A method of quantifying tidal non-stationarity

Abstract

The application provides a method for quantifying tidal non-stationarity, and belongs to the technical field of natural science research based on nonlinear data analysis; first, obtaining hourly tidal observation data of a target sea area for not less than 1 month; then, separating non-tidal low-frequency and high-frequency tidal changes in the data by using an empirical mode decomposition method to extract tidal components; then, returning the tidal components by a classical harmonic analysis (CHA) model to obtain CHA returned tides; then, calculating the correlation coefficient, mean absolute error and root mean square error of the tidal components and the CHA, and dividing the mean absolute error and the root mean square error by the average value of the observed absolute value respectively to obtain normalized indicators; finally, taking (1, 0, 0) as a reference point and (CC, NMAE, NRMSE) as an observation point, calculating a tidal non-stationarity index (NS) according to a two-point distance formula. The application can be widely applied to global sea areas, and is suitable for different tidal type sea areas and different tidal range sea areas.

Description

Technical Field

[0001] This invention belongs to the field of natural science research technology based on nonlinear data analysis, and particularly relates to a method for quantifying tidal nonstationarity. Background Technology

[0002] Deep-sea tides are relatively stable and predictable, primarily driven by the astronomical tidal forces of the Moon and Sun. Their tidal parameters, namely amplitude and phase lag, are commonly referred to as tidal constants because they are considered time-invariant. However, due to the combined effects of various non-astronomical processes such as sea ice, ocean circulation, runoff variations, climate modes, and human activities, observed tidal amplitude and phase lag can vary across multiple timescales, from intraseasonal to interannual and even multi-decade scales. Therefore, even in the deep sea, completely stable ocean tides do not actually exist.

[0003] As ocean tides propagate towards nearshore waters, they are significantly influenced by complex coastal topography, leading to a marked amplification of tidal range. As nearshore tides further into river channels, they gradually weaken and deform under the influence of runoff, significantly enhancing their non-stationarity. In some extreme high-runoff events, the tides within the river channel may even disappear within a short period. Overall, due to the modulation by non-stationary runoff, estuarine tides typically exhibit typical non-stationary characteristics. Therefore, the classical harmonic analysis (CHA) method, based on the assumption of complete stationarity, often performs poorly in backcalculating and predicting estuarine tides.

[0004] Internal tides are another typical type of non-stationary tide, arising from the interaction between barotropic tides and changing seabed topography (such as seamounts and continental slopes) in a stratified ocean. Compared to estuarine tides, internal tides are generally more complex in structure and dynamic processes, mainly in three aspects: First, estuarine tides can be approximated as a one-dimensional process, while internal tides are essentially three-dimensional; second, estuarine tides are mainly controlled by runoff, while internal tides may be influenced by a variety of physical processes, including large-scale ocean circulation, mesoscale eddies, and near-inertial internal waves; third, observational data for estuarine tides are relatively abundant, while observational data for internal tides is relatively scarce due to high observation costs and technical difficulties.

[0005] Over the past few decades, non-stationary tides have become a research hotspot due to their significant scientific value and practical implications. Estuarine areas are often densely populated, and a deep understanding and accurate forecasting of estuarine tides are fundamental to human activities such as water resource management and flood control. Internal tides are a crucial transitional link in the tidal-to-turbulent cascade process and play a vital role in maintaining global deep-sea circulation through density stratification mixing. To date, theoretical and numerical models for estuarine tides and internal tides have been established, successfully reproducing their non-stationary evolution characteristics. However, the non-stationarity of estuarine tides and internal tides has not yet been quantitatively characterized, which hinders comparative studies of different types of non-stationary tides on a global scale. Summary of the Invention

[0006] To address the aforementioned problems, this invention aims to propose a comprehensive index for quantitatively characterizing the non-stationarity of non-stationary tides. This index allows for the comparison of non-stationarity of tides across different tidal categories (estuarine tides and internal tides), different tidal range scales (spring tides, mesoproterozoic tides, and neap tides), and different tidal types (diurnal tides, mixed diurnal tides, mixed semi-diurnal tides, and semi-diurnal tides) within a unified framework. It is hoped that this index will deepen our understanding of the spatiotemporal evolution characteristics of non-stationary tides.

