CN105842672B - A kind of broad band low frequency Space-borne SAR Imaging ionosphere phase error compensation method - Google Patents

A kind of broad band low frequency Space-borne SAR Imaging ionosphere phase error compensation method Download PDF

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CN105842672B
CN105842672B CN201610339752.XA CN201610339752A CN105842672B CN 105842672 B CN105842672 B CN 105842672B CN 201610339752 A CN201610339752 A CN 201610339752A CN 105842672 B CN105842672 B CN 105842672B
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ionosphere
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王成
陈亮
刘露
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China Academy of Space Technology CAST
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    • G—PHYSICS
    • G01—MEASURING; TESTING
    • G01S—RADIO DIRECTION-FINDING; RADIO NAVIGATION; DETERMINING DISTANCE OR VELOCITY BY USE OF RADIO WAVES; LOCATING OR PRESENCE-DETECTING BY USE OF THE REFLECTION OR RERADIATION OF RADIO WAVES; ANALOGOUS ARRANGEMENTS USING OTHER WAVES
    • G01S7/00—Details of systems according to groups G01S13/00, G01S15/00, G01S17/00
    • G01S7/02—Details of systems according to groups G01S13/00, G01S15/00, G01S17/00 of systems according to group G01S13/00
    • G01S7/40—Means for monitoring or calibrating
    • G—PHYSICS
    • G01—MEASURING; TESTING
    • G01S—RADIO DIRECTION-FINDING; RADIO NAVIGATION; DETERMINING DISTANCE OR VELOCITY BY USE OF RADIO WAVES; LOCATING OR PRESENCE-DETECTING BY USE OF THE REFLECTION OR RERADIATION OF RADIO WAVES; ANALOGOUS ARRANGEMENTS USING OTHER WAVES
    • G01S13/00—Systems using the reflection or reradiation of radio waves, e.g. radar systems; Analogous systems using reflection or reradiation of waves whose nature or wavelength is irrelevant or unspecified
    • G01S13/88—Radar or analogous systems specially adapted for specific applications
    • G01S13/89—Radar or analogous systems specially adapted for specific applications for mapping or imaging
    • G01S13/90—Radar or analogous systems specially adapted for specific applications for mapping or imaging using synthetic aperture techniques, e.g. synthetic aperture radar [SAR] techniques

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Abstract

A kind of broad band low frequency Space-borne SAR Imaging ionosphere phase error compensation method, based on Legendre's orthogonal basis, series is carried out to ionosphere phase error and decomposes to cubic term, obtained each order error is mutually orthogonal, solves the problems, such as each order coupling of existing appraisal procedure.It is then based on each order expression formula that this orthogonal model is derived, qualitative assessment broadening of the zero degree phase error to orientation image, the translation that phase error is adjusted the distance to image, pulse stretching caused by quadratic phase error, and pulse distortion caused by phase error three times.Ionosphere total electron content TEC information can obtain based on this influence, and then ionosphere effect compensation is carried out to image.This method can more influence of the true ionosphere phase error of accurate response to broad band low frequency Space-borne SAR Imaging, therefore improve the compensation precision of ionosphere effect.

Description

A kind of broad band low frequency Space-borne SAR Imaging ionosphere phase error compensation method
Technical field
The present invention relates to a kind of broad band low frequency Space-borne SAR Imaging ionosphere phase error compensation method, belong to synthetic aperture Field of radar and ionospheric radio propagation field, it is mainly used in improving broad band low frequency Space-borne SAR Imaging ionosphere compensation precision.
Background technology
It is conceived in global range to demands such as the high-resolution of forest cover and underground vanishing target identifications, is operated in VHF/ One of low frequency satellite-borne synthetic aperture radar (Synthetic Aperture Radar, SAR) System Development trend of uhf band is Designed towards bigger relative bandwidth.However, because it is worked on or in ionosphere, the phase of echo of these frequency ranges is not Can avoid being affected, and carrier frequency is lower, bandwidth is bigger, influence more serious so that the service behaviour of SAR system drastically under Drop.In order to more thoroughly suppress the influence in ionosphere, it is premise and key that Accurate Model and assessment are carried out to it.Therefore, model The service behaviour of broad band low frequency Spaceborne SAR System after the compensation of ionosphere is directly determined with the accuracy of assessment.
