This study investigates the dynamics of daily Urmia Lake level (ULL) changes using spectral analysis tools to discover fluctuating patterns in the ULL series. Therefore, in the present research, the empirical mode decomposition (EMD), variational mode decomposition (VMD), empirical wavelet transform (EWT), and empirical Fourier decomposition (EFD) were used to analyze the ULL signal. ULL series were decomposed into subseries, and the optimized outcome was used. All methods concluded that the ULL series has a steep downward trend. Signal reconstruction was performed, and it was inferred that EFD could not estimate the ULL series appropriately and had root-mean-square error (RMSE) = 12.26. Different from EFD, other methods performed better signal construction according to RMSE and error analysis. The mode-mixing issue was the last step in verifying the capabilities of signal-analyzing methods. Based on the power spectral density (PSD), it was seen that EMDs had mode-mixing problems and limitations in signal decomposition, whereas VMD and EWT did not have these issues. Results demonstrated that the present study has some limitations. Overall, it was concluded that VMD performed better in terms of RMSE, error analysis, reconstruction, mode-mixing problems, and PSD analysis while decomposing and extracting features from the ULL signal.

  • Urmia Lake level (ULL) time series presented a steep downward trend, making it hard for signal processing methods to analyze accurately.

  • Variational mode decomposition (VMD) performed better than other methods in analyzing the ULL signal.

  • Empirical Fourier decomposition did not perform appropriately and was the weakest among other methods.

  • VMD had no mode-mixing issues contrary to empirical mode decompositions.

As one of the frequent environmental issues, endorheic saline lakes have declined or are almost extinct in dry areas of the world due to rising water demand, anthropogenic influences, and occasionally climate change (Alivernini et al. 2018). Salt lakes worldwide, with a volume of about 8,300 km3 representing about 40% of all lakes, are found in arid zones (Wurtsbaugh et al. 2017). Because lakes play critical roles in natural systems, it is critical to investigate the core causes of this problem (Ye et al. 2020). Large salinity lakes rapidly decline worldwide, endangering habitats, human health, and economic activity. Similarly, saline lakes have declined in Iran. The most important example is the one currently occurring in the northwest of Iran, the Urmia Lake.

In the present research, we have studied the Urmia Lake level (ULL), Iran's most prominent and the second-largest saltwater lake on Earth, which has declined significantly in the past two decades. It would be informative to use a methodology to capture the nonlinear alteration in the lake level. Also, such an outcome could be useful in examining the drought in the basin (e.g., Amirataee et al. 2018; Lashkari et al. 2021). According to several studies, human activity on water and land is the basis of the problem (e.g., Khazaei et al. 2019; Foroumandi et al. 2021). However, a report that discovered a similar pattern in headwater areas disproved the claim made in earlier studies that dam construction was the primary reason for the reduction (Fathian et al. 2014). Drought, climate change, and anthropogenic activity are the leading causes of the issues affecting salty lakes. The Great Salt Lake, Mono Lake, Walker Lake (the United States), the Dead Sea (the Jordan region), and the Aral Sea (Kazakhstan in the north and Uzbekistan) all have similar occurrences. When a lake's level dropped, it harmed human health, culture, economy, and more. Lake restoration necessitates integrating many fields of science, technology, engineering, management, and governance to avoid the fragmentation of research on these interdependent human–natural systems (Hart 1996; White et al. 2015; Elmore et al. 2016; Micklin 2016; Edwards & Null 2019; HDR 2020). Lake Urmia has shrunk 7 m and lost almost 90% of its volume since 2000 due to agricultural water development. Because of this reduction, the salinity has increased, and the density of the salts was changed, causing environmental effects such as decreasing the population of brine shrimp, namely, Artemia SPP. which is believed to be the birds' (especially flamingos) primary source of feeding (Abbaspour & Nazaridoust 2007). The lake level's decrease distanced resorts from water deep enough for leisure (Sima et al. 2021), resulting in the closure of tourist resorts. A further risk to the neighboring population's health occurred when the subsequent lake-level dropped, revealing that the lakebed caused the release of a complicated mixture of aerosols. It is crucial to highlight the stakeholders affected by Urmia Lake's water level drop and elaborate on the potential impact on agriculture, industry, and the local population.

The pattern of Urmia Lake's water level drop creates a significant problem for its stakeholders, particularly the agricultural (e.g., farmers) and industrial sectors (e.g., Ministry of Energy) and local population (e.g., Urmia City). A major contribution to the identification of the stakeholders and their different roles is very important (Cong et al. 2017; Shahid et al. 2018). To this end, a workshop in the city of Urmia was held to address the Ecosystem Approach to the Management of Urmia Lake. About 200 representatives of different stakeholder groups were identified as follows: provincial offices (East and West Azerbaijan provinces) (50%); ministries and governmental organizations (20%); universities, education, and research centers (20%); and NGOs (farmers and local population included) (10%). Based on these facts, it is important to propose approaches considering various concepts of lake-level decline (e.g., time series, spectral analysis, and so on).

Taking advantage of the trend in the water level's decline is one strategy to deal with the aforementioned issue. Therefore, by examining the dynamics of the water level changes around Urmia Lake, the current study looks into potential causes for the issue. In this study, the proper methodologies are used to identify trends and fluctuations in the ULL series. Applying spectral methods can produce good results when examining the intricacy and variations of the ULL time series, taking into account the seasonality of the hydrological processes. Any survival strategy and Urmia Lake restoration project must start by using effective mathematical methods to assess the ULL series components (e.g., Khazaei et al. 2019; Foroumandi et al. 2022).

