The Role of Nonlinear Forcing Singular Vector Tendency Error in Causing the “Spring Predictability Barrier” for ENSO

With the Zebiak–Cane model, the present study investigates the role of model errors represented by the nonlinear forcing singular vector(NFSV) in the "spring predictability barrier"(SPB) phenomenon in ENSO prediction. The NFSV-related model errors are found to have the largest negative effect on the uncertainties of El Nio prediction and they can be classified into two types: the first is featured with a zonal dipolar pattern of SST anomalies(SSTA), with the western poles centered in the equatorial central–western Pacific exhibiting positive anomalies and the eastern poles in the equatorial eastern Pacific exhibiting negative anomalies; and the second is characterized by a pattern almost opposite to the first type. The first type of error tends to have the worst effects on El Nin?o growth-phase predictions, whereas the latter often yields the largest negative effects on decaying-phase predictions. The evolution of prediction errors caused by NFSVrelated errors exhibits prominent seasonality, with the fastest error growth in spring and/or summer; hence,these errors result in a significant SPB related to El Nin?o events. The linear counterpart of NFSVs, the(linear) forcing singular vector(FSV), induces a less significant SPB because it contains smaller prediction errors. Random errors cannot generate an SPB for El Nio events. These results show that the occurrence of an SPB is related to the spatial patterns of tendency errors. The NFSV tendency errors cause the most significant SPB for El Nio events. In addition, NFSVs often concentrate these large value errors in a few areas within the equatorial eastern and central–western Pacific, which likely represent those areas sensitive to El Nio predictions associated with model errors. Meanwhile, these areas are also exactly consistent with the sensitive areas related to initial errors determined by previous studies. This implies that additional observations in the sensitive areas would not only improve the accuracy of the initial field but also promote the reduction of model errors to greatly improve ENSO forecasts.     

Role of Parameter Errors in the Spring Predictability Barrier for ENSO Events in the Zebiak–Cane Model

ABSTRACT The impact of both initial and parameter errors on the spring predictability barrier （SPB） is investigated using the Zebiak Cane model （ZC model）. Previous studies have shown that initial errors contribute more to the SPB than parameter errors in the ZC model. Although parameter errors themselves are less important, there is a possibility that nonlinear interactions can occur between the two types of errors, leading to larger prediction errors compared with those induced by initial errors alone. In this case, the impact of parameter errors cannot be overlooked. In the present paper, the optimal combination of these two types of errors [i.e., conditional nonlinear optimal perturbation （CNOP） errors] is calculated to investigate whether this optimal error combination may cause a more notable SPB phenomenon than that caused by initial errors alone. Using the CNOP approach, the CNOP errors and CNOP-I errors （optimal errors when only initial errors are considered） are calculated and then three aspects of error growth are compared： （1） the tendency of the seasonal error growth; （2） the prediction error of the sea surface temperature anomaly; and （3） the pattern of error growth. All three aspects show that the CNOP errors do not cause a more significant SPB than the CNOP-I errors. Therefore, this result suggests that we could improve the prediction of the E1 Nifio during spring by simply focusing on reducing the initial errors in this model.     

What Kind of Initial Errors Cause the Severest Prediction Uncertainty of El Nino in Zebiak-Cane Model

With the Zebiak-Cane(ZC)model,the initial error that has the largest effect on ENSO prediction is explored by conditional nonlinear optimal perturbation(CNOP).The results demonstrate that CNOP-type errors cause the largest prediction error of ENSO in the ZC model.By analyzing the behavior of CNOP- type errors,we find that for the normal states and the relatively weak El Nino events in the ZC model,the predictions tend to yield false alarms due to the uncertainties caused by CNOP.For the relatively strong El Nino events,the ZC model largely underestimates their intensities.Also,our results suggest that the error growth of El Nino in the ZC model depends on the phases of both the annual cycle and ENSO.The condition during northern spring and summer is most favorable for the error growth.The ENSO prediction bestriding these two seasons may be the most diffcult.A linear singular vector(LSV)approach is also used to estimate the error growth of ENSO,but it underestimates the prediction uncertainties of ENSO in the ZC model.This result indicates that the different initial errors cause different amplitudes of prediction errors though they have same magnitudes.CNOP yields the severest prediction uncertainty.That is to say,the prediction skill of ENSO is closely related to the types of initial error.This finding illustrates a theoretical basis of data assimilation.It is expected that a data assimilation method can filter the initial errors related to CNOP and improve the ENSO forecast skill.     

