INTRODUCTION

Measurement of the spectra of sea wave elevation is important for many fundamental and applied tasks of current oceanology. Information on such spectra is used for finding the pollution of water areas [1–3] (including monitoring the anthropogenic impact on sea water areas [1, 2, 4]) and for studies [4–6] and modeling [1, 7] of various hydrophysical processes that occur at the ocean–atmosphere boundary, as well as in other areas.

Retrieval of the two-dimensional spatial spectra of sea wave elevation over vast (including difficult-to-access) areas of seas and oceans is promising with the use of remote sensing methods, which are based on processing of images from air carriers [8, 9], as well as high spatial resolution space imagery [9–12]. The estimation of spectra of surface waves by optical imagery from aerospace carriers combines one-moment coverage of vast marine areas and possible remote registration of a wide range of wave components [9–13].

The retrieval of spectra of the sea wave elevation by optical aserospace imagery uses retrieving operators, which are functions for the transformation of the brightness fields (registered on similar images) to the spectra of sea wave slopes: the gradients of the elevation fields for certain directions, which are determined by the laws of geometrical optics [8–10, 14, 16]. These operators are built using numerical modeling, which takes into account the various formation conditions of aerospace images and characteristics of instruments of remote sensing [8–12]. The adequate retrieval of elevation spectra by the spectra of aerospace imagery is based on their validation using the results of in situ measurements [10–12, 15, 19].

The full retrieval of spatial spectra of sea wave elevation fields requires information on two or more spectra of slopes corresponding to various directions [8, 17]. This is possible if we use a multiposition method that involves the simultaneous processing of two or more images, which have distinct formation conditions of the brightness field and, thus, different gradients of the surface lightening [8]. The multiposition method cannot be applied for processing of single satellite images, because the survey height significantly exceeds the size of the studied sea surface area and the geometric conditions change insignificantly within these images. In these cases, it is impossible to retrieve completely the two-dimensional elevation spectra within the narrow angular sectors with a deficit of information on the sea elevation [9, 11, 15, 18]. Thus, the development of methods that eliminate these restrictions and determine the angular distribution of wave energy even by single satellite images is urgent. This paper is dedicated to this problem.

MATERIALS AND METHODS

A link between two-dimensional spatial spectra of the sea elevation with spectra of the brightness field registered by satellite optical images can be presented as an equation [9]:

$$\Phi ({\mathbf{k}}) = {\mathbf{R}}S({\mathbf{k}}),$$
(1)

where \(\Phi ({\mathbf{k}})\) is the spatial spectrum of sea wave slopes, \({\mathbf{k}}\) is the wave vector, \(S({\mathbf{k}})\) is the spatial spectrum of the satellite image, and \({\mathbf{R}}\) is the mathematic operator that retrieves the slope spectra by the spectra of satellite optical imagery and thus is named the retrieving operator [9].

Retrieving operator R is typically presented as a transfer function (a spatial-frequency filter), which is elementwise applied to the spectral density values of the satellite image calculated using a discrete Fourier transformation [9, 10].

Works [9, 10, 12, 19] elaborated an approach to retrieve the spatial spectra of slopes and elevations of the surface waves by aerospace optical imagery, which takes into account nonlinear modulation of the brightness fields by sea surface slopes and includes parametrization of the retrieving operator \({\mathbf{R}}\) depending on the observation conditions. For adequate application of the multiparametric retrieving operator \({\mathbf{R}}\), its parameters were preliminarily refined taking into account various wave formation conditions [9, 12, 13]. We used the results of comparison of spectra of the disturbed sea surface retrieved by space imagery with the results of measurements by sensors, which were placed both on a stationary hydrophysical platform and on floating buoys [11–13, 19]. This comparison yielded a preliminary correction of the parameters of the retrieving operator in the area of a power drop in the frequency spectrum. This correction corresponded to the wave formation conditions in water areas that showed a mixed elevation including both wind waves and ripples [12, 13]. The spectrum of the sea wave elevations could be found from the following equation [9, 10]:

$$\Psi ({\mathbf{k}}) = {\mathbf{R}}({\mathbf{W}})S({\mathbf{k}}){\text{/}}{{\left( {{{k}_{x}} \cdot \cos {{\phi }_{C}} + {{k}_{y}} \cdot \sin {{\phi }_{C}}} \right)}^{2}},$$
(2)

where \({{\phi }_{C}}\) is the mean direction of the gradient of lightening of the sea surface in the horizontal plane.

