Improving the spaceborne GNSS-R altimetric precision based on the novel multilayer feedforward neural network weighted joint prediction model

Yiwen Zhang , Wei Zheng , Zongqiang Liu

a Qian Xuesen Laboratory of Space Technology, China Academy of Space Technology, Beijing 100094, China

b China Academy of Aerospace Science and Innovation, Beijing 100176, China

c State Key Laboratory of Space-Ground Integrated Information Technology, Beijing Institute of Satellite Information Engineering, Beijing 100194, China

Keywords:GNSS-R satellite constellations Sea surface altimetric precision Underwater navigation Multilayer feedforward neural network

ABSTRACT Global navigation satellite system-reflection (GNSS-R) sea surface altimetry based on satellite constellation platforms has become a new research direction and inevitable trend, which can meet the altimetric precision at the global scale required for underwater navigation.At present, there are still research gaps for GNSS-R altimetry under this mode, and its altimetric capability cannot be specifically assessed.Therefore,GNSS-R satellite constellations that meet the global altimetry needs to be designed.Meanwhile, the matching precision prediction model needs to be established to quantitatively predict the GNSS-R constellation altimetric capability.Firstly, the GNSS-R constellations altimetric precision under different configuration parameters is calculated, and the mechanism of the influence of orbital altitude,orbital inclination,number of satellites and simulation period on the precision is analyzed,and a new multilayer feedforward neural network weighted joint prediction model is established.Secondly,the fit of the prediction model is verified and the performance capability of the model is tested by calculating the R2 value of the model as 0.9972 and the root mean square error(RMSE)as 0.0022,which indicates that the prediction capability of the model is excellent.Finally, using the novel multilayer feedforward neural network weighted joint prediction model, and considering the research results and realistic costs, it is proposed that when the constellation is set to an orbital altitude of 500 km, orbital inclination of 75° and the number of satellites is 6,the altimetry precision can reach 0.0732 m within one year simulation period, which can meet the requirements of underwater navigation precision, and thus can provide a reference basis for subsequent research on spaceborne GNSS-R sea surface altimetry.

1.Introduction

Sea surface altimetry not only plays an important role in natural environment and ecological economy, but also is one of the most direct means to obtain information on ocean dynamics and marine gravity field[1,2],which is crucial for establishing a high-precision global gravity field model in combined underwater inertial/gravity navigation systems [3-5].The existing conventional sea surface altimetry means mainly include tide gauge altimetry and satellite radar altimetry, but the tide gauge altimetry cannot meet the demand of high precision altimetry at global scale, and these two means are costly and have low spatial and temporal resolution.

In recent years, with the gradual improvement of the construction of the GNSS system,the use of global GNSS-R technology for ocean remote sensing and sea surface altimetry has become a new research concept since Martin-Neira proposed the concept of Passive Reflectometry and Interferometry System (PARIS) in 1993[6].The current sea surface altimetry methods using GNSS-R technology mainly include: code phase estimation [7-11], carrier phase estimation [8,12-14], and signal-to-noise-ratio (SNR) estimation [15,16], and currently experiments are mainly conducted using shore-borne[17]and airborne platforms[10,18].Code phase estimation model is simple and easy to implement, so it is widely used; carrier phase estimation requires coherent signal, but in the case of high satellite altitude, the reflected signal from the sea surface is usually an incoherent scattered signal; single antenna signal receivers are commonly used for SNR estimation, which are widely used and effectively reduce the cost, and the inversion precision is usually at the decimeter level.In terms of shore-borne observations, in 1995, Anderson et al.[19] presented the first GPS satellite signal interferometric results, and after comparing them with the tide gauge, they found that the measurement accuracy was about 12 cm;Johan et al.[20]proposed a tide gauge based on GNSS signals with inversion accuracy up to 4 cm using the SNR estimation.In terms of airborne observations,Ruffini et al.[18]used a low-altitude aircraft platform to collect GPS reflection signals under eddy sea conditions with a wind speed of 10 m/s and an effective wave height of 2 m.The experimental data were processed with code phase tracking, and the inverse sea surface height accuracy reached the decimeter level with a spatial resolution of 20 km; Carreno-Luengo et al.[21] demonstrated through airborne test that the sea surface height(SSH)measurement using GPS P(Y)code will increase the altimetry precision by 1.4-2.4 times compared with GPS C/A code.

