Error Sensitivity Analysis in 10–30-Day Extended Range Forecasting by Using a Nonlinear Cross-Prediction Error Model

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  • Corresponding author: Zhiye XIA, xiazhiye@cuit.edu.cn
  • Funds:

    Supported by the National Natural Science Foundation of China (41505012 and 41471305), Open Research Fund of Plateau Atmosphere and Environment Key Laboratory of Sichuan Province (PAEKL-2017-Y1), Scientific Research Fund of Chengdu University of Information Technology (J201613 and KYTZ201607), Innovation Team Fund (16TD0024), and Elite Youth Cultivation Project of Sichuan Province (2015JQ0037)

  • doi: 10.1007/s13351-017-6098-2

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  • Extended range forecasting of 10–30 days, which lies between medium-term and climate prediction in terms of timescale, plays a significant role in decision-making processes for the prevention and mitigation of disastrous meteorological events. The sensitivity of initial error, model parameter error, and random error in a nonlinear cross-prediction error (NCPE) model, and their stability in the prediction validity period in 10–30-day extended range forecasting, are analyzed quantitatively. The associated sensitivity of precipitable water, temperature, and geopotential height during cases of heavy rain and hurricane is also discussed. The results are summarized as follows. First, the initial error and random error interact. When the ratio of random error to initial error is small (10–6–10–2), minor variation in random error cannot significantly change the dynamic features of a chaotic system, and therefore random error has minimal effect on the prediction. When the ratio is in the range of 10–1–2 (i.e., random error dominates), attention should be paid to the random error instead of only the initial error. When the ratio is around 10–2–10–1, both influences must be considered. Their mutual effects may bring considerable uncertainty to extended range forecasting, and de-noising is therefore necessary. Second, in terms of model parameter error, the embedding dimension m should be determined by the factual nonlinear time series. The dynamic features of a chaotic system cannot be depicted because of the incomplete structure of the attractor when m is small. When m is large, prediction indicators can vanish because of the scarcity of phase points in phase space. A method for overcoming the cut-off effect (m > 4) is proposed. Third, for heavy rains, precipitable water is more sensitive to the prediction validity period than temperature or geopotential height; however, for hurricanes, geopotential height is most sensitive, followed by precipitable water.
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  • Fig. 3.  Diagonal nonlinear cross-prediction error (for the case in Chongqing, China on 21 July 1996).

    Fig. 1.  DNCPE with random error of different magnitudes (for the case in Chongqing, China on 21 July 1996).

    Fig. 2.  Correlation dimension D (for the case in Chongqing, China on 21 July 1996).

    Fig. 4.  DNCPE of PWAT, TEMP, and HGT for the heavy rain case in Worcester, USA on 20 July 2007.

    Fig. 5.  DNCPE of PWAT, TEMP, and HGT for the hurricane case in New Orleans, USA during 29–31 August 2005.

    Table 1.  Gaussian noise of different orders added into the NCPE model

    Standard deviation of Gaussian noise (δ')Magnitude size (E)
    δ' = 10–6δ10–6
    δ' = 10–5δ10–5
    δ' = 10–4δ10–4
    δ' = 10–3δ10–3
    δ' = 10–2δ10–2
    δ' = 10–1δ10–1
    δ' = 0.5δ0.5
    δ' = 2δ2
    Download: Download as CSV

    Table 2.  Parameters calculated for PWAT, TEMP, and HGT during two cases

    Case (time, place, weather)PWATTEMPHGT
    IoTmIoTmIoTm
    20 July 2007, Worcester, heavy rain502244620447234
    29–31 August 2005, New Orleans, hurricane511945220449244
    Download: Download as CSV
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Error Sensitivity Analysis in 10–30-Day Extended Range Forecasting by Using a Nonlinear Cross-Prediction Error Model

    Corresponding author: Zhiye XIA, xiazhiye@cuit.edu.cn
  • 1. College of Resources and Environment, Chengdu University of Information Technology, Chengdu 610225
  • 2. Plateau Atmosphere and Environment Key Laboratory of Sichuan Province, Chengdu 610225
  • 3. Key Laboratory of Middle Atmosphere and Global Environment Observation, Institute of Atmospheric Physics, Chinese Academy of Sciences, Beijing 100029
Funds: Supported by the National Natural Science Foundation of China (41505012 and 41471305), Open Research Fund of Plateau Atmosphere and Environment Key Laboratory of Sichuan Province (PAEKL-2017-Y1), Scientific Research Fund of Chengdu University of Information Technology (J201613 and KYTZ201607), Innovation Team Fund (16TD0024), and Elite Youth Cultivation Project of Sichuan Province (2015JQ0037)

