1 Introduction

Internal solitary wave (ISW) is a ubiquitous phenomenon in the world’s oceans, particularly in continental and marginal waters [4, 10, 13, 28]. ISWs are generated at the mixed layer beneath the ocean surface and show less obvious manifestations on the ocean surface. ISWs can travel hundreds of kilometers while maintaining their waveform or amplitude, owing to nonlinear and dispersion effects. The resort force of ISWs is the reduced gravity, which promises the generation of large-amplitude ISWs. ISWs in the South China Sea (SCS) were observed with amplitude over 240 m. The length of wave crest (LWC) of ISWs can also extend to several hundreds of kilometers. ISWs are found to travel across the whole northern SCS, the Andaman Sea, and the Sulu-Celebes Sea within a few days.

Ocean habitats, off-shore engineering, ocean military, ocean mixing, and sediment resuspension can all be affected by the propagation and breaking of ISWs. While the wave crest of ISWs can be extended to hundreds of kilometers, the ISW scale across the wave crest only ranges from several hundreds of meters to several thousand. ISW propagation speed ranges between 2.0–3.0 m/s in the deep ocean and 1.0–2.0 m/s on the continental shelf. The across wave crest features of ISWs means the ISW will pass by a fixed location within a few tens of minutes. Considering the fast propagating and large amplitudes of ISWs, the ISWs are extremely dangerous to submarine or underwater vehicles. The propagation of ISWs will be accompanied by ISW-induced currents, and severe shear forces will endanger the safety of off-shore equipment, such as oil rigs. The propagation of large amplitude ISWs will induce strong vertical mixing in the ocean and affect the distribution of suspension or sediments [8]. The importance of ISW makes the forecast of ISW propagation a meaningful but challenging task.

Global observation, random generation, fast propagation, and significant impact on the ocean have made ISWs a hot topic for decades. Various methods have been used to study ISWs, such as numerical models, in-situ observations, and satellite observations. Remote sensing techniques have developed rapidly and show significant advantages in ISW studies [10, 11]. To describe ISW propagations, various theories were developed, such as the Korteweg-de Vries equation (KdV) equation, the Benjamin-Ono (BO) equation, and the numerical models. The KdV equation for the propagation of ISWs is given by

$$\begin{aligned} \eta _{t}+c_{0}\eta _{x}+\alpha \eta \eta _{x}+\gamma \eta \eta _{xxx}=0 \end{aligned}$$
(1)

Here \(\eta \) is the amplitude of the solitary wave and \(\text {c}_{\text {0}}\) is the linear phase speed. When the nonlinear term \(\alpha \) is balanced with the dispersion term \(\gamma \), one gets the solitary wave with an analytical form

$$\begin{aligned} \eta \left( x,t\right) =\eta _{0}sech^{2}\left[\left( x-ct \right) /L \right]\end{aligned}$$
(2)
$$\begin{aligned} c=c_{0}\left( 1+\frac{\eta _{0}}{2h} \right) \end{aligned}$$
(3)
$$\begin{aligned} L=\sqrt{\frac{12\gamma }{\alpha \eta _{0}}} \end{aligned}$$
(4)

where \(\eta _{0}\) is the maximum amplitude, L is the characteristic length, \(\text {h}\) is the water depth, and \(\text {g}\) is the acceleration due to gravity. The KdV equation is commonly used to describe the propagation speed of ISWs, but the ISW amplitude needs to serve as preliminary information. Different theories have been developed for ISW propagations in different ocean areas, each with its advantages and disadvantages. Here we introduce a new data-driven model to forecast ISW propagation. The forecast model was trained using big data collected from multi-source remote sensing imageries. The model performance shows better results than the traditional equations and is more robust for errors included in the model inputs.

In the following chapter, we will first briefly overview the achievement of ISW studies using satellite observations. Then, machine learning techniques applied to the ISW studies will be introduced, and the establishment of the ISW forecast model will be presented. The model discussions and future works will be introduced in the last part of this chapter.