The denominator of the equation tends to zero in the direction orthogonal to the direction \({{\phi }_{C}}\). The angular sector in the area of this direct ion is the area of a deficit of information on the spectral density of the elevation field, because the application of Eq. (2) in this sector leads to significant errors in estimation of the elevation spectrum. To deal with this deficit, we elaborated the multiposition method described in [8]. Its application for space imagery is technically difficult due to the peculiarities of the space survey (a narrow view field of satellite instruments, the complexity of synchronous survey from two and more positions, etc.) [9–11, 19]. Thus, the solution of the task with respect to remote space sensing requires an advanced method of retrieval of two-dimensional spectra of the sea elevation by the spectra of satellite imagery, which describe the distribution of wave energy by directions.

The flowchart shown in Fig. 1 demonstrates the remote measurement of the angular energy distribution by two-dimensional spatial spectra of wind waves, which were retrieved by the spectra of satellite images. The calculation operations of this method are conditionally divided into blocks: (1) data preparation, (2) calculation of the spectral density of wave slopes and elevations, and (3) retrieval of the angular distribution of wave energy and estimation of the two-dimensional elevation spectra.

Fig. 1.
figure 1

Flowchart of remote measurement of the angular distribution of energy by the two-dimension spatial spectra of wind waves retrieved by the spectra of satellite images.

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The operations of data preparation (Block 1) include the choice of areas of interest in the satellite image and calculation of their spatial spectra, as well as the formation of parameters of the retrieving operator using metadata of the satellite imagery.

In Block 2, based on the prepared data, the wave spectra are retrieved using the retrieving operator, which takes into account nonlinear modulation of the brightness field by surface slopes. The operations in Block 2 result in two-dimensional spectra of slopes and elevations at all values of the wave vectors that did not fit the angular sector of the information deficit.

Block 3 retrieves the angular distribution of the energy and two-dimensional elevation spectrum using an advanced method, which is described below. The main direction of the advanced method is providing the possibility of retrieval of the angular distribution of wave energy in the two-dimensional spectrum by single satellite images. For this, we use an approach that was previously applied in processing of frequency-directed elevation spectra determined by the data of wave recorders [19–21, 23]. In accordance with this approach, the two-dimensional spectrum of elevation is presented in polar coordinates \((k,\phi )\), where \(k\) is the wave number (magnitude of wave vector \({\mathbf{k}}\)) and \(\phi \) is the wave azimuth characterizing the direction of propagations of wave harmonics with wave vector \({\mathbf{k}} = [k \cdot \cos \phi \), \(k \cdot \sin \phi ]\).

The two-dimensional spectrum will be considered as the product

$$\Psi (k,\phi ) = \chi (k)D(k,\phi ),$$
(3)

where \(\chi (k)\) is the integral one-dimensional spectrum and \(D(k,\phi )\) is the dimensionless function of the angular distribution of sea wave energy.

The angular distribution of sea wave energy \(D(k,\phi )\) is determined by the two-dimensional elevation spectrum \(\Psi (k,\phi )\) using the equation

$$D(k,\phi ) = \Psi \left( {k,\phi } \right){\text{/}}\int\limits_\pi ^\pi {\Psi (k,\phi )d\phi } .$$
(4)

The method of remote measurement of the spatial spectra \(\chi (k)\) for various wave formation conditions was previously validated in [11, 12, 19]. Thus, this work is dedicated to the development of the retrieval of the angular distribution of energy \(D(k,\phi )\) as the main characteristic of the two-dimensional spectrum of sea elevation.

Three new steps are added after the application of retrieving operator \({\mathbf{R}}\) and the transition to the spectrum of sea elevation. First, the spectrum is linearly interpolated in the area of the elevation spectrum information deficit. This area has two symmetrical (relative to the center) sectors with an angular measure of ~40°. Its central directions are orthogonal to the average gradient of the brightness field. The approximation is conducted in the tangential direction using values of the retrieved spectrum close to the deficit zone. Further normalization is carried out following Equation (4).