Abstract: Extended range forecasting of 10–30 days, which lies between medium-term and climate prediction in terms of timescale, plays a significant role in decision-making processes for the prevention and mitigation of disastrous meteorological events. The sensitivity of initial error, model parameter error, and random error in a nonlinear cross-prediction error (NCPE) model, and their stability in the prediction validity period in 10–30-day extended range forecasting, are analyzed quantitatively. The associated sensitivity of precipitable water, temperature, and geopotential height during cases of heavy rain and hurricane is also discussed. The results are summarized as follows. First, the initial error and random error interact. When the ratio of random error to initial error is small (10–6–10–2), minor variation in random error cannot significantly change the dynamic features of a chaotic system, and therefore random error has minimal effect on the prediction. When the ratio is in the range of 10–1–2 (i.e., random error dominates), attention should be paid to the random error instead of only the initial error. When the ratio is around 10–2–10–1, both influences must be considered. Their mutual effects may bring considerable uncertainty to extended range forecasting, and de-noising is therefore necessary. Second, in terms of model parameter error, the embedding dimension m should be determined by the factual nonlinear time series. The dynamic features of a chaotic system cannot be depicted because of the incomplete structure of the attractor when m is small. When m is large, prediction indicators can vanish because of the scarcity of phase points in phase space. A method for overcoming the cut-off effect (m > 4) is proposed. Third, for heavy rains, precipitable water is more sensitive to the prediction validity period than temperature or geopotential height; however, for hurricanes, geopotential height is most sensitive, followed by precipitable water.

    • Extended range forecasting of 10–30 days lies between medium-term and climate prediction in terms of time-scale. The limit of atmospheric predictability is about two weeks in mid–high latitudes (Lorenz, 1963), and 10–30-day extended range forecasting needs to go beyond this limit. Extended range forecasting is difficult and is also a blind spot in operational forecasting because of the lack of objective forecasting methods.

      The atmosphere is a complex nonlinear dynamic system and it is inherently chaotic. Atmospheric predictabi-lity can be divided into two kinds based on the prediction errors (Lorenz, 1965, 1969). The first kind is determined by initial error, whose growth leads to the uncertainty of atmospheric prediction. This uncertainty can be addressed through suppressing the growth of initial error, such as what is being practiced in daily weather forecast. The second kind is determined by external forcing, which is also called random error or boundary condition error. It often comes from inaccurate representation of the ocean, land, or cryosphere. Uncertainty occurs when external forcing dominates even with the effect of initial error being minimal, such as the case in climate prediction. In fact, uncertainty of extended range forecasting lies in between these two kinds of errors on different timescales, that is, initial and random errors may be equally important. The extended range prediction is simultaneously disturbed by these two prediction errors. Thus, improving accuracy for extended range forecasting is highly challenging.

      The three main approaches for carrying out 10–30-day extended range forecasting include the dynamical model approach, classical statistical model approach, and big-data model approach (Yang, 2015a). In the dynamical model approach, ensemble numerical prediction is utilized (Miller et al., 2010; Abhilash et al., 2014; Joseph et al., 2015); however, the dynamical description of numerical models is highly sensitive to errors such as initial error due to the chaotic feature of atmosphere (Ding and Li, 2008a), leading to great uncertainty in the results of extended range forecasting. Basically, each atmospheric variable can be decomposed into two components: one is sensitive to initial state and the other is not (Krishnamurti et al., 2000; Pattanaik and Das, 2015). Chou et al. (2010) and Zheng et al. (2012, 2013) applied different procedures including those that are statistics based for these two different components in extended range forecasting. They suggested that combining dynamics and statistics is necessary for improving extended range forecasting (Lee et al., 2015).

      In the classical statistical models, detecting the evolution of atmospheric low-frequency oscillation (Waliser et al., 1999; Webster et al., 2010; Janjic et al., 2011; He et al., 2013; Johnson et al., 2014) is a main method. Baldwin et al. (2003) employed an empirical statistical model in the extended range forecasting of monthly-mean Arctic oscillation.