The development of the above sea surface altimetry methods and the promotion of experiments on shore-borne and airborne platforms have fully verified the feasibility of using GNSS-R technology for SSH measurement,and the development of shore-borne and airborne platforms will also promote the research and application of GNSS-R spaceborne platforms.With the successful launches of the UK TechDemoSat-1(TDS-1)satellite[22,23],the US CYGNSS constellation [24,25], and the Chinese BF-1 A/B satellites[26], GNSS-R sea surface altimetry using the spaceborne platform has gradually entered the research vision.For single-satellite GNSSR sea surface altimetry, on the international side, in 2002, Lowe et al.[27] published the SNR and corresponding high accuracy measurements obtained using GPS reflection signals in the Pacific Ocean using different spaceborne platforms (SAC-C and CHAMP)for the first time;in 2016,Clarizia et al.[28]used GPS-R data from TDS-1 for inversion and preliminary studies of SSH in the South Atlantic and North Pacific, with the aid of Danmarks Tekniske Universitet 10 (DTU10) global mean sea surface model for validation; in 2018, Mashburn et al.[29] explored the performance of TDS-1 data in sea surface altimetry inversion,analyzed the sources of inversion errors caused by transmitter and receiver orbits,ionospheric and tropospheric delay models, antenna baseline offsets and so on,then compared the experimental results with DTU10 mean sea surface model.On the domestic side, in 2016, Liu et al.[30]systematically analyzed the GNSS reflection signals received by the UK-DMC satellite receiver and verified the feasibility of using GNSS-R signals for sea surface height inversion by processing the reflection signals from three different reflective surfaces: sea surface, land surface, and ice surface; in 2020, Zhang et al.[31]designed a sea surface altimetry error inversion model using the spaceborne GPS-R measured delay-doppler map (DDM) data of TDS-1 to improve the precision of the spaceborne sea surface height inversion.

The sea surface altimetry research on GNSS-R constellation is currently focused on the CYGNSS constellation,which consists of 8 low earth orbit (LEO) satellites with an orbital altitude of 500 km and an orbital inclination of 35°, with the predicted average searching time of up to 6 h.It can provide all-weather, gap-free coverage of the global ocean between 35°north and south latitudes[32].In 2019, Li et al.[33] processed CYGNSS raw data to generate reflected waveforms of different GNSS band signals,performed SSH inversion by applying the delay correction model, and compared the inversion results with the DTU18 global mean sea surface model; in 2020, Jake et al.[34] used DDM data from CYGNSS to perform sea surface height inversion in the seas of Indonesia and proposed priority design factors for future GNSS-R altimetry missions;Alex et al.[35]improved the orbit error of CYGNSS satellites and combined with the new ionospheric delay model to reduce the inversion error of sea surface height.These results provide effective support for the development of GNSS-R missions devoted to sea surface altimetry applications in the future.

The current spaceborne GNSS-R sea surface altimetry mode is mainly limited to the following aspects:(i)A single satellite cannot achieve uninterrupted detection in a specific range;(ii)The use of a constellation of multiple satellites can improve the GNSS-R global average sea surface altimetric precision, but there is a lack of existing simulation research results to make a specific assessment of the precision in this mode.Considering the above problems,the spaceborne GNSS-R altimetry platform based on constellations can take the following observational advantages[36,37]:(i)Realize the continuous all-weather detection in a specific region.Compared with a single star, the joint collaboration among satellites using constellation can help to achieve global or certain regional observation and data collection, and meet the precision of 5-8 cm sea surface altimetry required for tasks such as underwater navigation;(ii) With a large number of signal sources.Navigation satellites in orbit or planned provide abundant free public signals, which is conducive to the realization of a large range of high spatial resolution, short return period data acquisition and surface parameter inversion; (iii) Severe weather such as rainfall and fog have less impact on the L-band signal,which is conducive to long continuous all-weather observations;(iv)The heterogenous observation mode is adopted without the need of transmitter,so the complexity of the volume, cost and quality of the observation device are reduced.

It can be seen that the high-precision GNSS-R altimetry constellation networking is the inevitable trend of future GNSS-R sea surface altimetry, and the establishment of a prediction model of sea surface altimetry precision based on GNSS-R satellite constellations can fill the research gap in this mode and specifically assess its altimetry capability.With the rapid development of machine learning[38,39],the artificial neural network technology has received renewed attention.This technology has high nonlinear mapping capability, so combining the prediction model of spaceborne GNSS-R sea surface altimetric precision with artificial neural network can certainly improve the preciseness and practicality of the prediction model.Some domestic and foreign scholars have already used this technique jointly with GNSS-R sea surface altimetry, for example, in 2021, Wang et al.[40] analyzed the airborne delay waveform dataset of the Baltic Sea and constructed a new machine learning weighted average fusion feature extraction method for SSH inversion,with an RMSE of about 0.23 m compared with DTU15.This is the first research result of airborne GNSS-R SSH inversion based on machine learning method,which also provides experience and paves the way for the research of spaceborne altimetry based on neural network; in the same year, Zhang et al.[41] proposed two models of spaceborne SSH inversion based on TDS-1 data,Principal Component Analysis combined with Support Vector Regression (PCA-SVR) and Convolutional Neural Networks(CNN), and the MAE of inversion results of both models are below 1 m.