What Kind of Initial Errors Cause the Severest Prediction Uncertainty of El Nino in Zebiak-Cane Model

With the Zebiak-Cane （ZC） model, the initial error that has the largest effect on ENSO prediction is explored by conditional nonlinear optimal perturbation （CNOP）. The results demonstrate that CNOP-type errors cause the largest prediction error of ENSO in the ZC model. By analyzing the behavior of CNOPtype errors, we find that for the normal states and the relatively weak E1 Nifio events in the ZC model, the predictions tend to yield false alarms due to the uncertainties caused by CNOP. For the relatively strong E1 Nino events, the ZC model largely underestimates their intensities. Also, our results suggest that the error growth of E1 Nifio in the ZC model depends on the phases of both the annual cycle and ENSO. The condition during northern spring and summer is most favorable for the error growth. The ENSO prediction bestriding these two seasons may be the most difficult. A linear singular vector （LSV） approach is also used to estimate the error growth of ENSO, but it underestimates the prediction uncertainties of ENSO in the ZC model. This result indicates that the different initial errors cause different amplitudes of prediction errors though they have same magnitudes. CNOP yields the severest prediction uncertainty. That is to say, the prediction skill of ENSO is closely related to the types of initial error. This finding illustrates a theoretical basis of data assimilation. It is expected that a data assimilation method can filter the initial errors related to CNOP and improve the ENSO forecast skill.     

The Combined Effect of Initial Error and Model Error on ENSO Prediction Uncertainty Generated by the Zebiak-Cane Model

Initial errors and model errors are the source of prediction errors. In this study, the authors compute the conditional nonlinear optimal perturbation （CNOP）-type initial errors and nonlinear forcing singular vector （NFSV）- type tendency errors of the Zebiak-Cane model with respect to El Nifio events and analyze their combined effect on the prediction errors for E1 Nino events. The CNOP- type initial error （NFSV-type tendency error） represents the initial errors （model errors） that have the largest effect on prediction uncertainties for E1 Nifio events under the perfect model （perfect initial conditions） scenario. How- ever, when the CNOP-type initial errors and the NFSV- type tendency errors are simultaneously considered in the model, the prediction errors caused by them are not am- plified as the authors expected. Specifically, the predic- tion errors caused by the combined mode of CNOP-type initial errors and NFSV-type tendency errors are a little larger than those caused by the NFSV-type tendency er- rors. This fact emphasizes a need to investigate the opti- mal combined mode of initial errors and tendency errors that cause the largest prediction error for E1 Nifio events.     

Impact of SST Anomaly Events over the Kuroshio-Oyashio Extension on the “Summer Prediction Barrier”

The "summer prediction barrier"(SPB) of SST anomalies(SSTA) over the Kuroshio–Oyashio Extension(KOE) refers to the phenomenon that prediction errors of KOE-SSTA tend to increase rapidly during boreal summer, resulting in large prediction uncertainties. The fast error growth associated with the SPB occurs in the mature-to-decaying transition phase,which is usually during the August–September–October(ASO) season, of the KOE-SSTA events to be predicted. Thus, the role of KOE-SSTA evolutionary characteristics in the transition phase in inducing the SPB is explored by performing perfect model predictability experiments in a coupled model, indicating that the SSTA events with larger mature-to-decaying transition rates(Category-1) favor a greater possibility of yielding a more significant SPB than those events with smaller transition rates(Category-2). The KOE-SSTA events in Category-1 tend to have more significant anomalous Ekman pumping in their transition phase, resulting in larger prediction errors of vertical oceanic temperature advection associated with the SSTA events. Consequently, Category-1 events possess faster error growth and larger prediction errors. In addition, the anomalous Ekman upwelling(downwelling) in the ASO season also causes SSTA cooling(warming), accelerating the transition rates of warm(cold) KOE-SSTA events. Therefore, the SSTA transition rate and error growth rate are both related with the anomalous Ekman pumping of the SSTA events to be predicted in their transition phase. This may explain why the SSTA events transferring more rapidly from the mature to decaying phase tend to have a greater possibility of yielding a more significant SPB.     