[0007] This invention provides a method for quantifying tidal nonstationarity, comprising the following steps:

[0008] S1, acquire tidal observation data of the target sea area, the observation data including hourly tidal observation data, and the observation period is not less than 1 month;

[0009] S2, The empirical mode decomposition method is used to separate the non-tidal low-frequency variation and high-frequency tidal variation in the observed data to obtain the tidal component;

[0010] S3, the tidal components are returned based on the classical harmonic analysis CHA model to obtain the CHA returned tide;

[0011] S4, calculate the correlation coefficient CC, mean absolute error MAE, and root mean square error RMSE between the tidal components and the CHA report results;

[0012] S5. Divide the mean absolute error (MAE) and root mean square error (RMSE) by the average of the observed absolute values ​​to obtain the normalized mean absolute error (NMAE) and the normalized root mean square error (NRMSE).

[0013] S6. Calculate the tidal non-stationarity index NS based on the distance formula between two points, where the coordinates of the reference point are (1, 0, 0) and the coordinates of the observation point are (CC, NMAE, NRMSE). The larger the NS value, the stronger the tidal non-stationarity.

[0014] Preferably, the target sea area in S1 includes estuary areas, shallow seas and deep seas, the tidal types include diurnal tides, mixed diurnal tides, mixed semi-diurnal tides and semi-diurnal tides, and the tidal range includes strong tides, medium tides and weak tides.

[0015] Preferably, the specific process of S2 is as follows: using empirical mode decomposition to decompose hourly tidal observations into different modes, counting the number of maxima in each mode, dividing the data length by the number of maxima to obtain the average period of each mode; merging modes with an average period greater than 30 hours and defining them as non-tidal low-frequency variations; merging modes with an average period less than 30 hours and defining them as tidal components.

[0016] Preferably, the specific process of S3 is as follows: the amplitude H and lag angle G of each tidal component are estimated from the observed tidal components using the least squares method, and then the returned tide is obtained through the classical tidal harmonic analysis model.

[0017] Preferably, the specific process of S4 is as follows:

[0018] The calculation process for the correlation coefficient CC, mean absolute error MAE, and root mean square error RMSE is as follows:

[0019]

[0020] MAE range: [0, +∞], optimal value: 0;

[0021]

[0022] RMSE range: [0, +∞], optimal value: 0;

[0023]

[0024] CC value range: [-1, 1], optimal value: 1;

[0025] N is the number of observation points; P i For tidal return data; O i This is tidal observation data; This represents the average of the tidal return data; This represents the average value of tidal observation data.

[0026] Preferably, the specific process of decomposing hourly tidal observations into different modes using empirical mode decomposition is as follows: First, the original tidal observations are screened. By identifying local maxima and minima and fitting upper and lower envelopes, the difference between the envelope mean and the original signal is calculated to obtain the intrinsic mode function (IMF) with a positive instantaneous frequency. If the difference signal does not meet the IMF condition, it is used as a new signal and the above steps are repeated until a series of IMF components reflecting the characteristics of different time scales of tidal observations are decomposed. Finally, the remaining stationary residuals that cannot be further decomposed are the trend terms of the tidal observations.

[0027] Preferably, step S6 specifically involves: defining the CC, NMAE, and NRMSE of the harmonic analysis report as the X-axis, Y-axis, and Z-axis in three-dimensional space, respectively; using the ideal steady tidal report as a reference point, i.e., (1, 0, 0); and the actual tidal observation point coordinates as (CC, NMAE, NRMSE); and calculating the distance between the two points. NS is defined as the tidal nonstationarity index. The larger the NS, the stronger the tidal nonstationarity. NS of 0 represents tidal complete stationarity.