2 Satellite Observations of ISWs

Space-borne synthetic aperture radar (SAR) and optical satellite images are commonly used in ISW studies. SAR is an active microwave radar with all-weather, all-time, long-distance, and high-resolution detection advantages. The imaging principle of ISWs on SAR images is the Bragg backscattering mechanism [1]. Currents induced by ISW (usually refers to the first-mode depression ISW) modulate micro-scale waves on the sea surface, making convergent and divergent regions appear in the front and back of the wave, respectively. The sea surface roughness increases in the convergent region where the Bragg backscatter signal is enhanced and appears as a bright stripe on SAR images. The sea surface roughness decreases in the divergent region where the Bragg backscatter signal becomes weaker and appears dark on the SAR image. Figure 1a and b show that ISWs manifest as bright-dark stripes on SAR images. Although spatial resolutions of SAR images are relatively high, their time resolution is relatively low. The limited swath of SAR images also imposes restrictions on ISW studies. SAR images are generally used to study the mechanism of ISWs, such as inversion of amplitude [26], propagation speed [15], and energy analysis [17].

Fig. 1
figure 1

Satellite detection of ISWs in the Andaman Sea a, South China Sea b, Sulu Sea c, and Celebes Sea d. a composite map of Sentinel-1 images acquired on 10 March 2019; b ENVISAT ASAR image acquired on 5 May 2004; c MODIS image acquired on 14 March 2020; d MODIS image acquired on 28 March 2020

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The imaging mechanism of ISW on optical images is the quasi-specular reflection. Optical remote sensing uses sunlight as the light source and receives ocean information from the sunlight reflected by the sea surface. The characteristics of ISW in optical images are more complicated than SAR images. Similarly, for depression ISW, the characteristics of different locations on the optical image are different. They may appear as bright-dark or dark-bright strips in sun-glint areas or non-sun-glint areas. Figure 1c and 1d show satellite observations of ISWs in the Sulu-Celebes Sea. Multiple ISW packets can be observed propagating in different directions with long wave crests.

Optical remote sensing images have advantages of high time resolution and wide swath. Take Moderate-Resolution Imaging Spectroradiometer (MODIS) image as an example, its swath can reach 2330 km, and the same ocean area can be observed twice in one day. In addition, the spatial resolution of optical satellites launched in recent years has also been greatly improved. For example, the highest spatial resolution of GF-1 remote sensing images of China’s high-resolution series can reach 2 m, and the spatial resolution of GF-2 images has reached less than 1 m. However, the optical satellite images are heavily affected by the weather conditions, such as clouds and rain, which will limit its observation capability. Benefiting from the wide swath and high temporal resolution of optical satellite images, the temporal and spatial distribution characteristics of ISWs were studied in different ocean areas.

SAR and optical satellite images provide rich data sources for the research of ISWs. The generation mechanism, distribution, and propagation of the ISWs have been reported. Since the forecast of ISWs is significant for ocean environments, engineering, mixing, and military, the forecast of ISWs is meaningful. The ISW forecast is mainly conducted using empirical or numerical models [30]. In the SCS, ISWs have been predicted based on the west propagating barotropic tide in the Luzon Strait. ISW positions can be predicted according to the distance to the Luzon Strait. Based on the relationship between the ISW generation and the tide information, [6] forecasted the possible ISW occurrence in the north region of the Andaman Sea. The ISWs are generated at multiple sources and have complex patterns as revealed in Fig. 1. The complex pattern of ISWs makes the forecast more difficult. The complexity of multiple sources and wave crest merging during the propagation make it hard to apply the empirical method. The numerical model needs large computation resources and is difficult to setup for ISWs propagating in the coastal areas. To overcome these difficulties, we proposed a machine learning model to forecast ISW propagation in different oceans with different ISW characteristics.

