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This paper presents a data adaptive approach for the analysis of climate variability using bivariate empirical mode decomposition (BEMD). The time series of climate factors: daily evaporation, maximum and minimum temperatures are taken into consideration in variability analysis. All climate data are collected from a specific area of Bihar in India. Fractional Gaussian noise (fGn) is used here as the reference signal. The climate signal and fGn (of same length) are combined to produce bivariate (complex) signal which is decomposed using BEMD into a finite number of sub-band signals named intrinsic mode functions (IMFs). Both of climate signal as well as fGn are decomposed together into IMFs. The instantaneous frequencies and Fourier spectrum of IMFs are observed to illustrate the property of BEMD. The lowest frequency oscillation of climate signal represents the annual cycle (AC) which is an important factor in analyzing climate change and variability. The energies of the fGn's IMFs are used to define the data adaptive threshold to separate AC. The IMFs of climate signal with energy exceeding such threshold are summed up to separate the AC. The interannual distance of climate signal is also illustrated for better understanding of climate change and variability.

The climate variability and change (CVC) element emphasizes research to improve descriptions and understanding of past and current climate, as well as to advance national modeling capabilities to simulate climate and project how climate and related Earth systems may change in the future. The CVC refers to shifts in the mean state of the climate or in its variability, persisting for an extended period (decades or longer). Research under this element encompasses time scales ranging from short-term climate variations of a season or less to longer-term climate changes occurring over decades to centuries. The CVC element places a high priority on improving understanding and predictions of phenomena that may cause high impacts on society, the economy, and the environment. Climate variability refers to variations in the mean state of climate on all temporal and spatial scales beyond that of individual weather events. It may be due to natural changes or to persistent anthropogenic changes in the composition of the atmosphere or in land use. A lot of people have been observing drastic climate changes throughout certain portions of the world for the past few decades.

There is a perception that extreme natural disasters such as floods, droughts, and heat waves, have become more frequent. This change in climate plays an important role in the Earth’s sustainability. The impact of hydrological change as a result of global warming caused by anthropogenic emissions of greenhouse gases is a fundamental concern [

The features are consistent with previous studies. In the tropical Pacific, temperature rise is greater in the central-east part than in the western part. Such an anomaly pattern is observed in the El Nino years. The warming patterns among the scenarios are similar although the magnitudes are different [

The surface runoff changes had regional differences and both the increase and decrease were suggested in summer. It might increase the risk of mismatch between water demand and water availability in the agricultural region. Under the global warming, both temperature and humidity were projected to increase in Asian region. The increases of annual mean surface runoff and its large fluctuations were projected in a lot of Asian regions. In some regions, the projected seasonal changes of hydrological cycles under the global warming potentially increase the risk of droughts and floods [

Sampling is an important factor for modeling and analysis of climate data. Any climate signal

A new nonlinear technique, empirical mode decomposition (EMD), has recently been pioneered by Huang et al. [

Climate signals in the atmosphere are often lost among the noise and other data collection and assimilation problems. Although climate signals are mainly recognized by their time scales, previous analysis, such as the use of empirical orthogonal functions (EOFs), relies on spatial decompositions to remove excess noise. In some cases this is useful, but in general, there is no a priori reason to expect temporal signals to be defined in terms of spatial patterns ordered by their variances (as required by EOF analysis). In fact, time series analysis is more appropriate for the decomposition of climate signals. The traditional time series analysis tools usually rely on Fourier transforms in one way or another. However, Fourier transforms lead to inconclusive interpretations due mainly to the global nature (in the time domain) of the transforms.