[0028] Compared with the prior art, the present invention has the following beneficial effects:

[0029] (1) Quantifying tidal nonstationarity: Currently, there is a lack of quantitative indicators for tidal nonstationarity, and only qualitative descriptions of nonstationary tides are possible, which hinders in-depth research on the interaction between tidal and non-tidal processes. An index NS based on the traditional harmonic analysis return performance is proposed to quantify tidal nonstationarity, filling this gap. It can quantitatively characterize the changes in nonstationarity during tidal propagation and lay the foundation for in-depth exploration of tidal dynamic processes.

[0030] (2) Application of multi-dimensional statistical indicators: The root mean square error (RMSE), mean absolute error (MAE), and correlation coefficient (CC) are used in combination to evaluate the performance of non-stationary tides in traditional harmonic analysis. CC, RMSE, and MAE describe the performance of returns from different perspectives, avoiding the problems that may exist in quantifying tidal non-stationarity by relying on a single statistical indicator, and can comprehensively reflect the overall performance of the tidal non-stationarity indicator NS.

[0031] (3) Universality of tidal nonstationarity index: The proposed tidal nonstationarity index is based on the traditional harmonic analysis performance. It quantifies tidal nonstationarity by calculating three statistical indicators: root mean square error (RMSE), mean absolute error (MAE), and correlation coefficient (CC). Its calculation is not affected by the geographical location of the sea area and the characteristics of tidal dynamics. It can be widely applied to global sea areas and is suitable for sea areas with different tidal types (diurnal tides, mixed diurnal tides, mixed semi-diurnal tides, and semi-diurnal tides) and sea areas with different tidal ranges (strong tides, intermediate tides, and weak tides). Attached Figure Description

[0032] Figure 1 This is a flowchart illustrating the overall technical route of the present invention.

[0033] Figure 2 This is a map showing the eastward tidal current velocity at the Timor Strait (122.9598°E, 11.3683°S) station in Indonesia.

[0034] Figure 3 This is a map of the Timor Strait in Indonesia; the red dot (122.9598°E, 11.3683°S) in the map represents the station analyzed in this invention.

[0035] Figure 4 The eastward flow velocity at a depth of 150 m in the Timor Strait is observed hourly (blue line); the subtidal variation (pink line) and tidal variation (black line) are separated using the EMD method, and the back-calculated tidal current results of CHA are represented by the red line. Detailed Implementation

[0036] This invention proposes a method for quantifying tidal nonstationarity, and the overall technical route flowchart is as follows: Figure 1 As shown, the method includes:

[0037] S1. Obtain tidal observation data for the target sea area. The observation data includes hourly tidal observation data, and the observation period is no less than one month. The target sea area can be an estuary, shallow sea or deep sea. The tidal type can be diurnal tide, mixed diurnal tide, mixed semi-diurnal tide or semi-diurnal tide, and the tidal range includes strong tide, intermediate tide and weak tide. Figure 2 The tidal current observation (eastward direction) in the Timor Strait of Indonesia shown serves as a concrete example of the implementation of this method. Figure 3 This refers to the spatial location for observing this tidal current.

[0038] The Timor Strait is located in the southwestern part of the Indonesian archipelago, north of the Timor Sea. As a long and deep trench, the Timor Strait has an average depth exceeding 2000 m and plays a crucial role in regional ocean circulation. The well-known Indonesian Throughflow transports warm, low-salinity seawater from the Pacific Ocean to the Indian Ocean through the Timor Strait. The complex coastline morphology and dramatically changing seabed topography make the Indonesian waters one of the most active areas for intidal formation globally. This invention analyzes a mooring observation point located in the southeastern part of the Timor Strait (…). Figure 3 The data consists of hourly eastward velocity observations at five water depths (50 m, 150 m, 350 m, 750 m, and 1000 m) during the period from January 1, 2004 to December 31, 2005 (using the central red dot). Figure 2It is evident that significant background currents exist in all water depth velocity observations. The tidal current constants at the analysis point (122.9598°E, 11.3683°S), obtained based on classical harmonic analysis (CHA), are listed in Table 1. The eastward tidal current is predominantly semi-diurnal, with the M2 constituent exhibiting the largest amplitude, reaching 10.07 ± 0.40 cm / s.