      The big-data model includes methods based on multiple lagged regressive or complex autoregressive (Yang, 2014, 2015b), neural network (Borah et al., 2013), low-frequency weather chart (Sun et al., 2013), and Maddan–Julian Oscillation (MJO) (Love et al., 2010; Liang and Ding, 2012). Methods based on MJO and low-frequency weather chart are reviewed and it is concluded that more details associated with these methods need to be worked out and further improved (Ma et al., 2012).

      Xia et al. (2015) developed a nonlinear cross-prediction error (NCPE) model on the basis of chaotic theory. The local dynamic variation characteristics of an attractor in chaotic weather system in phase space were analyzed for 100 global disastrous weather events. The results showed that 4, 22, and 74 cases exhibited prediction validity periods of 1–2, 3–9, and 10–30 days, respectively, without false alarms or omissions, which is an encouraging result in extended range forecasting.

      Jung et al. (2010) identified the origin of errors in extended range forecasting and studied the influences of lower boundary conditions, the tropical atmosphere, and the Northern Hemisphere stratosphere. Nicolis (2016) derived a set of general expressions for the dynamical behavior of prediction errors of time-averaged observables involved in extended range forecasts and analyzed the effect of chaotic dynamics in the process of extended range forecasts. However, quantitative analyses on the influence of initial, random, and model parameter errors on the stability of extended range forecasts are quite limited (Keller et al., 2007). The present study intends to fill this gap. It should be noted that Xia et al. (2015) focused on the basic theoretical development of NCPE and its validation using real heavy rain samples for extended range forecasting, whereas the current study focuses on the influences of random error, initial error, and model parameter error on forecasting results.

      Based on Xia et al. (2015), this paper investigates the sensitivity of prediction errors to initial error and random error using the NCPE model. The paper is organized as follows. Section 2 discusses function of the NCPE model for 10–30-day extended range forecasting. Section 3 quantitatively analyzes the sensitivity of initial, model parameter, and random error to the extended range forecasting, and the sensitivity of different meteorological elements to different kinds of disastrous weather. Conclusions are given in Section 4.

    2.   NCPE model
    • The NCPE model was developed over the reconstructed phase space of a single meteorological variable according to chaotic dynamics theory (Xia et al., 2015). The NCPE model may be used to calculate nonlinear cross-prediction error of attractor local pairs, depict the local dynamical change features in the attractors, and evaluate the development mechanism for a heavy rain chaotic process, through combined dynamic and statistical analyses (Xia et al., 2015). It was tested for 100 global rainstorm cases, based on the chaotic single-variable nonlinear time series of precipitable water (PWAT).

      Criteria of the prediction eigen-peak (a prediction indicator) for 10–30-day extended range forecasting using the NCPE model were given in Xia et al. (2015). First, the eigen-peak is defined based on an error threshold of about 1.5, that is, values of the diagonal nonlinear cross-prediction error (DNCPE) greater than 1.5 could be regarded as prediction eigen-peak. Second, the DNCPE peaks before segment 5 were ignored because of the uncertainty effects of initial error. Third, the first emerged DNCPE peak after segment 5 was identified as prediction eigen-peak (Xia et al., 2015). The eigen-peaks are then used to analyze the forecasting validity period. The results show that 4, 22, and 74 cases obtained prediction validity periods of 1–2, 3–9, and 10–30 days, respectively, without false alarm or omission.

      In the DNCPE errors figure (see Fig. 3 of Xia et al., 2015), the x-axis represents the segment divided by the trajectory of an entire attractor that was projected in the phase space, where one segment corresponds to a certain time interval [i.e., one day in Xia et al. (2015)]. The y-axis is DNCPE, representing local relative dynamic error in the chaotic evolution of a weather system. The eigen-peak for the prediction indicator of heavy rain is regarded as mutation or the most unstable transition zone in the trajectory of the entire attractor.

      Figure 3.  Diagonal nonlinear cross-prediction error (for the case in Chongqing, China on 21 July 1996).