Is Model Parameter Error Related to a Significant Spring Predictability Barrier for El Nio events？ Results from a Theoretical Model

Within a theoretical ENSO model, the authors investigated whether or not the errors superimposed on model parameters could cause a significant ``spring predictability barrier' (SPB) for El Nino events. First, sensitivity experiments were respectively performed to the air--sea coupling parameter, α and the thermocline effect coefficient μ. The results showed that the uncertainties superimposed on each of the two parameters did not exhibit an obvious season-dependent evolution; furthermore, the uncertainties caused a very small prediction error and consequently failed to yield a significant SPB. Subsequently, the conditional nonlinear optimal perturbation (CNOP) approach was used to study the effect of the optimal mode (CNOP-P) of the uncertainties of the two parameters on the SPB and to demonstrate that the CNOP-P errors neither presented a unified season-dependent evolution for different El Nino events nor caused a large prediction error, and therefore did not cause a significant SPB. The parameter errors played only a trivial role in yielding a significant SPB. To further validate this conclusion, the authors investigated the effect of the optimal combined mode (i.e. CNOP error) of initial and model errors on SPB. The results illustrated that the CNOP errors tended to have a significant season-dependent evolution, with the largest error growth rate in the spring, and yielded a large prediction error, inducing a significant SPB. The inference, therefore, is that initial errors, rather than model parameter errors, may be the dominant source of uncertainties that cause a significant SPB for El Nino events. These results indicate that the ability to forecast ENSO could be greatly increased by improving the initialization of the forecast model.     

Investigating the Initial Errors that Cause Predictability Barriers for Indian Ocean Dipole Events Using CMIP5 Model Outputs

By analyzing the outputs of the pre-industrial control runs of four models within phase 5 of the Coupled Model Intercomparison Project, the effects of initial sea temperature errors on the predictability of Indian Ocean Dipole events were identified. The initial errors cause a significant winter predictability barrier(WPB) or summer predictability barrier(SPB).The WPB is closely related with the initial errors in the tropical Indian Ocean, where two types of WPB-related initial errors display opposite patterns and a west–east dipole. In contrast, the occurrence of the SPB is mainly caused by initial errors in the tropical Pacific Ocean, where two types of SPB-related initial errors exhibit opposite patterns, with one pole in the subsurface western Pacific Ocean and the other in the upper eastern Pacific Ocean. Both of the WPB-related initial errors grow the fastest in winter, because the coupled system is at its weakest, and finally cause a significant WPB. The SPB-related initial errors develop into a La Ni ?na–like mode in the Pacific Ocean. The negative SST errors in the Pacific Ocean induce westerly wind anomalies in the Indian Ocean by modulating the Walker circulation in the tropical oceans. The westerly wind anomalies first cool the sea surface water in the eastern Indian Ocean. When the climatological wind direction reverses in summer, the wind anomalies in turn warm the sea surface water, finally causing a significant SPB. Therefore, in addition to the spatial patterns of the initial errors, the climatological conditions also play an important role in causing a significant predictability barrier.     

Influence of positive and negative Indian Ocean Dipoles on ENSO via the Indonesian Throughflow: Results from sensitivity experiments

The role of the Indonesian Throughflow(ITF) in the influence of the Indian Ocean Dipole(IOD) on ENSO is investigated using version 2 of the Parallel Ocean Program(POP2) ocean general circulation model. We demonstrate the results through sensitivity experiments on both positive and negative IOD events from observations and coupled general circulation model simulations. By shutting down the atmospheric bridge while maintaining the tropical oceanic channel, the IOD forcing is shown to influence the ENSO event in the following year, and the role of the ITF is emphasized. During positive IOD events,negative sea surface height anomalies(SSHAs) occur in the eastern Indian Ocean, indicating the existence of upwelling.These upwelling anomalies pass through the Indonesian seas and enter the western tropical Pacific, resulting in cold anomalies there. These cold temperature anomalies further propagate to the eastern equatorial Pacific, and ultimately induce a La Nia-like mode in the following year. In contrast, during negative IOD events, positive SSHAs are established in the eastern Indian Ocean, leading to downwelling anomalies that can also propagate into the subsurface of the western Pacific Ocean and travel further eastward. These downwelling anomalies induce negative ITF transport anomalies, and an El Nio-like mode in the tropical eastern Pacific Ocean that persists into the following year. The effects of negative and positive IOD events on ENSO via the ITF are symmetric. Finally, we also estimate the contribution of IOD forcing in explaining the Pacific variability associated with ENSO via ITF.     