Due to the nature of random variation, stochastic dynamics can account for the apparent irregularity and unpredictability of lake-level behavior (de Domenico et al. 2013). To determine the degree to which each component affects the lake level, time series decomposition is a valuable tool. Long-term fluctuations have drawn attention and can be examined independently if necessary or excluded from calculations (Soomere et al. 2015). By separating the time series, we could identify the nonlinear trend in a particular phenomenon. A recently popular approach to decomposing nonstationary series is called empirical mode decomposition (EMD). An intrinsic mode function (IMF) is captured when EMD processes a signal. As an illustration, 60 years of Chinese sea level were modeled using EMD to eliminate long-term fluctuation from gauge records to reduce the uncertainty surrounding long-term sea-level variation in the calculation of sea-level acceleration (Jevrejeva et al. 2008; Sallenger et al. 2012; Calafat & Chambers 2013; Haigh et al. 2014; Cheng et al. 2016; Dehghan et al. 2022). Due to mode mixing issues in EMD, Wu & Huang (2009) proposed a noise-aided technique, namely, ensemble EMD (EEMD), to solve this problem. The EEMD approach was used by Devlin et al. (2020) to examine multiresolution variation in the tides of the Indian Ocean. Results indicated that, similarly, the EEMD results show the mode-mixing issue when decomposing nonlinear time series. To overcome this issue, Torres et al. (2011) suggested complete ensemble empirical mode decomposition adaptive noise (CEEMDAN). The processing of signals with nonlinear and nonstationary properties can be effectively and adaptively done using CEEMDAN (Cao et al. 2019). The decomposition of a complicated wideband time series into IMFs, resulting in a relatively modest subcomponent with various temporal features, is generally done using a binary filter. CEEMDAN delivers temporal data considering the original signal properties and captures adaptive IMFs. These IMFs have different properties of temporal scales and frequencies ranging from high to low and residual components, which extract the trend component associated with signal oscillation. Strong regularity in the decomposed IMFs component sequence is advantageous for model learning and raises prediction accuracy. The EMD family (EMDs), consisting of the EMD, EEMD, and CEEMDAN, will be utilized to extract signal features together in the present research.

As a data-driven approach, EMD captures IMFs from a signal of interest independent of its characteristics. After synthesizing each component's instantaneous information, the original signal's entire time and frequency information can be retrieved (Huang et al. 1998). Liu et al. (2020) mentioned that mode mixing and over-envelope, under-envelope, and end-point impact issues are some drawbacks in EMD. To prevail over these problems, the variational mode decomposition (VMD) method was devised by Dragomiretskiy & Zosso (2013), in which a signal with nonlinear and nonstationary features could be decomposed into numerous subseries having a specific bandwidth and center frequency by imposing variational restrictions. VMD is widely utilized in the processing of vibrations and diagnosis of faults (Li et al. 2019). However, the VMD parameters must be adjusted beforehand. Several academics have offered techniques for enhancing the VMD parameters, and significant progress has been made in the related field (Miao et al. 2019).

Gilles (2013) presented the empirical wavelet transform (EWT), similar to the wavelet transform, a unique decomposition method that may isolate a complicated signal into numerous subseries. In the EWT method, the segmentation of the frequency spectrum is performed based on the method of frequency band division derived from data, and each segmented interval is then reconstructed by creating orthogonal wavelet filter banks by applying the functions of scaling and wavelet (Shi et al. 2021; Zheng et al. 2022). EWT separates signals into components called empirical wavelet functions (EWFs). It must be noted that the decomposition of EWT relies on the spectrum segmentation border. Zheng et al. (2021) have recently presented the empirical Fourier decomposition (EFD) for nonstationary signal analysis based on fast Fourier transform (FFT) and adaptive spectrum segmentation technology. However, the spectrum division boundary of EFD must be determined in advance based on users' experience (Zheng et al. 2022).

The world's large saltwater lakes, including Lake Urmia, are confronting numerous issues. As there is an immediate demand for water resources administration, the present research used signal feature extraction analysis to analyze Urmia Lake's environmental and hydrological problems. To the authors' knowledge, crucial information extraction of daily ULL signal is provided for the first time utilizing both spectrum and frequency analysis based on our methodology. Using EMDs, VMD, EWT, and EFD, the primary purpose of this study is the spectral and temporal analysis of the daily ULL series from 1990 to 2021. The decomposition and trend analysis results are further reviewed and contrasted to determine their ability to process signals in the time–frequency domain, their progress in mode-mixing circumstances, and their benefits and drawbacks. In addition, the reconstruction of the signal will be investigated. The document arrangement is as follows: the Materials and Methods section describes the materials and methods used, study area, and data used in this investigation. In the Results and Discussion section, the spectrum and frequency analysis findings are displayed and discussed. Finally, this study is concluded in the Conclusion section.