      The extended range forecasting based on the NCPE model is intended for analysis of the local relative dynamic error in the phase space and for exploration of the chaotic characteristics of a weather system. The NCPE matrix depicts the errors between any two segments in phase space, which reflects the local dynamic features of a chaotic weather system. The NCPE matrix has six typical locations, and their meanings have been explained in detail in Xia et al. (2015).

      The remaining part of this paper discusses the sensitivities of the initial, model parameter, and random errors to the NCPE model prediction results and the stability of the validity period of the extended range forecasting. Moreover, sensitivities of different meteorological elements, such as precipitable water (PWAT), temperature (TEMP), and geopotential height (HGT) to different disastrous weathers are also investigated. These sensitivities analyses about the three kinds of errors and different meteorological elements based on the NCPE model are also validated by examining the above 100 global disastrous weather cases. Some common features about the 10–30-day extended range forecast of the NCPE model are analyzed and concluded.

    3.   Sensitivity analysis
    • The selection rule of raw data for sensitivity analysis in this paper is as follows. The raw data length is 31 days or more, which needs to cover the weather case period, and the 7–10 days after the weather case is also included in the raw data length. This rule contributes to observing the local dynamic characteristics before, during, and after the weather system in the phase space. The data for NCPE model calculation are a single variable time series (i.e., PWAT, TEMP, or HGT). These raw data are all from the four-time-daily NCEP–NCAR reanalysis dataset on a horizontal resolution of 2.5° × 2.5°.

      Data preprocessing steps include reconstruction of phase space based on Takens theorem and calculations of optimal interpolation times Io based on fractal interpolation and metric entropy, time delay T based on mutual information, correlation dimension D with G–P algorithm, and embedding dimension m based on false nearest neighbor method. Details on these steps can be found in Xia et al. (2015). The processed data are used as input to the NCPE model.

    • As an example, a heavy rain case that occurred in Chongqing, China on 21 July 1996 was chosen. The total rainfall of this case is 206 mm, and the raw data length of PWAT is 31 × 4 = 124. The parameters calculated through preprocessing are as follows: optimal interpolation time Io = 43, time delay T = 16, and embedding dimension m = 4.

      The standard deviation of the interpolated data obtained through atmospheric predictability analysis (Li and Ding, 2011; Shi et al., 2012) is δ = 7.9356. Thus, the initial perturbation of raw data is at a magnitude of $\delta (0) = 7.9356$, that is, initial error of the PWAT adopted in this case is at magnitude of $\delta (0) = 7.9356$.

      A hypothesis is made regarding a random error, which can be simulated by Gaussian noise. Random error is also called external forcing, which often originates from ocean, land, or cryosphere. As shown in Table 1, the random error δ' is set at different orders (or magnitudes) with E depicted as $E{\rm{ = }}\delta '/\delta $, where magnitude E represents the relationship between random error and initial error, and E ranges from 10–6 to 2.

      Standard deviation of Gaussian noise (δ')Magnitude size (E)
      δ' = 10–6δ10–6
      δ' = 10–5δ10–5
      δ' = 10–4δ10–4
      δ' = 10–3δ10–3
      δ' = 10–2δ10–2
      δ' = 10–1δ10–1
      δ' = 0.5δ0.5
      δ' = 2δ2

      Table 1.  Gaussian noise of different orders added into the NCPE model

      A new state can be configured as the initial state (obtained after data preprocessing) superimposed with random errors of different magnitudes of E. The new state data are then applied to the NCPE model, the relationship between random error and initial error is quantitatively analyzed, and the sensitivity of the extended range forecast performance to the above types of errors can be found out.

      Figure 1 [cf., Fig. 3 in Xia et al. (2015)] shows the DNCPE calculated from the NCEP model coupled with random error of different magnitudes as prescribed in Table 1 for the Chongqing heavy rain case. The heavy rain occurred at segment 21. The DNCPE values fluctuate, and the eigen-peak at segment 11 is also altered, with different magnitudes of E. When the random error is smaller than the initial error, that is, magnitude E varies from 10–6 to 10–2, DNCPE slightly alters, and the prediction characteristics remain nearly stable, regardless of the changes of E. As the heavy rain occurred at segment 21, the prediction eigen-peak of this heavy rain case is always at segment 11, and the prediction validity period t = 21 – 11 = 10 days, for E values within the range of 10–6 –10–2.

      Figure 1.  DNCPE with random error of different magnitudes (for the case in Chongqing, China on 21 July 1996).