At the final stage, the angular distribution of wave energy is approximated using the truncated Fourier transform, which omits all members above the second harmonics:

$$\begin{gathered} {{D}_{a}}(k,\phi ) = \frac{1}{\pi }\left( {\frac{1}{2} + {{a}_{1}} \cdot \cos \phi + {{b}_{1}} \cdot \sin \phi } \right. \\ \left. {{{{^{{^{\,}}}}}_{{_{{_{{_{{_{{_{{_{{_{{_{{}}}}}}}}}}}}}}}}}}} + \;{{a}_{2}} \cdot \cos 2\phi + {{b}_{2}} \cdot \sin 2\phi } \right). \\ \end{gathered} $$
(5)

The representation of (5) is also used in processing of the sea wave recorder data [19]. The comparison of the results of processing of the sea wave recorder data and the spectra retrieved by satellite imagery should take into account that the first harmonics in the spectra of these images are absent due to invariance of the imagery on the optical images of waves with opposite directions, i.e., \({{a}_{1}} = {{b}_{1}} = 0\). At the same time, parameters \({{a}_{1}},{{b}_{1}}\) for the wave recorder data differ from zero.

RESULTS OF EXPERIMENTAL STUDIES

The results of comprehensive experiments conducted on September 20, 2021, in the Black Sea water area were used for testing this method [19]. The experiment involved a satellite survey of the studied water area with a high spatial resolution (~1 m).

The sea truth measurements were conducted from the Black Sea Research Platform of the Marine Hydrophysical Institute, Russian Academy of Sciences, which is located ~500 m from the coast [18, 20, 23]. These measurements included the registration of waves using an array of six string wave recorders [20, 22, 25] and meteoparameters (speed and direction of wind, air temperature and humidity, temperature of water of the upper sea level), as well as video registration of the sea surface for estimation of the current velocity vector for the subsurface water layer[19, 24].

The speed of the southwesterly wind was 8.34 m/s, and the wave height was 0.61 m at the moment of the satellite flight. In the period of 8:00–9:00 LT, which spans the moment of the satellite flight, the wind was stable and its speed remained the same. During the experiment, the wave field evolved (the height of waves increased from 0.50 to 0.65 m) at a stable speed of the wind [19].

The sea truth data were processed following the methods described in [19]. Figure 2 shows the smoothed angular distributions of the sea wave energy calculated using Eq. (5) on the basis of in situ measurements using an array of string wave recorders (Fig. 2a), as well as using elevation spectra retrieved by fragments of the satellite image 1024 × 1024 pixels in size, which was taken near the marine platform (Fig. 2b). The angular function was visualized in coordinates (\(k\cos \phi \), \(k\sin \phi \)), where \(\phi \) is the direction of wave propagation. This presentation provided obvious information on the angular distribution of the sea wave energy.

Fig. 2.
figure 2

Smoothed angular distribution of the sea wave energy measured by (a) data from the array of string wave recorders and (b) using elevation spectra retrieved from a satellite image. 

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For more detailed analysis of the correspondence of the results of remote and in situ measurements of the angular distribution of energies, we compared them for the waves of various lengths Λ from 2.6 to 28.4 m. The results of this comparison are shown in Fig. 3, which demonstrate the unidimensional sections of angular distributions shown in Fig. 2 for various wavelengths. The sections that were retrieved by the spectra of fragments of the satellite images and approximated by Eq. (4) are shown by red lines, whereas the sections measured by string wave recorders are shown by blue lines.

Fig. 3.
figure 3

Angular distributions of the energy of sea waves of various lengths estimated by various methods: retrieved by spectra of fragments of a satellite image and approximated by Eq. (5) (red lines) and estimated by data from an array of string wave recorders (blue lines) at different wavelengths: Λ: (а) 28.4, (b) 14.1, (c) 9.3, (d) 7.1, (e) 4.0, (f)  2.6 m.

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The coordinate axes in Fig. 3 show the wave azimuth \(\phi \) and angular distribution \(D = {{D}_{a}}(k,\phi )\) calculated by formula (5) at \(k = 2\pi {\text{/}}\Lambda \).

For numerical comparison of the angular distributions, which resulted from processing of in situ data and was retrieved by fragments of the satellite imagery near the marine platform, we used a Mean Absolute Percentage Error (MAPE) [19]

$${\text{M}}({{\Psi }_{c}},{{\Psi }_{N}}) = {\text{E}}\left[ {\left| {1 - {{\Psi }_{N}}{\text{/}}{{\Psi }_{c}}} \right|} \right],$$
(6)

where \({{\Psi }_{c}}\) and \({{\Psi }_{N}}\) are the measured spectra resulting from in situ and remote measurements and \(\operatorname{E} [\,.\,]\) is the operator of mathematical expectation.

