CN102169174B - Method for focusing geo-synchronization orbit synthetic aperture radar in high precision - Google Patents

Abstract

The invention relates to a method for focusing a geo-synchronization orbit synthetic aperture radar (GEO SAR) in high precision, and belongs to the technical field of radar signal processing. An error of a 'Stop-and-Go' supposition is taken into consideration and applied to a deduction process of an algorithm. Aiming at the problem that a typical low earth orbit synthetic aperture radar (LEO SAR) algorithm cannot process a false equivalent linear model, a norm is adopted to express an inclined distance and high-precision approximation is executed. By the method, the imaging precision is high; the method is applicable to imaging at an ultralong synthetic aperture time; high distance migration and boundedness of the equivalent linear model are overcome; and the problem that the LEO SAR imaging algorithm cannot process the 'Stop-and-Go' supposition is solved.

Description

A kind of geostationary orbit synthetic-aperture radar high precision focus method

Technical field

The present invention relates to a kind of geostationary orbit synthetic-aperture radar (GEO SAR) high precision focus method, belong to the Radar Signal Processing technical field.

Background technology

Present synthetic aperture radar (SAR) satellite is low orbit satellite, and orbit altitude is no more than 1000km, to being generally 3 to 5 days the coverage cycle of particular locality, when carrying out orbit maneuver, also needs at least 1 day time; Therefore, low, solution of emergent event long problem retardation time of low rail SAR life period resolution.For example, after the Wenchuan violent earthquake takes place, after more than a day, just obtained first batch of diameter radar image by Japan and gondola satellite, and the SAR satellite of China obtained associated picture after two days, failing provides information service the most timely to disaster relief work.The method of head it off is geostationary orbit synthetic-aperture radar (GEO SAR).Various countries are also just carrying out the research about seismic monitoring at present, and wherein the ground globe lens plan of the U.S. is the most noticeable, and this plan is divided into four, and wherein one is exactly about GEO SAR.

At present low rail synthetic-aperture radar (LEO SAR) imaging algorithm is a lot; As based on the classics of Fresnel approximation apart from Doppler (RD) algorithm, analysis of spectrum (SPECAN) algorithm etc., based on secondary range compression (SRC) algorithm, the linear frequency modulation of equivalent straight line model become mark (CS) algorithm, nonlinear frequency modulation becomes mark (NCS) algorithm etc.Fresnel approximation and equivalent straight line model can only describe that the oblique distance course is the para-curve that opening makes progress between satellite and the target; And in GEO SAR, have two class targets oblique distance courses: one type is the target oblique distance course (is representative with perigee and equatorial positions) of the approximate parabolic path that makes progress of opening; Another kind of is the target oblique distance course (is representative with the apogee) of the approximate parabolic path that Open Side Down.For the target oblique distance course that Open Side Down, utilize the straight path model to approach to occur the imaginary value of velocity equivalent and equivalent front bevel angle degree, can't be from the physical significance explanation of getting on.

And be based upon " Stop-and-Go " of these LEO SAR algorithms supposes down.In LEO SAR, because oblique distance is shorter, and " Stop-and-Go " supposes it is more rational, be 36000km at GEO SAR middle orbit height still, its oblique distance reaches the magnitude of 40000km, and the round trip propagation delay is near second-time; The speed of satellite is also very fast, and near 3000m/s, " Stop-and-Go " supposes just to be false.

Analysis through above draws, and LEO SAR imaging algorithm can not be used for GEO SAR, therefore needs research GEO SAR high precision focus method.

Summary of the invention

The objective of the invention is to handle the equivalent straight line model situation that is false in order to solve traditional LEO SAR algorithm; And traditional LEO SAR algorithm do not consider the problem that " Stop-and-Go " supposes, proposed a kind of geostationary orbit synthetic-aperture radar high precision focus method.

The objective of the invention is to realize through following technical scheme.

A kind of geostationary orbit synthetic-aperture radar high precision focus method of the present invention the steps include:

1) GEO SAR being set up the signal model that accurate oblique distance model also obtains GEO SAR thus, also is the echo data of GEO SAR.The expression formula of the signal model of GEO SAR does

$s(t_r,\mathrm{nT})=\mathrm{\σ}\·a_r\left(\frac{t_r-2\·r_n/c}{T_p}\right)\·a_n\left(\mathrm{nT}\right)\·\exp [j\·\mathrm{\π}\·\mathrm{\β}\·{(t_r-\frac{2\·r_n}{c})}^2]\·\exp (-j\·\frac{4\·\mathrm{\π}\·r_n}{\mathrm{\λ}})---\left(1\right)$

Wherein, t _rFor the distance to the fast time, nT be the orientation to the slow time, s (t _r, be nT) at the moment (t _r, the echo that nT) receives, total echo are that fast time and slow time stack are obtained.σ is the target backscattering coefficient, a _r() be the distance to envelope function, a _n() be the orientation to envelope function, determine by antenna radiation pattern.T _pBe the radar pulsewidth, c is the light velocity, β be the distance to frequency, λ is a radar wavelength, r _nBe the one way oblique distance of GEO SAR, also promptly launch oblique distance and receive the half the of oblique distance sum.

Here can not be applicable to the apogean situation of GEO SAR in order to solve equivalent straight line model, not adopt equivalent straight line model to express oblique distance r _n, but the form of employing norm, promptly

With

Be respectively that satellite and target are at nT position vector constantly.Suppose invalid problem in order to solve " Stop-and-Go ", introduce a Δ r _nRepresent that " Stop-and-Go " supposes error.Therefore the expression formula of the accurate oblique distance of GEO SAR does

$r_n=\left|\right|{\stackrel{\→}{r}}_{\mathrm{gn}}-{\stackrel{\→}{r}}_{\mathrm{sn}}\left|\right|+{\mathrm{\Δr}}_n---\left(2\right)$

Δ r wherein _nExpression formula do

$\mathrm{\Δ}{r}_n=\frac{({\stackrel{\→}{v}}_{\mathrm{sn}}-{\stackrel{\→}{v}}_{\mathrm{gn}})\·{({\stackrel{\→}{r}}_{\mathrm{sn}}-{\stackrel{\→}{r}}_{\mathrm{gn}})}^T}{c}---\left(3\right)$

Where

and , respectively, for the satellite and the target velocity vector at time nT;

Formula (2) Taylor expansion is obtained

r _n＝(r+Δr)+(k ₁+Δk ₁)·(nT)+(k ₂+Δk ₂)·(nT) ²+(k ₃+Δk ₃)·(nT) ³+k ₄·(nT) ⁴+… (4)

Wherein r, k ₁～k ₄For 0～4 rank Taylor expansion coefficient, and Δ r, Δ k ₁～Δ k ₃Be Δ r _n0～3 rank Taylor expansion coefficient;

2) the GEO SAR echo data that step 1) is generated carries out distance to the FFT computing, is carrying out distance to compression apart from frequency domain then, and its process is:

The echo of setting up in the step 1) is carried out distance to FFT, obtain echo in expression formula apart from frequency domain orientation time domain

$s(f_r,\mathrm{nT})=\mathrm{\σ}\·A_r\left(f_r\right)\·a_n\left(\mathrm{nT}\right)\·\exp \·(-j\·\mathrm{\π}\·\frac{f_r^2}{\mathrm{\β}})\·\exp [-j\·\frac{4\·\mathrm{\π}\·r_n}{c}\·(f_r+f_c)]---\left(5\right)$

Wherein, f _rFor the distance to frequency, A _r(f _r) be the expression formula of distance behind envelope function FFT, f _cBe the radar center frequency.