3 Machine-Learning-Based ISW Forecast Model

Machine learning techniques are fast evolving and have already demonstrated tremendous promise in oceanographic research [5, 23, 24]. Liu et al. [16] explored extracting coastal inundation mapping information from SAR imagery by applying deep learning techniques. Machine learning also allows for the creation of connections between multi-dimensional data. Pan et al. [19] employed textural information taken from optical satellite images and ocean environmental factors to determine the amplitude of ISWs using the back-propagation (BP) algorithm. Li et al. [12] used a U-net-based method to obtain ISW wave crest from satellite images and conducted a thorough evaluation of the use of machine learning approaches in satellite image information mining. Machine-learning approaches have previously been proven to offer benefits in maritime applications due to their high nonlinear mapping ability and multi-dimensional data processing.

The propagation of ISW is influenced by ISW features as well as the ocean influencing factors, such as topography and seasonal variations. Machine learning techniques are an excellent option to manage multi-dimensional impacting elements with no defined relationship, making them a strong choice for ISW propagation model construction.

3.1 Model Establishment

We use a fully connected neural (FCN) network to build relationships between the ISW propagations and its impacting factors [25, 29]. Figure 2 depicts a sketch map of the FCN network. We use the error back-propagation (BP) technique to train the FCN network. The desired output is calculated by the forward calculation procedure, and the back-propagation of errors between the model and the desired output is used to adjust the model. The weights of the neural network can be automatically adjusted based on the errors inversely fed into the model. The built dataset is categorized into the training dataset and validation dataset. The validation dataset watches the training procedure while the training dataset is employed for model training. The model will be trained for several epochs after being initialized randomly. The model will stop training and be further validated by a test dataset once it achieves the best validation results.

Fig. 2
figure 2

Sketch map of an FCN network. The different thickness of lines connecting different neural indicate different weights

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The forecast model’s input parameters are made up of ISW property-related inputs, i.e., the Peak-to-Peak (PP) distance and the LWC [31], as well as ocean environment-related inputs such as longitude, latitude, mixed layer depth, density difference, and water depth. Satellite images may be used to measure an ISW’s LWC and PP distance. The water depth can be interpolated from the ETOPO1 dataset. The World Ocean Atlas (WOA) 2018 dataset, where the temperature and salinity can be obtained, may be used to estimate ocean stratification. The buoyance frequency peak corresponds to the depth of the mixed layer.

There are two modules in the model: propagation speed and direction (PS and PD) module. Seven input parameters are included and the output layer is the propagation speed and direction. The initial propagation direction is an extra input parameter in the PD module that is used to resolve occasions with cross-propagating ISWs. Figure 3 depicts the model framework for the ISW prediction model. Locations, ocean parameters, and ISW characteristics of an initial ISW location found from satellite images may all be gathered and used as model inputs. We may run the model for several time steps to get the expected ISW positions at each time step using the model’s predicted ISW propagation velocity and direction. The Levenberg-Marquardt algorithm-based training function ‘trainlm’ was chosen for its quick convergence rate [22]. The hyperbolic tangent sigmoid transfer (tansig) function is used to activate hidden layers, whereas the linear transfer (pureline) function is used to activate hidden and output layers. We used an early stopping strategy to combat the problem of over-fitting [20].

Fig. 3
figure 3

Model structure for the ISW forecast model

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3.2 Model Training

Both optical and SAR images can be applied to extract the training data. The MODIS sensors are onboard the National Aeronautics and Space Administration (NASA) satellites Terra and Aqua. The MODIS image has a swath of 2330 km and the highest spatial resolution of 250 m. The Ocean and Land Color Instrument (OLCI) has five camera modules on board Sentinel-3. The OLCI has a swath of 1440 km and the best spatial resolution of 300 m. We use 123 MODIS images and 33 OLCI images in the Andaman Sea and 149 MODIS images and 8 VIIRS images in the Sulu-Celebes Sea to build the dataset.

