brightness data.
The model is tested daily over a western North Pacific region for July-August 1971. Results for an 8-day representative period are presented and discussed. In general for 25% of the days tested, the model produces a flow field which accurately resembles the major features of the streamfunction field analyzed by the National Meteorological Center. In another 30% of the days it provides some valuable information about the flow patterns which would be difficult to obtain from boundary information alone. Experiments are also performed for two days in which the brightness data are enhanced by time-interpolated satellite infrared data. The resultant flow fields bear better resemblance to the NMC analysis. It is thus suggested that improved results may be expected when infrared and other types of advanced satellite data are available.
Abstract
The period July–December 1964 in the tropical western Pacific was marked by strong fluctuations in the meridional wind component with periods ranging from 4–6 days. This paper describes an intensive study of these disturbances, using spectrum-analysis techniques on time series of radiosonde data.
Two types of disturbances appear to be involved. One of these, prevalent at Canton Island, is characterized by upward phase propagation in the lower troposphere and downward phase propagation above 200 mb. It has previously been suggested that this wave may be the tropospheric manifestation of the mixed Rossby- gravity wave. A second type of disturbance, prevalent at stations further west, is marked by an absence of vertical phase propagation. Latent heat release may be important in its energetics.
Abstract
The period July–December 1964 in the tropical western Pacific was marked by strong fluctuations in the meridional wind component with periods ranging from 4–6 days. This paper describes an intensive study of these disturbances, using spectrum-analysis techniques on time series of radiosonde data.
Two types of disturbances appear to be involved. One of these, prevalent at Canton Island, is characterized by upward phase propagation in the lower troposphere and downward phase propagation above 200 mb. It has previously been suggested that this wave may be the tropospheric manifestation of the mixed Rossby- gravity wave. A second type of disturbance, prevalent at stations further west, is marked by an absence of vertical phase propagation. Latent heat release may be important in its energetics.
Abstract
The relative roles of clouds, surface evaporation, and ocean heat transport in limiting maximum sea surface temperatures (SSTs) in the western Pacific warm pool are investigated by means of simple and intermediate coupled ocean–atmosphere models. The authors first take an analytical approach by constructing a conceptual two-box model that contains dynamic coupling among the Walker circulation, SST, and ocean thermocline and thermodynamic coupling, which includes shortwave and longwave cloud forcing and latent and sensible heat fluxes at the ocean surface. In a realistic parameter regime, the three mechanisms mentioned above are all essential in limiting the SSTs within the observed range. The lack of any one mechanism would lead to an equilibrium SST that is too high, although unstable warming due to the super greenhouse effect would not occur. The analysis of the surface heat balance from the simple box model indicates that in the western Pacific warm pool, cloud reflection has a dominant effect, followed by evaporation and ocean dynamics.
The simple model results are further evaluated numerically by using an intermediate coupled ocean–atmosphere model. With the forcing of the annual-mean solar radiation, this model is capable of simulating a realistic annual mean climate in the tropical Pacific. The authors then introduce an initial SST perturbation and examine how the perturbation evolves with time in the presence of clouds, surface evaporation, and ocean dynamic processes. Four experiments have been designed. In the first three experiments, each of the three processes is studied separately; in the last experiment, they are combined. The intermediate model results indicate that in the western Pacific warm pool, the largest negative feedback comes from the cloud shortwave radiation forcing, followed by the surface evaporation and ocean heat transport. The sensitivity of the model to various initial SST perturbation patterns is also investigated.
Abstract
The relative roles of clouds, surface evaporation, and ocean heat transport in limiting maximum sea surface temperatures (SSTs) in the western Pacific warm pool are investigated by means of simple and intermediate coupled ocean–atmosphere models. The authors first take an analytical approach by constructing a conceptual two-box model that contains dynamic coupling among the Walker circulation, SST, and ocean thermocline and thermodynamic coupling, which includes shortwave and longwave cloud forcing and latent and sensible heat fluxes at the ocean surface. In a realistic parameter regime, the three mechanisms mentioned above are all essential in limiting the SSTs within the observed range. The lack of any one mechanism would lead to an equilibrium SST that is too high, although unstable warming due to the super greenhouse effect would not occur. The analysis of the surface heat balance from the simple box model indicates that in the western Pacific warm pool, cloud reflection has a dominant effect, followed by evaporation and ocean dynamics.