Need carry out distance to matched filtering to echo apart from frequency domain orientation time domain, obtain distance to the matched filtering function according to formula (5)

$H_1=\exp (j\·\mathrm{\π}\·\frac{f_r^2}{\mathrm{\β}})---\left(6\right)$

After matched filtering, echo becomes in the expression formula apart from frequency domain orientation time domain through distance

$s(f_r,\mathrm{nT})=\mathrm{\σ}\·A_r\left(f_r\right)\·a_n\left(\mathrm{nT}\right)\·\exp [-j\·\frac{4\·\mathrm{\π}\·r_n}{c}\·(f_r+f_c)]---\left(7\right)$

3) to step 2) carry out the orientation to the FFT computing through the echo of distance after matched filtering, obtain the two-dimensional frequency expression formula of echo, its process is:

When formula (7) is carried out the orientation to FFT; Can not adopt widely used resident phase place principle among traditional LEO SAR; Because resident phase place principle is difficult to find the solution the above frequency spectrum of second order; And the accurate oblique distance of deriving is up to quadravalence, and therefore resident phase place principle is no longer suitable, can only adopt the progression inversion principle to find the solution the 2-d spectrum expression formula.After the utilization progression inversion principle, obtain the 2-d spectrum expression formula of echo

$s(f_r,f_a)=\mathrm{\σ}\·A_r\left(f_r\right)\·A_a\left(f_a\right)\·$

$\exp \{j\·2\·\mathrm{\π}\·\left[\begin{array}{c}-\frac{2\·(f_r+f_c)}{c}\·(r+\mathrm{\Δr})\\ +\frac{1}{4\·(k_2+\mathrm{\Δ}{k}_2)}\·\left(\frac{c}{2\·(f_r+f_c)}\right)\·{[f_a+\frac{2\·(k_1+\mathrm{\Δ}{k}_1)\·(f_r+f_c)}{c}]}^2\\ +\frac{k_3+\mathrm{\Δ}{k}_3}{8\·{(k_2+\mathrm{\Δ}{k}_2)}^3}\·{\left(\frac{c}{2\·(f_r+f_c)}\right)}^2\·{[f_a+\frac{2\·(k_1+\mathrm{\Δ}{k}_1)\·(f_r+f_c)}{c}]}^3\\ +\frac{9\·{(k_3+\mathrm{\Δ}{k}_3)}^2-4\·(k_2+\mathrm{\Δ}{k}_2)\·k_4}{64\·{(k_2+\mathrm{\Δ}{k}_2)}^5}\·{\left(\frac{c}{2\·(f_r+f_c)}\right)}^3\·{[f_a+\frac{2\·(k_1+\mathrm{\Δ}{k}_1)\·(f_r+f_c)}{c}]}^4\end{array}\right]\}---\left(8\right)$

Wherein, f _aFor the orientation to frequency.

4) echo after handling through step 3) is carried out range migration correction and secondary range compression, its process is:

In step 3), obtained the 2-d spectrum expression formula of echo, i.e. formula (8).Can not directly carry out matched filtering to it for formula (8), reason is that two-dimensional frequency can't handle the space-variant shape of orientation to compression function, can only accomplish fine compensation to a bit, and can there be error in other points, and these errors can cause and defocus.Therefore need launch to obtain the range migration correction function to formula (8), secondary range compression function and orientation be to compression function.And range migration correction and secondary range compression can only be handled in two-dimensional frequency, and reason is apart to serious to coupling with the orientation, can't the two be decomposed.

The range migration correction function that echo is carried out in the range migration correction process does

H ₂＝exp(-j·φ ₁(f _r，f _a)) (9)

Wherein,

${\mathrm{\φ}}_1(f_r,f_a)=2\·\mathrm{\π}\·f_r\·\left[\begin{array}{c}\frac{1}{4\·(k_{20}+\mathrm{\Δ}{k}_{20})}\·(\frac{2\·{(k_{10}+\mathrm{\Δ}{k}_{10})}^2}{c}-\frac{c}{2f_c^2}\·f_a^2)+\\ \frac{k_{30}+\mathrm{\Δ}{k}_{30}}{8\·{(k_{20}+\mathrm{\Δ}{k}_{20})}^3}\·(\frac{2\·{(k_{10}+\mathrm{\Δ}{k}_{10})}^3}{c}-\frac{3\·(k_{10}+\mathrm{\Δ}{k}_{10})\·c}{2f_c^2}\·f_a^2-\frac{c^2}{2f_c^3}\·f_a^3)+\\ \frac{9\·{(k_{30}+\mathrm{\Δ}{k}_{30})}^2-4\·(k_{20}+\mathrm{\Δ}{k}_{20})\·k_{40}}{64\·{(k_{20}+\mathrm{\Δ}{k}_{20})}^5}\·\left(\begin{array}{c}\frac{2\·{(k_{10}+\mathrm{\Δ}{k}_{10})}^4}{c}-\frac{3\·{(k_{10}+\mathrm{\Δ}{k}_{10})}^2\·c}{f_c^2}{f}_a^2\\ -\frac{2\·(k_{10}+\mathrm{\Δ}{k}_{10})\·c^2}{f_c^3}\·f_a^3-\frac{3\·c^3}{8\·f_c^4}\·f_a^4\end{array}\right)\end{array}\right]---\left(10\right)$

The secondary range compression function that echo is carried out in the secondary range compression process does

H ₃＝exp(-j·φ ₂(f _r，f _a)) (11)

Wherein,

In formula (10) and formula (12), k ₁₀～k ₄₀, Δ k ₁₀～Δ k ₃₀It is the coefficient that launches in the accurate oblique distance at RP place.

5) echo after handling through step 4) is carried out distance to the IFFT computing, and compensating to compression and residual phase it being carried out the orientation apart from the Doppler territory, its process is:

After the step 4) processing, the echo data distance is to tentatively focusing on.2-d spectrum disassembles in step 4); Wherein two is range migration function and secondary range compression function; Remaining two be the orientation to compression function and residual phase function, these two functions handle distance to space-variant shape, therefore handling apart from the Doppler territory.Therefore at first the echo data that passes through after step 4) is handled be carried out distance to IFFT.Carry out the orientation then to compression.