Even wavelet analysis, developed to deal with non stationary and local frequency changes, produces confusing and sometimes contradicting results when applied to climate signals [

Empirical mode decomposition (EMD) can be a useful time series analysis tool. One example where EMD is found to be particularly useful is in analyzing climate records of the atmosphere beyond annual time scales. Global climate phenomena are often separated in temporal, rather than spatial, scales. Therefore, time series analysis is more appropriate for the initial decomposition of climate data than spatial methods that have previously been used to find climate components. The nonstationary and nonlinear nature of climate signals make the EMD appropriate to be used in analyzing such signals. One difficulty encountered when using this method is the sensitivity to end point treatments. The envelopes are calculated using a cubic spline; however, splines are notoriously sensitive to end points. It is important to make sure that the end effects do not propagate into the interior solution. The BEMD is employed here for robust and data adaptive analysis of climate variability.

The empirical mode decomposition (EMD) is a signal processing decomposition technique that decomposes the signal into waveforms modulated in both amplitude and frequency by extracting all of the oscillatory modes embedded in the signal [

The Complex Empirical Mode Decomposition (Complex-EMD) is an extension of the basic EMD suitable for dealing with complex signals [

The Bivariate Empirical Mode Decomposition (BEMD) is more generalized extension of the EMD to complex signals. The main difference between the BEMD and the Complex-EMD is that the latter uses the basic EMD to decompose complex signals, whereas the BEMD adapts the rationale underlying the EMD to a bivariate framework [

For

project

extract the maxima of

interpolate the set of points

compute the mean of all tangents:

subtract the mean to obtain

test if

if yes, repeat the procedure from the step (

if not, replace

The Bivariate-EMD can now be expressed as

Here

BEMD of climate signal (daily evaporation) associated with the fGn. The left and right columns represent the IMFs corresponding to climate signals (daily evaporation) and fGn, respectively.

It is well known that the EMD of fGn is acting as dyadic filter bank [

The Fourier spectra of the IMFs of (a) fractional Gaussian noise (fGn) and (b) climate signal (daily evaporation).

Instantaneous frequency (IF) represents the signal’s frequency at an instance. The concept of IF is physically meaningful only when applied to monocomponent signals. In order to apply the concept of IF to arbitrary signals, it is necessary to decompose the signal into a series of monocomponent contributions. In the recent approaches [

The IF of individual IMF given in Figure

Instantaneous frequencies of the IMFs shown in Figure

One of the useful properties of EMD as well as BEMD is that the first IMF contains the signal component with the highest oscillation frequency and the

Three subband signals obtained by applying BEMD on daily evaporation data.

BEMD is an adaptive method to decompose any complex/bivariate signal into a finite set of complex intrinsic mode function (IMF) components, which become the basis functions representing the data. As the bases are fully data adaptive, it usually offers a physically meaningful representation of the underlying processes. Also because of the adaptive nature of the bases, there is no need for harmonics consideration; therefore, BEMD is ideally suited for analyzing data from nonstationary and nonlinear processes. Even with these nice properties, BEMD still cannot resolve signal from noise in the most complicated cases, when the processes are nonlinear and the noise also has the same time scale as the signal; their separation becomes difficult. Nevertheless, BEMD offers a totally different approach to data decomposition, and we apply it to the study of the characteristics of white noise. We will show that with this approach, we can offer some measure of the information content of climate signals.

The Indian Statistical Institute (ISI), Kolkata, India has a large database of weather parameters. The daily evaporation, maximum temperature and minimum temperature, data are collected from that database. The study area is the place near Giridhi of Bihar, India and the data are acquired for the period of 1989 to 2004. All the weather parameters are measured with instruments (without any touch of human factors) specified by Indian Meteorological Department, Government. of India. Climatologically, the area under study is located in the tropical Indian monsoon region. The climate of the area before the monsoon is characterized by a hot summer seasons; this is called premonsoon season. However, in early March, the area also experiences the impact of western disturbances. The data are collected from a single weather station and hence it is very local for a specific area. The (uniform) sampling period of the data is one day that is, the weather parameters are acquired once a day. No global data is used in this study.