Study area and data

Lake Urmia is located in northwestern Iran (N 37.5°, E 45.5°) (Figure 1(a)). Elevations in the basin vary from 1,280 to 4,885 m, spanning over 52,000 km2 (Eimanifar & Mohebbi 2007). Average annual precipitation is predicted to be 350 mm. The lake is nourished by 13 perennial rivers, of which the two most significant rivers, the Zarrineh and Simineh, originate in the southern Kurdistan Province's Zagros Mountains. The inflow into the lake is sourced mainly from these two rivers (Farajzadeh & Alizadeh 2018). According to www.ulrp.ir/fa, river discharge varies seasonally, with flows from spring runoff being roughly 50 times more than flows from late summer (Parsinejad et al. 2022). More than 20% of the lake's water comes from direct precipitation. According to RSRC (2018) and ULRP (2015), lake evaporation ranges from 580 to 2,000 mm/annual (Sadra 2004; Sima & Tajrishy 2015; Safaie et al. 2021). It is reported that 41 irrigation reservoirs, both big and small, totaling 2.109 km3, have been built since 1970. Up to 1990, the lake received more than 5.109 km3 of water annually. The flow has reduced significantly from 2000 (Tajrishi 2014). Although legal and illicit groundwater extraction also helps with agricultural productivity, canals provide a substantial percentage of the water supply. In the basin, the growth season lasts almost seven months. Almost 6 million people reside in the basin, and 40% are concentrated in Tabriz and Urmia (capital cities of East and West Azerbaijan, respectively), which are very close to the lake. The lake-level fluctuation was reported to be between 1,274 and 1,278 m (highest elevation) from 1967 to 1999 due to the cycle of inherent climate and intensive water supply. From the beginning of 2021, the level of the lake had dropped to almost 1,271 m, causing a rise in the salinity (Abbaspour & Nazaridoust 2007). To supply water to the lake more effectively than to nearby marshes, the government of Iran has decided to use channelized rivers at their deltas (ULRP 2018). A severe drought in the lake occurred. The ULL's time series is shown in Figure 1(b), used in the current investigation. The duration of the ULL series is from 1990 to 2021 on a daily scale. The daily ULL statistical properties are as follows: max = 1,278.41 m, min = 1,270.04 m, and mean = 1,273.32 m.
Figure 1

(a) Geographic location of Urmia Lake and (b) daily time series of ULL.

Figure 1

(a) Geographic location of Urmia Lake and (b) daily time series of ULL.

Close modal

Empirical mode decomposition, ensemble EMD, and complete EEMD with adaptive noise

The decomposition of a time series into a finite number of specific oscillatory mode functions known as IMFs is done using the entirely adaptive EMD approach. Through the sifting process, these IMFs reveal the oscillation modes in the data. IMFs are generated using an iterative method and meet the following two criteria: (1) the total number of extrema and zero-crossings must either be equal or differ by no more than one, and (2) the local average of the upper and lower envelopes must always be zero. The following describes how the EMD algorithm functions:

  1. Identify the series' local maxima and minima (or extrema).

  2. Create emax(t) and emin(t) using the cubic spline interpolation approach (upper and lower envelopes, respectively).

  3. The mean value m1(t) = (emax(t) +emin(t))/2 is defined.

  4. Determine the emax(t) +emin(t)/2 as the mean value (m1(t)).

  5. To get the first nominee for the IMF, h1(t), the mean value m1(t) is subtracted from the signal y(t), so h1(t) =y(t) − m1(t).

If h1(t) meets the requirements for an IMF property, it is verified. If so, it counts as the first IMF, c1(t) =h1(t). It is then removed from the original signal, and in step 1, the first residual, r1(t) =y(t) − c1(t), is regarded as the new series. If not, the sifting procedure is repeated as often as necessary to produce an IMF.

Choosing the halting criteria for the sifting process and controlling the endpoints for the cubic splines interpolation in the EMD process should be addressed during implementation. Due to the signal discrepancy, different applications call for different stopping criteria (Niang et al. 2010). After the decomposition is complete, the original signal can be displayed as follows (Lu et al. 2014):
(1)
In Equation (1), the letters n and ci(t), i= 1, 2,…, n stand for the number of IMFs and the values of each IMF, respectively. According to a dyadic filter bank property of the EMD algorithm, the number of IMF modes is computed as follows:
(2)
where L is the number of points in the length of the data (Flandrin et al. 2004; Wu & Huang 2004; Huang et al. 2008). The EMD approach frequently yields monotone trends (r(t)). As a result, it lacks any oscillation with a fixed period (Roushangar & Alizadeh 2019).

The key steps of the EEMD approach are as follows:

  1. Create a white noise sequence and apply it to the target signal. Decompose the signal with additional noise into the specified number of IMFs using the EMD technique.

  2. Repeat step 1 several times. Note that each of the added white noise series is distinct.

  3. Calculate the final EEMD values by averaging the respective IMFs.

As a result of the EEMD decomposition, the added time series of white noise cancel each other out when combined (Alizadeh et al. 2019). The mean IMFs of EEMD maintain the beneficial characteristics of the EMD technique; however, the strength of the mode mixing in EEMD reduces relative to the EMD model (Wu & Huang 2009).

Torres et al. (2011) proposed the CEEMDAN, a revolutionary method of modification based on EMD. By incorporating an additional noise coefficient to regulate the noise level during each decomposition, it is possible to achieve precise signal reconstruction. The following is a description of the CEEMDAN method:
(3)
where the white noise of module variance with zero average (ηi(t)) is gathered for the ith time (i= 1, 2, …, N). The coefficient of white noise amplitude is represented by ε0, and the EMD's IMF component with the jth order is demonstrated by Ej(⋅).
The first residual signal should be determined as follows:
(4)
Obtain the second component by combining the first residual signal with white noise:
(5)
The residual signal is captured when the process can no longer degrade:
(6)
Reconstruction of the signal is captured as follows:
(7)

Variational mode decomposition

VMD is a powerful nonrecursive signal decomposition method (Dragomiretskiy & Zosso 2013). VMD can decompose a complex signal into a series of subsignals (i.e., modes) with specific bandwidth in the frequency domain to mine characteristics of complex nonlinear signals (Upadhyay & Pachori 2015). Recently, VMD has been employed to solve many engineering problems, including the diagnosis of faults in mechanical engineering (Wang et al. 2015), the processing of economic data signals (Lahmiri 2016), value estimation in metals (Liu et al. 2020), sea-level signal analysis (Dehghan et al. 2022), and processing signals in biomedicine (Lahmiri 2014).