Compared with the previous research results, this paper proposes a spaceborne GNSS-R sea surface altimetric precision index,aims to build a reasonable observation constellation configuration and improve the spaceborne GNSS-R sea surface altimetric precision, and reveals the influence mechanism of constellation configuration parameters on the altimetric precision;by combining multilayer feedforward neural network and adaptive momentum(ADAM) optimization algorithm, a novel multilayer feedforward neural network weighted joint prediction model is built, and an optimized constellation configuration scheme is proposed to meet the required altimetric precision for underwater navigation.

2.Data sets

2.1.GNSS precision ephemeris

Based on the precise ephemeris provided by the Wuhan University analysis centers(WHU)of multi-GNSS experiment (MGEX)[42], the 3D coordinates of GPS, GLONASS, GALILEO and BDS satellites are acquired respectively.The IGS precision ephemeris is in SP3 format,which mainly includes satellite position,satellite clock information, satellite velocity and so on.Since the SP3 precision ephemeris provides the satellite position every 15 min [43], in order to unify with the sampling time of LEO satellites we set, we need to fit the satellite orbit coordinates by the Legendre polynomial algorithm [44], thus obtaining the high-precision positioning of GNSS satellite.

2.2.Satellite simulation conditions

This paper draws on the TDS-1 satellite,which was successfully launched into orbit by Surrey Satellite Technology Ltd (SSTL), UK,on July 8, 2014, for providing in-orbit technology validation services.The satellite is in a sun-synchronous orbit at the altitude of 635 km and is primarily carrying a new generation of on-board GNSS remote sensing instrument receivers (SGR-ReSI) that can track, record and process four surface reflections from GPS L1, L2C and other navigation satellites [23].Simulation analysis is performed based on the satellite orbit parameters and antenna observation patterns.We use the control variable method to keep the other configuration parameters constant when analyzing the effect of a constellation configuration parameter on the spaceborne GNSS-R sea surface altimetric precision.

3.Construction of the novel multilayer feedforward neural network weighted joint prediction model

The rapid fluctuation of the ocean surface has an impact on the precision of sea surface altimetry, and its uncertainty is mainly reflected in the uncertainty of the waveform.In the process of inversion of sea surface height information by sea surface reflection signals,the broadening and deformation of the power waveform of scattered signal generated by the rough sea surface are the main factors affecting the precision of the altimetry.Moreover, when GNSS signal scattering occurs on the rough sea surface,the receiver will receive reflected signals from several different directions, and the reflection area here is the glistening zone.The glistening zone near the specular point has multiplicative noise called speckle noise, which is also a key factor affecting the precision of the altimetry.Therefore, by improving the SNR of the single specular point,the precision of spaceborne GNSS-R sea surface altimetry can be improved.

In addition,as a spaceborne platform for GNSS-R receivers,it is difficult for a single satellite to achieve uninterrupted detection of a specific range, and its coverage area always changes with flight time, and such changes are strictly influenced by orbital parameters, such as orbital altitude, orbital inclination, and other factors.Therefore, it is usually difficult to achieve global or certain area observation and data collection with only one satellite.The satellites in the constellation are deployed in space and form a relatively stable spatial configuration, and there is also a relatively stable spatial and temporal relationship between satellites.Through the interoperability of satellites, the satellite constellation can greatly improve the ground coverage and the efficiency of communication and navigation on the basis of a single satellite.Therefore, by revealing the influence mechanism of the simulation period and constellation configuration parameters such as orbital inclination,orbital altitude,and number of satellites on the spaceborne GNSS-R altimetric precision, it is possible to construct a satellite constellation configuration scheme based on the typical Walker constellation to meet the altimetry requirements, and thus predict the different altimetric precision that can be obtained under different conditions.

3.1.Sea surface altimetry geometrical model

For the antenna observation mode, the spaceborne GNSS-R payload typically includes left-hand circularly polarized antenna(LHCP), right-hand circularly polarized antenna (RHCP), and specialized GNSS-R receiver [45].The LHCP antennae are used to receive reflected signals,so the beamwidth and face direction of the LHCP antennae determine the antenna coverage.The antennae beamwidth for TDS-1 at GPS L1 is 34°×35°, while the nadir offpointing is 6°in-X axis.In this study,we also use the same antenna observation mode as the TDS-1 satellite.

