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Long-term seasonal rainfall forecasting: efficiency of linear modelling technique

  • Iqbal Hossain
  • H. M. Rasel
  • Monzur Alam Imteaz
  • Fatemeh Mekanik
Original Article

Abstract

Using the lagged (past) climate indices, including El Nino–Southern Oscillation (ENSO) and Indian Ocean Dipole (IOD) as input parameters and long-term spring rainfall as outputs, calibration and validation of the linear multiple regression (MR) models have been performed. Since Australian rainfall varies both temporally and spatially, the analysis on the linear MR models was performed on regional scale. These models show the capability of linear MR technique for long-term predictions of Western Australian spring rainfall. The emphasis was given to assess the statistical correlations between Western Australian spring rainfall and dominating large-scale climate modes. The efficiency of linear modelling technique was evaluated to predict seasonal rainfall forecasting. At the same time, the Pearson correlation (R), mean absolute error, root-mean-square error and Willmott index agreement (d) were used to assess the capability of MR models. The models which fulfilled the limits of statistical significances were used for the prediction of future spring rainfall using independent data set. The results indicate that during calibration periods maximum achievable correlations varied from 0.47 to 0.53 for the selected stations. In regard to predict peaks and troughs of rainfall time series, it was found that correlations between predicted and actual peaks varied from 0.82 to 0.94 and between predicted and actual troughs varied from 0.53 to 0.91.

Keywords

Seasonal rainfall Linear modelling Climate indices ENSO–IOD Rainfall forecasting 

Introduction

Rainfall is considered as the most important part of water cycle that falls from the cloud due to gravity and responsible for surface runoff. Throughout the world, many of the landscape characteristics and occurrence of flood have been determined by the rainfall (Tarboton 2003). Rainfall is also considered as one of the most important components of sustainable design. Therefore, improved understanding of rainfall and runoff process is predominantly important in water-sensitive urban design approach. Consequently, understanding and modelling of rainfall have become essentially important to solve many flood and water engineering problems and to maintain a balanced agro-economic system fulfilling the requirements of sustainable development.

Advanced determination of rainfall is prerequisite in planning, design and development of storm water resources management strategies. An accurate long-term prediction of rainfall will help watershed management authorities to adopt appropriate storm water management options. In the regions where variable hydro-climatic variability is higher, such as Australia (Peel et al. 2001), reliable rainfall prediction will also help to adopt appropriate strategies for the management of land, water catchments and water systems (Anwar et al. 2008). The water balance between the future water supply and demand can be investigated from the reliable prediction of rainfall, hence ensuring demand. Subsequently, rainfall forecasting has become one of the primary goals to water resource mangers.

However, it is well recognized that rainfall is complex global atmospheric phenomena that varies both temporally and spatially, which is difficult to predict accurately. Numerous studies have identified that large-scale climatic phenomena are responsible for rainfall. For example, Wang and Hendon (2007) considered El Nino–Southern Oscillation (ENSO); Ashok et al. (2003) highlighted Indian Ocean Dipole (IOD) and Madden–Julian oscillation (MJO); Hendon et al. (2007), Meneghini et al. (2007) and Rasel et al. (2015) identified Southern Annular Mode (SAM) as potential climate indices that is responsible for rainfall variability in Australia. SAM is also considered as the prime mode of climate variability in the mid and high latitudes. Other researchers, e.g. (Xie et al. 2009), established that sea surface temperature and sea level pressure in the Pacific and Indian Oceans have considerable effects on Australian rainfall. Many researchers, such as Ashok et al. (2004) and Houston (2006), already established the effect of sea surface temperature and sea level pressure on rainfall pattern and intensity. Changes in the weather patterns also affect the rainfall variability. In addition, local characteristics such as soil moisture and native vegetation may have effects on Australian rainfall variability. According to Risby et al. (2009), the remote driver variables have the better control on the prediction of rainfall since they vary at low frequencies that are suitable to modulate rainfall process. Nevertheless, there exists lack of analysis on the rainfall forecasting with these changes.