Currently for the ionosphere effect of the low frequency and narrow bandwidth SAR signals such as PALSAR and BIOMASS SAR, because its phase is missed Difference it is secondary and above it is every it is smaller can ignore, Taylor series expansion turns into analysis and compensates each order error is imaged matter to SAR Amount influences the most frequently used method.It is secondary and quickly increase can not for error term three times however, when low frequency SAR signal bandwidths increase Ignore, if still ionospheric error is analyzed and compensated with every non-orthogonal Taylor series, due to being included in high-order term Low order item composition also accordingly increase, will now produce obvious errors, detailed problems are as follows：
First, according to Fourier transform property, zero degree item error can't have an impact to SAR image distance to compression, But orientation image resolution ratio can be caused to decline.Generally, orientation picture quality is more sensitive to zero degree phase error, because This needs more accurate assessment.And with the increase of bandwidth, because existing method is not considered in quadratic phase error Zero degree component, therefore the influence that zero degree phase error is brought can not be compensated well for.
Second, a phase error can cause image distance to translation, although for scene single-point situation, image shift is not Image image quality can be influenceed, but actual scene is made up of many point targets, and this skew can cause local resolution to decline. As bandwidth increases, existing compensation method does not have the component of degree n n considered three times in phase error, therefore also result in can not The error ignored.
3rd, quadratic phase error can cause distance to cause resolution ratio to decline, due to existing to pulse stretching (after pulse pressure) The quadratic phase error that method is assessed contains zero degree item, therefore need to improve its precision.
4th, three times phase error can cause distance to asymmetrical distortion after pulse pressure, for PALSAR or BIOMASS SAR Deng narrowband systems, phase error magnitude very little can be ignored three times.And when analyzing big bandwidth, first order in existing method into Branch causes obvious error.
The content of the invention
The purpose of the present invention：A kind of overcome the deficiencies in the prior art, there is provided broad band low frequency Space-borne SAR Imaging ionosphere phase Error compensating method, each order phase error coupled problem for overcoming Taylor series expansion to obtain, effectively increases ionosphere shadow Ring the precision of compensation.
The technical solution adopted by the present invention is：
A kind of broad band low frequency Space-borne SAR Imaging ionosphere phase error compensation method, comprises the following steps：
(1) the broad band low frequency satellite-borne SAR echo-signal under ionosphere effect is obtained, is built for the satellite-borne SAR echo-signal The ionosphere zero degree of the Legendre's orthogonal series that is based on expansion is to phase error three times；
(2) influence of the ionosphere zero degree phase error to orientation image quality is determined, that is, determines ionosphere zero degree phase Orientation maximum secondary phase error caused by error, and then determine the first vertical total electron content value TEC1；
(3) influence that the phase error of ionosphere one time is adjusted the distance to image quality is determined, that is, determines the phase of ionosphere one time Image distance caused by error determines the second vertical total electron content value TEC to translational movement2；
(4) determine the influence that ionosphere quadratic phase error is adjusted the distance to image quality, that is, determine ionosphere caused by away from Descriscent maximum secondary phase error, and then determine the 3rd vertical total electron content value TEC3；
(5) determine the ionosphere influence that phase error is adjusted the distance to image quality three times, that is, determine ionosphere caused by away from Descriscent maximum phase error, and then determine the 4th vertical total electron content value TEC three times4；
(6) according to the four TEC values determined in step (2)~(5), vertical total electron content average value TEC is calculatedIt is average；
(7) according to the TEC obtained in step (6)It is average, determine ionosphere zero degree in step (1) to phase error three times； (illustrate in embodiment by TECIt is averageSubstitute into step 1 formula)
(8) according to the ionosphere zero degree determined in step (7) to phase error three times, satellite-borne SAR image is compensated, Obtain removing the borne SAR image of ionosphere effect.
Zero degree is followed successively by Δ φ to phase error three times in the step (1)0Le(fτ)、Δφ1Le(fτ)、Δφ2Le (fτ)、Δφ3Le(fτ), obtained especially by equation below：
Wherein, f0For carrier frequency, B is transmitted bandwidth, A0For constant, and A0=40.28；fτ∈ [- B/2, B/2] is SAR signals Spectral range, TEC are vertical total electron content value.