      When the random error is nearly equal or slightly greater than the initial error, that is, the magnitude E varies from 10–1 to 2, the prediction characteristics of NCPE model are almost independent on initial error, and random error dominates. The prediction eigen-peak for the heavy rain disappears. Attention should be paid to the random error instead of only the initial error in this condition. Because the random error is simulated by Gaussian noise in NCPE model here, the DNCPE values always fluctuate around 1.

      When the magnitude E of random error to initial error alters from 10–2 to 10–1, the prediction characteristics of NCPE model is between the above two scenarios. That is, the prediction eigen-peak may either remain stable or disappear. This is because both the random error and the initial error are comparable in magnitude, they contribute to the phase structure together and then affect the prediction validity period, which brings great uncertainty to the extended range forecasting results. De-noising is needed for cases of E in this magnitude range.

      The results from additional cases with the same conclusion are provided in the online supplemental material. Ding and Li (2008a, b) compared the effects of initial, random, and model parameter errors on predictability using the simulated Lorenz system. They obtained conclusions similar to those drawn above in this study. It is believed that understanding the above relationship between the errors of different sources and their coupled effects on the extended-range forecasts with the NCPE model has important implications for operational forecasts.

    • The essence of traditional numerical weather prediction or the prediction using nonlinear dynamic approaches for extended range forecasting are based on function methods. Therefore, the stability of prediction is influenced by model parameters. While simplified parameterization schemes of various physical processes in the atmosphere are usually adopted in the prediction model, imperfect descriptions of these physical processes could lead to model parameter errors. For example, parameters such as the spatial pattern, thickness of clouds, and so on are used in numerical weather prediction. In extreme weather studies based on chaos theory, Ding and Li (2007) used chaotic approaches to analyze the spatiotemporal distribution of atmospheric predictability with the nonlinear local Lyapunov exponent (NLLE) model. Xia et al. (2015) introduced a new algorithm, that is, NCPE, for the study of extended range forecasting of heavy rain.

      Parameter embedding dimension m in the phase space is a critical parameter in the chaotic analysis approaches. It is inevitable in the phase space reconstruction, because all the subsequent calculations in the NCPE model are performed in the phase space.

      The embedding dimension must satisfy the condition $m\; \geqslant \;2D\; + \;1$ (Wolf et al., 1985), where D is correlation dimension. The length of vector data in the phase space will decrease if the embedding dimension m is too large. As a result, the phase points in the entire attractor will become sparse, which will make the calculation more difficult. Moreover, the prediction error will increase due to the redundancies of embedding dimension. However, if the embedding dimension m is too small, dynamic features of chaos system cannot be fully depicted because of the incomplete structure of attractor in phase space.

      At present, there is no method to determine the embedding dimension accurately except certain experiences. However, the structure of attractor varies with different chaotic systems and can be different even for the same type of weather system. The structure of attractor is also different with different meteorological elements as inputs, which may induce prediction errors. Thereby, the embedding dimension m is critical. In this section, the effect of embedding dimension m on the stability of prediction over the forecast validity period is investigated based on the NCPE model.

      Figure 2 shows the correlation dimension D = 3.44 calculated by the G–P algorithm (Takens, 1981) using the PWAT data described above for the heavy rain case in Chongqing. The correlation dimension is calculated by $D = \ln c(r)/\ln r$. Where m is set from 1 to 30, c(r) is the correlation sum, and r is distance. D is calculated by the stable slope of the straight parts of all the curves, which is denoted by the black solid line in Fig. 2.

      Figure 2.  Correlation dimension D (for the case in Chongqing, China on 21 July 1996).

      Figure 3 shows the diagonal nonlinear cross-prediction error based on different embedding dimensions. The relative errors (y-axis) calculated by NCPE model fluctuate with m. The prediction eigen-peak for the heavy rain also changes with m. Heavy rain occurred at segment 21. When m = 3, unlike that in other conditions, the error is smooth and no eigen-peak exists. Although the error is 1.202 at segment 2, the peak at segment 2 cannot be regarded as the prediction indicator based on the criterion of the prediction eigen-peak in Section 2 (Xia et al., 2015). When m = 4, the eigen-peak clearly shows up at segment 11, and the prediction validity period is about 10 days (t = 21 – 11 = 10 days), so m = 4 can also be regarded as a cut-off. When m = 5 or 6, the eigen-peak at segment 11 remains clear, but the eigen-peak also appears at other segments, such as segment 4 when m = 5, or segments 4, 9, and 18 when m = 6. These redundant characteristics cause confusions for the prediction validity period. When m = 7, 8, 9, or 10, no eigen-peak exists, and errors are relatively small at all the segments.