The simple model results are further evaluated numerically by using an intermediate coupled ocean–atmosphere model. With the forcing of the annual-mean solar radiation, this model is capable of simulating a realistic annual mean climate in the tropical Pacific. The authors then introduce an initial SST perturbation and examine how the perturbation evolves with time in the presence of clouds, surface evaporation, and ocean dynamic processes. Four experiments have been designed. In the first three experiments, each of the three processes is studied separately; in the last experiment, they are combined. The intermediate model results indicate that in the western Pacific warm pool, the largest negative feedback comes from the cloud shortwave radiation forcing, followed by the surface evaporation and ocean heat transport. The sensitivity of the model to various initial SST perturbation patterns is also investigated.
Abstract
A limited-area numerical model designed specifically for forecasting typhoon tracks has been operational at the Central Weather Bureau (CWB) in Taipei, Taiwan, since January 1990. It is a primitive equation model with nine σ levels and a grid size of 70 km. The model domain of 8500 km × 6000 km is centered near Taiwan, and covers the western part of the Pacific Ocean and southeast China. A model-balanced vortex is bogussed into the analysis to initialize the forecast. To ensure the maintenance of the vortex circulation throughout the forecast period, artificial heating options are incorporated to supplement the Kuo-type cumulus parameterization in the model.
The statistics of track errors for all forecast cases conducted during the development and operational checkout period (before December 1989) and during 1990, the first year of real-time operation, are reported. During the operational checkout period, 12 typhoons were forecasted, with an average 48-h track error of 415 km (62 forecast cases). For the 1990 season, there were 11 typhoons, with an average 48-h error of 392 km (63 forecast cases). The errors are compared with the One-Way Interactive Tropical Cyclone Model (OTCM), which is considered as the best long-term operational numerical track model for the western Pacific, using a homogeneous sample. The results indicate that the two models have similar average errors. The model had larger errors than the climatology and persistence (CLIPER) method. However, for all three typhoons with erratic movements, the model outperformed the CLIPER.
The model was modified in several ways prior to the beginning of the 1990 season. The most beneficial modification appears to have been the enlargement of the forecast domain. However, the domain was still not large enough to cover important synoptic fields for Typhoon Marian, which was the westernmost typhoon forecasted by the model. Postoperational experiments were conducted and the forecast track of Typhoon Marian improved when the model domain was expanded to the west. Examination of the synoptic patterns indicates that the track forecast depends closely on the forecast of the subtropical high circulation.
Abstract
A limited-area numerical model designed specifically for forecasting typhoon tracks has been operational at the Central Weather Bureau (CWB) in Taipei, Taiwan, since January 1990. It is a primitive equation model with nine σ levels and a grid size of 70 km. The model domain of 8500 km × 6000 km is centered near Taiwan, and covers the western part of the Pacific Ocean and southeast China. A model-balanced vortex is bogussed into the analysis to initialize the forecast. To ensure the maintenance of the vortex circulation throughout the forecast period, artificial heating options are incorporated to supplement the Kuo-type cumulus parameterization in the model.
The statistics of track errors for all forecast cases conducted during the development and operational checkout period (before December 1989) and during 1990, the first year of real-time operation, are reported. During the operational checkout period, 12 typhoons were forecasted, with an average 48-h track error of 415 km (62 forecast cases). For the 1990 season, there were 11 typhoons, with an average 48-h error of 392 km (63 forecast cases). The errors are compared with the One-Way Interactive Tropical Cyclone Model (OTCM), which is considered as the best long-term operational numerical track model for the western Pacific, using a homogeneous sample. The results indicate that the two models have similar average errors. The model had larger errors than the climatology and persistence (CLIPER) method. However, for all three typhoons with erratic movements, the model outperformed the CLIPER.
The model was modified in several ways prior to the beginning of the 1990 season. The most beneficial modification appears to have been the enlargement of the forecast domain. However, the domain was still not large enough to cover important synoptic fields for Typhoon Marian, which was the westernmost typhoon forecasted by the model. Postoperational experiments were conducted and the forecast track of Typhoon Marian improved when the model domain was expanded to the west. Examination of the synoptic patterns indicates that the track forecast depends closely on the forecast of the subtropical high circulation.
Abstract
In an effort to improve the tropical cyclone track forecast, two preprocessing procedures are applied to an operational baroclinic forecast system at the Central Weather Bureau (CWB) in Taipei. The first replaces the environmental wind field near the storm by the previous 6-h.movement vector of the storm. The second incorporates a wavenumber-1 asymmetry constructed by matching the flow at the center of the asymmetry with the previous 6-h storm movement. Applying both processes to the 32 typhoon casts archived at the CWB in 1990 reduces the averaged 48-h forecast distance error from 474 to 351 km.