The conducted studies to investigate the effect of prevailing climate drivers, including ENSO, IOD and SAM, on Australian rainfall focused on the identification of relationships between the climatic modes and daily, monthly and seasonal rainfall. However, most of these studies considered only one climatic driver on rainfall variability. There are also studies that considered the interactions of two variables. For example, Meyers et al. (2007) considered the interaction between ENSO and IOD and their effects on Australian rainfall. Again some of the researchers studied regionally, while others conducted as a whole Australia. For example, Cai et al. (2011), Kirono et al. (2010) and Risbey et al. (2009) considered whole Australia in their study, whereas Mekanik et al. (2013) focused on South-East Australia; Ummenhoffer et al. (2008) emphasized on South-Western Australia; Verdon et al. (2004) emphasized on East Australia; Hossain et al. (2015) studied Western Australia; Evans et al. (2009) and Rasel et al. (2016) explored South Australia. Most of the previously conducted research studies found that there exists much complex relationship between climate predictors and actual rainfall, and hence, single predictors alone are unable to forecast Australian rainfall accurately. However, the maximum seasonal rainfall predictability of these region was achieved only 30%. Murphy and Timbal (2008) found the maximum correlation between spring rainfall and spring Nino3.4 only 0.37. Due to the complexity of the relationship between large-scale climate modes and Australian rainfall, many aspects of the phenomena remain undiscovered.

Moreover, the majority of the previous studies did not consider the effect of lagged climatic modes for the prediction of rainfall. Some of the researchers, such as Hossain et al. (2015), Mekanik et al. (2013), Abbot and Marohasy (2012), Schepen et al. (2012), Kirono et al. (2010) and Drosdowsky and Chmabers (2001), investigated the relationship between lagged climate indices and Australian rainfall. The research study done by Kirono et al. (2010) considered the relationship between 2 months’ average climate mode and Australian rainfall. For the prediction of seasonal rainfall in Queensland, Australia, Abbot and Marohasy (2012) used 1–2-month lagged Dipole Mode Index (DMI), Southern Oscillation Index (SOI), a sea surface temperature-based index of ENSO (Nino3.4) and Pacific Decadal Oscillation (PDO). However, till now long-term forecasting of rainfall remains challenge. Consequently, hydrological managers strive to develop options for the long-term prediction of rainfall for many years.

Different techniques have been used by many researchers to investigate the relationships between seasonal rainfall and large-scale climate modes. For example, Schepen et al. (2012) used Bayesian joint probability approach for seasonal rainfall forecasting. Many researchers have used the linear MR modelling technique for rainfall forecasting (He et al. 2014) and flood forecasting (Latt and Wittenberg 2014). Ihara et al. (2007) applied the linear MR modelling technique to assess the relationship between IOD and ENSO with summer monsoon rainfall. Rasel et al. (2015) used the linear MR models and demonstrated the influences of lagged ENSO and SAM as the potential climate predictors for the long-term rainfall forecasting in South Australia. Mekanik et al. (2013) examined the influence of lagged ENSO and IOD on Victorian rainfall using the linear MR modelling technique. However, Hudson et al. (2011) found that the predictive capability of the currently available rainfall forecasting models more than 1 week and shorter than a season is still questionable. Vitart (2004) concluded that the usual forecasting systems lost information from the atmospheric initial conditions, which forms the basis of weather forecasting. In addition, rainfall forecasting using climate indices and incorporating local geographical factors will make the model very complex.