      Embedding dimension m should be set to m = 8 theoretically as in Wolf et al. (1985). However, the rule does not have to be met since appropriate m can be determined by quantitative calculations (Heathcote and Elliott, 2005). In fact, m used in the NCPE model in this study is determined based on the idea of Heathcote and Elliott (2005).

      The influence of model parameter error on heavy rain extended range forecasting must be considered. Regarding the NCPE model, selection of the embedding dimension m should be determined through quantitative analysis. If the embedding dimension is small, the dynamic features of the chaotic system cannot be depicted because of the incomplete structure of the attractor in the phase space. The case with m = 3 shown in Fig. 3 is a good example of this situation. If the embedding dimension is too large, the prediction indicators will vanish because of the decreasing length of vector data or the sparsity of phase points in phase space, Furthermore, new noises can even be introduced into the system and affect the forecast, as the cases shown in Fig. 3 with m = 7, 8, 9, and 10. The effect of the model parameter should be considered when the embedding dimension is around the cut-off (i.e., close to 4). Figure 3 shows that chaotic dynamic features in the phase space change significantly when the model parameter m exceeds the cut-off.

      To the chaotic single variable system of PWAT, TEMP, or HGT, the embedding dimension m = 4 as an input parameter for NCPE model is probably appropriate based on the calculations of 100 disastrous weather cases, including heavy rain events and hurricanes. However, the embedding dimension m = 4 may not be a good option for a multiple variable system based on the NCPE model calculation.

    • A heavy rain event and a hurricane case are selected for test study with the NCPE model. Three meteorological elements, that is, PWAT, TEMP, and HGT are investigated. For the heavy rain event that occurred in Worcester, USA, the selection rule of PWAT data is the same as that for the Chongqing case (Fig. 1), and TEMP and HGT are selected at level 500 hPa. For Hurricane Katrina, which occurred during 29–31 August 2005, TEMP and HGT at level 300 hPa are selected for study, considering that the center of tropical hurricane usually has a warm core that forms in the upper–middle troposphere (Liu et al., 2012). Table 2 shows parameters of the PWAT, TEMP, and HGT, including the optimal interpolation time Io calculated based on fractal interpolation and metric entropy, time delay T based on mutual information, and embedding dimension m based on false nearest neighbor method.

      Case (time, place, weather)PWATTEMPHGT
      IoTmIoTmIoTm
      20 July 2007, Worcester, heavy rain502244620447234
      29–31 August 2005, New Orleans, hurricane511945220449244

      Table 2.  Parameters calculated for PWAT, TEMP, and HGT during two cases

      Figures 4 and 5 present the diagonal nonlinear cross-prediction error of the heavy rain event in Worcester and the Hurricane Katrina by the NCPE model, respectively. The results for the heavy rain case is shown in Fig. 4, which indicates that heavy rain in Worcester occurred at segment 26. Note that one segment corresponds to one day on timescale. The DNCPE for TEMP and HGT at 500 hPa are smooth and without eigen-peak. However, the eigen-peak is obvious at segment 8 for PWAT, and the prediction validity period is 18 days (t = 26 – 8 = 18 days). It is worth noting that the error of 2.33 at segment 1 cannot be considered as a prediction eigen-peak based on the identification criterion discussed in Section 2. According to Simmons et al. (1995), the effect of the initial state errors is significant at the initial stage of modern NWP models, but the impact of errors from the model itself will become more important along with the increase in validity forecast period.

      Figure 4.  DNCPE of PWAT, TEMP, and HGT for the heavy rain case in Worcester, USA on 20 July 2007.

      Figure 5.  DNCPE of PWAT, TEMP, and HGT for the hurricane case in New Orleans, USA during 29–31 August 2005.

      The results show that the NCPE model is more sensi-tive to PWAT, and prediction validity period for PWAT is stable compared to that for the other two elements. This is closely related to the attractor structures of the above three meteorological elements mapped in the phase space.