The climate variables are always treated as discrete time series. The nonlinear trend, nonstationary nature, irregular frequencies, and amplitude modulation which are inherently present in climate signals. Such types of properties of climate signal requires a data adaptive signal analysis tool which is suitable to analyze the time series of nonlinear and nonstationary systems. The annual cycle (AC) of climate signal is effectively separated using BEMD-based time domain filtering. Fourier transform represents any stationary signal as a combination of predefined basis functions (also called harmonics), that is, the analyzing signal is fitted with a set of harmonics. Sometimes it makes a distortion of the original signal to properly fit with the bases. The wavelet transform is a slightly improved version to analyze nonstationary signals. It is also troublesome to select the basis wavelet depending on the analyzing signal. To overcome the mentioned limitations, BEMD is applied here to implement data adaptive time domain filtering. It does not require any predefined basis vector. It derives the bases directly from the data and hence it is called data adaptive approach.

Before examining any results, it is necessary to list the properties of an IMF as follows: an IMF is any function having symmetric envelops defined by the local maxima and minima separately, and also having the same number of zero-crossing and extrema. Based on this definition, we can determine the mean period of the function by counting the number of the peaks (local maxima) of the function. The mean period

The mean periods of the IMFs of daily evaporation and fGn.

An interesting property of EMD is that the averaged period of any IMF component almost exactly doubles that of the previous one, suggesting that the EMD is a dyadic filter. This finding is consistent with the recent result by Flandrin et al. [

Period statistics of daily evaporation data.

IMFs | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 |
---|---|---|---|---|---|---|---|---|---|---|---|

No. of maxima | 1992 | 1148 | 656 | 366 | 209 | 117 | 63 | 29 | 15 | 8 | 3 |

Mean period (day) | 2.80 | 4.90 | 8.50 | 15.30 | 26.90 | 47.91 | 89.23 | 189.12 | 364.75 | 773.34 | 1579.50 |

Period statistics of daily maximum temperature data.

IMFs | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 |
---|---|---|---|---|---|---|---|---|---|---|---|

No. of maxima | 1980 | 1104 | 606 | 332 | 182 | 104 | 47 | 25 | 14 | 7 | 3 |

Mean period (day) | 2.81 | 5.13 | 9.20 | 16.94 | 30.72 | 53.23 | 118.67 | 225.81 | 400.70 | 767.51 | 1363.1 |

Period statistics of daily minimum temperature data.

IMFs | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 |
---|---|---|---|---|---|---|---|---|---|---|---|

No. of maxima | 2045 | 1165 | 628 | 344 | 198 | 103 | 51 | 22 | 13 | 7 | 4 |

Mean period (day) | 2.74 | 4.82 | 8.90 | 16.22 | 28.23 | 54.41 | 108.85 | 238.0 | 446.26 | 816.32 | 1409.3 |

The annual cycle (AC) is an important factor to take into consideration, while the climate variability is to be analyzed [

The effect of annual cycle is considered as the low frequency trend of climate signals. The trends of the recorded climate signals are detected using the energy distribution of the signal over the individual IMF. The analyzing signal

Trend detection noisy speech signal. (a) Standardized empirical mean of the fine-to-coarse EMD reconstruction, evidencing

The detrending method described in [

The energy of the IMF of climate exceeds the confidence limit as it represents the trend of the signal. Some reprocessing on the raw climate signal is performed to make peaks of the annual cycle prominent. Any climate signal

Preprocessing of the raw climate signal. The original climate signal (a), the Hilbert envelope (b), and the smoothed version of the Hilbert envelope (c).

The BEMD is applied to the complex signal

Use climate signal say

Decompose

Compute the energy of each of imaginary IMFs and also its 99% confidence limits.

Compute the energy of each of real IMFs. Find which IMF (i.e., its energy) exceeds the upper confidence limits of the energies of fGn, that is, the staring index (

The annual cycle effect is separated by summing up the real IMFs starting from

(a) The energy distribution of fGn over the IMFs, its upper and lower bounds with 99% confidence interval. The energies of individual IMF of daily evaporation signal indicate that the 8th one is the starting IMF to represent the annual cycle. (b) The separation of annual cycle of daily evaporation. The preprocessed climate signal (top), extracted annual cycle (middle), and the residual signal (bottom).