Therefore, from the motivation of better understanding the relationship between large-scale climatic modes and long-term rainfall, this study investigated Western Australian spring rainfall as a case study. The primary objective of this research was the investigation of remote drivers on Australian rainfall variability. Subsequently, the influence of ENSO and IOD on spring rainfall has been assessed as potential predictors. Since both ENSO and IOD are responsible for the formation of rainfall in this region, this research study emphasizes the investigation of the relationship between combined ENSO–IOD and Western Australian spring rainfall. To achieve this objective, the efficiency of linear modelling technique MR was adopted and assessed. Three rainfall stations located in Western Australia were chosen as a case study. Attempts were made to predict the spring rainfall by applying the constructed linear MR models that satisfied the statistical significance.

Data collection and study area

For the likelihood estimation of water resource and hydrological variables, the linear MR modelling technique is commonly used. The present study emphasizes the efficiency of MR modelling technique on seasonal rainfall forecasting. However, the construction and application of the MR models in the current research require the extraction of two types of key parameters: seasonal rainfall and climate indices.

Rainfall data are considered as the primary input for any water resource projects. Australian Bureau of Meteorology (www.bom.gov.au/climate/data/) collects and stores historical rainfall data from nearly 18,000 locations. This research concentrated on three rainfall stations located in Western Australia as a case study. The specific locations of the stations are shown in Fig. 1. The stations were nominated considering the recorded length of data which have fewer missing rainfall values. For the selected stations, monthly observed rainfall data in millimetres were extracted from January 1957 to December 2013.
Fig. 1

A map of the study area showing rainfall stations

The second type of key parameter for the study is climate indices. The climate indices are the parameters which can be explained by the sea surface temperature and sea level pressure anomalies around the globe. They can be used to describe the state and changes in the global atmospheric phenomena, e.g. seasonal rainfall. Statistical study, such as analysis and comparison of time series data, and their averages, extremes and trends can be performed using the climate indices.

Since previous studies identified ENSO and IOD as potential predictors of Australian seasonal rainfall variability, they have been selected in this study. Two different types of indicators represent the ENSO: SOI and sea surface temperature anomalies. SOI is the measure of the sea level pressure anomalies from Darwin to Tahiti, whereas sea surface temperature anomalies are measured in the form of Nino3 (5°S to 5°N, 150°W to 90°W), Nino4 (5°S to 5°N, 160°W to 150°W) and Nino3.4 (5°S to 5°N, 170°W to 120°W) in the equatorial Pacific Ocean. The measure of IOD is considered as the Dipole Model Index (DMI). The difference in the average sea surface temperature anomalies from the tropical eastern Indian Ocean (10°S to equator, 90°E to 110°E) to the tropical Western Indian Ocean (10°S to 10°N, 50°E to 70°E) is measured by the DMI.

It is well established that the effects of ENSO on Australian seasonal rainfall forecasting are strongest in the world. Moreover, IOD which is the by-product of Indian Ocean sea surface temperature significantly influences the Western Australian seasonal rainfall. Therefore, ENSO and IOD have been used and analysed as potential predictors of seasonal rainfall forecasting in this research. The combined interaction between ENSO and IOD (climate indices from different oceanic-atmospheric phenomena) would be the advanced findings on seasonal rainfall variability in this region.

For this study, the climate indices data were extracted from Climate Explorer website (http://climexp.knmi.nl). All of the extracted data (seasonal rainfall, ENSO and IOD) were divided into two sets to be used for the construction of linear MR models and validation of the developed models. The MR models were assembled using the data from 1957 to 2008. The performance and efficiency of the developed linear models were tested using the data from 2009 to 2013.

Methodology

To assess the efficiency of linear modelling techniques, the linear MR modelling technique was used in this study. The MR is linear statistical modelling technique which uses the least square method to find out the best correlation between a variable (seasonal rainfall) and several other variables (climate indices). The general equation for MR models can be expressed according to Eq. 1.
$$ R_{i} = c_{0} + a_{1} X_{\text{lag}} + a_{2} X_{\text{lag}} + e $$
(1)
where R i is the spring rainfall; Xlag and Xlag are the variables of the linear MR equations (ENSO and IOD in this study); a 1 and a 2 are the coefficients of the corresponding variables; c 0 is the constant; and ‘e’ is the error of the linear MR analysis. The effects of lagged climate indices were considered by adopting the spring rainfall of year ‘n’ and monthly (Decn−1, Jann to Augn) values of the climate indices.