The decomposition process utilizing VMD may be converted into the following optimization subject by the presumption that each mode-component's frequency, uk, is focused close to a center frequency, ωk, during the breaking-down process. Finding K modes with the minimum combined bandwidth for all modes is the aim of optimization. In addition, it must be ensured that the sum of all modes equals the initial signal y(t) (signal reconstruction). The following are some in which that the optimization problem can be changed into a variational problem:
(8)
where y(t) represents the original input signal, δ(t) represents the Dirac distribution function, t represents time, ‘‘*’’ represents the convolution operation, and uk and ωk denote the kth mode and the corresponding center frequency of the mode, respectively; {uk} denotes the mode set {u1, u2,…, uk}, {ωk} denotes the set of center frequencies {ω1, ω2,…, ωk}, and K is the total number of modes. The quadratic penalty and Lagrange multiplier are introduced to transform the aforementioned constrained variational problem into an unconstrained optimization problem, namely:
(9)

In this problem, the Lagrange multiplier (λ(t)) is utilized as a constraint constrictor, α is the parameter of balancing the data constancy constraint, and is the term of quadratic penalty to perform the convergence rate acceleration. The mentioned optimization in Equation (2) is settled by locating the saddle point of the augmented Lagrange equation using the alternating direction method of multipliers (Hestenes 1969).

Empirical wavelet transforms

A novel method for creating adaptive wavelets is presented by Gilles' EWT method (Gilles 2013). The fundamental concept is to create a wavelet filter bank that is suitable for the process and is capable of extracting the various signal modes. EWT has been utilized in signal processing and time series modeling extensively since its inception (Amezquita-Sanchez & Adeli 2015; Xin et al. 2019). In contrast to discrete wavelet transform and EMD, EWT performs an FFT on the time series and then directly analyzes it in the Fourier domain before using band-pass filtering to execute a spectral separation (Flandrin et al. 2004).

The first stage in the EWT is to divide the Fourier spectrum of the signal to be decomposed using an adaptive segmentation technique, and the second is to build a wavelet filter bank (Daubechies 1992). Assume that the spectrum is specified on the range [−π, π] of normalized frequencies. The following are descriptions of the segmentation method and wavelet filter bank for the frequency range [0, π], whereas those for the frequency range [−π, 0] can be inferred from the Hermitian symmetry of the Fourier spectrum in the normalized frequency range [−π, π].

The method of local maxima, which divides the spectrum in [0, π] into N continuous frequency segments, is one method for segmenting the EWT (Gilles 2013). Sn= [ωn-1, ωn] represents each segment in the n [1, N] domain, ω0 and ωN are equal to 0 and π, respectively, and ωn values are calculated based on the first N 1 greatest local maxima of the magnitude of the spectrum. The re-indexed frequencies in the descending order are marked by the symbol [Ω1, Ω2, …, ΩN-1], where Ω1<Ω2<…<ΩN-1, and are those that only match the identified maxima. Furthermore, Ω0 is equal to zero. The ωn value is identified by:
(10)
which concludes the local maxima technique. As an alternative to the local maxima technique, the lowest minima technique was developed for the EWT: the spectrum division and frequency re-indexing procedures, which are the same as those in the local maxima technique, are first carried out. Then the minimum of the spectrum magnitude in the frequency range [Ωn-1, Ωn] is identified, and the value of ωn is determined where the local maxima method is completed. The lowest minima method was created for the EWT as an alternative method to the local maxima method. Prior to the local maxima, spectrum division and frequency re-indexing operations are performed. The value of ωn is then calculated by finding the minimum of the spectrum's magnitude in the frequency range [Ωn-1, Ωn] as follows:
(11)
(12)
where represents the FT of a function, the circular frequency is represented by ω, an ideal function by β, and the transition phase size determination parameter by τn interdependent with the nth and (n+ 1)th segments; the range of transition phase in [ωn−τn, ωn+τn] is represented (Gilles 2013). Here, variable x is applied to present the frequently used form of β in Equations (13) and (14) (Daubechies 1992):
(13)
Calculation of τn is performed as follows:
(14)
where variable γ is a small enough value to avoid the bounds of nonzero and from overlapping. A prerequisite for a value that is acceptable is given as follows:
(15)
Determination of γ for all n values is presented as follows:
(16)
where R is the data number in the decomposed time series; the and variables are utilized to construct the filter bank of the wavelet. The filter bank is structured behind the use of the segmentation approach.

Following is a stepwise explanation of the EWT for a signal y(t).

  1. Apply the FT to determine the Fourier spectrum of y(t).

  2. Use a segmentation approach, such as the lowest minima or local maxima techniques, to divide the spectrum in step 1.

  3. Create a wavelet filter bank using the frequency segments from step 1 as a foundation for decomposing.

  4. Based on the wavelet filter bank from step 3, express approximation and detail coefficient functions.

  5. Breakdown y(t) and reconstruct it. The EWT can produce reliable decomposition results when y(t) has widely separated modes. However, a mode-mixing issue may arise because of the transition period when y(t) modes are tightly spaced.

Empirical Fourier decomposition

Similar to the EWT, the EFD consists of two critical steps: an improved segmentation technique and the construction of a zero-phase filter bank. In the EFD, the Fourier spectrum of a signal to be decomposed is defined on a normalized frequency range [−π, π], and the improved segmentation technique and construction of a zero-phase filter bank for the spectrum in the frequency range [0, π] are described in this section (Zhou et al. 2022).