Multiexisting typhoons may have interactions among themselves that depend on relative intensity. Proper representation of the intensities in the initial bogus is important for the track forecast. Experiments with different initial bogus intensities are conducted on a case of dual typhoons-Nat and Mireille in 1991. The forecast using different bogus vortices according to the estimated intensities of each typhoon gives substantially smaller errors than that using identical bogus vortices. The impact of initial bogus vortex intensity on the track forecast for single typhoon cases is also illustrated.
Abstract
In an effort to improve the tropical cyclone track forecast, two preprocessing procedures are applied to an operational baroclinic forecast system at the Central Weather Bureau (CWB) in Taipei. The first replaces the environmental wind field near the storm by the previous 6-h.movement vector of the storm. The second incorporates a wavenumber-1 asymmetry constructed by matching the flow at the center of the asymmetry with the previous 6-h storm movement. Applying both processes to the 32 typhoon casts archived at the CWB in 1990 reduces the averaged 48-h forecast distance error from 474 to 351 km.
Multiexisting typhoons may have interactions among themselves that depend on relative intensity. Proper representation of the intensities in the initial bogus is important for the track forecast. Experiments with different initial bogus intensities are conducted on a case of dual typhoons-Nat and Mireille in 1991. The forecast using different bogus vortices according to the estimated intensities of each typhoon gives substantially smaller errors than that using identical bogus vortices. The impact of initial bogus vortex intensity on the track forecast for single typhoon cases is also illustrated.
Abstract
The tropical Atlantic interhemispheric gradient in sea surface temperature significantly influences the rainfall climate of the tropical Atlantic sector, including droughts over West Africa and Northeast Brazil. This gradient exhibits a secular trend from the beginning of the twentieth century until the 1980s, with stronger warming in the south relative to the north. This trend behavior is on top of a multidecadal variation associated with the Atlantic multidecadal oscillation. A similar long-term forced trend is found in a multimodel ensemble of forced twentieth-century climate simulations. Through examining the distribution of the trend slopes in the multimodel twentieth-century and preindustrial models, the authors conclude that the observed trend in the gradient is unlikely to arise purely from natural variations; this study suggests that at least half the observed trend is a forced response to twentieth-century climate forcings. Further analysis using twentieth-century single-forcing runs indicates that sulfate aerosol forcing is the predominant cause of the multimodel trend. The authors conclude that anthropogenic sulfate aerosol emissions, originating predominantly from the Northern Hemisphere, may have significantly altered the tropical Atlantic rainfall climate over the twentieth century.
Abstract
The tropical Atlantic interhemispheric gradient in sea surface temperature significantly influences the rainfall climate of the tropical Atlantic sector, including droughts over West Africa and Northeast Brazil. This gradient exhibits a secular trend from the beginning of the twentieth century until the 1980s, with stronger warming in the south relative to the north. This trend behavior is on top of a multidecadal variation associated with the Atlantic multidecadal oscillation. A similar long-term forced trend is found in a multimodel ensemble of forced twentieth-century climate simulations. Through examining the distribution of the trend slopes in the multimodel twentieth-century and preindustrial models, the authors conclude that the observed trend in the gradient is unlikely to arise purely from natural variations; this study suggests that at least half the observed trend is a forced response to twentieth-century climate forcings. Further analysis using twentieth-century single-forcing runs indicates that sulfate aerosol forcing is the predominant cause of the multimodel trend. The authors conclude that anthropogenic sulfate aerosol emissions, originating predominantly from the Northern Hemisphere, may have significantly altered the tropical Atlantic rainfall climate over the twentieth century.
Abstract
The frequencies flown on the Special Sensor Microwave Imager (SSM/I) are sensitive to liquid water near the earth's surface. These frequencies are primarily atmospheric window channels, which receive the majority of their radiation from the surface. Liquid water near the surface depresses the emissivity as a function of wavelength. The relationship between brightness temperatures at different frequencies is used to dynamically derive the amount of liquid water in each SSM/I observation at 1/3° resolution. These data are averaged at 1° resolution throughout the globe for each month during the period of 1992–97, and the 6-yr monthly means and the monthly anomalies of the wetness index are computed from this base period. To quantify the relationship between precipitation and surface wetness, these anomalies are compared with precipitation anomalies derived from the Global Precipitation Climate Program. The analysis was performed for six agricultural regions across six continents. There is generally a good correspondence between the two variables. The correlation generally increases when the wetness index is compared with precipitation anomalies accumulated over a 2-month period. These results indicate that the wetness index has a strong correspondence to the upper layer of the soil moisture in many cultivated areas of the world. The region in southeastern Australia had the best relationship, with a correlation coefficient of 0.76. The Sahel, France, and Argentina showed that the wetness index had memory of precipitation anomalies from the previous months. The memory is shorter for southeastern Australia and central China. The weakest correlations occurred over the southeastern United States, where the surface is covered by dense vegetation. The unique signal, strengths, and weaknesses of the wetness index in each of the six study regions are discussed.