For any established model, evaluation is considered as an essential part to determine whether the initiative is worthwhile in terms of providing the anticipated outputs. The general tendency for the evaluation of empirical models is performed based on statistical correlation tests. In this research study, the performances of the constructed linear MR models were assessed by applying several error indices and statistical performance tests. To check the efficiency of the adopted linear modelling technique, the outputs of the developed linear models were assessed applying the extensively used statistical methods, such as Pearson correlation coefficients (R), root-mean-square error (RMSE), mean absolute error (MAE) and Willmott index of agreement (d). In MR modelling technique, investigations on the independency of residuals are solely important to validate the accurate modelling outcomes. The MR technique is reliable for the residuals where there is no autocorrelation amongst them. In the case of autocorrelation within the residuals, MR models are not capable of capturing all the interactions between the inputs and the outputs. Therefore, to check the existence of autocorrelation amongst the samples, Durbin–Watson (D/W) statistical test was applied according to Field (2009). Multicollinearity amongst the data sets is another serious problem in the validation, interpretation and analysis of MR models (Lin 2008). To identify the multicollinearity problems amongst the predictors, two indicators can be used: tolerance (T) and variance inflation factor (VIF). Lin (2008) identified that the tolerance value less than 0.10–0.20 indicates the presence of multicollinearity problem. In this study, statistical ‘T’ test was performed for the assessment of the multicollinearity of the data sets.

Results and discussions

In this study, the ability of ENSO and IOD has been investigated in forecasting Western Australian seasonal rainfall. The individual correlation between spring rainfall and monthly climate indices, e.g. DMI, Southern Oscillation Index (SOI), Nino3, Nino4 and Nino3.4, was analysed. Since the highest correlation between spring rainfall and climate predictors could be obtained using SOI and 2–3 months’ average values of sea surface temperature (Chiew et al. 1998), lagged values of the climate indices were used in this study. The rainfall values which have statistically significant correlations with climate indices were further analysed using linear MR modelling technique.

The analysis revealed that Western Australian spring rainfall has significant correlations with DMI of January, May, August and December; SOI, Nino4 and Nino3.4 of June, July and August; and Nino3 of June and July. Pearson correlations of the individual climate predictors with the rainfall are shown in Table 1a and b. According to Table 1a, Western Australian spring rainfall significantly affected by ENSO particularly DMI and SOI comparing with the other considered indices (Nino3, Nino4 and Nino 3.4) for Sturt Creek and Quanbun Downs rainfall stations. Although Marradong’s spring rainfall is significantly affected by DMI, there is no statistical significant correlation with SOI. The maximum significant correlation between the rainfall and DMI was observed in December for all the three rainfall stations. The observed correlations were 0.29, 0.41 and 0.36 for Marradong, Quanbun Downs and Sturt Creek, respectively. However, the maximum significant correlations between SOI and spring rainfall could not be detected on the same month. For the Sturt Creek station, the maximum significant correlation between spring rainfall and SOI was found to be 0.42 in July, whereas the same correlation for the Quanbun Downs station was observed as 0.32 in August as evidenced in Table 1a. Nevertheless, there negative correlation was discovered between spring rainfall and Nino3, Nino4 and Nino3.4 as shown in Table 1b. The analysis supports the complex nature of atmospheric rainfall formation. Only single climatic driver is not sufficient to predict long-term seasonal rainfall.
Table 1

Significant correlation between spring rainfall and lagged climate indices

(a) Significant correlation between spring rainfall and lagged climate indices for DMI and SOI

