A variational optimization algorithm for Secchi disk depth based on Multi-satellite data

A Secchi depth is a basic parameter to describe the optical properties of water, which is related to the composition and content of chlorophyll, suspended solids and yellow substances in water, and is also closely related to solar radiation on the surface of water, physical and chemical properties of water and meteorological conditions. The most direct way to obtain the spatiotemporal distribution of water Secchi depth is to use ships to regularly measure the Secchi depth of the base-stations, but this method can only obtain the Secchi depth of the measurement station state, and it is impossible to obtain the Secchi depth characteristics of seawater with large spatiotemporal distribution. As a brand-new observation method, remote sensing technology can obtain the distribution characteristics of ocean parameters in large time and space. In recent years, with the rapid development of remote sensing technology, especially the development of water-colored remote sensors and the improvement of the accuracy of inversion algorithms, many remote sensing products for water Secchi depth have been provided. However, due to the difference in orbit operation and observation parameters of different remote sensing loads, water Secchi depth products are affected by clouds and meteorological climate environment, and there is no same standard for verification and evaluation, which limit the promotion and application of water Secchi depth products. At the same time, data integrated technology has been widely recognized by many disciplines and has been greatly developed in recent years, and pixel-level, feature-level, decision-level integrated technology and development are also popular research directions in the world. Therefore, based on an improved variational optimization algorithm, this article integrates SDD retrieval products on multiple satellite data t such as SNPP, MODIS, and MERIS, and verifies them with ship survey data. The correlation is better than 0.9, proving that this method can improve the accuracy of SDD inversion products.


Introduction
Secchi disk depth (SDD) is a basic parameter to describe the optical properties of water, which is related to the composition and content of chlorophyll, suspended matter and yellow matter in water, and also closely related to the solar radiation on the sea surface, physical and chemical properties of water and meteorological conditions [1][2][3].SDD refers to the degree of turbidity in seawater, reflecting the extinction characteristics of seawater.It is a fundamental optical parameter of the ocean that varies with region, season, and depth.It is a fundamental element for measuring the optical properties of seawater in current ocean survey work.The horizontal and vertical distribution of seawater transparency is an important factor in military activities such as underwater submarine communication, submarine stealth and anti submarine, mine deployment and detection, and seabed mapping, et al.
The traditional SDD observation mainly includes type method, cross disk method and Secchi disk method, but manual observation can not guarantee time synchronization and measurement accuracy, and has been developed from manual visual observation to automatic observation.At present, it is the most common method to measure SDD of each station with ship-borne instruments, but this method can only obtain SDD of measuring points, and it is impossible to obtain the characteristics of SDD with large spatial and temporal distribution.
As a new observation method, ocean remote sensing technology can obtain the distribution characteristics of ocean parameters in large space and time.In recent years, the remote sensing technology of ocean water color has developed rapidly, especially the improvement of the precision of water color sensor technology and related inversion algorithm, which makes the remote sensing of SDD possible.According to the research results of He Xianqiang and Pan Delu et al. [1,2], the remote sensing inversion algorithms of SDD can be divided from inversion approaches, mainly including direct inversion method and indirect inversion method.SDD remote sensing inversion algorithm can be divided into direct remote sensing inversion algorithm and indirect remote sensing inversion algorithm.The direct remote sensing inversion algorithm is to obtain SDD directly by using remote sensing inversion of water radiance or remote sensing reflectivity.The indirect remote sensing inversion algorithm refers to the inversion of the concentration of water color elements or the optical properties of water from the water radiance, and then the inversion of SDD.
Most of the satellites are polar orbit satellites, and single sensor data is severely limited by the gap between detector scanning orbits, resulting in limited daily coverage.Additionally, satellite sensors are susceptible to phenomena such as clouds, solar flares, and thick aerosols, resulting in a significant decrease in effective sampling rates.Data integrated can fill in gaps between data, expand the coverage of SDD data, and make the data continuous in time and space.The integrated data not only protects the temporal continuity of the data, but also expands the spatial effective range of the data, which can better reflect the continuous distribution and changes of SDD in time and space.Another important significance of data fusion is to improve the credibility of the extracted optical parameter data.The fused data protects the temporal continuity of the data while optimizing the spatial effective range of the data.The data fusion of multiple satellite sensors can use the attributes of different sensors such as spectrum, space, time, and ground coverage characteristics.The application of data from multiple sensors can also achieve the advantages of centralizing various sensors.By increasing the sampling rate at each point, the credibility of extracting optical parameters is enhanced.It can be seen that in order to meet the long-term continuous observation of the distribution and changes of SDD, it is necessary to fuse the data of multiple satellite sensors [3,4].
No matter which algorithm is used, due to the difference of orbit operation and observation parameters of different remote sensing loads, SDD products are affected by cloud cover and meteorological and climatic environment, and there is no unified verification and evaluation standard, which limits the promotion and application of SDD products [5,6].At the same time, data fusion technology has been widely recognized by many disciplines, and has achieved great development in recent years.The integration technology and development of pixel level, feature level and decision level are also hot research directions in the world.Therefore, based on variational optimization algorithm, this paper constructs a multi-load SDD product fusion algorithm, and evaluates and validates it with ship survey data.