Orientation maximum secondary phase error caused by determining ionosphere zero degree phase error in the step (2), enters And determine the first vertical total electron content value TEC1, it is specially：
Wherein LsFor length of synthetic aperture, TEC0It is defined as thunder Up to TEC values during distance objective minimum oblique distance, TEC0=TEC1/ cos θ, θ are radar downwards angle of visibility, f0For carrier frequency, B is transmitting band Width, c are the light velocity, ΔΦa2LeFor orientation maximum secondary phase error.
The step (3) determines image distance caused by the phase error of ionosphere one time to translational movement, and then determines second Vertical total electron content value TEC2, it is specially：
Wherein, Δ LrLeIt is image distance to flat Shifting amount,
The step (4) determines distance caused by ionosphere to maximum secondary phase error, and then determines that the 3rd is vertical total Electron content value TEC3, it is specially：
Wherein, ΔΦr2LeIt is distance to maximum secondary phase error.
The step (5) determines distance caused by ionosphere to maximum phase error three times, and then determines that the 4th is vertical total Electron content value TEC4, it is specially：
Wherein, ΔΦr3LeIt is distance to maximum phase error three times.
TEC in step (6)It is average=(TEC1+TEC2+TEC3+TEC4)/4。
The present invention can use to obtain following technique effect compared with prior art：
(1) the inventive method is decomposed based on Legendre's orthogonal basis to ionosphere phase error, perfect can distinguish each rank Influence of the secondary error to SAR image, solves each order coupled problem of existing Taylor series expansion appraisal procedure.
(2) present invention derives that zero degree is imaged matter to ionosphere phase error three times to SAR image based on this orthogonal model The mathematical modeling influenceed is measured, because now higher order term error is free of lower term composition, therefore obtained mathematical modeling is compared with Taylor's level Number method of deploying obtains more accurate.Based on these influence models and then four total electron content TEC values are can obtain, final To its average value.Now the satellite-borne SAR image by ionosphere effect is compensated using obtained TEC average values.Due to This method can the perfect influence for distinguishing each order error to SAR image, solve that each order of existing appraisal procedure is not independent to ask Topic, improves ionosphere effect compensation precision.
Brief description of the drawings
Fig. 1 is the flow chart of the inventive method；
Fig. 2 is the phase error simulation result schematic diagram three times that Taylor series expansion obtains；
Fig. 3 is the phase error simulation result schematic diagram three times that the present invention obtains；
Fig. 4 is the simulation result schematic diagram that the ionosphere phase error that the present invention obtains is compensated true error, wherein, figure 4 (a) is the one-dimensional result of satellite-borne SAR image ionosphere effect compensation；Upper figure in Fig. 4 (a) is to be tied by ionosphere effect Fruit, shown in middle figure is to utilize Legendre's level for the SAR image using Taylor series expansion compensation by ionosphere effect, figure below SAR image of the number expansion compensation by ionosphere effect；Fig. 4 (b) is two-dimensional simulation result, Fig. 4 corresponding to upper figure in Fig. 4 (a) (c) it is two-dimensional simulation result corresponding to middle figure；Fig. 4 (d) is two-dimensional simulation result corresponding to figure below.
Embodiment
Below in conjunction with the accompanying drawings and give an actual example, to the spaceborne image quality of the broad band low frequency by ionosphere effect according to the present invention Appraisal procedure is described in detail, and used main radar and Ionospheric Parameters are as shown in table 1.