Station

DMI

SOI

Dec

Jan

May

Aug

Jun

Jul

Aug

Marradong

0.29*

− 0.28*

Quanbun Downs

0.41**

0.30*

0.30*

0.32*

Sturt Creek

0.36**

0.28*

0.28*

0.28*

0.42**

0.34**

(b) Significant correlation between spring rainfall and lagged climate indices for Nino3, Nino4 and Nino3.4

Station

Nino3

Nino4

Nino3.4

 

Jun

Jul

Jun

Jul

Aug

Jun

Jul

Aug

Marradong

− 0.29*

− 0.30*

Quanbun Downs

−  0.304*

− 0.37**

− 0.29*

− 0.30*

− 0.30*

Sturt Creek

− 0.34*

− 0.27*

− 0.35**

− 0.50**

− 0.47**

− 0.46**

− 0.46**

− 0.42**

*Statistical correlation is significant at 0.05 level

**Statistical correlation is significant at 0.01 level

Since both the ENSO and IOD have profound influence on Western Australia, combined effects of these indices were investigated. To assess this combined effect, the climate drivers with significant Pearson correlations months were further organized to apply the linear MR modelling technique. IOD–ENSO input sets were analysed as potential predictors of Western Australian spring rainfall for all the three rainfall stations. The combined IOD–ENSO models used for linear MR analysis are shown in Table 2.
Table 2

Combined climate predictors model input sets for spring rainfall

Station

DMIx–SOIy

DMIx–Nino3y

DMIx–Nino4y

DMIx–Nino3.4y

Marradong

Dec–Jun, Dec–Aug Aug–Jun, Aug–Aug

Quanbun Downs

Dec–Jul, Dec–Aug Jan–Jul, Jan–Aug

Dec–Jul, Dec–Aug, Jan–Jul, Jan–Aug

Dec–Jun, Dec–Jul, Dec–Aug, Jan–Jun, Jan–Jul, Jan–Aug

Sturt Creek

Dec–Jun, Dec–Jul, Dec–Aug, Jan–Jun, Jan–Jul, Jan–Aug, May–Jun, May–Jul, May–Aug

Dec–Jun, Dec–Jul, Jan–Jun, Jan–Jul, May–Jun, May–Jul

Dec–Jun, Dec–Jul, Dec–Aug, Jan–Jun, Jan–Jul, Jan–Aug, May–Jun, May–Jul, May–Aug

Dec–Jun, Dec–Jul, Dec–Aug, Jan–Jun, Jan–Jul, Jan–Aug, May–Jun, May–Jul, May–Aug

The linear MR technique was applied to explore the predictive capability of spring rainfall using DMI–SOI, DMI–NINO3, DMI–NINO4 and DMI–NINO3.4 combinations. The analysis was performed to find out the potential combined predictors of spring rainfall. Amongst the established linear forecasting models of the MR analysis, the models that have the minimum errors were designated as the dominant models for Western Australian spring rainfall forecasting. The summary of the best MR models (regression coefficients, D/W statistics and tolerances) for all the selected three rainfall stations along with the values of the regression coefficients is shown in Table 3.
Table 3

Summary of the best developed multiple regression models (statistically significant at 0.05 level)

Station

Models

Const.

Coefficient

D/W

T

DMI (Dec)

SOI (Jul)

SOI (Aug)

Nino3.4 (Jun)

Nino3.4 (Jul)

Nino3.4 (Aug)

Marradong

DMI(Dec)–Nino3.4(Aug)

43.23

11.50

− 6.25

2.36

0.93

Quanbun Downs

DMI(Dec)–SOI(Aug)

11.09

21.81

1.94

1.84

0.93

 

DMI(Dec)–Nino3.4(Jun)

10.70

21.26

− 4.18

1.86

0.94

Sturt Creek

DMI(Dec)–SOI(Jul)

15.02

16.63

6.79

1.33

0.95

 

DMI(Dec)–SOI(Aug)

15.42

17.18

4.34

1.67

0.93

 

DMI(Dec)–Nino3.4(Jun)

14.60

15.49

− 10.19

1.52

0.94

 