Application of Conditional Nonlinear Optimal Perturbation to Targeted Observation Studies of the Atmosphere and Ocean

This paper reviews progress in the application of conditional nonlinear optimal perturbation to targeted observation studies of the atmosphere and ocean in recent years, with a focus on the E1 Nifio-Southern Oscillation （ENSO）, Kuroshio path variations, and blocking events. Through studying the optimal precursor （OPR） and optimally growing initial error （OGE） of the occurrence of the above events, the similarity and localization features of OPR and OGE spatial structures have been found for each event. Ideal hindcasting experiments have shown that, if initial errors are reduced in the areas with the largest amplitude for the OPR and OGE for ENSO and Kuroshio path variations, the forecast skill of the model for these events is significantly improved. Due to the similarity between patterns of the OPR and OGE, additional observations implemented in the same sensitive region would help to not only capture the precursors, but also reduce the initial errors in the predictions, greatly increasing the forecast abilities. The similarity and localization of the spatial structures of the OPR and OGE during the onset of blocking events have also been investigated, but their application to targeted observation requires further study.     

Diagnosing SST Error Growth during ENSO Developing Phase in the BCC_CSM1.1(m) Prediction System

In this study, the predictability of the El Nino-South Oscillation(ENSO) in an operational prediction model from the perspective of initial errors is diagnosed using the seasonal hindcasts of the Beijing Climate Center System Model,BCC;SM1.1(m). Forecast skills during the different ENSO phases are analyzed and it is shown that the ENSO forecasts appear to be more challenging during the developing phase, compared to the decay phase. During ENSO development, the SST prediction errors are significantly negative and cover a large area in the central and eastern tropical Pacific, thus limiting the model skill in predicting the intensity of El Nino. The large-scale SST errors, at their early stage, are generated gradually in terms of negative anomalies in the subsurface ocean temperature over the central-western equatorial Pacific,featuring an error evolutionary process similar to that of El Nino decay and the transition to the La Nina growth phase.Meanwhile, for short lead-time ENSO predictions, the initial wind errors begin to play an increasing role, particularly in linking with the subsurface heat content errors in the central-western Pacific. By comparing the multiple samples of initial fields in the model, it is clearly found that poor SST predictions of the Nino-3.4 region are largely due to contributions of the initial errors in certain specific locations in the tropical Pacific. This demonstrates that those sensitive areas for initial fields in ENSO prediction are fairly consistent in both previous ideal experiments and our operational predictions,indicating the need for targeted observations to further improve operational forecasts of ENSO.     

Decadal and interannual variability of the Indian Ocean Dipole

This study investigates the decadal and interannual variability of the Indian Ocean Dipole （IOD）. It is found that the long-term IOD index displays a decadal phase variation. Prior to 1920 negative phase dominates but after 1960 positive phase prevails. Under the warming background of the tropical ocean, a larger warming trend in the western Indian Ocean is responsible for the decadal phase variation of the IOD mode. Due to reduced latent heat loss from the local ocean, the western Indian Ocean warming may be caused by the weakened Indian Ocean westerly summer monsoon. The interannual air-sea coupled IOD mode varies on the background of its decadal variability. During the earlier period （1948-1969）, IOD events are characterized by opposing SST anomaly （SSTA） in the western and eastern Indian Ocean, with a single vertical circulation above the equatorial Indian Ocean. But in the later period （1980-2003）, with positive IOD dominating, most IOD events have a zonal gradient perturbation on a uniform positive SSTA. However, there are three exceptionally strong positive IOD events （1982, 1994, and 1997）, with opposite SSTA in the western and eastern Indian Ocean, accompanied by an El Nifio event. Consequently, two anomalous reversed Walker cells are located separately over the Indian Ocean and western-eastern Pacific; the one over the Indian Ocean is much stronger than that during other positive IOD events.     