On satellite images, ISWs appear as bright-dark bands. ISW locations can be saved as the GIS formatted file which is used to extract the spatial position. The LWC can be obtained using ISW labels and the PP distance equals the positive and negative peaks of the ISW profiles [31]. The propagation direction of the input ISW wave crest is introduced to solve cross-propagating ISW problems. A detailed procedure of how to build the training dataset is shown in Fig. 4.

Fig. 4
figure 4

Procedure to build the training dataset from satellite images. a A subset of the MODIS image showing three ISW packets, the PP distance can be measured from the satellite image. The insert map shows the extracted profile of the ISW and how the PP distance is measured; b extracted locations of ISW, the LWC was calculated from extracted wave crests, and how the initial propagation direction is obtained; c subset of the extracted dataset, the brown shaded area shows model input parameters and the green shaded area indicates model output parameters

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We estimate the phase speed of ISWs based on the difference of ISW locations and image acquisition time. One utilizes the location and time difference of the ISWs to get the ISW propagation speed if an ISW was detected on two quasi-synchronous images. We assume the time difference between two ISWs in the same satellite image equals the period of the semi-diurnal tide [7, 9]. In the Andaman Sea, 1189 samples were extracted, while in the Sulu-Celebes Sea, 1546 samples were extracted. The training and independent test datasets were created from these samples, which were split by 80/20%.

Figure 5shows the results of PS and PD modules for ISWs in the Andaman Sea. The Sulu-Celebes Sea forecast model yields similar results. The root mean square (RMSE) of the training (test) datasets for the PS module is 0.19 m/s (0.20 m/s), while the correlation coefficients (CC) are 0.90 (0.88). The results demonstrate that the loss of the forecast decreases over time, and the PS module’s mean square error (MSE) obtains its optimal validation performance at epoch 37. The PD module has an RMSE of 10\(^{\circ }\), and the CCs are over 0.99. The gradient was steadily reduced, and at epoch 16, the PD module obtained its optimal performance. The MSE does not decrease and validation tests rise at epoch 16, the model stops training.

Fig. 5
figure 5

Loss of the propagation velocity module aand the propagation direction module b for ISW forecast model

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ISWs in various ocean locations have varied features. In the Andaman Sea, we observe cross-propagating ISWs, while in the Sulu-Celebes Sea, we observe ISWs propagating in the opposite direction. We trained the model 30 times with and without initial primary propagation directions to see how they affected the predicted outcomes. Models having an initial major propagation direction as input performed better, with a reduced RMSE and more stable model performance, as illustrated in Fig. 6. The model without initial primary propagation directions had large deviations and lower correlation coefficients, indicating lower model generalizability. For circumstances with cross-propagating ISW patterns, it is required to add the propagation direction of the ISW wave crest in the model inputs.

Fig. 6
figure 6

Model tests with (blue lines) or without (orange lines) the propagation direction of ISW wave crests

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3.3 Model Validation

The forecast model was validated for ISWs in the Andaman Sea and Sulu-Celebes Sea. The ISWs created by successive semi-diurnal tides in the Andaman Sea is depicted in Fig. 7. On the MODIS image, three ISW wave packets propagating eastward can be seen. IW1 and IW2 have LWCs of 146.41 and 242.26 km. IW1 and IW2 have PP distances of 1391.48 and 731.54 m. IW1 (IW2) parameters are used as model inputs, while IW2 (IW3) acts as model validation. In Fig. 7b and c, the time step is 6.21 h, and ISW positions after one time step are depicted with dashed lines. The model results (satellite observations) after one semi-diurnal tide are indicated by the black (red) lines in Fig. 7b and c. The model predicted results appear to be in good agreement with satellite data.