      As shown in Fig. 5, Hurricane Katrina occurred at segment 28. For PWAT, the prediction eigen-peak is at segment 7, and the prediction validity period is 21 days (t = 28 – 7 = 21 days). For HGT, the prediction eigen-peak is at segment 6, and the prediction validity period 22 days (t = 28 – 6 = 22 days). Apparently the validity prediction periods for HGT and PWAT are basically consistent.

      Moreover, the DNCPE for the HGT field is more significant and the prediction features at segments 6, 16, 21, and 26 are evident, since the center of a tropical hurricane often corresponds to a warm core in the upper–middle troposphere, Thereby, the local relative dynamic variation of HGT field at 300 hPa is evident. Multiple eigen-peaks also qualitatively depict the complicated dynamic features of HGT for the hurricane in phase space.

      However, no eigen-peak exists for TEMP. This is possibly attributed to the fact that the TEMP field at 300 hPa is selected for the study without considering its vertical variation between 200 and 350 hPa. An exact processing method is needed to smooth the TEMP from 200 to 350 hPa (Liu et al., 2006).

    4.   Discussion and conclusions
    • The sensitivity of the NCPE model performance to initial, model parameter, and random errors and the stability of validity period of extended range forecasting are studied. The sensitivity of three meteorological elements in a heavy rain event and the case of Hurricane Katrina is also discussed. Sensitivity analysis of three types of errors and different meteorological elements in the NCPE model are based on 100 global disastrous weather cases, including heavy rain/snow events and hurricanes. Some common features about the 10–30-day extended range forecast using the NCPE model are summarized below.

      First, influences of the initial errors and random errors on the prediction validity period of extended range forecasting of heavy rain and these errors’ mutual relationships are quantitatively analyzed based on the NCPE model. When the random error is smaller than the initial error, for example, the magnitude of their difference varies from 10–6 to 10–2, a small variation of the random error cannot change the dynamic features of the chaotic system, and it only has a minimal effect on the prediction validity period. However, when the magnitude is around the cut-off point (10–2–10–1), influences of both the initial error and random error must be considered. Their mutual effects could result in great uncertainties in the extended range forecasting, and thus de-noising is necessary. When the magnitude ratio E of random error to initial error is within the range from 10–1 to 2, the effect of random error would become dominant. Under this condition, more attention should be paid to the random error instead of only exploring the effect of the initial condition.

      Second, the influence of model parameter error on the prediction validity period for heavy rain extended range forecasting is analyzed quantitatively based on the NCPE model. Embedding dimension m should be determined by the factual nonlinear time series. When the embedding dimension is small, the dynamic features of chaotic system cannot be depicted because of the incomplete structure of the attractor in the phase space. The prediction indicators would vanish because of decreasing length of the vector data or sparsity of phase points when the embedding dimension is large. When the embedding dimension is around the cut-off, such as when it is equal to 4, the influence of the parameter must be considered. When the model parameter exceeds the cut-off, local dynamic characteristics could significantly change and induce confusions in the prediction.

      Third, the sensitivities of three meteorological elements, PWAT, TEMP, and HGT, in extended range forecasting based on the NCPE model are discussed. In the heavy rain case, PWAT is more sensitive to prediction validity period compared to the other two elements. However, HGT is the most sensitive in the hurricane case, followed by PWAT.

      The sensitivity study of initial, model parameter, and random errors to the prediction validity period of 10–30-day extended range forecasting is important, and analysis of the mutual interaction mechanism among these errors is useful for further improvement of the extended range forecasting model by determining appropriate para-meters.

      However, analyses in the present study are all based on single chaotic variable. Theoretically, the extended range forecasting model, when coupled with multiple variables, is less sensitive to noises and requires fewer data during phase reconstruction. How can multivariable chaotic elements be integrated into the NCPE model to improve extended range prediction accuracy? How to identify the model parameters when using multiple chaotic variable? How can the NCPE model be applied to the forecast of other disastrous weather and even extreme weather? These concerns will be discussed in our next paper. The model coupled with multiple variables is expected to show great advantages for the 10–30-day extended range forecasting.

      Acknowledgments. We would like to thank the anonymous reviewers and editors for their valuable comments that have greatly improved the paper.

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