The ionosphere of table 1 and radar parameter
As shown in figure 1, it is based on Legendre's orthogonal series expansion progress broad band low frequency Space-borne SAR Imaging for one kind of the present invention Ionosphere effect compensation method flow chart, what this method included comprises the following steps that：
(1) the broad band low frequency satellite-borne SAR echo-signal under ionosphere effect is obtained, is built for the satellite-borne SAR echo-signal The ionosphere zero degree of the Legendre's orthogonal series that is based on expansion is to phase error three times；
Ionosphere is a kind of dispersive medium, and the time delay of different spectral is not when SAR signals are propagated wherein, in signal bandwidth Together, therefore the extra phase term of echo can be caused, its two-dimentional echo-signal is represented by：
In above formula, τ, η represent SAR fast time and slow time variable respectively.wa[η] is orientation envelope, W (fτ) be away from The Fourier transformation form of descriscent envelope, KaFor orientation chirp rate, R (η) be in slow time η, radar and target away from It is the light velocity from, c, fτ∈ [- B/2, B/2] is SAR signal spectrum scopes, and B is signal bandwidth.Δφiono(fτ) it is to be ionized The true additive phase that layer influences, its expression formula can be written as：
Wherein A0=4 π 40.28/c, f0For signal carrier frequency, TEC is ionosphere total electron content (Total Electron Content).If being deployed with traditional Taylor series, above formula can be approximately
Wherein, Δ φ0Ta,Δφ1Ta,Δφ2Ta, and Δ φ3TaIt is the zero degree that Taylor series expansion obtains respectively, once, Secondary and phase error three times.Typically for the ionosphere effect of low frequency and narrow bandwidth SAR signals, because its phase error is secondary And above it is every it is smaller can ignore, therefore above formula can accurately be assessed and influenceed caused by ionosphere enough.However, when relative When bandwidth increases, because Taylor expansion is every and non-orthogonal, high-order term error contains low order composition, such as cubic term contains one Secondary item, it will now produce the error that can not ignore.Fig. 2 show the parameter using table 1, obtained phase error Δ three times φ3TaSimulation result, it can be seen that linear segment causes larger assessment errors enough.Therefore, for broad band low frequency satellite-borne SAR Signal ionosphere effect is assessed, it is necessary to establish every orthogonal series expansion method, and the present invention is entered based on Legendre's orthogonal basis Row series decomposes, can be again approximate to true phase error：
Wherein, Δ φ0Le(fτ),Δφ1Le(fτ),Δφ2Le(fτ), and Δ φ3Le(fτ) it is to be based on Legnedre series exhibition Obtained optimal zero degree is opened, once, secondary, phase error expresses formula three times.LnFor Legendre's basic function, it is represented by：
And anFor corresponding coefficient
Therefore, the expression formula of each term coefficient can be written as：
Finally, each order optimum phase error expression for deploying to obtain based on Legnedre series can be written as：
Similar Fig. 2 simulation parameter, shown in Fig. 3 for phase error af three times3LeSimulation result, can now see Go out due to orthogonality, Δ φ3LeIn do not include linear segment, therefore ionosphere phase error three times can correctly be described The influence come to picture strip.Similarly, other order errors are Optimal Error statement.
(2) influence of the ionosphere zero degree phase error to orientation image quality is determined, that is, determines ionosphere zero degree phase Orientation maximum secondary phase error caused by error, and then determine the first vertical total electron content value TEC1；
When only considering zero degree phase error, broad band low frequency satellite-borne SAR echo expression formula can be written as：
When the vertical TEC values in scene are invariable, now Δ φ0LeChange along the orientation slow time is due to every Caused by oblique TEC values difference suffered by individual slow time echo, and tiltedly the change of TEC values is become by distance between radar and target Caused by change, i.e.,：
Wherein TEC0It is defined as TEC values during distance by radar target minimum oblique distance, TEC0=TEC1/ cos θ, θ are under radar Visual angle.Quadratic term in the result of above formula on the slow time can cause orientation pulse stretching, i.e. resolution ratio declines, general with most Big quadratic phase error describes the spreading characteristic after pulse pressure, and its expression formula can be written as：
Wherein Ls=TaV is length of synthetic aperture.ΔΦa2LeIt is calculated using adaptive iteration backoff algorithm, is Common sense in the field.When it is determined that after orientation maximum secondary phase error caused by the zero degree phase error of ionosphere, and then can be true Fixed first vertical total electron content value TEC1。
(3) influence that the phase error of ionosphere one time is adjusted the distance to image quality is determined, that is, determines the phase of ionosphere one time Image distance caused by error determines the second vertical total electron content value TEC to translational movement2；
Image distance caused by the phase error of ionosphere one time is determined to translational movement, and then determines that second vertical total electronics contains Value TEC2, it is specially：
Wherein, Δ LrLeIt is image distance to translational movement, can be obtained by image strong point coordinate, existing Taylor series expansion Obtained translation expression formula is unrelated with bandwidth, and as can be seen from the above equation, image translation caused by linear term is that have with bandwidth Close.