The compression function of orientation in compression process does

H ₄＝exp(-j·φ ₃(f _a)) (13)

Wherein,

${\mathrm{\φ}}_3\left(f_a\right)=\frac{\mathrm{\π}}{2\·(k_2+\mathrm{\Δ}{k}_2)}\·(2\·(k_1+\mathrm{\Δ}{k}_1)\·f_a+\frac{\mathrm{\λ}}{2}\·f_a^2)+$

$\frac{\mathrm{\π}\·(k_3+\mathrm{\Δ}{k}_3)}{4\·{(k_2+\mathrm{\Δ}{k}_2)}^3}\·(3\·{(k_1+\mathrm{\Δ}{k}_1)}^2\·f_a+\frac{3\·\mathrm{\λ}\·(k_1+\mathrm{\Δ}{k}_1)}{2}\·f_a^2+\frac{{\mathrm{\λ}}^2}{4}\·f_a^3)+---\left(14\right)$

$\mathrm{\π}\·\frac{9\·{(k_3+\mathrm{\Δ}{k}_3)}^2-4\·(k_2+\mathrm{\Δ}{k}_2)\·k_4}{32\·{(k_2+\mathrm{\Δ}{k}_2)}^5}\·\left(\begin{array}{c}4\·{(k_1+\mathrm{\Δ}{k}_1)}^3\·f_a+3\·\mathrm{\λ}\·{(k_1+\mathrm{\Δ}{k}_1)}^2\·f_a^2\\ +(k_1+\mathrm{\Δ}{k}_1)\·{\mathrm{\λ}}^2\·f_a^3+\frac{{\mathrm{\λ}}^3}{8}{f}_a^4\end{array}\right)$

Residual phase function in the residual phase compensation process does

H ₅＝exp(-j·φ ₄(f _a)) (15)

${\mathrm{\φ}}_4=2\mathrm{\π}\begin{array}{c}\left[\begin{array}{c}-\frac{2f_c}{c}(r_0+\mathrm{\Δ}{r}_0)+\frac{\mathrm{\λ}\·{(k_1+{\mathrm{\Δk}}_1)}^2}{8\·(k_2+\mathrm{\Δ}{k}_2)}+\frac{\mathrm{\λ}\·(k_3+\mathrm{\Δ}{k}_3)\·{(k_1+\mathrm{\Δ}{k}_1)}^3}{16\·{(k_2+{\mathrm{\Δk}}_2)}^3}\\ +\frac{9\·{(k_3+{\mathrm{\Δk}}_3)}^2-4\·(k_2+{\mathrm{\Δk}}_2)\·k_4}{128\·{(k_2+{\mathrm{\Δk}}_2)}^5}\·{(k_1+{\mathrm{\Δk}}_1)}^3\·\mathrm{\λ}\end{array}\right]---\left(16\right)\end{array}$

6) echo after handling through step 5) is carried out the orientation to the IFFT computing, the target image that obtains focusing on.After the step 5) processing, target focuses on fully, but data need be carried out the orientation to IFFT with it still apart from the Doppler territory at this moment.

Finally, obtained focusing on good target image in two-dimensional time-domain.

Beneficial effect

A kind of geostationary orbit synthetic-aperture radar high precision focus method of the present invention; Imaging precision is high; Be suitable for the imaging of synthetic aperture under the time of overlength; Can overcome big range migration, also overcome the limitation of equivalent straight line model, solve low rail SAR imaging algorithm and can not handle the problem that " Stop-and-Go " supposes.

Description of drawings

Fig. 1 is a GEO SAR imaging algorithm schematic flow sheet in the preferred embodiment for the present invention;

Fig. 2 is the accurate oblique distance synoptic diagram of GEO SAR in the preferred embodiment for the present invention;

Fig. 3 is GEO SAR point target simulation result figure in the preferred embodiment for the present invention;

Fig. 4 is a GEO SAR point target two dimension contour map in the preferred embodiment for the present invention.

Embodiment

Below in conjunction with accompanying drawing and embodiment the present invention is further specified.

Embodiment

A kind of geostationary orbit synthetic-aperture radar high precision focus method, as shown in Figure 1, the steps include:

1) the accurate oblique distance model modeling of GEO SAR also obtains the accurate signal model of GEO SAR thus.

The key of GEO SAR signal model is to make up the oblique distance model.As shown in Figure 2; Among the figure satellite when transmitting at the A point; Be designated as

this moment target at the D point; Be designated as when signal arrives target; The position of satellite and target is respectively B and E; Note is made

and

when signal turns back to satellite respectively; Satellite position is at the C point; Position coordinates is the F point for the position of

target; Coordinate is

according to Fig. 2, and GEO SAR round trip oblique distance is written as

$R_n=R_1+R_2=\left|\right|{\stackrel{\→}{r}}_{\mathrm{sn}}-{\stackrel{\→}{r}}_{\mathrm{sn},t_1}\left|\right|+\left|\right|{\stackrel{\→}{r}}_{\mathrm{gn},t_1}-{\stackrel{\→}{r}}_{\mathrm{sn},t_2}\left|\right|---\left(17\right)$

To formula (17) Taylor expansion, and consider fast time variable t ₁And t ₂-t ₁Be same magnitude, and between gap very little, therefore replace with a variable, i.e. t=2t ₁≈ 2 (t ₂-t ₁), obtain

$R_n=2\·\left|\right|{\stackrel{\→}{r}}_{\mathrm{gn}}-{\stackrel{\→}{r}}_{\mathrm{sn}}\left|\right|+({\stackrel{\→}{v}}_{\mathrm{sn}}-{\stackrel{\→}{v}}_{\mathrm{gn}})\·{\stackrel{\→}{u}}_{\mathrm{gs},n}^T\·t---\left(18\right)$

Wherein, wherein

and is respectively satellite and target in nT velocity constantly;

${\stackrel{\→}{u}}_{\mathrm{gs},n}=({\stackrel{\→}{r}}_{\mathrm{sn}}-{\stackrel{\→}{r}}_{\mathrm{sn}})/\left|\right|{\stackrel{\→}{r}}_{\mathrm{sn}}-{\stackrel{\→}{r}}_{\mathrm{sn}}\left|\right|.$

For the convenience of subsequent treatment, formula (18) is rewritten as one way, promptly

$r_n=\left|\right|{\stackrel{\→}{r}}_{\mathrm{gn}}-{\stackrel{\→}{r}}_{\mathrm{sn}}\left|\right|+\frac{({\stackrel{\→}{v}}_{\mathrm{sn}}-{\stackrel{\→}{v}}_{\mathrm{gn}})\·{\stackrel{\→}{u}}_{\mathrm{gs},n}^T}{2}\·t---\left(19\right)$

Order

${\mathrm{\Δr}}_n=\frac{({\stackrel{\→}{v}}_{\mathrm{sn}}-{\stackrel{\→}{v}}_{\mathrm{gn}})\·{\stackrel{\→}{u}}_{\mathrm{gs},n}^T}{2}\·t---\left(20\right)$

After further deriving, formula (20) is written as

${\mathrm{\Δr}}_n=\frac{({\stackrel{\→}{v}}_{\mathrm{sn}}-{\stackrel{\→}{v}}_{\mathrm{gn}})\·{({\stackrel{\→}{r}}_{\mathrm{gn}}-{\stackrel{\→}{r}}_{\mathrm{sn}})}^T}{c}---\left(21\right)$

Therefore GEO SAR signal oblique distance is written as

$r_n=\left|\right|{\stackrel{\→}{r}}_{\mathrm{gn}}-{\stackrel{\→}{r}}_{\mathrm{sn}}\left|\right|+\mathrm{\Δ}{r}_n---\left(22\right)$