[0039] Table 1. Tidal amplitude (cm / s) and lag angle (degrees) of the four main tides at different depths.

[0040]

[0041] S2, the non-tidal low-frequency variations and high-frequency tidal variations in the observed data are separated using the empirical mode decomposition method to obtain the tidal components. The specific process of decomposing hourly tidal observations into intrinsic mode functions (IMFs) of different periods using empirical mode decomposition is as follows: First, the original tidal observations are screened by identifying local maxima and minima and fitting upper and lower envelopes. The difference between the envelope mean and the original signal is calculated to obtain the IMF with a positive instantaneous frequency. If the difference signal does not meet the IMF condition, it is used as a new signal and the above steps are repeated until a series of IMF components reflecting the characteristics of tidal observations at different time scales are decomposed. Finally, the remaining stationary residuals that cannot be further decomposed are the trend terms of the tidal observations.

[0042] The number of maxima in each mode is counted, and the data length is divided by the number of maxima to obtain the average period of each mode. Modes with an average period greater than 30 hours are merged and defined as non-tidal low-frequency variations; modes with an average period less than 30 hours are merged and defined as high-frequency tidal variations, i.e., tidal components. Figure 4 As shown, the pink curve represents the non-tidal low-frequency variation, and the black curve represents the tidal component (the main average period is approximately 12 hours and 24 hours).

[0043] S3, the tidal components are returned based on the classical harmonic analysis CHA model to obtain the CHA returned tide. As shown in formula (1), the classical harmonic tidal analysis model assumes that the tide level is formed by the linear superposition of a series of tides, and each tide is a cosine function with constant amplitude and lag angle that do not change with time.

[0044] (1)

[0045] J 0 is the mean sea level. J(t) It is to observe the tide level. w n , h n , g n , fn , u n , v n is the frequency, amplitude, lag angle, intersection factor, intersection correction angle, and initial phase angle corresponding to the nth tidal constituent. N is the total number of tidal constituents, which depends on the length of the observation data. In the solution process, Equation 1 is generally linearized, i.e., Equation (2).

[0046] (2)

[0047] in:

[0048] A n = h n cos g n , B n = h n sin g n (3)

[0049] (4)

[0050] If there are M observations, formula (2) can be written in matrix form, i.e., formula (5). J is the observation data matrix, K is the known model parameter matrix, and P is the parameter matrix to be solved.

[0051] J=KP(5)

[0052] (6)

[0053] (7)

[0054] (8)

[0055] According to the least squares method, the optimal estimate of the unknown parameter P is given by formula (9):

[0056] P=(K T K) -1 K T J(9)

[0057] The amplitude H and lag angle G of each tidal component are estimated from the observed tidal components using formula (9), and then the returned tide is obtained using the classical tidal harmonic analysis model formula (1). Figure 4 As shown, the red curve represents the tides of the harmonic analysis model's returns.

[0058] S4. Calculate the mean absolute error (MAE) (Formula 10), root mean square error (RMSE) (Formula 11), and correlation coefficient (CC) (Formula 12) between the tidal components and the CHA return results: By calculating the correlation coefficient CC, mean absolute error (MAE), and root mean square error (RMSE) between the observed tidal components and the harmonic analysis return tides, the performance of the classical tidal harmonic analysis model returns is quantified. The larger the CC, the smaller the MAE, and the smaller the RMSE, the closer the harmonic analysis return tides are to the observed tides, i.e., the better the return performance.

[0059] As shown in Table 2, the correlation coefficient (CC) of the tidal components observed at a water depth of 150 meters and the tidal feedback reported by harmonic analysis is 0.71, the mean absolute error (MAE) is 0.05 m / s, and the root mean square error (RMSE) is 0.06 m / s.

[0060] (10)

[0061] MAE value range: [0, +∞], optimal value: 0.