In this paper,we need to define an antenna coverage range and select the appropriate GNSS signal for each LEO satellite.Fig.1 shows the schematic diagram of the GNSS-R antenna coverage range,where point P and P′are both specular reflection points,and point O is the center of the Earth;β is the elevation beamwidth of the antenna, φ is the geocentric angle between the GNSS-R LEO satellite and the specular reflection point,θ is the geocentric angle between the GNSS-R LEO satellite and the GNSS satellite,αminis the minimum elevation angle of the antenna,h is the orbital altitude of the GNSS-R LEO satellite, H is the orbital altitude of the GNSS satellite,and R is the mean radius of the Earth.The area between P and P′′refers to the antenna reflection ground coverage area(see Fig.2).

Fig.1.Schematic diagram of GNSS-R antenna coverage.

Fig.2.Structure diagram of three layers feedforward neural network.

Using the geometric model combined with the trigonometric relationship, we can now obtain [46].

it can be seen that the magnitude of the angle θ between GNSS satellites and LEO satellites depends mainly on H and α, that is, θ varies with the altitudes of different GNSS satellites.Table 1 shows the different GNSS satellite orbital altitudes(H)and the geocentric angles between GNSS satellites and LEO satellites (θ), and the available GNSS signals are selected by the range of geocentric angles.

3.2.Calculation of specular reflection points

The current specular reflection point algorithms mainly include S∙C∙Wu algorithm [47], Gleason algorithm [48], and dichotomy of the line-segment algorithm [49].In this paper, the dichotomy algorithm [50] is chosen to calculate the specular reflection point.

The difference between the dichotomous algorithm and the S∙C∙Wu algorithm is mainly in the way of searching for M points.The basic idea of the dichotomous algorithm is similar to the idea of using the dichotomous method to solve the problem by continuously dichotomizing the line segment RT to find the M points that satisfy the conditions.The dichotomous algorithm is simple andstraightforward, with fewer iterations, and as the search accuracy increases, the advantage of fast convergence of the dichotomous algorithm is more obvious compared to the S∙C∙Wu and Gleason algorithms.

Table 1 Geocentric angles between the LEO and the different GNSS satellites.

The dichotomous algorithm is used to iteratively solve the angle between the receiver and the specular reflection point and the angle between the GNSS satellite and the specular reflection point that satisfy the Fresnel condition,so that the results obtained from the iteration can be substituted into the GNSS-R geometric relationship to calculate the specific specular reflection point.

3.3.Calculation of reflected signal correlation power

Due to the presence of noise, the scattered signal correlation power model usually refers to the first-order statistical average as a function of two variables, time delay τ and Doppler frequency fc.Here we choose the Z-V model to simulate the reflected signal correlation power waveform at the receiver side,and we can obtain the theoretical expression of the scattered signal power at any moment [51]:

where PTrepresents the power of the GNSS transmitter, λ represents the electromagnetic wavelength of the GNSS signal, Tirepresents the integration time,GT(ρ)and GR(ρ)represent the antenna gain of the GNSS transmitter and the satellite-based receiver,respectively, RTPand RPRrepresent the geometric distances from the navigation satellite and LEO satellite to the specular reflection point, respectively, Δτ(ρ) and Δf(ρ) represents the time delay difference and the Doppler frequency difference between local replica signal and arrival signal, and σ0(ρ) is the normalized bistatic scattering coefficient, Λ representing the Woodward ambiguity function (WAF) [52].σ0(ρ) can be expressed as [53]

where PPDF(•)is the probability density function of the sea surface slope.From Eqs.(2)and(3),it can be seen that the contribution of the scattered signal mainly comes from the intersection of four spatial regions: the antenna coverage region determined by GT(ρ)and GR(ρ), the iso-range zone determined by the Λ2function characteristic, the iso-Doppler zone determined by, and the glistening zone determined by the PPDFrelated to the sea surface roughness.

3.4.Calculation of spaceborne GNSS-R sea surface altimetric precision

The precision of sea surface altimetry σsshat the single specular reflection point is calculated as [52]

where PZ(0)and PZ(0)′are the amplitude of the average power and the slope of the power waveform at the specular reflection point,c is the speed of light in vacuum, cPZ(0)/PZ(0)′is defined as the altimetry sensitivity,which indicates the variation of the altimetry results with the magnitude of the reflected signal correlation power and the ratio of its change rate.From the altimetric model,it can be seen that the σsshis related to the SNR of signal correlation power,the number of incoherent accumulations (Nincoh), the elevation angle of the specular reflection point (εele), and the altimetry sensitivity.When only thermal noise is considered,the SNR can be expressed as [54,55]

where k is Boltzmann constant, T is the equivalent temperature of the receiver, B is the signal bandwidth of the receiver.