Fig. 7
figure 7

MODIS images acquired on 9 May, 2017. The ISW locations extracted from satellite images are represented with red lines. The solid (dashed) black lines represent forecast positions after 12.42 h (6.21 h)

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We established three parameters, namely root-mean-square error (RMSE), Fréchet distance (FD), and CC, to qualitatively examine the effectiveness of the constructed ISW forecast model. The FD is a strict evaluation factor that considers the placement and order of points of the ISW wave crest. IW1 and IW2 have RMSEs of 6.10 and 2.50 km; the FDs is 18.28 and 9.06 km, and the CCs are 0.96 and 0.89, respectively. In the Andaman Sea, we examined 8 examples for ISWs, including distinct locations with different ISW features. Table 1 shows the statistical findings. The average CC value is 0.95, and the average FD is 11.46 km, showing that the model-predicted ISW positions and satellite observations have a good degree of agreement.

Table 1 Statistical results of forecast model in the Andaman Sea
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Fig. 8
figure 8

Forecast model validation cases in the Sulu-Celebes Sea. MODIS image acquired on 18 May 2015 and 29 October 2019 (upper left and right) and corresponding forecast results (lower left and right)

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The results of the model validation for ISWs propagating in the Sulu-Celebes Sea are shown in Fig. 8. Three ISW wave packets have been detected moving northward (southward) in the Sulu (Celebes) Sea. For these three wave packets, the leading ISWs are called IW1, IW2, and IW3. The model input is the wave crest IW1 (IW2), and the model validation is IW2 (IW3). With solid black lines, the model predicted results are illustrated in Fig. 8. We can observe that the model outputs and satellite data are typically in agreement. Table 2 shows the statistical results of the nine validation instances that we gathered. The RMSE is 12.92 km, the FD is 18.73 km, and the CC is 0.98, which indicates the model performs well.

ISWs are consecutively generated by semi-diurnal cycles, so more than one tidal cycle can be observed by satellite images. We also tested the forecast model on these ISWs. The model runs iteratively to estimate ISW positions, the model predicted ISW locations is the input for the estimation of the next tidal cycles. The time step is 6.21 h, three MODIS images were utilized to assess model performance. The predicted ISW positions are displayed as solid black lines. The forecast model estimated ISW positions were consistent with satellite observations, as shown in Fig. 9a and b. The deviations between satellite observations and model estimations are particularly pronounced in Fig. 9c. Because of the complex terrain in the north region of the Andaman Sea, significant variations in the PP distance of ISWs may impact model estimations.

The forecast model estimates were tested after two semi-diurnal tide cycles in the Sulu-Celebes Sea using a MODIS image collected on 29 October 2019 with distinct ISW signals. Figure 10a depicts the results. The Sulu-Celebes Sea ISW sites identified as IW1 are utilized as model input. In Fig. 10b, the ISW prediction is displayed after three semi-diurnal tidal cycles. After two or three semi-diurnal tidal cycles, the model estimations coincide well with satellite measurements.

4 Influence Factors on the ISW Forecast Model

While the model is validated and shows high accuracy as described above, some affecting factors will be discussed. When utilizing the estimated ISW positions of the first semi-diurnal tidal cycle as the model input for the following forecast, errors may be included. Based on the ISW locations, the locations and ocean environment characteristics could be changed, and the LWC could be computed. The ISW PP distance will remain constant in the forecast model’s subsequent iterative runs. The predicted outcome deviations will accumulate, resulting in more severe discrepancies in the subsequent iterative prediction. Influences of the time step, input parameter errors, the influence of seasonal variations, and comparison with the KdV equation were discussed in this section.

Table 2 Statistical results for validation cases in the Sulu-Celebes Sea
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Fig. 9
figure 9

Model results for ISW propagation after two semi-diurnal tidal cycles

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Fig. 10
figure 10

Iterative forecast results for ISWs propagating after a two and b three semi-diurnal tidal cycles. ISWs are extracted from MODIS images acquired on a 29 October 2019 and b 25 February 2015. Black lines: model forecast results. Red lines: satellite observations of the ISW locations. Dashed lines: ISW locations every semi-diurnal tidal cycle