So, just obtain the expression formula of GEO SAR echoed signal, for

$s(t_r,\mathrm{nT})=\mathrm{\σ}\·a_r\left(\frac{t_r-2\·r_n/c}{T_p}\right)\·a_n\left(\mathrm{nT}\right)\·\exp [j\·\mathrm{\π}\·\mathrm{\β}\·{(t_r-\frac{2\·r_n}{c})}^2]\·\exp (-j\·\frac{4\·\mathrm{\π}\·r_n}{\mathrm{\λ}})---\left(23\right)$

Wherein, t _rFor the distance to the fast time, nT is that the orientation is the target backscattering coefficient to slow time σ, a _r() is respectively distance to envelope function, a _n() is that the orientation is to envelope function, by antenna radiation pattern decision, T _pBe the radar pulsewidth, c is the light velocity, β be the distance to frequency, λ is a radar wavelength, r _nBe the accurate one way oblique distance of GEO SAR course, confirm, after its Taylor expansion, obtain by formula (22)

r _n≈(r+Δr)+(k ₁+Δk ₁)·(nT)+(k ₂+Δk ₂)·(nT) ²+(k ₃+Δk ₃)·(nT) ³+k ₄·(nT) ⁴+… (24)

Wherein,

$r=\left|\right|{\stackrel{\→}{r}}_{s0}-{\stackrel{\→}{r}}_{g0}\left|\right|---\left(25\right)$

$\mathrm{\Δr}=\frac{({\stackrel{\→}{v}}_{s0}-{\stackrel{\→}{v}}_{g0})\·{({\stackrel{\→}{r}}_{g0}-{\stackrel{\→}{r}}_{s0})}^T}{c}---\left(26\right)$

$k_1=\frac{({\stackrel{\→}{v}}_{s0}-{\stackrel{\→}{v}}_{g0})\·{({\stackrel{\→}{r}}_{s0}-{\stackrel{\→}{r}}_{g0})}^T}{\left|\right|{\stackrel{\→}{r}}_{s0}-{\stackrel{\→}{r}}_{g0}\left|\right|}---\left(27\right)$

$\mathrm{\Δ}{k}_1=\frac{({\stackrel{\→}{a}}_{s0}-{\stackrel{\→}{a}}_{g0})\·{({\stackrel{\→}{r}}_{s0}-{\stackrel{\→}{r}}_{g0})}^T+({\stackrel{\→}{v}}_{s0}-{\stackrel{\→}{v}}_{g0})\·{({\stackrel{\→}{v}}_{s0}-{\stackrel{\→}{v}}_{g0})}^T}{c}---\left(28\right)$

$k_2=\frac{({\stackrel{\→}{a}}_{s0}-{\stackrel{\→}{a}}_{g0})\·{({\stackrel{\→}{r}}_{s0}-{\stackrel{\→}{r}}_{g0})}^T+{\left|\right|{\stackrel{\→}{v}}_{s0}-{\stackrel{\→}{v}}_{g0}\left|\right|}^2}{2\·\left|\right|{\stackrel{\→}{r}}_{s0}-{\stackrel{\→}{r}}_{g0}\left|\right|}-\frac{{[({\stackrel{\→}{v}}_{s0}-{\stackrel{\→}{v}}_{g0})\·{({\stackrel{\→}{r}}_{s0}-{\stackrel{\→}{r}}_{g0})}^T]}^2}{2\·{\left|\right|{\stackrel{\→}{r}}_{s0}-{\stackrel{\→}{r}}_{g0}\left|\right|}^3}---\left(29\right)$

$\mathrm{\Δ}{k}_2=\frac{({\stackrel{\→}{b}}_{s0}-{\stackrel{\→}{b}}_{g0})\·{({\stackrel{\→}{r}}_{s0}-{\stackrel{\→}{r}}_{g0})}^T+3\·({\stackrel{\→}{a}}_{s0}-{\stackrel{\→}{a}}_{g0})\·{({\stackrel{\→}{v}}_{s0}-{\stackrel{\→}{v}}_{g0})}^T}{2\·c}---\left(30\right)$

$k_3=\frac{({\stackrel{\→}{b}}_{s0}-{\stackrel{\→}{b}}_{g0})\·{({\stackrel{\→}{r}}_{s0}-{\stackrel{\→}{r}}_{g0})}^T+3\·({\stackrel{\→}{a}}_{s0}-{\stackrel{\→}{a}}_{g0})\·{({\stackrel{\→}{v}}_{s0}-{\stackrel{\→}{v}}_{g0})}^T}{6\·\left|\right|{\stackrel{\→}{r}}_{s0}-{\stackrel{\→}{r}}_{g0}\left|\right|}-\frac{{[({\stackrel{\→}{v}}_{s0}-{\stackrel{\→}{v}}_{g0})\·{({\stackrel{\→}{r}}_{s0}-{\stackrel{\→}{r}}_{g0})}^T]}^3}{2\·{\left|\right|{\stackrel{\→}{r}}_{s0}-{\stackrel{\→}{r}}_{g0}\left|\right|}^5}---\left(31\right)$

$-\frac{({\stackrel{\→}{v}}_{s0}-{\stackrel{\→}{v}}_{g0})\·{({\stackrel{\→}{r}}_{s0}-{\stackrel{\→}{r}}_{g0})}^T\·[({\stackrel{\→}{a}}_{s0}-{\stackrel{\→}{a}}_{g0})\·{({\stackrel{\→}{r}}_{s0}-{\stackrel{\→}{r}}_{g0})}^T+{\left|\right|{\stackrel{\→}{v}}_{s0}-{\stackrel{\→}{v}}_{g0}\left|\right|}^2]}{2\·{\left|\right|{\stackrel{\→}{r}}_{s0}-{\stackrel{\→}{r}}_{g0}\left|\right|}^3}$

${\mathrm{\Δk}}_3=\frac{({\stackrel{\→}{d}}_{s0}-{\stackrel{\→}{d}}_{g0})\·{({\stackrel{\→}{r}}_{s0}-{\stackrel{\→}{r}}_{g0})}^T+4\·({\stackrel{\→}{b}}_{s0}-{\stackrel{\→}{b}}_{g0})\·{({\stackrel{\→}{v}}_{s0}-{\stackrel{\→}{v}}_{g0})}^T+3\·{\left|\right|{\stackrel{\→}{a}}_{s0}-{\stackrel{\→}{a}}_{g0}\left|\right|}^2}{6\·c}---\left(32\right)$

$k_4=\frac{({\stackrel{\→}{d}}_{s0}-{\stackrel{\→}{d}}_{g0})\·{({\stackrel{\→}{r}}_{s0}-{\stackrel{\→}{r}}_{g0})}^T+4\·({\stackrel{\→}{b}}_{s0}-{\stackrel{\→}{b}}_{g0})\·{({\stackrel{\→}{v}}_{s0}-{\stackrel{\→}{v}}_{g0})}^T+3\·{\left|\right|{\stackrel{\→}{a}}_{s0}-{\stackrel{\→}{a}}_{g0}\left|\right|}^2}{24\·\left|\right|{\stackrel{\→}{r}}_{s0}-{\stackrel{\→}{r}}_{g0}\left|\right|}-\frac{k_2^2+2\·k_1\·k_3}{2\·\left|\right|{\stackrel{\→}{r}}_{s0}-{\stackrel{\→}{r}}_{g0}\left|\right|}---\left(33\right)$

Formula (25) is to formula (33);

be respectively satellite and target in the orientation to aperture center position vector constantly;

is respectively satellite and target in aperture center velocity constantly; Correspondingly for satellite at constantly acceleration, acceleration vector of aperture center, add the acceleration vector, and

for target at constantly acceleration, acceleration vector of aperture center, add the acceleration vector;

2) the GEO SAR echo data that step 1) is generated carries out distance to the FFT computing, is carrying out distance to compression apart from frequency domain then, and its process is:

At first to carry out distance to formula (23) to the FFT computing, adopt resident phase place principle.