After the calculation of the altimetric precision of single specular reflection point, the earth is divided into 0.2°×0.2°(20 km×20 km) grids, and all the calculated specular reflection points are cast into the divided grids according to their latitude and longitude coordinates.After the division is completed,the number of points in each grid and the single point altimetric precision are counted to obtain the mean precision of each grid.

3.5.Novel multilayer feedforward neural network weighted joint prediction model

Currently, the feasibility of spaceborne GNSS-R sea surface altimetry based on the constellation platform has been verified,but a specific assessment of its altimetric capability is lacking.Moreover,due to the large amount of simulation work,if large-scale and small-step simulation is conducted under a long simulation period, the time consumed will be measured in years.Therefore,this paper chooses to establish a prediction model of altimetric precision using multilayer feedforward neural networks to facilitate the subsequent quantitative evaluation of the satellite constellation's altimetric capability, so as to select the best constellation configuration and simulation period that can meet the current altimetric precision requirements.

A multilayer feedforward neural network consists of an input layer, a hidden layer, and an output layer, which obtains the expected output result by modifying the weight and threshold between the neural network nodes.The multilayer feedforward neural network learns by transmitting the information layer by layer in a forward manner and by returning the error between the output result and the expected result layer by layer in a reverse manner, and then adjusting the connection weights between the neuron nodes.Through such iterative cyclic processing, the multilayer feedforward neural network is trained,and finally the optimal multilayer feedforward neural network model can be obtained.

The number of neurons in the input layer of the multilayer feedforward neural network is determined by the number of input data variables.In this paper, the configuration parameters of the constellation (orbital altitude, orbital inclination and number of satellites) and the simulation period are used as the input nodes;the final calculated spaceborne GNSS-R sea surface altimetric precision is used as the output node.

For the selection of the training function, we choose the LM(Levenberg Marquardt)algorithm considering the disadvantages of slow convergence and the tendency to fall into local minima of the neural network.The LM algorithm is a combination of the gradient descent method and the Gauss-Newton method, which combines the local convergence of the Gauss-Newton method and the global features of the gradient method, and has the advantages of fast convergence and high stability in the case of relatively few network parameters, thus reducing the step of network iterations [56]; For the selection of transfer function, the nonlinear transfer function"tansig" is adopted for the input layer and hidden layer of the network in this paper,and the linear function"purelin"is adopted for the output layer to maintain the range of output.

To address the shortcomings of the gradient descent method,the ADAM optimized algorithm is introduced in this paper.This algorithm combines the advantages of GDM and RMSprop optimizer, which can dynamically adjust the adaptive learning rate of different parameters and update the multilayer feedforward neural network weights and thresholds.ADAM optimized algorithm modifies the first-order moment deviation and second-order moment deviation by calculating the first-order moment estimation mtand second-order moment estimation vtof gradient g,and finally modifies the weight and threshold of the network.Among them, mtand vtare calculated as follows [57]:

where β1and β2represent the exponential attenuation rate;dk and dk2represent the gradient value and the square of the gradient value of the neural network weights or thresholds,respectively.The final updated weights and thresholds obtained can be expressed as

where wtand btare the updated neural network weights and thresholds;α is the learning rate, and δ is the smoothing term.

To evaluate the performance of the final established prediction model, the holistic evaluation index R2, which is the coefficient of determination, is introduced to evaluate the generalization ability of the network model.Its definition equation is as follows [58]:

where ^yi(i=1,2,…,n) is the predicted data of the ith sample;yi(i=1,2,…,n)is the actual data of the ith sample;n is the number of samples.When R2is closer to 1,it indicates that the performance of the novel multilayer feedforward neural network weighted joint prediction model is better.

Table 2 Related satellite parameters.

4.Results and discussions

For the satellite constellation, when discussing the time scale larger than one orbit repetition period, the right ascension of ascending node, argument of perigee and true anomaly have almost no effect on the final results.Therefore, we select the satellite orbital altitude, orbital inclination and the number of satellites as the constellation configuration parameters, and consider the simulation period to discuss the influence on the spaceborne GNSS-R sea surface altimetric precision.After the influence mechanism of each parameter is clarified,a multilayer feedforward neural network is used to establish a prediction model of altimetric precision based on satellite constellation.

In this paper,the analysis and discussion are based on the TDS-1 satellite orbit and antenna parameters, and the specific parameter settings used are shown in Table 2.

The sampling time of each satellite is 1 s(see Fig.3).The GPS L1 C/A, GLONASS L1OC, GALILEO E1A and BDS-3 B1I signals are used for a month-long simulation analysis.According to the calculation Eq.(4) of the single-point sea surface altimetric precision, the results of the analysis of different GNSS satellite signals are shown in Fig.4, that is, the altimetric precision of different GNSS satellite signals is different,and the precision increases with the increase of the elevation angle of the specular reflection point.

