Constructing predictive models for seismic oscillation parameters using covariance functions and Doppler effect phenomena: A case study of InSight mission V2 data

An ability to construct predictive models for identifying seismic oscillation parameters by using the mathematics of covariance functions and Doppler effect phenomena is examined in this work. In the calculations, the Mars seismic oscillations measurement data from InSight Mission V2, observed in the months May, June and July of 2019, was used. To analyze the observation data arrays the Doppler phenomena and the expressions of covariance functions were employed. The seismic oscillations trend's intensity vectors were assessed by least squares method, and the random errors of measurements at the stations were eliminated partially as well. The estimates of the vector's auto-covariance and cross-covariance functions were derived by altering the quantization interval on the general time scale while varying the magnitude of the seismic oscillation vector on the same time scale. To detect the mean values of z —the main parameter of Doppler expression— we developed a formula by involving the derivatives of cross-covariance functions of a single vector and algebraic sum of the relevant vectors.

Introduction.-In this work, the covariations of seismic field strength components and their impact on the covariant variation of field strength are analyzed.The temporal variation of the field strength component covariance is used to determine the correlation dependence between the temporal and spatial variation of the seismic field strength.Using the data from the seismic observations, the equations are developed to compute the estimates of the covariance matrices and covariance functions of the seismic field strength components.The accuracy of each calculated parameter is evaluated as well.
instruments [1][2][3][4][5][6].That enables it to focus on the deep interior of the planet, investigating the processes of terrestrial planet formation and evolution.InSight mission attempts to answer basic questions about the planet: what is thickness of the crust; structure of the mantle; size and density of the core; distribution of seismicity?Apart from the Earth and the Moon, Mars is the only planet on which the seismometer has been operated for an extended period of time -nearly 4 years.A number of efforts was made to investigate the Mars as planet interior structure and processes [7][8][9], to detect the marsquakes [4,[10][11][12] and craters [13][14][15].First of all, InSight has provided a lot of results of surface-based geophysical measurements applying seismology and precision tracking.
After few years of observations it was stated that Mars is seismically active: for example, in 2019 more than 170 significant (M between 3.0 and 4.0) marsquakes were identified, and the total number of registered seismic events is over 1300 [16,17].The first not very significant (M 2.2) marsquake was detected in March, 2019 [4,11], and at last, in May 2022 InSight has detected the largest magnitude seismic event, whose moment magnitude was about M 4.7 [7,18,19].
It was found that crust thickness is 24 to 72 km, most likely about 40 km with a very deep lithosphere near to 500 km [2,5,16,20,21].By analysis of the timing of S waves from 11 low-frequency marsquakes it was detected that core radius is about 1830 km, and its density is about 6 g/cm 3 [7,22], and median seismic velocity at the top of the core is about 4.9 to 5.0 km/s [22].
During the last two decades the modern data processing methods were proposed, which enabled seismologists to detect meaningful signals from passive seismic records of random vibrations generated by natural phenomena like winds and earthquakes.These methods are named as seismic interferometry.The experimental conditions of In-Sight are very specific because only one seismometer is employed.So, the most direct observables are autocorrelation functions.Thus, it is expected that autocorrelation analysis is suitable for Martian subsurface imaging.
In [23] authors used data from the InSight seismometer to estimate the microtremor parameters and detect underground structures that influence the propagation of these microtremors.Firstly, they applied the polarization analysis of the ambient seismic wave field to estimate temporal variations of the ambient noise on Mars.It was noted that local noise peaks correspond to the elastic resonances of the lander excited by the wind.These results demonstrate that the amplitude of ambient noise is strongly associated with the wind strength.Secondly, they did the autocorrelation analysis of the vertical and horizontal motions of the seismometer record to estimate the geological structure at the InSight landing site.
In [24] authors used ambient noise autocorrelation to analyze the available vertical component seismic data to recover the reflectivity beneath the Insight lander.They identified the noise that is approximately periodic with the Martian soil as daily lander operations and the diurnal variation in Martian weather and tides.Also research suggested the composition of Martian crust is composed of basaltic and andesitic rocks.The authors preprocessed the raw data and used different frequency bands to detect discontinuities at different depths.
The autocorrelation analysis of seismic data was performed also in [25].The authors applied the seismic interferometry method to two types of diffuse wavefields -ambient vibrations and the diffuse part of seismic events-and investigated the possibility to retrieve the deep vertical reflection response of the Martian crust beneath InSight.
Unfortunately due to dust-covered solar arrays the electric power situation had become worse and the InSight's Seismic Experiment for Interior Structure (SEIS) instrument was to be periodically switched off leading to the fragmental data of seismic observations.Finally, the InSight stopped functioning from December 2022.However, a huge amount of different kinds of data was collected and should be analysed and evaluated.The successful In-Sight mission has renewed the interest in seismic measurements as a natural part of landed missions to other planets, especially the establishing of seismic stations networks is welcome.
As an example, we will analyse the Mars seismic field intensity ϕ measurement data which were gathered in the months May, June and July of 2019 [26].The graphical views of observation data vectors centered according to the intensity component are shown in fig. 1.We can see the quite different changes in intensity vectors of the seismic station in it.The accuracy of intensity components of individual seismic station is defined by the estimates of their standard deviations: s ϕ = (8743593.3,5825505.8,5732518.0)cnt.
To predict the spread of seismic oscillations, we assume that seismic oscillations from their epicenter travel as harmonic oscillations in all the directions with decreasing amplitudes.Hence, the structure of the seismic vibrations both in seismic stations closer to the epicenter and more distant seismic stations is the same.For analysis of the observations data of the seismic stations, we will introduce the mathematics of covariance functions and the formula of the Doppler phenomena.
The theoretical base of the observations data analysis is maintained by the statement of a stationary function, assuming that the observations errors of seismic oscillation parameters are random and are of similar precision, that is, the mean of random errors M Δ = const → 0, the dispersion DΔ = const, and covariance functions of observed digital signals depend only on the difference in parameters, that is, on the quantization interval τ on the general time scale.