The improved segmentation technique is proposed based on the lowest minima technique described in Study Area and Data section. In the improved segmentation technique, [0, π] is divided into N contiguous frequency segments. Unlike the local maxima and lowest minima techniques, ω0 and ωN are not necessarily equal to 0 and π, respectively, and their values are determined in an adaptive sorting process. In the sorting process, Fourier spectrum magnitudes at ω= 0 and ω=π and their local maxima are identified and extracted into a series. All magnitudes in the series are sorted in the descending order. Frequencies corresponding to the first N largest values in the sorted series are denoted by [Ω1, Ω2,…, ΩN]. In addition, Ω0= 0 and ΩN+1=π are defined. The boundaries of each segment are determined by (Zhou et al. 2022):
(17)
where denotes the Fourier spectrum magnitudes between Ωn and Ωn+1, which concludes the improved segmentation technique.
A zero-phase filter bank in EWT consists of constructing a filter bank. In the EWT, a wavelet filter bank is formed by the empirical scaling function and wavelet functions. In the EFD, the Hilbert transform filter bank is constructed based on the Fourier spectrum of the analytical signal associated with a signal to be decomposed. In the EFD, a zero-phase filter bank is constructed based on frequency segments obtained by the improved segmentation technique. In each frequency segment, a zero-phase filter (Zhou et al. 2022) is a band-pass filter with ωn-1 and ωn serving as its cutoff frequencies, and it has no transition phases. Hence, the zero-phase filter retains the major Fourier spectrum component in the segment, and all other Fourier spectrum components beyond the segment are excluded. The Fourier transform of a signal to be decomposed y(t) is expressed as follows:
(18)
A zero-phase filter bank can be constructed by :
(19)
where 1 ⩽ nN and values of ωn are determined by Equation (17). Filtered signals that correspond to are calculated by:
(20)
Decomposed components in the time domain can be obtained using the inverse Fourier transform:
(21)
The reconstructed signal is calculated as a summation of all decomposed components:
(22)
Step 1. Obtain the Fourier spectrum of a signal to be decomposed y(t) using the Fourier transform.

Step 2. Determine boundaries of segment ωn using the improved segmentation technique based on the Fourier spectrum obtained in step 1.

Step 3. Construct a zero-phase filter bank based on ωn obtained in step 2.

Step 4. Obtain filtered signals in the frequency domain using obtained in step 3.

Step 5. Obtain decomposed components yn(t) in the time domain using inverse Fourier transforms of obtained in step 4.

Proposed ULL signal analysis

A concise explanation of our proposed method is presented, which serves as the roadmap for this research. The EMD family of models (EMD, EEMD, and CEEMDAN), the VMD model, the EWT model, and the EFD model are each used to analyze the lake-level measurements. These models extract the time-dependent amplitudes and phases of ULL, which is an appropriate approach to characterize the influence of both external and internal parameters. These amplitudes and phases can be employed in a variety of applications. In addition, the levels of the sub-lakes modeled by these methods are compared with one another. After that, a comparison of the signal reconstruction results is carried out. In conclusion, a Fourier analysis is carried out on the ULL series to determine whether models' outputs exhibit any mode mixing. Figure 2 represents the methodology of signal processing in this study.
Figure 2

Flowchart of signal processing in this study.

Figure 2

Flowchart of signal processing in this study.

Close modal

This section reports the results of the decomposing of the ULL time series, and the analysis of the results of the six models is reported. First, preprocessing and data analysis are presented. Next, the ULL time series analysis based on the EMD family will be discussed. VMD, EWT, and EFD are used to analyze the ULL signal. Finally, an evaluation of the results and comparative performance of these tools will be presented.

As depicted in Figure 3, the ULL time series exhibits irregularity but also periodic features. The downward trend is evident, but recent initiatives have halted this downward trend. The signal is transformed from the time domain to the time–frequency–power domain using the continuous wavelet transform (CWT) with the Morlet mother wavelet to reveal the periodic characteristics of ULL. When the translation and scale parameters of the wavelets are continuously varied, the CWT is a proper instrument for representing a signal in its entirety. Torrence & Compo (1998) and Roushangar et al. (2018a, 2018b) are some resources for more information on CWT. The wavelet parameters were adjusted to s0 = 2 days (the minor scale of the wavelet) since s = 2t (s is the wavelet scale) and j = 1/12 (the spacing between discrete scales) to capture 12 suboctaves per octave and j1 = 11/j (the number of scales minus one) for the CWT analysis, as the data were recorded daily. A total of (j1 + 1) scales are available, each having a set of 11 powers of two and a corresponding j sub-octave range. The CWT broke down the ULL time series into 133 subseries (namely coefficients) using the aforementioned parameters. The first 13 coefficients are used to illustrate the two-day band. The four-day band is represented in the 14th through 25th coefficients, whereas the eight- and 16-day bands are represented in the 37th–49th and 49th–61st coefficients. The 37th–49th and 49th–61st coefficients also have a corresponding monthly period (Figure 3). A seasonal period is captured in the 61st–73rd and 73rd–85th coefficients. Annual and bi-annual (256–512-day) periods can be seen in the ULL wavelet power spectrum, as shown in Figure 3. This band (256–512 days) is the only one that represents a continuous feature of the ULL. In other words, annual and bi-annual periods are dominant in the ULL series. Despite its advantages, CWT has significant drawbacks, such as adding redundant data and a high computational cost. CWT does not provide any information on the investigated signal's phase. The CWT coefficients cannot be used to reconstruct the original signal. Discrete signal analysis is necessary for signal reconstruction in applications.
Figure 3

Wavelet power spectrum using the Morlet mother wavelet.