Echo in the expression formula apart from frequency domain does

$s(f_r,\mathrm{nT})=\mathrm{\σ}\·A_r\left(f_r\right)\·a_n\left(\mathrm{nT}\right)\·\exp \·(-j\·\mathrm{\π}\·\frac{f_r^2}{\mathrm{\β}})\·\exp \left(-j\·\frac{4\·\mathrm{\π}\·r_n}{c}\·(f_r+f_c)\right)---\left(34\right)$

Wherein, A _r(f _r) for passing through the distance of distance after the FFT computing to the spectrum expression formula, f _rFor the distance to frequency, f _cBe the radar center frequency.

First exponential term in the formula (34) is apart to modulating function, constructs apart to the spectral filtering function doing based on this

$H_1=\exp (j\·\mathrm{\π}\·\frac{f_r^2}{\mathrm{\β}})---\left(35\right)$

Second exponential term in the formula (34) be the orientation to distance to coupling terms, following step is handled this exactly;

3) to step 2) carry out the orientation to the FFT computing through the echo of distance after matched filtering, obtain the two-dimensional frequency expression formula of echo.Its process is:

Process step 2) matched filtering is being carried out preliminary compression (coupling terms also is untreated) apart from the target that makes progress.It is to want to handle well second exponential term of step (2) Chinese style (34) that target has focused on key fully, with its independent taking-up, for

$\mathrm{ss}(f_r,\mathrm{nT})=\exp (-j\·\frac{4\·\mathrm{\π}\·r_n}{c}\·(f_r+f_c))---\left(36\right)$

In order to obtain the 2-d spectrum of distance after matched filtering, need the orientation is carried out to FFT in (36).And classical resident phase place principle can not be handled the above expansion of second order, therefore can only adopt the progression reversal process.So adopt the progression counter-rotating for ease, (36) write as

$\mathrm{ss}(f_r,\mathrm{nT})={\mathrm{ss}}_1(f_r,\mathrm{nT})\·\exp (-j\·\frac{4\·\mathrm{\π}\·(f_r+f_c)}{c}\·(k_1+\mathrm{\Δ}{k}_1)\·\left(\mathrm{nT}\right))---\left(37\right)$

Ss ₁(f _r, nT) with respect to ss (f _r, nT) removed linear phase, obtain ss ₁(f _r, behind 2-d spectrum nT), utilize formula (38)～(40) to try to achieve ss (f _r, 2-d spectrum nT), thus further obtain through the echo 2-d spectrum after step (2) processing, for

${\mathrm{ss}}_1(f_r,\mathrm{nT})\⇔{\mathrm{ss}}_1(f_r,f_a)---\left(38\right)$

${\mathrm{ss}}_1(f_r,\mathrm{nT})\·\exp (-j\·\frac{4\·\mathrm{\π}\·(f_r+f_c)}{c}\·(k_1+{\mathrm{\Δk}}_1)\·\left(\mathrm{nT}\right))\⇔{\mathrm{ss}}_1(f_r,f_a+\frac{2\·(k_1+\mathrm{\Δ}{k}_1)\·(f_r+f_c)}{c})---\left(39\right)$

$\mathrm{ss}(f_r,\mathrm{nT})\⇔\mathrm{ss}(f_r,f_a)={\mathrm{ss}}_1(f_r,f_a+\frac{2\·(k_1+{\mathrm{\Δk}}_1)\·(f_r+f_c)}{c})---\left(40\right)$

To ss ₁(f _r, nT) carry out the orientation to FFT, then by long-pending phase place done

$\mathrm{\Θ}(f_r,\mathrm{nT})=-\frac{4\·\mathrm{\π}\·r_n^{\′}}{c}\·(f_r+f_c)-2\·\mathrm{\π}\·f_a\·\left(\mathrm{nT}\right)---\left(41\right)$

Wherein,

$r_n^{\′}=r_n-(k_1+{\mathrm{\Δk}}_1)\·\left(\mathrm{nT}\right)=(r+\mathrm{\Δr})+(k_2+{\mathrm{\Δk}}_2)\·{\left(\mathrm{nT}\right)}^2+(k_3+\mathrm{\Δ}{k}_3)\·{\left(\mathrm{nT}\right)}^3+k_4\·{\left(\mathrm{nT}\right)}^4+\·\·\·\left(42\right)$

Differentiate obtains to formula (41),

$2\·(k_2+{\mathrm{\Δk}}_2)\·\left(\mathrm{nT}\right)+3\·(k_3+\mathrm{\Δ}{k}_3)\·{\left(\mathrm{nT}\right)}^2+4\·k_4\·{\left(\mathrm{nT}\right)}^3=-\frac{c\·f_a}{2\·(f_r+f_c)}---\left(43\right)$

Order

$\mathrm{nT}=A_1\·[-\frac{c\·f_a}{2\·(f_r+f_c)}]+A_2\·{[-\frac{c\·f_a}{2\·(f_r+f_c)}]}^2+A_3\·{[-\frac{c\·f_a}{2\·(f_r+f_c)}]}^3+\·\·\·\left(44\right)$

(44) substitution (43) is obtained

$A_1=\frac{1}{2\·(k_2+\mathrm{\Δ}{k}_2)},$ $A_2=-\frac{(k_3+{\mathrm{\Δk}}_3)}{8\·{(k_2+{\mathrm{\Δk}}_2)}^3},$ $A_3=\frac{18\·{(k_3+\mathrm{\Δ}{k}_3)}^2-8\·(k_2+\mathrm{\Δ}{k}_2)\·k_4}{32\·{(k_2+\mathrm{\Δ}{k}_2)}^5}---\left(45\right)$

Therefore obtain ss ₁(f _r, nT) two-dimensional frequency expression formula,

${\mathrm{ss}}_1(f_r,f_a)=\exp \left\{j2\mathrm{\π}\left[\begin{array}{c}-\frac{2\·(f_r+f_c)}{c}\·(r+\mathrm{\Δr})+\frac{1}{4\·(k_2+\mathrm{\Δ}{k}_2)}\·(\frac{c}{2\·(f_r+f_c)}\·f_a^2)\\ +\frac{k_3+\mathrm{\Δ}{k}_3}{8\·{(k_2+{\mathrm{\Δk}}_2)}^3}\·{\left(\frac{c}{2\·(f_r+f_c)}\right)}^2\·f_a^3\\ +\frac{9\·{(k_3+\mathrm{\Δ}{k}_3)}^2-4\·(k_2+\mathrm{\Δ}{k}_2)\·k_4}{64\·{(k_2+\mathrm{\Δ}{k}_2)}^5}\·{\left(\frac{c}{2\·(f_r+f_c)}\right)}^3\·f_a^4\end{array}\right]\right\}---\left(46\right)$