(4) determine the influence that ionosphere quadratic phase error is adjusted the distance to image quality, that is, determine ionosphere caused by away from Descriscent maximum secondary phase error, and then determine the 3rd vertical total electron content value TEC3；
Quadratic phase error can cause distance to cause resolution ratio to decline, maximum caused by it to the expansion after image pulse pressure Quadratic phase error expression formula is：
Determine distance caused by ionosphere to maximum secondary phase error ΔΦr2LeAfterwards, and then determine that the 3rd is vertical total electric Sub- content value TEC3.Wherein, ΔΦr2LeIt is distance to maximum secondary phase error, it is using adaptive iteration backoff algorithm meter Obtain, be common sense in the field.
(5) determine the ionosphere influence that phase error is adjusted the distance to image quality three times, that is, determine ionosphere caused by away from Descriscent maximum phase error, and then determine the 4th vertical total electron content value TEC three times4；
Phase error can cause distance asymmetrical distortion situation, this asymmetrical distortion occur to the pulse after pulse pressure three times The appearance of false target can be caused when serious, the phase error form three times derived based on Legendre, maximum can be obtained three times The expression formula of phase error：
Determine distance caused by ionosphere to maximum phase error ΔΦ three timesr3LeAfterwards, and then determine that the 4th is vertical total electric Sub- content value TEC4.Wherein, ΔΦr3LeIt is distance to maximum phase error three times.Adaptive iteration backoff algorithm can also be used in it It is calculated.
(6) according to the four TEC values determined in step (2)~(5), vertical total electron content average value TEC is calculatedIt is average；
TECIt is average=(TEC1+TEC2+TEC3+TEC4)/4 (15)
(7) according to the TEC obtained in step (6)It is average, determine ionosphere zero degree in step (1) to phase error three times；
The TEC that will be obtainedIt is averageThe ionosphere zero degree that Legnedre series are deployed to obtain in average value substitution step (1) is to three times In phase error, the phase error estimated.
(8) according to the ionosphere zero degree determined in step (7) to phase error three times, satellite-borne SAR image is compensated, Obtain removing the borne SAR image of ionosphere effect；
In order to preferably describe appraisal procedure superiority proposed by the present invention, We conducted point target simulating, verifying, such as Shown in Fig. 4, simulation parameter used is as shown in table 1, peak sidelobe ratio (the Peak Side-lobe Ratio of simulation result:PSLR) With integration secondary lobe ratio (Integrate Side-lobe Ratio:ISLR) as shown in table 2.The upper figures of Fig. 4 (a) are by true electricity Absciss layer phase error (i.e. Δ φiono) obtained pulse pressure result, it can be seen that now picture quality degradation, Fig. 4 (b) for pair The two-dimensional simulation result answered.Shown in middle figure is utilization Taylor series expansion compensation by the image of ionosphere effect, Fig. 4 (c) Corresponding two-dimensional simulation result, picture quality still not up to requires after compensating as can be seen from Table 2, illustrates now to be based on Taylor's exhibition The approximation opened can not react true phase error situation well.It should be noted that this difference can not pass through addition Higher order time error (such as four times, five times, it is six inferior) improved.Being compensated for utilization Legnedre series expansion shown in figure below By the image of ionosphere effect, Fig. 4 (d) is corresponding two-dimensional simulation result, and the image after now compensating has returned to ideal Situation, illustrate that each order error based on Legnedre series expansion can be good at reacting true ionospheric error situation.
The emulating image quality assessment result of table 2
Image evaluation PSLR(dB) ISLR(dB)
Fig. 4 (c) -8.46 -8.36
Fig. 4 (d) -13.55 -10.74
The content not being described in detail in description of the invention belongs to the known technology of those skilled in the art.