The Roles of Spatial Locations and Patterns of Initial Errors in the Uncertainties of Tropical Cyclone Forecasts

In this study,a series of sensitivity experiments were performed for two tropical cyclones (TCs),TC Longwang (2005) and TC Sinlaku (2008),to explore the roles of locations and patterns of initial errors in uncertainties of TC forecasts.Specifically,three types of initial errors were generated and three types of sensitive areas were determined using conditional nonlinear optimal perturbation (CNOP),first singular vector (FSV),and composite singular vector (CSV) methods.Additionally,random initial errors in randomly selected areas were considered.Based on these four types of initial errors and areas,we designed and performed 16 experiments to investigate the impacts of locations and patterns of initial errors on the nonlinear developments of the errors,and to determine which type of initial errors and areas has the greatest impact on TC forecasts.Overall,results from the experiments indicate the following:(1) The impact of random errors introduced into the sensitive areas was greater than that of errors themselves fixed in the randomly selected areas.From the perspective of statistical analysis,and by comparison,the impact of random errors introduced into the CNOP target area was greatest.(2) The initial errors with CNOP,CSV,or FSV patterns were likely to grow faster than random errors.(3) The initial errors with CNOP patterns in the CNOP target areas had the greatest impacts on the final verification forecasts.     

Temporal Structures of the North Atlantic Oscillation and Its Impact on the Regional Climate Variability

In this study, the temporal structure of the variation of North Atlantic Oscillation （NAO） and its impact on regional climate variability are analyzed using various datasets. The results show that blocking formations in the Atlantic region are sensitive to the phase of the NAO. Sixty-seven percent more winter blocking days are observed during the negative phase compared to the positive phase of the NAO. The average length of blocking during the negative phase is about 11 days, which is nearly twice as long as the 6-day length observed during the positive phase of the NAO. The NAO-related differences in blocking frequency and persistence are associated with changes in the distribution of the surface air temperature anomaly, which, to a large extent, is determined by the phase of the NAO. The distribution of regional cloud amount is also sensitive to the phase of the NAO. For the negative phase, the cloud amounts are significant, positive anomalies in the convective zone in the Tropics and much less cloudiness in the mid latitudes. But for the positive phase of the NAO, the cloud amount is much higher in the mid-latitude storm track region. In the whole Atlantic region, the cloud amount shows a decrease with the increase of surface air temperature. These results suggest that there may be a negative feedback between the cloud amount and the surface air t.emperature in the Atlantic region.     

Can the Uncertainties of Madden–Jullian Oscillation Cause a Significant “Spring Predictability Barrier” for ENSO Events?

With the Zebiak-Cane model and a parameterized stochastic representation of intraseasonal forcing, the impact of the uncertainties of Madden–Jullian Oscillation (MJO) on the “Spring Predictability Barrier (SPB)” for El Ni?no–Southern Oscillation (ENSO) prediction is studied. The parameterized form of MJO forcing is added physically to the Zebiak-Cane model to obtain the so-called Zebiak-Cane-MJO model and then the effects of initial error, stochastic model error, and their joint error mode on the SPB associated with El Ni?no prediction are estimated. The results show that the model errors caused by stochastic MJO forcing could hardly lead to a significant SPB while initial errors can do; furthermore, the joint error mode of initial error and model error associated with the stochastic MJO forcing can also lead to a significant SPB. These demonstrate that the initial error is probably the main error source of the SPB, which may provide a theoretical foundation of data assimilation for ENSO forecasts.     

Simulation of the Electrification of a Tropical Cyclone Using the WRF-ARW Model：An Idealized Case

Evolution of the electrifi cation of an idealized tropical cyclone （TC） is simulated by using the Advanced Weather Research and Forecasting （WRF-ARW） model. The model was modifi ed by addition of explicit electrifi cation and a new bulk discharge scheme. The characteristics of TC lightning is further examined by analyses of the electrifi cation and the charge structure of the TC. The fi ndings thus obtained are able to unify most of the previous inconsisitent observational and simulation studies. The results indicate that the TC eyewall generally exhibits an inverted dipole charge structure with negative charge above the positive. In the intensifi cation stage, however, the extremely tall towers of the eyewall may exhibit a normal tripole structure with a main negative region between two regions of positive charge. The outer spiral rainband cells display a simple normal dipole structure during all the stages. It is further found that the diff erences in the charge structure are associated with diff erent updrafts and particle distributions. Weak updrafts, together with a coexistence region of diff erent particles at lower levels in the eyewall, result in charging processes that occur mainly in the positive graupel charging zone （PGCZ）. In the intensifi cation stage, the occurrence of charging processes in both positive and negative graupel charging zones is associated with strong updraft in the extremely tall towers. In addition, the coexistence region of graupel and ice crystals is mainly situated at upper levels in the outer rainband, so the charging processes mainly occur in the negative graupel charging zone （NGCZ）.     