Utilize formula (40), obtain

$\mathrm{ss}(f_r,f_a)=$

$\exp \{j\·2\·\mathrm{\π}\·\left[\begin{array}{c}-\frac{2\·(f_r+f_c)}{c}\·(r+\mathrm{\Δr})\\ +\frac{1}{4\·(k_2+\mathrm{\Δ}{k}_2)}\·\left(\frac{c}{2\·(f_r+f_c)}\right)\·{[f_a+\frac{2\·(k_1+\mathrm{\Δ}{k}_1)\·(f_r+f_c)}{c}]}^2\\ +\frac{k_3+\mathrm{\Δ}{k}_3}{8\·{(k_2+\mathrm{\Δ}{k}_2)}^3}\·{\left(\frac{c}{2\·(f_r+f_c)}\right)}^2\·{[f_a+\frac{2\·(k_1+\mathrm{\Δ}{k}_1)\·(f_r+f_c)}{c}]}^3\\ +\frac{9\·{(k_3+\mathrm{\Δ}{k}_3)}^2-4\·(k_2+\mathrm{\Δ}{k}_2)\·k_4}{64\·{(k_2+\mathrm{\Δ}{k}_2)}^5}\·{\left(\frac{c}{2\·(f_r+f_c)}\right)}^3\·{[f_a+\frac{2\·(k_1+\mathrm{\Δ}{k}_1)\·(f_r+f_c)}{c}]}^4\end{array}\right]\}---\left(47\right)$

Further obtain

$s(f_r,f_a)=\mathrm{\σ}\·A_r\left(f_r\right)\·A_a\left(f_a\right)\·$

$\exp \{j\·2\·\mathrm{\π}\·\left[\begin{array}{c}-\frac{2\·(f_r+f_c)}{c}\·(r+\mathrm{\Δr})\\ +\frac{1}{4\·(k_2+\mathrm{\Δ}{k}_2)}\·\left(\frac{c}{2\·(f_r+f_c)}\right)\·{[f_a+\frac{2\·(k_1+\mathrm{\Δ}{k}_1)\·(f_r+f_c)}{c}]}^2\\ +\frac{k_3+\mathrm{\Δ}{k}_3}{8\·{(k_2+\mathrm{\Δ}{k}_2)}^3}\·{\left(\frac{c}{2\·(f_r+f_c)}\right)}^2\·{[f_a+\frac{2\·(k_1+\mathrm{\Δ}{k}_1)\·(f_r+f_c)}{c}]}^3\\ +\frac{9\·{(k_3+\mathrm{\Δ}{k}_3)}^2-4\·(k_2+\mathrm{\Δ}{k}_2)\·k_4}{64\·{(k_2+\mathrm{\Δ}{k}_2)}^5}\·{\left(\frac{c}{2\·(f_r+f_c)}\right)}^3\·{[f_a+\frac{2\·(k_1+\mathrm{\Δ}{k}_1)\·(f_r+f_c)}{c}]}^4\end{array}\right]\}---\left(48\right)$

Wherein, the s (f here _r, f _a) be two-dimensional frequency expression formula through the echo of step (2) distance after matched filtering;

4) echo after handling through step 3) is carried out range migration correction and secondary range compression, its process is:

In step 3), tried to achieve distance to the 2-d spectrum expression formula behind FFT of the echo bearing after the matched filtering.But can not directly utilize formula (48) structure matched filtering function, need to utilize formula (49)～(51) to launch the exponential term in (48),

$\frac{1}{f_r+f_c}=\frac{1}{f_c}\·[1-\frac{f_r}{f_c}+{\left(\frac{f_r}{f_c}\right)}^2-{\left(\frac{f_r}{r_c}\right)}^3+\·\·\·]---\left(49\right)$

${\left(\frac{1}{f_r+f_c}\right)}^2=\frac{1}{f_c^2}\·[1-\frac{2\·f_r}{f_c}+3\·{\left(\frac{f_r}{f_c}\right)}^2-4\·{\left(\frac{f_r}{f_c}\right)}^3+\·\·\·]---\left(50\right)$

${\left(\frac{1}{f_r+f_c}\right)}^3=\frac{1}{f_c^3}\·[1-\frac{3\·f_r}{f_c}+6\·{\left(\frac{f_r}{f_c}\right)}^2-10\·{\left(\frac{f_r}{f_c}\right)}^3+\·\·\·]---\left(51\right)$

After the expansion, obtain

s(f _r，f _a)＝σ·A _r(f _r)·A _a(f _a)·

exp{j·[φ ₁(f _r，f _a)+φ ₂(f _r，f _a)+φ ₃(f _a)+φ ₄(f _a)]}

Wherein,

${\mathrm{\φ}}_1(f_r,f_a)=2\·\mathrm{\π}\·f_r\·\left[\begin{array}{c}\frac{1}{4\·(k_{20}+\mathrm{\Δ}{k}_{20})}\·(\frac{2\·{(k_{10}+\mathrm{\Δ}{k}_{10})}^2}{c}-\frac{c}{2f_c^2}\·f_a^2)+\\ \frac{k_{30}+\mathrm{\Δ}{k}_{30}}{8\·{(k_{20}+\mathrm{\Δ}{k}_{20})}^3}\·(\frac{2\·{(k_{10}+\mathrm{\Δ}{k}_{10})}^3}{c}-\frac{3\·(k_{10}+\mathrm{\Δ}{k}_{10})\·c}{2f_c^2}\·f_a^2-\frac{c^2}{2f_c^3}\·f_a^3)+\\ \frac{9\·{(k_{30}+\mathrm{\Δ}{k}_{30})}^2-4\·(k_{20}+\mathrm{\Δ}{k}_{20})\·k_{40}}{64\·{(k_{20}+\mathrm{\Δ}{k}_{20})}^5}\·\left(\begin{array}{c}\frac{2\·{(k_{10}+\mathrm{\Δ}{k}_{10})}^4}{c}-\frac{3\·{(k_{10}+\mathrm{\Δ}{k}_{10})}^2\·c}{f_c^2}{f}_a^2\\ -\frac{2\·(k_{10}+\mathrm{\Δ}{k}_{10})\·c^2}{f_c^3}\·f_a^3-\frac{3\·c^3}{8\·f_c^4}\·f_a^4\end{array}\right)\end{array}\right]---\left(52\right)$

${\mathrm{\φ}}_3\left(f_a\right)=\frac{\mathrm{\π}}{2\·(k_2+\mathrm{\Δ}{k}_2)}\·(2\·(k_1+\mathrm{\Δ}{k}_1)\·f_a+\frac{\mathrm{\λ}}{2}\·f_a^2)+$