Modelling of seismic oscillations parameters. -
Measurement data obtained at seismic stations are processed using mathematical statistical procedures, and efforts are made to obtain and exclude systematic and random errors.
In this part, we describe a method that employs the least squares mathematics to compute the most reliable values for seismic vibration vector trends.We suppose that the trend of the observations vector is a constant scalar.The usage of least squares mathematics partially removes random errors from vibrations and gives asymptotically valid values of the calculated parameters for a large number of measured data, even if the measured data are not normally distributed.
Each seismic field strength vector is assumed as a random function with random observations errors.Application of the least square's mathematics to the seismic field strength vector ϕ observed at seismic stations, we can calculate the weighted average ϕ, which will be the most reliable number of a trend.We can write the simple parametric equation for single value ϕ i of the observations vector as follows: Here ε i is the random error of the single value of the observations vector, ϕ i is the single value of the observations vector, φ is the calculated trend of the observations vector.
Applying the general condition of the least squares method, we can calculate the most reliable number of a trend of the seismic field strength vector ϕ, i.e., φ = ε T P ε = min; ( here P is a diagonal matrix (n×n) of observations weights p i of the vector's single values ϕ i .The weights of separate values ϕ i could be obtained by the formula where σ 0 is the standard deviation of the observation results ϕ 0 , whose weight is assumed to be equal to unit: p 0 = 1.In fact, the value of σ 0 could be chosen freely, which does not have any influence on the results of the calculations.Usually, we want to have weights p i to be close to unit, hence we should choose an appropriate value of ϕ 0 .Using an equation we will have σ ϕi = σ ui ϕ i . (5) Formula (5) shows that the result of σ ϕi depends on the value ϕ i .By introducing formula (4), we will obtain accepting the average value By zeroing the function's partial derivative of φ and resolving the resulting expression, the extremum of function (3) could be found: Further we obtain −e T P ε = 0 and e T P e φ − e T P ϕ = 0.
We will express the solution as follows: φ = e T P e −1 e T P ϕ = N −1 ω; ( 8 ) here N = (e T P e), ω = e T P ϕ .Application of the least square method gives the possibility to calculate the accuracy of the estimates of the parameters.For that we should calculate the covariance matrix K φ as follows: here σ 0 is the estimate of the standard deviation σ 0 , obtained from the expression The components of the seismic oscillations model.-We will apply the coefficient z of Doppler phenomena to the intensity vectors of seismic stations' oscillations to determine their auto-covariance and crosscovariance values, and the displacements between seismic oscillations.As is well known, the Doppler effect equation could be introduced to detect the speed of moving objects.In this equation the value of the coefficient z indicates the object's relative speed in comparison to the speed of light in vacuum c.We will use following form of the expression for the coefficient z [37,38]: here f e is the frequency of emitted oscillations, f o is the frequency of received oscillations.It is possible that changes in seismic oscillation intensities (strengths) are inversely proportional to phase shifts observed at seismic stations.Therefore, the algebraic addition of seismic oscillation phases (frequencies) is proportional to the algebraic sum of seismic oscillation intensities, i.e., δB ∼ δω, here δB, δω stand for changes of intensities and phases correspondingly, respectively.
The following formulas can be used to express changes in the algebraic sum of intensities and changes in the oscillation strength: here ω e = 2πf e , ω o = 2πf o , the starting phases ϕ 0 are accepted to be equal to zero, δa → δB.Now we can express the intensities of oscillations at the moment in time t i : here B ei is the intensity of emitted seismic waves from epicenter of event, B oi is the intensity of observed seismic waves at seismic stations, B ei ∼ ω ei , B oi ∼ ω oi .Now let us look into an expression of cross-covariance of the difference ΔB ei = B ei − B oi of two intensities B ei and B oi (observed and emitted) at the moment in time t i and a single intensity B oi : here δB oi = B oi − MB oi , δB ei = B ei − MB ei , δB ei = δB oi (z i + 1), MB oi , MB ei is the average values of the intensities of the relevant parameters of oscillations, δΔB ei = δB ei − δB oi , σ B is the standard deviation of the intensities.We suppose that the intensities of all oscillations have identical standard deviations.The mean value Mz i of the coefficient z at the moment in time t i will be calculated according to the formula Further we will use the mathematics of covariance functions to find the cross-covariance functions of the relevant vectors, assuming that each intensity vector is the random function [39,40]: and 16) here τ = s • Δ is the quantization interval, which is a variable quantity, Δ is the value of the oservations unit, u is the argument of vectors of intensities, s is the number observations units (or quantization intervals), T is the range of the vector's variation.The estimate K z (τ ) of the cross-covariance function based on observations data could be calculated as follows: here n is the number of discrete intervals in total.