DMI(Dec)–Nino3.4(Jul)

14.39

15.54

− 9.77

1.56

0.94

 

DMI(Dec)–Nino3.4(Aug)

14.38

15.75

− 7.67

1.41

0.93

It can be seen from Table 3 that the values of the D/W statistical tests for all the constructed MR models were found to be nearly two. This confirms that the residuals of the models developed by linear analysis have no autocorrelations and they are independent. Furthermore, the tolerances of the developed MR models were found to be close to one. The tolerance value less than 0.10 of the data sets indicates the presence of multicollinearity (Lin 2008). Therefore, our data sets are free from multicollinearity. Hence, the developed IOD–ENSO-based MR models are statistically valid and can be used for further analysis.

Various statistical performances of the developed MR models were assessed with Pearson correlations (R) and statistical errors such as RMSE and MAE. These performance results of the analysis are shown in Tables 4 and 5 for calibration and validation periods, respectively. It was observed from the developed MR models that IOD–Nino3.4-based combined predictor models established statistically significant results with good forecasting capability of Western Australian spring rainfall having R values of 0.47, 0.51 and 0.53 for Marradong, Quanbun Downs and Sturt Creek, respectively, during the calibration periods as shown in Table 4. The MR model for Sturt Creek in the validation stage showed very compatible predictive capability having the maximum correlation of 0.68. However, the same models for Marradong and Quanbun Downs showed incompatible performances having R values of − 0.55 and − 0.40 during the validation periods as evidenced in Table 5. It is to be noted here that during validation periods there were several unusual years (i.e. Millennium drought), which can be considered as statistical outliers and a simple liner model considering only two indices are really unable to predict such unusual phenomena as in reality rainfall is affected by few other local and regional factors. It is recommended that further linear and nonlinear analyses can be performed before conducting a generic conclusion.
Table 4

Performance evaluation of the developed multiple regression models during calibration

Region

Station

Models

R

RMSE

MAE

Western Australia

Marradong

DMI(Dec)–Nino3.4(Aug)

0.47

12.24

9.63

 

Quanbun Downs

DMI(Dec)–SOI(Aug)

0.49

12.57

9.21

  

DMI(Dec)–Nino3.4(Jun)

0.51

12.45

9.18

 

Sturt Creek

DMI(Dec)–SOI(Jul)

0.51

14.40

10.86

  

DMI(Dec)–SOI(Aug)

0.44

15.02

11.49

  

DMI(Dec)–Nino3.4(Jun)

0.53

14.26

10.73

  

DMI(Dec)–Nino3.4(Jul)

0.53

14.23

11.00

  

DMI(Dec)–Nino3.4(Aug)

0.50

14.55

11.06

Table 5

Performance evaluation of the developed multiple regression models during validation

Region

Station

Models

R

RMSE

MAE

Western Australia

Marradong

DMI(Dec)–Nino3.4(Aug)

− 0.55

18.53

15.32

 

Quanbun Downs

DMI(Dec)–SOI(Aug)

− 0.41

31.03

26.85

  

DMI(Dec)–Nino3.4(Jun)

− 0.40

31.39

27.00

 

Sturt Creek

DMI(Dec)–SOI(Jul)

0.31

8.24

7.27

  

DMI(Dec)–SOI(Aug)

0.52

6.29

4.33

  

DMI(Dec)–Nino3.4(Jun)

0.68

5.59

4.05

  

DMI(Dec)–Nino3.4(Jul)

0.58

5.80

4.14

  

DMI(Dec)–Nino3.4(Aug)

0.44

6.36

4.67

The RMSE of the developed MR model sets is also shown in Tables 4 and 5 for calibration and validation periods, respectively. It is clear from the tables that RMSE of the constructed linear MR models is reasonably lower during the calibration period for all the stations. However, RMSE of the validation period is lower only for Sturt Creek station comparing with the calibration period. It is also evidenced that negative correlation during the validation periods has the higher RMSE values. Therefore, application of linear MR models for the first two stations (Marradong and Quanbun Downs) may not be valid.