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Because ISWs are frequently generated by semidiurnal tidal cycles, the time step is set to 12.42 h by default. When the time step is changed, we will see how it affects the model outcomes. Figure 11 depicts the effect of the model running with various time increments. We compared the model forecast results with satellite observations using time increments from 1/4 to one semi-diurnal tidal cycle. ISWs propagate from IW1 to IW2, the depth of water changes from about 2,000 m to around 1,000 m. The results are poorer using a time step equals 12.42 h compared with smaller time steps. The model results were nearly the same when time steps were 3.11 and 4.14 h, and the disparity with the time step equals 6.21 h was similarly low. As a consequence, we infer that the ISWs cross over the steep isobaths when the terrain changes dramatically; a lower time step may enhance the prediction result. If the terrain changes gently, the time step may be set to be a large one.

Fig. 11
figure 11

Comparison of the model forecast results with different time steps

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There are eight input parameters for the model which were taken from satellite images or publicly accessible datasets. The forecast model’s outcomes will be influenced by the inaccuracy made in the input parameters. Except for the PP distance and initial propagation direction, all input parameters were modified when we ran the model repeatedly. In the following model predictions, initial ISW PP distance and propagation direction at the starting point will be used as corresponding inputs. When ISWs were not clearly spotted owing to unsatisfied imaging conditions, the PP distance may introduce errors. In four locations of the Andaman Sea, the effects of mistakes in PP distances and ISW propagation direction on the model estimations were studied. Figure 12 depicts the results.

An inaccuracy of \( \pm {10}^{\circ }\) was considered to analyze its impacts on the model predictions. The time step is 6.21 h, and the results are presented in Fig. 12a and b. The model predicted ISW positions that were near to each other and satellite data which promise the model tolerance on initial propagation direction errors.

Given that a one-pixel inaccuracy resulted in an error of \({\pm }300\) m to the PP distances, we compare the results with different PP distance inputs. In the Andaman Sea, two locations with significant and modest water depth fluctuations were examined. The results reveal that, despite the varying inaccuracies in the ISW PP distance, the predicted ISW positions were close to satellite measurements.

The results demonstrate that the proposed forecast model is extremely forgiving of inaccuracies in input parameters like the ISW PP distance and propagation direction of the input ISW wave crest. Minor inaccuracies in some input parameters had no effect on the model’s performance because the ISW propagation was defined by eight factors. Despite this, the model produced results that were close to satellite data. It’s important to remember that input parameter mistakes will accumulate over time when a model runs iteratively. However, the model’s predictions were remained valid after two or three tidal cycles, according to results presented in Fig. 10.

Fig. 12
figure 12

Influence of input parameter errors for initial main propagation directions (a, b) and PP distances (c, d)

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Fig. 13
figure 13

ISW forecast in different seasons and locations in the Andaman Sea

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The stratification of the ocean fluctuates because of precipitation and other reasons, ISW propagation is impacted by seasonal variations. The dry season in the Andaman Sea starts from January to April and the rainy season start from May to November. We estimate ISW propagation in four Andaman Sea locations throughout the dry and wet seasons to see how seasonal differences affected the model’s results.

March (August) was selected to represent the dry (rainy) season. The density information was calculated from the WOA2018 dataset. In the two seasons, there were differences in the depth and density of the mixed layer. The predicted outcomes of the forecast model are displayed in Fig. 13. The model-predicted ISW positions were near to each other and also close to the satellite observations in Cases 13a, 13b, and 13d. The ISW predictions in Fig. 13c exhibited more significant disparities between the outcomes. The findings suggest that ISW propagation shows small seasonal fluctuations in cases a, b, and d, whereas there are disparities in cases c.

Figure 13 depicts the buoyancy frequency of four locations in two seasons. The buoyancy frequency distribution in Fig. 13g and h showed more significant inconsistencies than in Figs. 13e and f. In the dry season, Fig. 13g and 13h had two peaks, while in the rainy season, there was only one peak. The most significant difference is found in Fig. 13g, indicating the most substantial fluctuations in ocean stratification. A larger buoyancy frequency peak indicates stronger ocean stratification. The ISW propagates faster with a larger density difference [27]. This explains why, in Fig. 13c, ISW spread quicker during the rainy season.