Claims (7)

  1. A kind of 1. broad band low frequency Space-borne SAR Imaging ionosphere phase error compensation method, it is characterised in that comprise the following steps：
    (1) the broad band low frequency satellite-borne SAR echo-signal under ionosphere effect is obtained, base is established for the satellite-borne SAR echo-signal In the ionosphere zero degree that Legendre's orthogonal series deploys to phase error three times；
    (2) influence of the ionosphere zero degree phase error to orientation image quality is determined, that is, determines ionosphere zero degree phase error Caused orientation maximum secondary phase error, and then determine the first vertical total electron content value TEC1；
    (3) influence that the phase error of ionosphere one time is adjusted the distance to image quality is determined, that is, determines the phase error of ionosphere one time Caused image distance determines the second vertical total electron content value TEC to translational movement2；
    (4) determine the influence that ionosphere quadratic phase error is adjusted the distance to image quality, that is, determine ionosphere caused by distance to Maximum secondary phase error, and then determine the 3rd vertical total electron content value TEC3；
    (5) determine the ionosphere influence that phase error is adjusted the distance to image quality three times, that is, determine ionosphere caused by distance to Maximum phase error three times, and then determine the 4th vertical total electron content value TEC4；
    (6) according to the four TEC values determined in step (2)~(5), vertical total electron content average value TEC is calculatedIt is average；
    (7) according to the TEC obtained in step (6)It is average, determine ionosphere zero degree in step (1) to phase error three times；(8) root According to the ionosphere zero degree determined in step (7) to phase error three times, satellite-borne SAR image is compensated, obtains removing ionization The borne SAR image that layer influences.
  2. 2. a kind of broad band low frequency Space-borne SAR Imaging ionosphere phase error compensation method according to claim 1, its feature It is：Zero degree is followed successively by Δ φ to phase error three times in the step (1)0Le(fτ)、Δφ1Le(fτ)、Δφ2Le(fτ)、 Δφ3Le(fτ), obtained especially by equation below：
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    <mrow> <msub> <mi>&amp;Delta;&amp;phi;</mi> <mrow> <mn>1</mn> <mi>L</mi> <mi>e</mi> </mrow> </msub> <mrow> <mo>(</mo> <msub> <mi>f</mi> <mi>&amp;tau;</mi> </msub> <mo>)</mo> </mrow> <mo>=</mo> <msub> <mi>A</mi> <mn>0</mn> </msub> <mo>&amp;CenterDot;</mo> <mi>T</mi> <mi>E</mi> <mi>C</mi> <mo>&amp;CenterDot;</mo> <msub> <mi>f</mi> <mi>&amp;tau;</mi> </msub> <mrow> <mo>(</mo> <mfrac> <mn>12</mn> <msup> <mi>B</mi> <mn>2</mn> </msup> </mfrac> <mo>-</mo> <mfrac> <mrow> <mn>12</mn> <msub> <mi>f</mi> <mn>0</mn> </msub> </mrow> <msup> <mi>B</mi> <mn>3</mn> </msup> </mfrac> <mi>l</mi> <mi>n</mi> <mo>(</mo> <mfrac> <mrow> <mn>2</mn> <msub> <mi>f</mi> <mn>0</mn> </msub> <mo>+</mo> <mi>B</mi> </mrow> <mrow> <mn>2</mn> <msub> <mi>f</mi> <mn>0</mn> </msub> <mo>-</mo> <mi>B</mi> </mrow> </mfrac> <mo>)</mo> <mo>)</mo> </mrow> </mrow>