The MAE of the developed MR models also supports this hypothesis. The MAE of the constructed models is shown in Tables 4 and 5 for calibration and validation periods, respectively. Like RMSE, the higher MAE values were observed during the validation period for the first two stations, Marradong and Quanbun Downs. The models with lower RMSE have the lower MAE. Hence, IOD–ENSO-based linear models are capable of forecasting spring rainfall with reasonable accuracy only for Sturt Creek station. Again linear models are not sufficient to predict the complex atmospheric phenomena like rainfall.

For further assessment of model accuracy, the statistical test, index of agreement (d) values were calculated according to Willmott (1984). The values of the ‘d’ for the developed models are shown in Table 6 for both calibration and validation periods. According to Table 6, IOD–ENSO-based MR models have the good agreement with Western Australian spring rainfall during the calibration period having ‘d’ closer to one for all the three stations. However, the performances the models reduced during the validation period for Marradong and Quanbun Downs stations. This confirmed that the linear models are not capable of forecasting Western Australia’s spring rainfall with reasonable accuracy for all areas. The linear model’s performance was better only for Sturt Creek station having ‘d’ values close to one during the validation period comparing with the calibration period as evidenced in Table 6. Hence, it can be concluded that rainfall is complex global phenomena and its prediction cannot be done with single model using single climatic variable. Different models are suitable for different areas. To come up for generic conclusion, further similar analysis need to be performed for other areas. Also nonlinear analysis should be performed to check the validity of this hypothesis.
Table 6

Comparison of the index of agreement (d) between model calibration and validation

Region

Station

Models

Results for calibration period (1957–2008)

Results for validation period (2009–2013)

D

Western Australia

Marradong

DMI(Dec)–Nino3.4(Aug)

0.58

0.22

 

Quanbun Downs

DMI(Dec)–SOI(Aug)

0.62

0.13

  

DMI(Dec)–Nino3.4(Jun)

0.64

0.11

 

Sturt Creek

DMI(Dec)–SOI(Jul)

0.64

0.73

  

DMI(Dec)–SOI(Aug)

0.57

0.71

  

DMI(Dec)–Nino3.4(Jun)

0.65

0.75

  

DMI(Dec)–Nino3.4(Jul)

0.65

0.78

  

DMI(Dec)–Nino3.4(Aug)

0.62

0.72

From the analysis of the data sets, the selection of the best predicted models was performed considering the models having higher ‘R’ values, lower errors and ‘d’ close to one. The best predicted models for Marradong, Quanbun Downs and Sturt Creek stations are shown in Eqs. 2, 3 and 4, respectively:
$$ {\text{Predicted Rainfall }} = \, 43.23 \, + \, 11.50 \, \, \times \,{\text{ DMI}}_{\text{Dec}} {-}\,6.25 \, \, \times \,{\text{ Nino}}3.4_{\text{Aug}} $$
(2)
$$ {\text{Predicted Rainfall }} = \, 10.70 \, + \, 21.26 \, \times {\text{ DMI}}_{\text{Dec}} {-} \, 4.18 \, \times {\text{ Nino}}3.4_{\text{Jun}} $$
(3)
$$ {\text{Predicted Rainfall }} = \, 14.39 \, + \, 15.54 \, \times {\text{ DMI}}_{\text{Jun}} {-} \, 9.77 \, \, \times \,{\text{ Nino}}3.4_{\text{Jul}} $$
(4)
Comparisons of the linear MR modelling outputs with observed rainfall are plotted for the whole period of study. Figure 2 presents the comparison between the observed rainfall and predicted rainfall obtained from the best models constructed by using the linear MR technique. It was discovered that linear models were not capable of re-producing the observed rainfall with reasonable accuracy. Moreover, the linear technique is not suitable to predict the extreme rainfall at all. As we know that rainfall is the final outcomes of complex global atmospheric global phenomena, the extreme rainfall prediction remains challenge. Instead of considering only two climate predictors, influences of other elements might be intense during the extreme rainfall years.
Fig. 2