4.1.Altimetric precision results for different orbital altitude

Fig.3.Flowchart for improving the precision of GNSS-R sea surface altimetry based on the novel multilayer feedforward neural network weighted joint prediction model.

Fig.4.The precision of single point sea surface altimetric precision with different GNSS signal sources varies with elevation angle.

When setting the discussion range of orbital altitude, the main considerations are the purpose of satellite application,the effect of space debris environment and the effect of link loss.When the satellite constellation is deployed at 800 km,1100 km and 1400 km,the space debris generated by their collisions will stay in the space debris environment for a long time and cause a more serious effect on the environment, while the satellite orbital altitude of 500 km has the least effect on the space debris environment [59]; at the same time, the long distance between satellites or between satellites and the earth leads to a large delay in information transmission and a growth in link loss.Therefore, when setting the orbital altitude of the constellation,we should take into account the altimetric precision requirements and the signal transmission loss.In this paper, the orbital altitude is discussed in the range of 400-800 km, and the step size is set to 50 km, the influence mechanism of the orbital altitude on the spaceborne GNSS-R sea surface altimetric precision of the single satellite is discussed when the orbital inclination is 75°.

The results of the calculation of the spaceborne GNSS-R sea surface altimetric precision of the single satellite at different orbital altitudes are shown in Table 3; the variations of the altimetric precision of the single satellite at different orbital altitudes in the latitude and longitude directions are shown in Figs.5 and 6,respectively.

As can be seen from the results in Figs.5 and 6:

In terms of latitude distribution, the spaceborne GNSS-R sea surface altimetric precision have relatively higher precision in the middle and low latitude areas, and lower precision in the high latitude areas.The range of the latitude distribution is related to the orbital inclination of the LEO satellite, and the peak part of the latitude distribution represents a surge in the number of specular reflection points in this region but all of them are at low elevation angles.

In terms of longitude distribution, the spaceborne GNSS-R sea surface altimetric precision varies less with longitude and is more evenly distributed in the longitude range, but there is a slight decrease in precision at about 100°E.The reason for this phenomenon is the presence of GEO and IGSO satellites of BeiDou-3 Navigation Satellite System.The longitude of the intersection of the three IGSO satellites is 118°E,and the phase difference between the three satellites is 120°; the three GEO satellites are fixed at 80°E,110.5°E, and 140°E.The presence of GEO and IGSO satellites enhances the coverage of the BeiDou Navigation Satellite System in East Asia, and therefore the distribution of specular reflection points in this region is relatively dense.

From the results of Table 3 combined with Figs.4-6, it can be seen that,it can be seen that there is an effect of orbital altitude on the spaceborne GNSS-R sea surface altimetric precision, but the effect is small.From the GNSS-R spatial geometry relationship, it can be seen that, under the current orbital altitude range, the higher the satellite orbital altitude is, the more specular reflection points with larger elevation angle can be generated, and the corresponding precision is higher.However, due to the long transmission distance of GNSS-R signal from transmitter to receiver,theeffects of ionosphere and troposphere also need to be taken into account.Therefore, based on the calculation results, the designed constellation orbit altitude of 500 km is finally chosen,considering the signal transmission path loss,space debris environment and the influence of the atmosphere on the satellite lifetime.

Table 3 Spaceborne GNSS-R sea surface altimetric precision of single satellite at different orbital altitudes.

Fig.5.Spaceborne GNSS-R sea surface altimetric precision as a function of latitude at different orbital altitudes.

Fig.6.Spaceborne GNSS-R sea surface altimetric precision as a function of longitude at different orbital altitudes.

4.2.Altimetric precision results for different orbital inclination

The discussion range of orbital inclination is mainly considered when setting the latitude coverage of the detection target.Finally,we set the discussion range of orbital inclination to 70°-95°with a step size of 5°.Based on the results of the above discussion of the orbital altitude,we discuss the influence mechanism of the orbital inclination on the spaceborne GNSS-R sea surface altimetric precision under the condition of only the single satellite at an orbital altitude of 500 km.

The results of the calculation of the spaceborne GNSS-R sea surface altimetric precision of the single satellite with different orbital inclinations are shown in Table 4; the variations of the altimetric precision of the single satellite with different orbital inclinations in the latitude and longitude directions are shown in Figs.7 and 8, respectively.