(a) The energy distribution of fGn over the IMFs, its upper and lower bounds with 99% confidence interval. The energies of individual IMF of daily max. temperature signal indicate that the 9th one is the starting IMF to represent the annual cycle. (b) The separation of annual cycle of daily maximum temperature. The preprocessed climate signal (top), extracted annual cycle (middle), and the residual signal (bottom).

(a) The energy distribution of fGn over the IMFs, its upper and lower bounds with 99% confidence interval. The energies of individual IMF of daily minimum temperature signal indicate that the 8th one is the starting IMF to represent the annual cycle. (b) The separation of annual cycle of daily maximum temperature. The preprocessed climate signal (top), extracted annual cycle (middle), and the residual signal (bottom).

It is difficult to implement the-model based approach described in [

There is another issue of analyzing the annual cycle (AC) of different climate signals as a function of climate variability and change. When there is a climate change, there is a good chance of nonstationary of the interannual distances of the climate signals. The interannual distances of the three climate signals (daily evaporation, maximum, and minimum temperature) are shown in Figure

Correlation Coefficients among the annual cycles (ACs) of three climate signals.

AC of evaporation | AC of max temperature | AC of min temperature | |
---|---|---|---|

AC of evaporation | 1 | 0.74208 | 0.34837 |

AC of max temperature | 0.74208 | 1 | 0.75252 |

AC of min temperature | 0.34837 | 0.75252 | 1 |

The interannual distances computed from the extracted annual cycles of of three climate signals.

It is expected that the interannual distance is close to 360 days, whereas such distances for the mentioned three climate signals are varied up to two months (as illustrated in Figure

The EMD method is highly data adaptive and efficient for nonlinear and nonstationary time series. Other decompositions, for example, Fourier-based filtering and wavelet transform are very much model dependent. There are some assumptions on data to be adapted with the required model and hence there occurs the change in original properties of the analyzing data. Such types of loss or gain of climate data affect the climate analysis, greatly whereas, the EMD-based filtering, being fully data adaptive, does not cause any loss of original data.

The main superiority of this method is to apply the EMD method yielding IMFs based on local properties of the signal and the instantaneous frequencies for complicated data sets. The use of BEMD makes easy to represent the nonstationary and nonlinear climate signals without considering as a collection of harmonics (as in Fourier transformation). The EMD is a new approach to many researchers in climate research. This study plays a vital role for analysis of the properties of nonlinear and nonstationary daily maximum and minimum temperature and evaporation time series data. This study focuses on the determination of climate change and variability based on three climate signals namely, daily maximum temperature, minimum temperature, and evaporation using EMD data analyzing method.

The proposed BEMD is the extension of ordinary EMD to generalize as bivariate decomposition. Using EMD, it is required to decompose the fGn and the climate signals separately and hence different number of IMFs could be produced. The frequency correspondence is difficult between such types of heterogeneous sets of IMFs. In BEMD, the climate signal and the reference signals (i.e., fGn) are considered as the real and imaginary components and decomposed simultaneously. There is no chance to produce different numbers of IMFs for one climate signal and the corresponding fGn. Hence it performs better and more appropriate than the ordinary EMD for the analysis of climate signals.

The IMFs partition the time series as a function of time-scale (frequency) in a statistically significant way. The residual series show that the data is overall fitted though a slight underprediction of extreme values which occurred due to small underlying trends caused climate change. Some further statistical research would be needed to address these problems. The IMFs, each carrying its own time scales, could be used in statistical prediction of future climate scenarios. The annual cycle extraction in a data adaptive way is the most significant achievement of this research. The climate variability is crystal clear from the scenario of the interannual distance. The deviation of the interannual distance from a year (360 days) illustrates the climate variability over the years.