Fig. 14
figure 14

Comparison of the forecast results of the proposed model and the KdV equation (upper panels) and sensitivity of the model results to the ISW amplitude (lower panels)

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The nonlinear propagation velocity of ISWs are described by the KdV equation [3, 18]:

$$\begin{aligned} C_{p} = C_{0} + \frac{\alpha }{3}A_{0}. \end{aligned}$$
(5)

Where \(C_{0}\) is the linear phase speed, \(\alpha \) is the nonlinear coefficient, and \(A_{0}\) is the ISW amplitude. The ISW propagation velocity is related to the ISW amplitude which is normally unknown [21]. Based on previous studies [2, 14], the amplitudes of ISWs in the Sulu-Celebes Sea vary from 30 to 90 m. To compare the predicted results, we set the ISW amplitude to 60 m. Figure 14 depicts the comparison results. The RMSE (FD) between the KdV-predicted ISW positions and satellite observations is 44.06 km (59.42 km). The developed model estimation has an RMSE (FD) of 14.67 km (28.35 km). The findings reveal that the proposed model’s predicted ISW positions are closer to satellite results. The KdV equation produces a larger ISW propagation velocity error as compared to satellite data.

ISW amplitudes range from tens to hundreds of meters. The anticipated ISW positions will deviate as a result of this uncertainty. Generally, we assume the linear propagation speed to 2.5 m/s, the mixed layer depth to 100 m, and the water depth is 3000 m. After one semi-diurnal tidal cycle, a 20 m uncertainty in the ISW amplitude results in an ISW position error of 32.42 km. Figure 14 shows the forecast results for three ISW amplitudes (30, 60, and 90 m) using the KdV equation. The results reveal that ISW locations predicted by the KdV equation are sensitive to ISW amplitudes. Because of the amplitude uncertainty, the predicted ISW positions will shift. Without any unknown characteristics, input parameters of the proposed model may be assessed from satellite observations or publically accessible datasets. When input parameters include errors, the proposed forecast model is more resilient.

5 Conclusions and Future Works

To forecast ISW propagations, the ISW forecast model was built using machine learning techniques in this chapter. The training dataset was built using samples extracted from satellite observations and the publicly accessible datasets ETOPO1 and WOA 2018. An FCN network with eight input parameters, which include ocean elements and ISW features, was proposed. The proposed forecast model can predict ISW positions after propagating several time steps when given an initial ISW position. The model’s estimation is close to satellite data.

The impact of the model’s time step on the predicted outcomes was investigated. When propagating ISWs pass over isobaths, a smaller time step yields better results. Measurements of the PP distance and propagation direction of the given ISW wave crest are easy to have errors. The PP distance and propagation direction of the given ISW wave crest is not modified in subsequent predictions when the forecast model runs repeatedly. The impact of input errors on model estimations was also investigated. The findings reveal that the proposed model is not sensitive to input parameter mistakes. An error of ±300 m on the PP distance and \(\text {10}^{\circ }\) on the propagation direction did not affect the model estimation greatly. This result demonstrates that the proposed forecast model can still provide reliable results with errors included. The impact of seasonal variation on ISW propagation was investigated. The findings suggest that differences were discovered as a result of seasonal fluctuations in ocean stratification. Comparison with the KdV equation indicates that the forecast model produced superior forecast results and was more resilient.

In contrast to numerical models, the forecast model does not need prior knowledge, a rigid boundary, or beginning conditions. Only satellite observations and publicly available datasets are used. Hence, the model provides an alternate but easy way to forecast ISW propagation and can be readily modified to apply to other ocean regions. The initial position of an ISW wave crest is all that is required to run our forecast model. The machine learning algorithms used here have a lot of potential in oceanographic research for multi-dimensional data processing and forecasting.