Combining the results in Table 4, Figs.7 and 8, it can be seen that:

Under different orbital inclinations, the distribution characteristics of the spaceborne GNSS-R sea surface altimetric precision ofsingle satellite in latitude and longitude directions are the same as above, while the larger the orbital inclination of LEO satellite, the wider the latitude range of its specular reflection point distribution.When the orbital inclination is 70°-90°, the altimetric precision decreases as the satellite orbital inclination increases,but the range of altimetry is wider; when the orbital inclination is 90°-95°, the altimetric precision increases as the satellite orbital inclination increases, but the range of altimetry shrinks.This result is mainly due to the fact that the coverage of global specular reflection points increases as the orbital inclination of the satellite increases, which leads to the decrease of the altimetric precision.

Table 4 Spaceborne GNSS-R sea surface altimetric precision of single satellite at different orbital inclinations.

Fig.7.Spaceborne GNSS-R sea surface altimetric precision as a function of latitude at different orbital inclinations.

Fig.8.Spaceborne GNSS-R sea surface altimetric precision as a function of longitude at different orbital inclinations.

Table 5 Spaceborne GNSS-R sea surface altimetric precision with different numbers of satellites.

Considering that the global ocean distribution is mainly located between 75°north and south latitudes, in polar regions and high latitude marine areas,the distribution of sea ice is more extensive.Therefore, combining the influence of the satellite inclination on the sea surface altimetric precision, the latitude range of the altimetry under different satellite inclination conditions and the global ocean distribution, we finally choose the design of the constellation orbit inclination of 75°.

4.3.Altimetric precision results for different orbital inclination

The discussion of the number of satellites in the constellation is set in the range of 2, 4 and 8 satellites.In combination with the above discussion,the satellite orbital altitude is set to 500 km and the orbital inclination is set to 75°.The spaceborne GNSS-R sea surface altimetric precision of three Walker constellation configurations (δ constellation, rose constellation and star constellation)with different numbers of satellites are calculated separately, and we have chosen a configuration with two satellites per plane and the phase factor of 0.It can be seen from Table 5 that different Walker constellation configurations have little effect on the altimetric precision,and the larger the number of satellites,the higher the altimetric precision.

4.4.Altimetric precision results for different simulation periods

According to the results in Table 5,when the simulation period is one month, the altimetric precision only reaches the decimeter level, which still cannot meet the demand of the precision of underwater navigation.Therefore,we extend the simulation period to 1 year and 3 years respectively, and only calculate the spaceborne GNSS-R sea surface altimetric precision for different simulation periods of δ constellation configuration, considering the small effect of Walker constellation configuration on the precision,and the results are shown in Table 6.It can be seen that the longer the simulation period, the higher the corresponding altimetric precision.

4.5.Validation and application of the novel multilayer feedforward neural network weighted joint prediction model

From the results in Table 6,it can be seen that when the number of satellites of the constellation is 8 and the simulation period is 1 year, it already meets the required precision of 5-8 cm for underwater navigation; therefore, we presume that the required precision can also be achieved when the number of satellites of the constellation is between 4 and 8,but due to the lack of simulation data, the quantitative prediction of the altimetric capability of the constellation cannot be performed.Therefore,we can train the four kinds of factors that influence the altimetric precision discussed above by using the novel multilayer feedforward neural network weighted joint prediction model, so that we can come up with an optimal constellation design that meets the altimetry demand.The sampling interval of different altimetric precision influencing factors will have an impact on the accuracy of the prediction model,the smaller the sampling interval is,the more the training samples are, the better its corresponding model accuracy is.However,considering that a smaller sampling interval will lead to a long presimulation period, the orbital altitudes selected here are 400-800 km with a step of 50 km; the orbital inclinations are 70°-97.5°with a step of 5°;the numbers of satellites are 1,2,4,and 8;and the simulation periods are 1 year and 3 years.Eighty percent of the data are randomly selected as training samples.

Fig.9 shows the comparison between the output of the spaceborne GNSS-R sea surface altimetric precision by the novel multilayer feedforward neural network weighted joint prediction model and the actual simulation precision results, and Table 7 shows the statistical indicators of the training error.It can be seen that in the test sample, the model prediction and the actual simulation are relatively close, and the coefficient of determination R2is 0.9972,indicating that the prediction model is precise in predicting the altimetric precision.Meanwhile,the results of mean absolute error(MAE), mean absolute percentage error (MAPE) and root mean square error (RMSE) are small, which also indicate that the multilayer feedforward neural network prediction model performs well.

The neural network training process is shown in Fig.10.It can be seen that the mean square error (MSE) gradually decreases and tends to the minimum value as the number of iterations increases,and the validation set has the lowest MSE when the number ofgenerations is 17, which means that the model works best at this time.Combined with the evaluation indexes discussed above, the novel multilayer feedforward neural network weighted joint prediction model can better evaluate the constellation altimetry capability for a given orbital altitude,orbital inclination,number of satellites and simulation period,so that the spaceborne GNSS-R sea surface altimetric precision can be quantitatively analyzed.In this case, the model expression is

Table 6 Spaceborne GNSS-R sea surface altimetric precision with different simulation periods.