And now, using formula (15), we can get an expression to determine the mean value of the coefficient z: here m is the number of components of the covariance function, σ 2 B → K z (0) is the estimate of dispersion of the vector of intensities.
The results of the experiment and the analysis.-Upon applying formula (17), the estimates of auto-covariance and cross-covariance functions of 3 seismic oscillations intensity vectors have been calculated.The quantization intervals of covariance functions vary in the range between 1 and n/2, here n = 30603 is the number of components of intensity vectors of seismic stations.Totally 9 graphical expressions of auto-covariance and crosscovariance functions were obtained.The most important graphical views of the covariance functions are shown in figs.2, 3.
Upon applying formula (18), the mean values of the coefficient z were detected.In the cases when z values of are relatively small (z < |0.1|) for calculating approximate speeds of seismic vectors, one is able to use the equation v = z • c, where v is the approximate speed of the reciprocal movement of vibrations.The positive values of the coefficient z point out that vectors recede from each other and the negative values of the coefficient z point out that vectors move to each other.In table 1, the speeds of seismic intensity vectors with respect to the Earth in different Mars months are provided.It may be seen from table 1 that seismic vectors recede from each other on their spreading.
The accuracy evaluation.-Let us find the standard deviation of the coefficient z for two cases: when z values are obtained on the base of observed frequencies of the emitted and observed vibrations, and when standard deviation is calculated on the base of intensities of the said oscillations.
We can write down the expression to detect the standard deviation of the coefficient z upon using eq.( 11): In formula (19), the estimate σ z of the standard deviation of the coefficient z was detected upon assuming that σ fe = σ fo , the mean value of z = 1 and σ fe f0 = 1.4 • 10 −8 .So, the relative error of the mean value of the coefficient z will be Let us evaluate the accuracy of the coefficient z for the case when the calculations are performed upon applying intensities of the corresponding members of the oscillations.Using formulas ( 12), ( 13) and ( 19), we obtain or here the calculations were performed upon assuming that σ Be = σ Bo , ωo and the mean value z = 1.Consequently, we found the relative error of the average value z B : The calculation results prove that the accuracy of coefficient z is the same in both cases: by applying frequencies (phases) of oscillations in procedures of their processing and on applying intensities (strengths) of the oscillations.
Let us find the accuracy of estimates of the speed of intensities vectors movement.Upon using the equation ln v = ln z + ln c, we can write an expression of the relative accuracy: Conclusions.-1) The Doppler effect formula and the mathematics of the cross-covariance functions of algebraic summing of seismic vectors' intensities and the intensities of a single vector were applied to detect values of the coefficient z.The value of the coefficient z is derived as a mean value obtained from all the values of the corresponding cross-covariance function.
2) In single Mars months, the expressions of the auto-covariance and cross-covariance functions of seismic vectors' intensities are the same.The probabilistic dependence of seismic oscillations intensity vectors in single months is strong, thus r → 1.0.
3) The values r of cross-covariance functions of seismic intensity vectors in Mars months vary in large ranges r → (0 : 1.0, 0 : −0.7) in the whole quantization interval.
4) In the period under research, the Mars seismic intensity vectors move with respect to the Earth and the range of their movement speeds in Mars months (May, June, July) are (1851.5,1853.0,1854.5)km/s.
5) The same expressions of the relevant auto-covariance and cross-covariance functions in three Mars months show that seismic oscillations of the Mars spread from a single source of them.* * *

Fig. 1 :
Fig. 1: The graphical image of variations of Mars seismic oscillations intensity vectors.

Fig. 2 :
Fig. 2: The shape of the normalised auto-covariance function of the Mars seismic oscillations intensity vector in May.

Fig. 3 :
Fig. 3: The shape of the normalised cross-covariance function of the Mars seismic oscillations intensity vector in May and June.

or σ v v ≈ 3 • 10 − 9 , where σ c c = 3 • 10 − 9 .
Further we have σ v = 3 • 10 −6 km/s, taking into account that v = 1000 km/s.The expressions of auto-covariance and cross-covariance functions of seismic oscillations intensity vectors in all other Mars months are the same.

Table 1 :
The speed of movement of the Mars seismic intensity vectors with respect to the Earth.The values of the coefficient z and speed v of a set of the vectors within three months