Retrieval Model
D Suppose that the observed geometry is shown in Figure 1.The total radiance received by the remote sensor is Here Lr is the direct solar reflection from water surface，tLwc is the contribution of surface foam reflection, Lr is the contribution of atmospheric molecular scattering (Rayleigh scattering), La+Lra is the contribution of aerosol and Rayley-aerosol scattering, there is ( ) Where the subscript x is r、a、ra、wc、w. 0 is the zenith Angle of the sun, 0 F is the vertical solar irradiance at the top of the atmosphere.Therefore, In general, remote sensing loads with side-view scanning ability, such as SeaWiFS, CZCS, etc. (COCTS hasn't the function), can be ignored.Therefore, the above equation can be simplified as： After atmospheric correction, According to the theory proposed by Tyler et al. [7] , the inversion formula of SDD can be represented by the vertical bright contrast attenuation function as follows value with the least attenuation in water.In this paper, the measured data are all the measurements of the Pearl River Estuary and Bohai Sea from 2010 to 2020, so the algorithm has a high degree of stability.
Taking Terra/MODIS data as an example, L2 level data of March and December 2022 data is used for inversion, and the results are shown in Figure 2 and 3.The figure shows that the sea water transparency in spring is obviously higher than that in winter, which is basically consistent with the change of sea water transparency in the Yellow Sea and Bohai Sea area.

Variational algorithm
From the direct inversion of SDD, multi-source satellite products can be obtained, but the fusion of products is a difficult problem in the application of satellite products.Therefore, based on the variational idea [4], the fusion of multi-source remote sensing products is carried out.The idea level of the so-called variational uses the background field and observed data, takes into account various comprehensive attenuation factors, and deduces the idea of the best inversion result.
Suppose that the inversion value of a satellite product is   , the background field is   , and the final fusion result is, i=1……N, stands for multiple channels of a satellite payload.Each bold and bold display variable is a matrix.Then there is where B is the background covariance matrix, R is the observation error covariance matrix, and MIN is the fusion accuracy control.In this study, the minimum value is limited to less than 2%.Hence there Further simplify to Then retranspose to So the fusion value is Thus, satellite multi-channel fusion can be carried out.