Numerical Experiments of the Effects of Initial Desert Moisture on the Climate Change

A numerical model with the p-sigma incorporated coordinate system and primitive equations is used to simulate the effect of initial soil moisture in desert areas on the climate change. The results show that the present deserts have a tendency to expand. When the initial soil moisture in the desert regions increases,the desert areas will shrink but can not disappear. The small deserts may not remain any longer when there are sources of water vapour around. Both the land-sea contrast and the topography are the background conditions of the present desert distribution through the mechanism of the downdrafts and the rare precipitation over the desert regions. The increase of the initial desert soil moisture will weaken the summer monsoon circulation and, consequently, the monsoonal precipitation.     

Time-Dependent Nonlinear Forcing Singular Vector-Type Tendency Error of the Zebiak-Cane Model

Based on the Zebiak-Cane model, the timedependent nonlinear forcing singular vector （NFSV）-type tendency errors with components of 4 and 12 （denoted by NFSV-4 and NFSV-12） are calculated for predetermined El Nifio events and compared with the constant NFSV （denoted by NFSV-1） from their patterns and resultant prediction errors. Specifically, NFSV-1 has a zonal dipolar sea surface temperature anomaly （SSTA） pattern with negative anomalies in the equatorial eastern Pacific and positive anomalies in the equatorial central-western Pa- cific. Although the first few components in NFSV-4 and NFSV-12 present patterns similar to NFSV-1, they tend to extend their dipoles farther westward; meanwhile, the positive anomalies gradually cover much smaller regions with the lag times. In addition, the authors calculate the predic- tion errors caused by the three kinds of NFSVs, and the results indicate that the prediction error induced by NFSV-12 is the largest, followed by the NFSV-4. However, when compared with the prediction errors caused by random tendency errors, the NFSVs generate significantly larger prediction errors. It is therefore shown that the spatial structure of tendency errors is important for producing large prediction errors. Furthermore, in exploring the tendency errors that cause the largest prediction error for E1 Nifio events, the timedependent NFSV should be evaluated.     

Extended-Range Forecasts of the Principal 20–30-Day Oscillation of the Circulation over East Asia During the Summer of 2002

Daily 850-hPa meridional wind fields in East Asia from March to September 2002 were used to establish a model of the principal oscillation pattern (POP). This model was then used to conduct independent extended-range forecasts of the principal temporal and spatial variations in the low-frequency meridional wind field on a time scale of 20-30 days. These variations affect the occurrence of heavy precipitation events in the lower reaches of the Yangtze River valley (LYRV). The results of 135 forecast experiments during the summer half year show that the predicted and observed anomalies are strongly correlated at a lead time of 20 days (mean correlation greater than 0.50). This strong correlation indicates that the model is capable of accurately forecasting the low-frequency variations in meridional wind that corresponded to the 3 heavy precipitation events in the LYRV during the summer of 2002. Further forecast experiments based on data from multiple years with significant 20-30-day oscillations show that these prediction modes are effective tools for forecasting the space-time evolution of the low-frequency circulation. These findings offer potential for improving the accuracy of forecasts of heavy precipitation over the LYRV at lead times of 3-4 weeks.     

ON TEMPERATURE INITIALIZATION IN PRIMITIVE EQUATION FORECAST MODELS

Four procedures of specifying model initial temperature were described and tested in the present study. It was found that the use of observed temperatures along with a proper vertical interpolation scheme was not only acceptable, but produced less error than the use of temperatures derived from geopotential height through the hydrostatic equation did. Use of the difference form of the hydrostatic equation would produce unacceptable errors in the initial temperatures, unrealistic horizontal and vertical distribution of temperature, and these errors would influence the calculalion of the pressure gradient force, resulting in substantial, artificial disturbances within the model domain.In addition, an approach to check the initial data was described. Taking advantage of the fact that the geostrophic wind in sigma coordinates should be nondivergent, geopotential height and temperature were used to calculate the pressure gradient force terms and an initial divergence of the geostrophic wind. This approach ca