Figure 3

Wavelet power spectrum using the Morlet mother wavelet.

Close modal
The trend and volatility of the ULL are captured using the EMD family approach in the following, as shown in Figure 4(a). Applying the EMDs could offer a functional and nonlinear technique to extract lake fluctuations since changes in the ULL series are typically not linear if the feature of interest has such qualities. White noise aids in establishing a dyadic reference frame in the time–frequency or timescale space, which is the primary distinction between EMD and EEMD. The EEMD uses every statistical property of the noise: it helps to perturb the signal and enables the EMD algorithm to visit all possible solutions in the finite neighborhood of the proper final answer. It also uses the zero mean of the noise to cancel out this noise background once it has served its purpose of providing the uniformly distributed frame of scales. The EEMD is capable of separating signals of varying scales without excessive mode-mixing (which will be investigated in the next section). White noise is introduced to the original signal in the EEMD and CEEMDAN methods for noise-assisted data analysis. In the current study, white noise was defined as noise with an amplitude equal to 0.1–0.2 of the SD of the ULL series, which corresponds to a suitable signal-to-noise ratio. The 1990–2021 daily ULL time series were broken into 11 IMFs by EMD, 13 IMFs by the EEMD method, and 14 IMFS by CEEMDAN, respectively, using ensemble sizes of 500–700 times for each EEMD and CEEMDAN ensemble member (Figure 4(b)). EMD IMFs range from 1 to 10, EEMD IMFs range from 1 to 12, and CEEMDAN IMFs range from 1 to 13, each of which represent an oscillation component with a distinct period and has frequencies ranging from high to low. After dealing with the oscillating components of the ULL series, the residual component (trend component) was obtained. The EMDs determined the declining ULL trend. Figure 4(b) and 4(c) show that the CEEMDAN modes 11–13 and the EEMD modes 9–12 have relatively low amplitudes and are not symmetric as one might assume. The reason for the low-energy problem is that all realizations were averaged in EEMD, even though there was a significant difference in the number of modes. The missing modes must be padded with zeros to accomplish the averaging, resulting in low amplitudes. Because each realization of residue plus noise is broken down until the first mode is reached and employs the EMD's halting criterion for the final mode, the CEEMDAN does not have this problem (Torres et al. 2011).
Figure 4

Decomposition of daily ULL series using (a) EMD, (b) EEMD, and (c) CEEMDAN.

Figure 4

Decomposition of daily ULL series using (a) EMD, (b) EEMD, and (c) CEEMDAN.

Close modal
The VMD model coefficients α, τ, and ε are defined as 2,000, 0, and 10–7. Through VMD, the lake levels at Urmia are broken down into five modes. From 3 to 15, the decomposition level (K) was chosen. Overall, K = 5 VMD-based decompositions outperformed VMD decompositions with other K. The aforementioned experiments are displayed in Figure 5 as the decomposed results of the ULL series using VMD.
Figure 5

Captured IMFs from daily ULL time series using VMD.

Figure 5

Captured IMFs from daily ULL time series using VMD.

Close modal

The settings K were found to produce the optimal decomposition for the VMD process in terms of least reconstruction error and best performance. As a result, Figure 5 might not have any hydrologic meaning. As seen, all modes separated from the ULL series are grouped in Figure 5 according to their frequencies, from lowest to highest. In the deconstructed results, VMD's imf1 is comparatively steady, but imf2 and imf3 exhibit subcomponents more often fluctuating. Compared with the first three IMFs, imf4 and imf5 are more complex and exhibit high volatility. In the ULL, the residual component exhibits a declination.

Each IMF component has a unique hydrological meaning as the ULL's structural component. The Res subseries, the main component of the ULL trend, fluctuates slowly around the ULL's long-term average values. It may be deduced by contrasting the original signal and Res in the corresponding decomposition outcomes shown in Figure 5. Each increase or decrease in the lower frequency components (such as imf2 and imf3) typically corresponds to significant occasions with an impact on the hydrological events. Small amplitudes and high frequencies characterize the higher frequency components (imf4 and imf5), primarily representations of the ULL's daily or short-term variations.

VMD is an appropriate instrument for signal analysis because the lake series primarily exhibits a nonstationary characteristic. Some modes are produced following the original ULL series' VMD decomposition, as depicted in Figure 5. The decomposed modes by VMD in Figure 5 exhibit more steady fluctuation, distinct regularity, and a simple structure when compared with the original series. All of the modes, except the Res component, are almost stationary, according to the results of the ULL time series' VMD decomposition. All these aspects of the VMD decomposition findings make the model design easier.

In EWT, the times of decomposition should be predetermined. Insufficient decomposition will result from too few instances and excessive decompositions from too many instances. (In our case, it was susceptible, which led to large numbers of decompositions, and therefore, we had to examine and find the best fitting parameters.) In addition, EWT divides the frequency range using the local-maximum method. The impact of either too few or too many decomposition times on the extraction result is then confirmed in more detail. Figure 6(a) displays the EWFs captured from spectrum segmentation using EWT. As shown in Figure 6(a), when the decomposition time is 3, the spectrum with a frequency between 0 and 2.4 Hz and higher will become a whole. As a result, all of this frequency band's impact components will be combined, which may make the feature extraction in the later stage difficult. On the other hand, if the number of decompositions exceeds the preceding number, the signal is over-decomposed because it is divided into various frequency bands at multiple locations where the energy impact is not immediately apparent. Furthermore, it treats the frequency band's dividing frequency at 2.4 Hz as submerging the information it contains.
Figure 6

(a) Spectrum segmentation by EWT method, (b) EWFs decomposed by EWT, and (c) filter on top of the magnitude spectrum of the input signal based on decompositions.