Technical framework
Footnotes should be avoided whenever possible.If required they should be used only for brief notes that do not fit conveniently into the text.On this basis, the fusion value of each satellite can be taken as the observed value, and the fusion process can be continued to obtain the multi-channel fusion of multi-source satellites.The variational fusion process includes four steps: (1) Data preprocessing Because the parameters of different satellites are different, the storage format and variable pages are different, so the data should be preprocessed first.The secondary data of multi-source satellites used in this paper include the Level 2B data of TERRA/MODIS, AQUA/MODIS and SNPP/VIIRS.The field observation data, including diffuse reflection coefficient and reflectance, were provided by the School of Oceanography, Sun Yat-sen University.Other auxiliary data to LAADS DAAC data download and https://www.earthdata.nasa.gov/reanalysis data.
(2) SDD Retrieval Single-channel, double-channel and multi-channel inversion algorithms are used respectively.This is a prerequisite for fusion.
(3) Calculation of observation covariance After inversion, the transparency inversion results were obtained and averaged daily.The average results of 10 days were output and the observed values of the day were fused to calculate the observation covariance.
(4) Background field calculation Calculate the average of the month before the date of fusion; Monthly average and reanalysis product average are used as background fields.
(5) Background covariance The background covariance matrix is calculated from the background field and the reanalysis product.
(6) Quality control conditions The calculation time should be controlled within three days, and the calculation grid should be reset after three days.If the fusion result is greater than 10km×10km, the fusion is terminated, and the average number of days is increased to recalculate the background field.
The above process is a single-satellite multi-channel fusion process.If multi-satellite fusion is carried out, it is carried out on the basis of this fusion.That is, single satellite fusion is used as input observation data.
The final output is the transparency result after fusion.

Calculation result
Taking Terra/MODIS as an example, this paper uses the idea of variational to realize the integration of transparency in the East China Sea, as shown in Figure 4. Figure 4 (1) is the calculated background field, (2) is the observed data, and (3) is the final fusion result.The spatial resolution of the fusion data is 4km. (

Quality assessment
The satellite data used in this paper are Terra/MODIS, AQUA/MODIS and SNPP.The field observation data used are 15 aerial survey data provided by Sun Yat-sen University, as shown in Figure 5 (1); the position data of aerial survey station are shown in Figure 5 (2); the comparison between fusion data and the observation data is shown in Figure 5 (3).In Figure (3), the horizontal axis is the field measurement value of SDD, and the vertical axis is the satellite fusion data value (1:1).By defining the time-space matching window, comparing the field observation data with the satellite observation data, and comparing the matching point data, calculating the root mean square (RMS), Average error (Bias), Average Ration, correlation coefficient square (R2) and other statistical parameters, and quantitatively evaluating the fusion data through statistical parameters.

Conclusions Calculation result Quality assessment
In recent years, with the rapid development of remote sensing technology, there are more and more remote sensing products for SDD.However, SDD products are greatly affected by load parameters and environment, so how to improve the application efficiency of SDD products is a top priority.Therefore, this paper constructs an improved variational optimization algorithm to achieve multi-load SDD product fusion and verifies them with ship survey data.The correlation is better than 0.9, proving that this method can improve the accuracy of SDD inversion products.The method in this article not
t  ,and the normalized water radiance wn L or remote sensing reflectance of the water surface rs R , and then the weighted average concentrations of chlorophyll (Chl), suspended matter (SPM) and yellow matter (CDOM) of the water surface can be retrieved from wn L and rs R .In the equation, other factor refers to the factors affecting SD other than water color elements (Chl,SPM,CDOM), mainly including sea state, surface illumination, and subjective factors of observers.Therefore, SD is linked with the information of water color elements t  , and theoretically, SD can be obtained by t  inversion.
the apparent and inherent contrast of the target object, and c are the beam attenuation coefficient.For completely diffused light, , the theoretical visible depth of the target object is obtained.As long as the value of the sum a and b b is obtained, the value d Z can be obtained.The actual value d Z is a function of the wavelength  , because the sum is a and d Z a function of both  , and SDD is the corresponding
) the calculated background field (2) the observed data(3) the final fusion result

( 1 )
The 15 survey observation data (2) the position of survey station(3) the comparison between fusion data and observation.