[0062] (11)

[0063] RMSE range: [0, +∞], optimal value: 0.

[0064] (12)

[0065] CC value range: [-1, 1], optimal value: 1.

[0066] N is the number of observation points; P i For tidal return data; O i This is tidal observation data; This represents the average of the tidal return data; This represents the average value of tidal observation data;

[0067] S5, Normalized Error: The mean absolute error (MAE) and root mean square error (RMSE) are divided by the average of the observed absolute values ​​to eliminate dimensions, yielding the normalized mean absolute error (MAE) (NMAE) and normalized root mean square error (RMSE) (NRMSE). As shown in Table 2, the NMAE for the tidal component observed at a water depth of 150 meters and the tidal response reported by harmonic analysis are 0.67 and 0.89, respectively.

[0068] Table 2 Calculation results of tidal nonstationarity index and other statistical evaluation indicators for each water depth

[0069]

[0070] S6, Quantifying Tidal Non-Stationarity: In the ideal case of perfectly stationary tides, the harmonic analysis report tides perfectly match the observed tides, hence the correlation coefficient CC is 1, the normalized mean absolute error (NMAE) is 0, and the normalized root mean square error (NRMSE) is 0. However, actual tides are non-stationary, therefore CC will be less than 1, NMAE will be greater than 0, and NRMSE will be greater than 0. The deviation between the actual performance and the ideal situation represents the magnitude of tidal non-stationarity, and this deviation can be calculated using the distance formula. The CC, NMAE, and NRMSE of the harmonic analysis report are defined as the X-axis, Y-axis, and Z-axis in three-dimensional space, respectively. The ideal stationary tidal report performance is used as a reference point, i.e., (1, 0, 0).

[0071] The tidal nonstationarity index NS is calculated based on the distance formula between two points, where the coordinates of the reference point are (1, 0, 0) and the coordinates of the observation point are (CC, NMAE, NRMSE). The distance between the two points is then calculated. The tidal nonstationarity index (NS) is defined as follows: the larger the NS value, the stronger the tidal nonstationarity; NS = 0 represents completely stationary tides. As shown in Table 2, the coordinates of the observation point at a water depth of 150 meters are (0.71, 0.67, 0.89), and the tidal nonstationarity index (NS) is 1.15, which is significantly greater than 0, indicating that the tides in this sea area are relatively nonstationary.

[0072] like Figure 4 As shown, significant non-tidal variations exist in the eastward current velocity at a depth of 150 m at 122.9598°E, 11.3683°S in the Timor Strait, with a dominant period significantly longer than one day (pink line). This invention uses the EMD method to identify and remove sub-tidal variations. Results show that strong background currents suppress tidal amplitude; while when the background current is weak, the tidal current may exhibit a significant but transient enhancement (…). Figure 4 (The black line) suggests the influence of other potential physical processes.

[0073] At a water depth of 150 m, the tidal nonstationarity index can reach as high as 1.15. The tidal nonstationarity at other water depths is less than that at 150 m, as shown in Table 2. Among them, at a water depth of 750 m, the tidal nonstationarity is only 0.58. The correlation coefficient (CC) of the moored observation points at various water depths ranges from 0.71 (150 m) to 0.93 (750 m).

[0074] A large tidal nonstationarity index reveals the high complexity of endotidal processes. The variation of tidal nonstationarity with water depth is not monotonic, which may be related to the multimodal structure of endotidal processes. Low-mode baroclinic tides have longer wavelengths, enabling them to propagate to distant areas, while high-mode baroclinic tides have shorter wavelengths and tend to dissipate rapidly locally. The vertical superposition of different baroclinic tide modes further complicates the vertical variation of tidal nonstationarity.

[0075] The above description is merely a preferred embodiment of this application and is not intended to limit this application. Various modifications and variations can be made to this application by those skilled in the art. Any modifications, equivalent substitutions, improvements, etc., made within the spirit and principles of this application should be included within the protection scope of this application.