Abstract
The frequencies flown on the Special Sensor Microwave Imager (SSM/I) are sensitive to liquid water near the earth's surface. These frequencies are primarily atmospheric window channels, which receive the majority of their radiation from the surface. Liquid water near the surface depresses the emissivity as a function of wavelength. The relationship between brightness temperatures at different frequencies is used to dynamically derive the amount of liquid water in each SSM/I observation at 1/3° resolution. These data are averaged at 1° resolution throughout the globe for each month during the period of 1992–97, and the 6-yr monthly means and the monthly anomalies of the wetness index are computed from this base period. To quantify the relationship between precipitation and surface wetness, these anomalies are compared with precipitation anomalies derived from the Global Precipitation Climate Program. The analysis was performed for six agricultural regions across six continents. There is generally a good correspondence between the two variables. The correlation generally increases when the wetness index is compared with precipitation anomalies accumulated over a 2-month period. These results indicate that the wetness index has a strong correspondence to the upper layer of the soil moisture in many cultivated areas of the world. The region in southeastern Australia had the best relationship, with a correlation coefficient of 0.76. The Sahel, France, and Argentina showed that the wetness index had memory of precipitation anomalies from the previous months. The memory is shorter for southeastern Australia and central China. The weakest correlations occurred over the southeastern United States, where the surface is covered by dense vegetation. The unique signal, strengths, and weaknesses of the wetness index in each of the six study regions are discussed.
Abstract
Observing systems consisting of a finite number of in situ monitoring stations can provide high-quality measurements with the ability to quality assure both the instruments and the data but offer limited information over larger geographic areas. This paper quantifies the spatial coverage represented by a finite set of monitoring stations by using global data—data that are possibly of lower resolution and quality. For illustration purposes, merged satellite temperature data from Microwave Sounding Units are used to estimate the representativeness of the Global Climate Observing System Reference Upper-Air Network (GRUAN). While many metrics exist for evaluating the representativeness of a site, the ability to have highly accurate monthly averaged data is essential for both trend detection and climatology evaluation. The calculated correlations of the monthly averaged upper-troposphere satellite-derived temperatures over the GRUAN stations with all other pixels around the globe show that the current 9 certified GRUAN stations have moderate correlations (r ≥ 0.7) for approximately 10% of the earth, but an expanded network incorporating another 15 stations would result in moderate correlations for just over 60% of the earth. This analysis indicates that the value of additional stations can be quantified by using historical, satellite, or model data and can be used to reveal critical gaps in current monitoring capabilities. Evaluating the value of potential additional stations and prioritizing their initiation can optimize networks. The expansion of networks can be evaluated in a manner that allows for optimal benefit on the basis of optimization theory and economic analyses.
Abstract
Observing systems consisting of a finite number of in situ monitoring stations can provide high-quality measurements with the ability to quality assure both the instruments and the data but offer limited information over larger geographic areas. This paper quantifies the spatial coverage represented by a finite set of monitoring stations by using global data—data that are possibly of lower resolution and quality. For illustration purposes, merged satellite temperature data from Microwave Sounding Units are used to estimate the representativeness of the Global Climate Observing System Reference Upper-Air Network (GRUAN). While many metrics exist for evaluating the representativeness of a site, the ability to have highly accurate monthly averaged data is essential for both trend detection and climatology evaluation. The calculated correlations of the monthly averaged upper-troposphere satellite-derived temperatures over the GRUAN stations with all other pixels around the globe show that the current 9 certified GRUAN stations have moderate correlations (r ≥ 0.7) for approximately 10% of the earth, but an expanded network incorporating another 15 stations would result in moderate correlations for just over 60% of the earth. This analysis indicates that the value of additional stations can be quantified by using historical, satellite, or model data and can be used to reveal critical gaps in current monitoring capabilities. Evaluating the value of potential additional stations and prioritizing their initiation can optimize networks. The expansion of networks can be evaluated in a manner that allows for optimal benefit on the basis of optimization theory and economic analyses.