For the analyzed range of wavelengths (2.6 to 30 m), the MAPE value was \(M = 0.3\). This value is close to the results of work [19], which compared the unidimensional integral spectra for the studied range if the wavelengths and the M values vary from 0.2 to 0.4 at various distances from the platform.

The advanced method of retrieval of spectra of the sea elevation by the satellite imagery spectra can be used for monitoring anomalous processes and events in the ocean. As an example, we provide the results of studies of an anomaly caused by a deep runoff to the water area of Mamala Bay described in [1, 2, 4, 5].

Figure 4a shows a panorama of the anomalous sea surface areas, presented in [1], which are related to the impact of turbulent jets caused by runoff through a submerged outfall. The anomalous sea surface areas (1.33 × 1.33 km2 in size) were distinguished by the appearance of additional narrow (the average broadening \(\Delta \bar {\Lambda }\) ~ 4 m) spectral harmonics in the two-dimensional spatial spectra of fragments of the satellite image of high spatial resolution. The average spatial period for these spectral components was \(\bar {\Lambda }\) = 90 m.

Fig. 4.
figure 4

Results of study of the angular distribution of wave energy in the area of deep runoff anomalies: (a) panorama of an anomalous sea surface area distinguished by spatial spectra of fragments of satellite images [1]; (b, c) angular distributions of elevation energy determined by spectra of sea waves retrieved by spectra of anomalous and background fragments of satellite images; (d) comparison of the background and anomalous angular distributions of energy of a sea wave with a length of 90 m determined by elevation spectra retrieved for anomalous fragment no. 11 (red curves) and background fragment no. 104 (blue curves) of satellite images.

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These spectral harmonics encounter the condition [1] \(\Delta \Lambda \ll \Lambda \), where Λ is the spatial period for these narrow spectral harmonics, which can be called “quasi-monochromatic.” They are caused by the surface disturbances related to the internal waves, which are generated by deep runoff turbulent jets [1]. No similar spectral harmonics are present in the background spectra. These effects are described in detail in [1].

Different colors in Fig. 4a show the fragments of the satellite image corresponding to sea surface areas 1.33 × 1.33 km2 in size with various intensity of anomalies, which are distinguished by the spectra of fragments of space images [1].

As an example Fig. 4b shows the angular distribution of sea wave energy by the two-dimensional elevation spectrum, which was retrieved using our method for fragment no. 11 of the satellite image located southwest of the diffuser of the outfall in the area of the sea surface anomaly caused by deep runoff. The angular distribution of energy evidently exhibits the “quasi-monochromatic” wave components with a spatial period of 90 generated by internal waves, which are caused by the impact of deep runoff turbulent jets [1]. These “quasi-monochromatic” spectral components are highlighted by red circles in Fig. 4b.

For comparison Fig. 4c shows the angular distribution of energy resulting from a similar method for the background water area (fragment 104) located at a distance of ~13 km south of the diffuser. This figure shows no spectral component similar to those in Fig. 4b, which correspond to the anomalous sea surface area. The surface areas with an angular distribution are shown by black squares in Fig. 4a.

Figure 4d shows the angular distributions of energy of sea waves with a length of 90 m for directions 0°–180° by the elevation spectra, which were retrieved using the advanced method, for the anomalous (no. 11, curve of red color) and background (104, curve by blue color) fragments of satellite images of the studied water area. The angular distribution of the wave energy at the anomalous water area (red curve) exhibits an evident local maximum in direction of ~170°, which is highlighted by a red circle.

These local maxima are absent in the angular distribution of the wave energy against the background water area (blue curve). These results correspond to those of spatial spectral processing of space images of high spatial resolution [1].

CONCLUSIONS

The advanced method of remote measurement of two-dimensional spatial spectra of wave elevation by satellite imagery of high spatial resolution (~1 m) allows estimation of the angular distribution of sea wave energy even in the presence of a limited angular sector and a deficit in spectral data due to the uniposition character of the satellite survey. These estimations based on remote measurements are of undoubted practical interest.

The angular wave distributions retrieved in this work by two-dimensional spatial spectra of satellite observations are consistent with the results of in situ measurements by string wave recorders. The average module of relative error was ~0.3 for a range of wavelengths from 2.6 to 30 m.

The results of processing of satellite data from an impact area on the surface of internal waves, which were created by deep runoff turbulent jets, showed the possibility of using of our method for the study and monitoring of anomalies that originated on the sea surface and in subsurface layers of seas and oceans and are manifested in changes in the elevation spectra.