Figure 6

(a) Spectrum segmentation by EWT method, (b) EWFs decomposed by EWT, and (c) filter on top of the magnitude spectrum of the input signal based on decompositions.

Close modal

Figure 6(b) shows the waveform of each EWT subseries after decomposition. Each component's x-axis in the figure indicates times, which is a unit equal to a day. The ordinate of ULL oscillations is amplitude, measured in metres. It demonstrates that the low-frequency interference and trend components – essentially devoid of periodicity – make up ewt1 in its entirety. The periodicity of each original signal can be accurately reflected by the excellent periodicity that ewt2 exhibits. The other component (ewt3) consists primarily of high-frequency portions that can effectively express the variation in ULL signals between states. Figure 6(c) plots each filter on top of the magnitude spectrum of the input signal based on decompositions. The plots are on the frequency interval [−π, π]; the magnitude spectrum is normalized.

EFD shares some similarities with EWT; however, there are also significant variances. The decomposition times should be predefined. Here, five is the optimal number of modes N for the EFD approach (based on the trial-and-error process). Figure 7(a) depicts the spectrum division limits of EFD, while Figure 7(b) displays the decomposition findings of the EFD approach. As shown in Figure 7(a), the EFD method is processed by spectral envelope, and the positions of the first two maximum points within the spectrum are determined according to the decreasing amplitude. Following the segmentation boundary of EFD presented in Figure 7(a), the interval generated by each boundary is the most noise-free spectrum interval. EFD divides three components into discrete frequency bands, as shown in Figure 7(b). Each frequency range, however, has additional noise. Figure 6(a) depicts the segmentation border of EWT, where the first component is separated into two spectrum bands.
Figure 7

(a) Spectrum segmentation by EFD method and (b) IMFs decomposed by EWT.

Figure 7

(a) Spectrum segmentation by EFD method and (b) IMFs decomposed by EWT.

Close modal

In contrast to EWT, the EFD approach can successfully separate each component by spectral envelope processing. Both EFD and EWT have adequate and comparable spectrum segmentation boundaries. Each of EFD and EWT's two ideal frequency bands contains the least amount of noise interference. As depicted in Figure 7(a), when the decomposition time is five, the spectrum with a frequency between 0 and 2.7 Hz and higher will constitute a single entity. Figure 7(b) depicts the decomposition results of EFD for the simulated ULL signal.

Signal reconstruction

Based on the methodology used herein, it was seen how the aforementioned methods can be used to analyze, or decompose, signals and time series. This process is called decomposition or signal breakdown (Alizadeh et al. 2019). The second half of the proposed methodology is how those broken-down parts may be put back together into the original signal without information being lost. This method is known as reconstruction or synthesis. Inverse analysis refers to the type of mathematical operation that impacts synthesis. Three separate meters were utilized for this purpose. First, all decomposed subseries for each approach were stacked and individually examined to reconstruct the signal. Second, the original and estimated ULL series' root-mean-square errors (RMSE) were compared (Roushangar et al. 2018a, 2018b). In the final step, the estimating error series was depicted, and the outcomes were examined and contrasted.

The reconstruction results for each method are shown in Figure 8. As can be seen, all methods, except for EFD, have successfully recreated the primary signal within a reasonable range of error. Since EFD's reconstruction flaws were intolerable, it will not be used in the remaining stages of the study. According to each approach's RMSE, values of the reconstructed signals are shown in Table 1. Except for EFD, all approaches worked well, but VMD outperformed them with RMSE = 8.05 × 10−4, according to the data. Performance for EMD, CEEMDAN, and EWT was quite similar. According to RMSE, EEMD performed almost inadequately in signal reconstruction. It must be emphasized that the signal's reconstruction via EFD was inaccurate.
Table 1

RMSE of original and reconstructed signals

Method
EMDCEEMDANEEMDVMDEWTEFD
RMSE 0.000864 0.00081 0.009553 0.000805 0.00081 12.26 
Method
EMDCEEMDANEEMDVMDEWTEFD
RMSE 0.000864 0.00081 0.009553 0.000805 0.00081 12.26 
Figure 8

(a) Reconstructed time series from EMDs, VMD, EWT, and EFD versus ULL series; (b) zooming into a specific period of time.

Figure 8

(a) Reconstructed time series from EMDs, VMD, EWT, and EFD versus ULL series; (b) zooming into a specific period of time.

Close modal
Figure 9 depicts the reconstruction error as a time series. All methods exhibit stationary error series, and except EFD, the errors in the middle of the series for all other approaches have become minimal. In the situations of VMD and EEMD, the faults were periodically variable and occasionally volatile. Maximum inaccuracy was seen along the edges of the time series, which may have resulted from the boundary effect. Overall, it can be concluded that VMD, EWT, and EMDs performed adequately during the decomposition procedure.
Figure 9

Error time series created during the reconstruction process by EMDs, VMD, EWT, and EFD.

Figure 9

Error time series created during the reconstruction process by EMDs, VMD, EWT, and EFD.