    <mfenced open = "" close = ""> <mtable> <mtr> <mtd> <mrow> <msub> <mi>&amp;Delta;&amp;phi;</mi> <mrow> <mn>2</mn> <mi>L</mi> <mi>e</mi> </mrow> </msub> <mrow> <mo>(</mo> <msub> <mi>f</mi> <mi>&amp;tau;</mi> </msub> <mo>)</mo> </mrow> <mo>=</mo> <msub> <mi>A</mi> <mn>0</mn> </msub> <mo>&amp;CenterDot;</mo> <mi>T</mi> <mi>E</mi> <mi>C</mi> <mrow> <mo>(</mo> <mfrac> <mrow> <mn>30</mn> <msubsup> <mi>f</mi> <mi>&amp;tau;</mi> <mn>2</mn> </msubsup> </mrow> <msup> <mi>B</mi> <mn>3</mn> </msup> </mfrac> <mo>-</mo> <mfrac> <mn>5</mn> <mrow> <mn>2</mn> <mi>B</mi> </mrow> </mfrac> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>&amp;CenterDot;</mo> <mrow> <mo>(</mo> <mo>-</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <mi>l</mi> <mi>n</mi> <mo>(</mo> <mfrac> <mrow> <mn>2</mn> <msub> <mi>f</mi> <mn>0</mn> </msub> <mo>+</mo> <mi>B</mi> </mrow> <mrow> <mn>2</mn> <msub> <mi>f</mi> <mn>0</mn> </msub> <mo>-</mo> <mi>B</mi> </mrow> </mfrac> <mo>)</mo> <mo>-</mo> <mfrac> <mrow> <mn>6</mn> <msub> <mi>f</mi> <mn>0</mn> </msub> </mrow> <mi>B</mi> </mfrac> <mo>+</mo> <mfrac> <mrow> <mn>6</mn> <msubsup> <mi>f</mi> <mn>0</mn> <mn>2</mn> </msubsup> </mrow> <msup> <mi>B</mi> <mn>2</mn> </msup> </mfrac> <mi>l</mi> <mi>n</mi> <mo>(</mo> <mfrac> <mrow> <mn>2</mn> <msub> <mi>f</mi> <mn>0</mn> </msub> <mo>+</mo> <mi>B</mi> </mrow> <mrow> <mn>2</mn> <msub> <mi>f</mi> <mn>0</mn> </msub> <mo>-</mo> <mi>B</mi> </mrow> </mfrac> <mo>)</mo> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> </mtable> </mfenced>
    <mfenced open = "" close = ""> <mtable> <mtr> <mtd> <mrow> <msub> <mi>&amp;Delta;&amp;phi;</mi> <mrow> <mn>3</mn> <mi>L</mi> <mi>e</mi> </mrow> </msub> <mrow> <mo>(</mo> <msub> <mi>f</mi> <mi>&amp;tau;</mi> </msub> <mo>)</mo> </mrow> <mo>=</mo> <msub> <mi>A</mi> <mn>0</mn> </msub> <mo>&amp;CenterDot;</mo> <mi>T</mi> <mi>E</mi> <mi>C</mi> <mrow> <mo>(</mo> <mfrac> <mrow> <mn>140</mn> <msubsup> <mi>f</mi> <mi>&amp;tau;</mi> <mn>3</mn> </msubsup> </mrow> <msup> <mi>B</mi> <mn>4</mn> </msup> </mfrac> <mo>-</mo> <mfrac> <mrow> <mn>21</mn> <msub> <mi>f</mi> <mi>&amp;tau;</mi> </msub> </mrow> <msup> <mi>B</mi> <mn>2</mn> </msup> </mfrac> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>&amp;CenterDot;</mo> <mrow> <mo>(</mo> <mo>-</mo> <mfrac> <mn>4</mn> <mn>3</mn> </mfrac> <mo>+</mo> <mfrac> <mrow> <mn>3</mn> <msub> <mi>f</mi> <mn>0</mn> </msub> </mrow> <mi>B</mi> </mfrac> <mi>ln</mi> <mo>(</mo> <mfrac> <mrow> <mn>2</mn> <msub> <mi>f</mi> <mn>0</mn> </msub> <mo>+</mo> <mi>B</mi> </mrow> <mrow> <mn>2</mn> <msub> <mi>f</mi> <mn>0</mn> </msub> <mo>-</mo> <mi>B</mi> </mrow> </mfrac> <mo>)</mo> <mo>+</mo> <mfrac> <mrow> <mn>20</mn> <msubsup> <mi>f</mi> <mn>0</mn> <mn>2</mn> </msubsup> </mrow> <msup> <mi>B</mi> <mn>2</mn> </msup> </mfrac> <mo>-</mo> <mfrac> <mrow> <mn>20</mn> <msubsup> <mi>f</mi> <mn>0</mn> <mn>3</mn> </msubsup> </mrow> <msup> <mi>B</mi> <mn>3</mn> </msup> </mfrac> <mi>l</mi> <mi>n</mi> <mo>(</mo> <mfrac> <mrow> <mn>2</mn> <msub> <mi>f</mi> <mn>0</mn> </msub> <mo>+</mo> <mi>B</mi> </mrow> <mrow> <mn>2</mn> <msub> <mi>f</mi> <mn>0</mn> </msub> <mo>-</mo> <mi>B</mi> </mrow> </mfrac> <mo>)</mo> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> </mtable> </mfenced>
    Wherein, f0For carrier frequency, B is transmitted bandwidth, A0For constant, and A0=40.28；fτ∈ [- B/2, B/2] is SAR signal spectrums Scope, TEC are vertical total electron content value.