[0076] While the specific embodiments of the present invention have been described above, they are not intended to limit the scope of protection of the present invention. Those skilled in the art should understand that various modifications or variations that can be made by those skilled in the art without creative effort based on the technical solutions of the present invention are still within the scope of protection of the present invention.

Claims

1. A method for quantifying tidal nonstationarity, characterized in that, Includes the following processes: S1, acquire tidal observation data of the target sea area, the observation data including hourly tidal observation data, and the observation period is not less than 1 month; S2, the non-tidal low-frequency variations and high-frequency tidal variations in the observed data are separated using the empirical mode decomposition method to obtain the tidal components; the specific process is as follows: Empirical mode decomposition was used to decompose hourly tidal observations into different modes. The number of maxima in each mode was counted, and the average period of each mode was obtained by dividing the data length by the number of maxima. Modes with an average period greater than 30 hours were merged and defined as non-tidal low-frequency variations. Modes with an average period less than 30 hours were merged and defined as tidal components. S3, the tidal components are returned based on the classical harmonic analysis CHA model to obtain the CHA returned tide; S4, calculate the correlation coefficient CC, mean absolute error MAE, and root mean square error RMSE between the tidal components and the CHA report results; S5. Divide the mean absolute error (MAE) and root mean square error (RMSE) by the average of the observed absolute values ​​to obtain the normalized mean absolute error (NMAE) and the normalized root mean square error (NRMSE). S6. Calculate the tidal non-stationarity index NS based on the distance formula between two points, where the coordinates of the reference point are (1, 0, 0) and the coordinates of the observation point are (CC, NMAE, NRMSE). The larger the NS value, the stronger the tidal non-stationarity.

2. The method for quantifying tidal nonstationarity as described in claim 1, characterized in that: The target sea area in S1 includes estuary areas, shallow seas and deep seas, and the tidal types include diurnal tides, mixed diurnal tides, mixed semi-diurnal tides and semi-diurnal tides. The tidal range includes strong tides, intermediate tides and weak tides.

3. The method for quantifying tidal nonstationarity as described in claim 1, characterized in that: The specific process of S3 is as follows: the amplitude H and lag angle G of each tidal component are estimated from the observed tidal components using the least squares method, and then the returned tide is obtained through the classical tidal harmonic analysis model.

4. The method for quantifying tidal nonstationarity as described in claim 1, characterized in that: The specific process of S4 is as follows: The calculation process for the correlation coefficient CC, mean absolute error MAE, and root mean square error RMSE is as follows: MAE range: [0, +∞], optimal value: 0; RMSE range: [0, +∞], optimal value: 0; CC value range: [-1, 1], optimal value: 1; N is the number of observation points; P i For tidal return data; O i This is tidal observation data; This represents the average of the tidal return data; This represents the average value of tidal observation data.

5. The method for quantifying tidal nonstationarity as described in claim 1, characterized in that: The specific process of decomposing hourly tidal observations into different modes using empirical mode decomposition is as follows: First, the original tidal observations are screened. By identifying local maxima and minima and fitting upper and lower envelopes, the difference between the envelope mean and the original signal is calculated to obtain the intrinsic mode function (IMF) with a positive instantaneous frequency. If the difference signal does not meet the IMF condition, it is used as a new signal and the above steps are repeated until a series of IMF components reflecting the characteristics of different time scales of tidal observations are decomposed. Finally, the remaining stationary residuals that cannot be further decomposed are the trend terms of the tidal observations.

6. The method for quantifying tidal nonstationarity as described in claim 1, characterized in that: Specifically, S6 involves defining the CC, NMAE, and NRMSE of the harmonic analysis return as the X-axis, Y-axis, and Z-axis in three-dimensional space, respectively. The ideal steady tidal return is used as a reference point, i.e., (1, 0, 0). The actual tidal observation point coordinates are (CC, NMAE, NRMSE), and the distance between the two points is calculated. NS is defined as the tidal nonstationarity index. The larger the NS, the stronger the tidal nonstationarity. NS of 0 represents tidal complete stationarity.

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