Comparison of the multiple regression modelling outputs with observed rainfall. a Marradong, b Quanbun Downs, c Sturt Creeks

The prediction results of Western Australian spring rainfall, several statistical assessment parameters as well as the statistical significances revealed the competency of developed IOD–ENSO-based combined linear models in forecasting Western Australian rainfall. The analysis demonstrated that sometimes the linear models overestimate rainfall and other times they underestimate from the actual observed rainfall. The reasons behind these variations in the predictions may be due to the effects of other climatic drivers and/or some local phenomena, which a linear model having only two independent variables is unable to replicate.

Further evaluations of the developed linear models were performed to the peaks and troughs. Plotted results of the evaluation in terms of producing peaks and troughs are shown in Fig. 3. Table 7 shows values of the correlation coefficients for the peaks and troughs for the prediction of Western Australian spring rainfall for all the selected three stations. According to Fig. 3 and Table 7, the linear models were capable of predicting the peaks with a correlation coefficient of 0.82–0.94. However, for troughs, this capture rate ranges from 0.53 to 0.91. Nevertheless, the rainfall pattern for all the stations was not the same, and hence, their response to the peaks and troughs is different.
Fig. 3

Capability of the developed multiple regression models to reproduce the peaks and troughs. a Marradong, b Quanbun Downs, c Sturt Creeks

Table 7

Correlation of the multiple regression models capturing the peaks and troughs

Station

Peak

Trough

Marradong

0.82

0.53

Quanbun Downs

0.94

0.80

Sturt Creek

0.86

0.91

Conclusions and recommendations

In this study, single and combined climate predictors (IOD and ENSO) have been used as potential predictors of Western Australian spring rainfall. More specifically, the climate indices DMI, Nino3, Nino4 and Nino3.4 were selected for the analysis. The analysis of the study revealed that Western Australian spring rainfall exhibits significant correlations with 3-month climate indices (June, July and August).

The climate indices which have the statistically significant correlation with rainfall were used to develop combined models for advance analysis using linear MR technique. The established MR models were assessed to examine the predictive capability of seasonal rainfall with separate data sets. It was observed that the statistical evaluation parameters (R, RMSE and MAE) of the developed MR models are showing low forecasting ability during the validation period comparing with the calibration period for two stations (Marradong and Quanbun Downs). For example, the Pearson correlations (R) were observed negative and model errors were found to be higher for these stations. Probably, reasons for these inferior predictions are discussed in the previous section. However, for the Sturt Creek station, there is a significant improvement of correlations and model error was found during the validation period. The linear MR models in the validation stage revealed very compatible predictive capability of seasonal rainfall for the Sturt creek having R close to 0.68 with lower model error. Moreover, all the ‘d’ values in the calibration stage are very much close to 0.60. This indicates that DMI–ENSO-based linear models should be improved to forecast Western Australian spring rainfall.

Nevertheless, the constructed MR models are not suitable for the first two stations, i.e. Marradong and Quanbun Down stations. The models also could not predict the extreme rainfall with reasonable accuracy. Therefore, the complex oceanic-atmospheric climate phenomena (rainfall) cannot be predicted using only linear analyses. Linear MR technique is not capable of forecasting long-term rainfall. Further investigation should be performed using both linear and nonlinear modelling techniques in this region to suggest a generalize model for seasonal rainfall prediction.

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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  • Iqbal Hossain
    • 1
  • H. M. Rasel
    • 1
  • Monzur Alam Imteaz
    • 1
  • Fatemeh Mekanik
    • 1
  1. 1.Department of Civil and Construction Engineering, Faculty of Science, Engineering and TechnologySwinburne University of TechnologyMelbourneAustralia

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