Close modal

Mode-mixing analysis

Mode mixing occurs when a single IMF/EWF comprises numerous intrinsic timescales or when similar intrinsic timescales are dispersed over many IMFs/EWFs. The aliasings of two nearby IMF/EWF waveforms influence each other, making them harder to distinguish. In other words, mode mixing can be characterized as a single IMF/EWF consisting of vastly different scales or similar scales living in various IMF/EWF components, and the physical meaning of each IMF/EWF in the mode-mixing area is uncertain (Wu & Huang 2009; Xu et al. 2016). In other words, the energy of the same lake-level species – a collection of lake-level elements with comparable frequencies – exists in more than one subseries, which is a reflection of the mode mixing in terms of lake levels. The mode-mixing situation is unavoidable, but it must be appropriately managed. This section addresses the presence of mode mixing and discovers which method handles the situation better.

To examine the mode-mixing phenomenon in EMDs, VMD, and EWT, spectral analysis of these models' IMFs/EWFs is performed. For the case of EMDs, it is envisaged that the IMFs from the first to the last (except the res component) will contain ULL signals with varying frequencies and times (based on Figure 4). For the mode-mixing study, power spectral density (PSD) analysis with semi-logarithmic frequency was performed (Figure 10). According to Figure 10(a)–10(c), spectral analysis of EMD, EEMD, and CEEMDAN reveals that the mode-mixing problem exists in the decomposed time series produced by these techniques. In EMD, for example, it is expected to find that the PSD of imf10 has a peak with a lower frequency and higher power/frequency attributes when compared with the other IMFs, and this peak must vary to a higher frequency and lower power/frequency when imf10 is compared with imf9 through imf1 (a similar pattern is expected for all IMFs/EWFs). It is clear from this statement that mode mixing occurred in imf10 through imf7. Imf7 has a higher power/frequency ratio than imf8, with nearly the same peak frequency, and a similar issue exists in certain other IMFs as well. EEMD and CEEMDAN have nearly identical mode-mixing issues. The problem of mode mixing is not solved because the EEMD and CEEMDAN modes still contain oscillations with significantly different timescales. In EEMD, for example, the subsequent imf11 is similarly distorted. Because of mode mixing, EMDs become unstable, and their subseries lack physical uniqueness.
Figure 10

Power spectral density analysis of IMFs captured by (a) EMD, (b) EEMD, (c) CEEMDAN, (d) VMD, and (e) EWT for mode-mixing analysis.

Figure 10

Power spectral density analysis of IMFs captured by (a) EMD, (b) EEMD, (c) CEEMDAN, (d) VMD, and (e) EWT for mode-mixing analysis.

Close modal

According to Dragomiretskiy & Zosso (2013), their development of VMD was able to solve the mode-mixing scenario of EMD. According to Figure 10(d) and 10(e), the PSD of VMD provides an almost perfect power/frequency versus frequency figure with distinct peaks and no mode-mixing issue. In the case of EWT, similarly, no mode mixing was found; however, it should be noted that just two EWFs were retrieved. Overall, it is possible to infer that VMD performs better in terms of RMSE, error series, reconstruction, mode-mixing problems, and PSD analysis while decomposing and extracting features from the ULL signal.

Determining lake-level variations is critical for the authenticity of management practices in rural areas, particularly those next to cities. Because of its unique natural and hydrological conditions, Urmia Lake's location, and proximity to the rural area, ULL has always attracted considerable attention in terms of the economy, tourism, and investment, among other things. On the other hand, the recent reduction in the water level of Urmia Lake is a severe threat to this region. This study advocated looking into the dynamics of daily ULL variations using appropriate methodologies for detecting patterns and oscillations in the ULL series. This work investigated the capacity of specific data-adaptive multiresolution approaches, notably EMD, EEMD, CEEMDAN (EMDs), VMD, EWT, and EFD, to break down a signal into physically relevant components to assess the structure of daily ULL data. As a result, data were fragmented into various band-limited IMFs. The use of such a technique for ULL data yielded encouraging results. To examine diverse signal-processing methods, the VMD, EWT, and EFD approaches have been used simultaneously. However, to the best of our knowledge, no such multiresolution analysis has been applied to the lake-level series. Based on these methodologies, data-adaptive analysis splits the ULL series into different modes. The signal reconstruction approach was used to test the capabilities of the data analysis method. Except for the EFD, all approaches were found to have an acceptable range of errors based on RMSE and the error analysis method. Mode mixing was also examined since capturing subseries with unique frequencies, periods, and physically relevant features is critical. All EMD approaches suffered from mode mixing, whereas VMD and EWT could capture the ideal time–frequency characteristics. The investigation findings determined that VMD performed the best in signal decomposition, accurate reconstruction, and feature extraction. As the first attempt to use data-adaptive methods to distinguish the lake level, this study discovered that VMD performed better when dealing with nonstationary ULL time series. Results demonstrated that the present study has some limitations. According to the analysis, it was deduced that the EMD family has limitations in the decomposition of the signal. As an illustration, mode-mixing issues and inaccurate temporal estimations, number of iterations, physically meaningless components, etc. are some of the EMD issues that might be faced when using this tool. On the other hand, EWT has some parameters to be tuned, and it is very sensitive, which can lead to unfavorable results. Reconstruction results approved that EFD is not capable of analyzing ULL signals. However, further studies are required to support EFD's capability. Despite VMD's best outcome, it also has some limitations. Parameter tuning and decomposition-level determination need careful settings.

This study considered the spectral analysis of lake-level time series and changes via various methods. For future work, it is suggested to use the outcome of the present study on ULL alterations considering climate conditions in different periods and the outcome could be compared to demonstrate the effect of climate change on lake systems.

This research was supported by the research grant of the University of Tabriz (number 212).

Farhad Alizadeh: methodology, software, and investigation. Validation. Kiyoumars Roushangar: supervision.

All relevant data are included in the paper or its Supplementary Information.

The authors declare there is no conflict.

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