$\frac{\mathrm{\π}\·(k_3+\mathrm{\Δ}{k}_3)}{4\·{(k_2+\mathrm{\Δ}{k}_2)}^3}\·(3\·{(k_1+\mathrm{\Δ}{k}_1)}^2\·f_a+\frac{3\·\mathrm{\λ}\·(k_1+\mathrm{\Δ}{k}_1)}{2}\·f_a^2+\frac{{\mathrm{\λ}}^2}{4}\·f_a^3)+---\left(54\right)$

$\mathrm{\π}\·\frac{9\·{(k_3+\mathrm{\Δ}{k}_3)}^2-4\·(k_2+\mathrm{\Δ}{k}_2)\·k_4}{32\·{(k_2+\mathrm{\Δ}{k}_2)}^5}\·\left(\begin{array}{c}4\·{(k_1+\mathrm{\Δ}{k}_1)}^3\·f_a+3\·\mathrm{\λ}\·{(k_1+\mathrm{\Δ}{k}_1)}^2\·f_a^2\\ +(k_1+\mathrm{\Δ}{k}_1)\·{\mathrm{\λ}}^2\·f_a^3+\frac{{\mathrm{\λ}}^3}{8}{f}_a^4\end{array}\right)$

${\mathrm{\φ}}_4=2\mathrm{\π}\begin{array}{c}\left[\begin{array}{c}-\frac{2f_c}{c}(r_0+\mathrm{\Δ}{r}_0)+\frac{\mathrm{\λ}\·{(k_1+{\mathrm{\Δk}}_1)}^2}{8\·(k_2+\mathrm{\Δ}{k}_2)}+\frac{\mathrm{\λ}\·(k_3+\mathrm{\Δ}{k}_3)\·{(k_1+\mathrm{\Δ}{k}_1)}^3}{16\·{(k_2+{\mathrm{\Δk}}_2)}^3}\\ +\frac{9\·{(k_3+{\mathrm{\Δk}}_3)}^2-4\·(k_2+{\mathrm{\Δk}}_2)\·k_4}{128\·{(k_2+{\mathrm{\Δk}}_2)}^5}\·{(k_1+{\mathrm{\Δk}}_1)}^3\·\mathrm{\λ}\end{array}\right]---\left(55\right)\end{array}$

Its Chinese style (52) is the range migration amount, and formula (53) is the secondary range modulation voltage, and formula (54) is the orientation modulation voltage, and formula (55) is a residual volume.

Correspondingly constructing distance to the migration correction function does

H ₂＝exp(-j·φ ₁(f _r，f _a)) (56)

The secondary range compression function does

H ₃＝exp(-j·φ ₂(f _r，f _a)) (57)

5) echo after handling through step 4) is carried out distance to the IFFT computing, and compensating to compression and residual phase it being carried out the orientation apart from the Doppler territory.

After the step 4) processing, range migration is calibrated good, and distance is to also compressing fully.Echo after need step 4) being handled carries out distance to IFFT, is carrying out the orientation to compression apart from the Doppler territory.Constructing the orientation according to (54) in the step 4) to compression function does

H ₄＝exp(-j·φ ₃(f _a)) (58)

Also need remove the influence of residual phase in addition, construct residual phase equally according to the formula (55) in the step (4) and remove function focusing on

H ₅＝exp(-j·φ ₄(f _a)) (59)

6) echo after handling through step 5) is carried out the orientation to the IFFT computing, the target image that obtains focusing on.

After the above-mentioned steps processing, the imaging results of point target is as shown in Figure 3.In Fig. 3 altogether emulation the individual point targets of 169 (13*13), the distance between the target is to being 5km with the orientation to the interval.Fig. 4 is the point target two-dimensional points spread function contour map that is positioned at the scene edge, and its two-dimentional secondary lobe is high-visible, and each item index reaches requirement, has also verified the correctness of geostationary orbit synthetic-aperture radar high precision focus method thus.

Claims (1)

1. geostationary orbit synthetic-aperture radar high precision focus method is characterized in that step is:

1) GEO SAR being set up the signal model that accurate oblique distance model also obtains GEO SAR thus, also is the echo data of GEO SAR, and the expression formula of the signal model of GEO SAR does

$s(t_r,\mathrm{nT})=\mathrm{\σ}\·a_r\left(\frac{t_r-2\·r_n/c}{T_p}\right)\·a_n\left(\mathrm{nT}\right)\·\exp [j\·\mathrm{\π}\·\mathrm{\β}\·{(t_r-\frac{2\·r_n}{c})}^2]\·\exp (-j\·\frac{4\·\mathrm{\π}\·r_n}{\mathrm{\λ}})$

Wherein, t _rFor the distance to the fast time, nT be the orientation to the slow time, σ is the target backscattering coefficient, a _r() is respectively distance to envelope function, a _n() be the orientation to envelope function, determine by antenna radiation pattern; T _pBe the radar pulsewidth, c is the light velocity, β be the distance to frequency, λ is a radar wavelength, r _nBe the accurate one way oblique distance of GEO SAR course, expression formula does

$r_n=\left|\right|{\stackrel{\→}{r}}_{\mathrm{gn}}-{\stackrel{\→}{r}}_{\mathrm{sn}}\left|\right|+\mathrm{\Δ}{r}_n$

Wherein, Be respectively that satellite and target are at nT position vector constantly, Δ r _nBe the error of supposing introducing by " Stop-and-Go ", Δ r _nExpression formula do

$\mathrm{\Δ}{r}_n=\frac{({\stackrel{\→}{v}}_{\mathrm{sn}}-{\stackrel{\→}{v}}_{\mathrm{gn}})\·{({\stackrel{\→}{r}}_{\mathrm{sn}}-{\stackrel{\→}{r}}_{\mathrm{gn}})}^T}{c}$

Where

and , respectively, for the satellite and the target velocity vector at time nT;

To r _nCarrying out Taylor expansion obtains

r _n＝(r+Δr)+(k ₁+Δk ₁)·(nT)+(k ₂+Δk ₂)·(nT) ²+(k ₃+Δk ₃)·(nT) ³+k ₄·(nT) ⁴+...

Wherein r, k ₁～k ₄Be 0～4 rank Taylor expansion coefficient, and Δ r, Δ k ₁～Δ k ₃Be Δ r _n0～3 rank Taylor expansion coefficient;

2) the GEO SAR echo data that step 1) is generated carries out distance to the FFT computing, is carrying out distance to compression apart from frequency domain then, and its process is:

The echo of setting up in the step 1) is carried out distance to the FFT computing, obtain echo in expression formula apart from frequency domain orientation time domain

$s(f_r,\mathrm{nT})=\mathrm{\σ}\·A_r\left(f_r\right)\·a_n\left(\mathrm{nT}\right)\·\exp (-j\·\mathrm{\π}\·\frac{f_r^2}{\mathrm{\β}})\·\exp [-j\·\frac{4\·\mathrm{\π}\·r_n}{c}\·(f_r+f_c)]$

Wherein, f _rFor the distance to frequency, A _r(f _r) be the expression formula of distance behind envelope function FFT, f _cBe the radar center frequency;

Apart from frequency domain orientation time domain echo is being carried out distance to matched filtering, obtaining distance to the matched filtering function

$H_1=\exp (j\·\mathrm{\π}\·\frac{f_r^2}{\mathrm{\β}})$

After matched filtering, echo becomes in the expression formula apart from frequency domain orientation time domain through distance