  3. 3. a kind of broad band low frequency Space-borne SAR Imaging ionosphere phase error compensation method according to claim 1, its feature It is：Orientation maximum secondary phase error caused by the zero degree phase error of ionosphere is determined in the step (2), and then really Fixed first vertical total electron content value TEC1, it is specially：
    Wherein LsFor length of synthetic aperture, TEC0Be defined as radar away from From target minimum oblique distance R0When TEC values, TEC0=TEC1/ cos θ, θ are radar downwards angle of visibility, f0For carrier frequency, B is transmitted bandwidth, c For the light velocity, ΔΦa2LeFor orientation maximum secondary phase error.
  4. 4. a kind of broad band low frequency Space-borne SAR Imaging ionosphere phase error compensation method according to claim 1, its feature It is：The step (3) determines image distance caused by the phase error of ionosphere one time to translational movement, and then determines that second is vertical Total electron content value TEC2, it is specially：
    Wherein, Δ LrLeIt is image distance to translational movement.
  5. 5. a kind of broad band low frequency Space-borne SAR Imaging ionosphere phase error compensation method according to claim 1, its feature It is：The step (4) determines distance caused by ionosphere to maximum secondary phase error, and then determines the 3rd vertical total electronics Content value TEC3, it is specially：
    Wherein, ΔΦr2Le It is distance to maximum secondary phase error.
  6. 6. a kind of broad band low frequency Space-borne SAR Imaging ionosphere phase error compensation method according to claim 1, its feature It is：The step (5) determines distance caused by ionosphere to maximum phase error three times, and then determines the 4th vertical total electronics Content value TEC4, it is specially：
    <mrow> <msub> <mi>&amp;Delta;&amp;Phi;</mi> <mrow> <mi>r</mi> <mn>3</mn> <mi>L</mi> <mi>e</mi> </mrow> </msub> <mo>=</mo> <mfrac> <mrow> <mn>2255.68</mn> <msub> <mi>&amp;pi;TEC</mi> <mn>4</mn> </msub> </mrow> <mrow> <mi>c</mi> <mi>B</mi> </mrow> </mfrac> <mo>&amp;CenterDot;</mo> <mrow> <mo>(</mo> <mrow> <mfrac> <mn>4</mn> <mn>3</mn> </mfrac> <mo>-</mo> <mfrac> <mrow> <mn>3</mn> <msub> <mi>f</mi> <mn>0</mn> </msub> </mrow> <mi>B</mi> </mfrac> <mi>ln</mi> <mrow> <mo>(</mo> <mfrac> <mrow> <mn>2</mn> <msub> <mi>f</mi> <mn>0</mn> </msub> <mo>+</mo> <mi>B</mi> </mrow> <mrow> <mn>2</mn> <msub> <mi>f</mi> <mn>0</mn> </msub> <mo>-</mo> <mi>B</mi> </mrow> </mfrac> <mo>)</mo> </mrow> <mo>-</mo> <mfrac> <mrow> <mn>20</mn> <msubsup> <mi>f</mi> <mn>0</mn> <mn>2</mn> </msubsup> </mrow> <msup> <mi>B</mi> <mn>2</mn> </msup> </mfrac> <mo>+</mo> <mfrac> <mrow> <mn>20</mn> <msubsup> <mi>f</mi> <mn>0</mn> <mn>3</mn> </msubsup> </mrow> <msup> <mi>B</mi> <mn>3</mn> </msup> </mfrac> <mi>ln</mi> <mrow> <mo>(</mo> <mfrac> <mrow> <mn>2</mn> <msub> <mi>f</mi> <mn>0</mn> </msub> <mo>+</mo> <mi>B</mi> </mrow> <mrow> <mn>2</mn> <msub> <mi>f</mi> <mn>0</mn> </msub> <mo>-</mo> <mi>B</mi> </mrow> </mfrac> <mo>)</mo> </mrow> </mrow> <mo>)</mo> </mrow> <mo>,</mo> </mrow>
    Wherein, ΔΦr3LeIt is distance to maximum phase error three times.
  7. 7. a kind of broad band low frequency Space-borne SAR Imaging ionosphere phase error compensation method according to claim 1, its feature It is：TEC in step (6)It is average=(TEC1+TEC2+TEC3+TEC4)/4。
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