Fig.9.Comparison of actual values and predicted values of the novel multilayer feedforward neural network weighted joint prediction model.

Table 7 Statistic indicators of the prediction model.

Thus, we can use the model to quantitatively analyze the performance of satellite constellation sea surface altimetry under unknown simulation parameters.According to the results in Table 6,we have analyzed that when the number of satellites is between 4 and 8 and the simulation period is one year, we can meet the requirements of underwater navigation precision,so here we use the model to predict the altimetric precision at different orbital altitudes and different number of satellites, where the results are shown in Fig.11.It is shown that when the number of satellites is 6 and 7, the constellation altimetric precision can meet the requirement of underwater navigation precision at any orbital altitude,and the precision can reach to 0.073 m when the number of satellites is 6; but when the number of satellites is 7, the precision is only improved by about 0.55 cm, which is a small improvement.Therefore, considering the actual cost, we believe that when the orbital altitude is 500 km,the orbital inclination is 75°,the number of satellites is 6, and the simulation period is 1 year, the satellite constellation can meet the centimeter-level precision required for underwater navigation.

5.Conclusions

The use of spaceborne GNSS-R based on constellation can significantly improve the altimetric precision.This observation mode can not only compensate for the high cost and low spatial and temporal resolution of traditional altimetry, but also solve the problem of single satellite with relatively low precision and inability to achieve uninterrupted detection in a specific range,thus meeting the precision required for underwater navigation.

Fig.10.The training process of neural network:(a) The best identification location is pointed out with the green circle;(b) The fitted value is regressed to the true value, and the higher the goodness of fit is, the better the fitting effect is.

(1) Considering the global sea distribution, the purpose of satellite application, the impact of space debris environment and power supply,when the satellite altitude is set at 800 km and the orbital inclination is 75°, the single satellite can achieve the highest precision of sea surface altimetry, and the higher the number of satellites and the longer the simulation period, the better the altimetric capability of the satellite constellation.However, considering the loss of signal on the transmission link and orbit maturity, the satellite orbital altitude should be set at about 500 km; In addition, in practical engineering applications, when the satellite orbit inclination is 97.5°of sun-synchronous orbit inclination, the satellite can operate under the same light conditions every day.While having a stable power supply,the fixed light can also bring a stable temperature environment to the satellite and reduce the design cost of the satellite in solving the temperature environment variation.Moreover, the satellite constellation can be composed of satellites with different orbital inclinations, but since the constellation configuration with the same orbital inclination is commonly chosen in previous studies,this paper also sets the same orbital inclination for different satellites in the constellation for the purpose of simplifying the simulation calculation and shortening the simulation time.In the subsequent study, the inclination should be further refined and discussed based on different mission objectives or hotspot areas of interest.

(2) The R2value of the novel joint multilayer feedforward neural network weighted joint prediction model constructed using multilayer feedforward neural network combined with ADAM optimization algorithm is 0.9972 and the RMSE is 0.0022,which represents its good prediction ability and can be used for subsequent quantitative evaluation of the constellation altimetry performance.

(3) Due to the large scale and longtime consumption of the simulation, we can use the novel multilayer feedforward neural network weighted joint prediction model to predict the precision of the satellite constellation for a given orbital altitude, orbital inclination, number of satellites, and simulation period, so as to give the optimal constellation configuration that meets the current altimetry requirements.According to the prediction results,when the satellite orbital altitude is 500 km,the orbital inclination is 75°,the number of satellites is 6 and the simulation period is 1 year, it is the optimal constellation simulation conditions to meet the demand of altimetric precision of underwater navigation,which can reach 0.0732 m.Furthermore,compared with the traditional simulation calculation, the prediction model solves the problems of time-consuming and complicated calculation methods of traditional simulation, and quickly calculate the sea surface altimetric precision under the current conditions according to different orbit parameters, and fill the research gap of the prediction of spaceborne GNSS-R sea surface altimetry capability based on satellite constellation under the current mode.

Fig.11.Prediction of spaceborne GNSS-R sea surface altimetric precision with orbital altitude for different numbers of satellites using the novel multilayer feedforward neural network weighted joint prediction model.

Funding

This work was supported in part by the National Natural Science Foundation of China under Grant (42274119), in part by the Liaoning Revitalization Talents Program under Grant(XLYC2002082),in part by National Key Research and Development Plan Key Special Projects of Science and Technology Military Civil Integration (2022YFF1400500), and in part by the Key Project of Science and Technology Commission of the Central Military Commission.

Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgements

We would like to thank MGEX for providing SP3 precision ephemeris.