$s(f_r,\mathrm{nT})=\mathrm{\σ}\·A_r\left(f_r\right)\·a_n\left(\mathrm{nT}\right)\·\exp [-j\·\frac{4\·\mathrm{\π}\·r_n}{c}\·(f_r+f_c)];$

3) to step 2) carry out the orientation to the FFT computing through the echo of distance after matched filtering, obtain the two-dimensional frequency expression formula of echo, for

$s(f_r,f_a)=\mathrm{\σ}\·A_r\left(f_r\right)\·A_a\left(f_a\right).$

$\exp \{j\·2\·\mathrm{\π}\·\left[\begin{array}{c}-\frac{2\·(f_r+f_c)}{c}\·(r+\mathrm{\Δr})\\ +\frac{1}{4\·(k_2+\mathrm{\Δ}{k}_2)}\·\left(\frac{c}{2\·(f_r+f_c)}\right)\·{[f_a+\frac{2\·(k_1+\mathrm{\Δ}{k}_1)\·(f_r+f_c)}{c}]}^2\\ +\frac{k_3+\mathrm{\Δ}{k}_3}{8\·{(k_2+\mathrm{\Δ}{k}_2)}^3}\·{\left(\frac{c}{2\·(f_r+f_c)}\right)}^2\·{[f_a+\frac{2\·(k_1+\mathrm{\Δ}{k}_1)\·(f_r+f_c)}{c}]}^3\\ +\frac{9\·{(k_3+\mathrm{\Δ}{k}_3)}^2-4\·(k_2+\mathrm{\Δ}{k}_2)\·k_4}{64\·{(k_2+\mathrm{\Δ}{k}_2)}^2}\·{\left(\frac{c}{2\·(f_r+f_c)}\right)}^3\·{[f_a+\frac{2\·(k_1+\mathrm{\Δ}{k}_1)\·(f_r+f_c)}{c}]}^4\end{array}\right]\}$

Wherein, f _aFor the orientation to frequency;

4) echo after handling through step 3) is carried out range migration correction and secondary range compression, the range migration correction function that wherein echo is carried out in the range migration correction process does

H ₂＝exp(-j·φ ₁(f _r，f _a))

In the following formula

${\mathrm{\φ}}_1(f_r,f_a)=2\·\mathrm{\π}\·f_r\·\left[\begin{array}{c}\frac{1}{4\·(k_{20}+\mathrm{\Δ}{k}_{20})}\·(\frac{2\·{(k_{10}+\mathrm{\Δ}{k}_{10})}^2}{c}-\frac{c}{{2f}_c^2}\·f_a^2)+\\ \frac{k_{30}+\mathrm{\Δ}{k}_{30}}{8\·{(k_{20}+\mathrm{\Δ}{k}_{20})}^3}\·(\frac{2\·{(k_{10}+\mathrm{\Δ}{k}_{10})}^3}{c}-\frac{3\·(k_{10}+\mathrm{\Δ}{k}_{10})}{{2f}_c^2}\·f_a^2-\frac{c^2}{{2f}_c^3}\·f_a^3)+\\ \frac{9\·{(k_{30}+\mathrm{\Δ}{k}_{30})}^2-4\·(k_{20}+\mathrm{\Δ}{k}_{20})\·k_{40}}{64\·{(k_{20}+\mathrm{\Δ}{k}_2)}^5}\·\left(\begin{array}{c}\frac{2\·{(k_{10}+\mathrm{\Δ}{k}_{10})}^4}{c}-\frac{3\·{(k_{10}+\mathrm{\Δ}{k}_{10})}^2\·c}{f_c^2}{f}_a^2\\ -\frac{2\·(k_{10}+\mathrm{\Δ}{k}_{10})\·c^2}{f_c^3}\·f_a^3-\frac{3\·c^3}{8\·f_c^4}\·f_a^4\end{array}\right)\end{array}\right]$

The secondary range compression function that echo is carried out in the secondary range compression process does

H ₃＝exp(-j·φ ₂(f _r，f _a))

In the following formula

K wherein ₁₀～k ₄₀, Δ k ₁₀～Δ k ₃₀It is the coefficient that launches in the accurate oblique distance at RP place;

5) echo after handling through step 4) is carried out distance to the IFFT computing, and apart from the Doppler territory it is being carried out the orientation to compression and residual phase compensation, wherein the compression function of orientation in compression process does

H ₄＝exp(-j·φ ₃(f _a))

Wherein

${\mathrm{\φ}}_3\left(f_a\right)=\frac{\mathrm{\π}}{2\·(k_2+\mathrm{\Δ}{k}_2)}\·(2\·(k_1+\mathrm{\Δ}{k}_1)\·f_a+\frac{\mathrm{\λ}}{2}\·f_a^2)+$

$\frac{\mathrm{\π}\·(k_3+\mathrm{\Δ}{k}_3)}{4\·{(k_2+\mathrm{\Δ}{k}_2)}^3}\·(3\·{(k_1+\mathrm{\Δ}{k}_1)}^2\·f_a+\frac{3\·\mathrm{\λ}\·(k_1+\mathrm{\Δ}{k}_1)}{2}\·f_a^2+\frac{{\mathrm{\λ}}^2}{4}\·f_a^3)+$

$\mathrm{\π}\·\frac{9\·{(k_3+\mathrm{\Δ}{k}_3)}^2-4\·(k_2+\mathrm{\Δ}{k}_2)\·k_4}{32\·{(k_2+\mathrm{\Δ}{k}_2)}^5}\·\left(\begin{array}{c}4\·{(k_1+\mathrm{\Δ}{k}_1)}^3\·f_a+3\·\mathrm{\λ}\·{(k_1+\mathrm{\Δ}{k}_1)}^2\·f_a^2\\ +(k_1+\mathrm{\Δ}{k}_1)\·{\mathrm{\λ}}^2\·f_a^3+\frac{{\mathrm{\λ}}^3}{8}{f}_a^4\end{array}\right)$

Residual phase function in the residual phase compensation process does

H ₅＝exp(-j·φ ₄(f _a))

Wherein ${\mathrm{\φ}}_4=2\mathrm{\π}\left[\begin{array}{c}-\frac{2f_c}{c}(r+\mathrm{\Δ\; r})+\frac{\mathrm{\λ}\·{(k_1+\mathrm{\Δ}{k}_1)}^2}{8\·(k_2+\mathrm{\Δ}{k}_2)}+\frac{\mathrm{\λ}\·(k_3+\mathrm{\Δ}{k}_3)\·{(k_1+\mathrm{\Δ}{k}_1)}^3}{16\·{(k_2+\mathrm{\Δ}{k}_2)}^3}\\ +\frac{9\·{(k_3+\mathrm{\Δ}{k}_3)}^2-4\·(k_2+\mathrm{\Δ}{k}_2)\·k_4}{128\·{(k_2+\mathrm{\Δ}{k}_2)}^5}\·{(k_1+\mathrm{\Δ}{k}_1)}^3\·\mathrm{\λ}\end{array}\right];$

6) echo after handling through step 5) is carried out the orientation to the IFFT computing, the target image that obtains focusing on.

Images