CN113063806A - Method for inverting bare soil moisture profile by using multichannel microwave radiation data - Google Patents

Method for inverting bare soil moisture profile by using multichannel microwave radiation data Download PDF

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CN113063806A
CN113063806A CN202110302226.7A CN202110302226A CN113063806A CN 113063806 A CN113063806 A CN 113063806A CN 202110302226 A CN202110302226 A CN 202110302226A CN 113063806 A CN113063806 A CN 113063806A
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soil
soil moisture
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张涛
王光辉
齐建伟
戴海伦
刘宇
艾萍
陆尘
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Ministry Of Natural Resources Land Satellite Remote Sensing Application Center
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Abstract

The invention relates to the technical field of microwave remote sensing, and discloses a method for inverting a bare soil moisture profile by using multi-channel microwave radiation data, which comprises the steps of inverting soil moisture of different channels respectively according to a soil moisture inversion algorithm by using single-channel microwave radiation data; calculating penetration depths under different observation conditions according to microwave radiation signal penetration depth calculation models with different observation frequencies and different observation angles; and calculating the soil moisture content of different depth layers on the vertical section of the soil by using the soil moisture of different channels and the penetration depth under different observation conditions. The invention fully utilizes observation information of different frequencies and different angles of passive microwave remote sensing, inverts the soil moisture of different depths of the vertical section of the soil according to the characteristic that microwave radiation signals of different observation channels have different penetrating abilities to the earth surface, and is beneficial to improving the applicability of passive microwave remote sensing soil moisture products of different depths in the land process mode and the global change research.

Description

Method for inverting bare soil moisture profile by using multichannel microwave radiation data
Technical Field
The invention relates to the technical field of microwave remote sensing, in particular to a method for inverting a bare soil moisture profile by using multichannel microwave radiation data.
Background
The microwave remote sensing earth observation technology obtains earth surface parameter information by measuring radiation or scattering signals of electromagnetic waves in earth surface microwave wave bands. Compared with the optical wave band, the electromagnetic wave of the microwave wave band can penetrate through cloud mist, has certain penetrating power to vegetation and soil on the ground surface, and can obtain information of the inside of a vegetation layer and a soil layer with a certain depth. The passive microwave remote sensing has the characteristics of relatively simple data processing flow and low spatial resolution, can acquire data of large regional scales of the world or the country and the like, and plays an important role in the inversion of parameters such as vegetation, soil, freeze thawing, snow accumulation and the like.
Because microwave remote sensing has certain penetrability to vegetation and soil, the signal that microwave remote sensing sensor obtained is the synthesis of earth's surface to a certain degree of depth soil horizon signal, and surface parameters such as soil moisture that the retrieval obtained is also the average value of soil horizon moisture content of certain thickness. In a traditional passive microwave remote sensing soil moisture inversion algorithm or a product, soil moisture inversion is mainly performed by using a certain frequency or comprehensively using observation signals of certain frequencies, and the result does not consider or blurs the thickness of a soil layer represented by a soil moisture inversion result to a certain extent. Because the remote sensing inversion soil moisture cannot be determined in the algorithm to be the value of a thick soil layer, a plurality of uncertainties are brought to the processing and application of passive microwave remote sensing data products.
Disclosure of Invention
The invention provides a method for inverting a bare soil moisture profile by using multi-channel microwave radiation data, thereby solving the problems in the prior art.
The invention provides a method for inverting a bare soil moisture profile by using multichannel microwave radiation data, which comprises the following steps of:
s1) establishing a single-channel microwave radiation data inversion soil moisture algorithm by using a microwave radiation transmission model, and respectively inverting the soil moisture of different channels according to the single-channel microwave radiation data inversion soil moisture algorithm;
s2) establishing microwave radiation signal penetration depth calculation models with different observation frequencies and different observation angles, and calculating penetration depths under different observation conditions according to the microwave radiation signal penetration depth calculation models with different observation frequencies and different observation angles;
s3) calculating the soil moisture content of different depth layers on the vertical section of the soil by using the soil moisture of different channels and the penetration depth under different observation conditions, and obtaining the information of the soil moisture section.
Further, in step S1), a single-channel microwave radiation data inversion soil moisture algorithm is established by using the microwave radiation transmission model, and soil moisture in different channels is inverted respectively according to the single-channel microwave radiation data inversion soil moisture algorithm, including the following steps:
s11) obtaining the H-polarization microwave radiation bright temperature TB (f, theta) of the bare soil, wherein theta represents an observation angle, and f represents observation frequency;
s12), obtaining the surface temperature T, calculating the rough surface emissivity E (f, theta) according to the H polarization microwave radiation brightness temperature TB (f, theta) of the bare soil and the surface temperature T, and calculating the rough surface emissivity
Figure BDA0002986709490000021
Calculating a rough surface reflectivity R (f, θ) from the rough surface emissivity E (f, θ), the rough surface reflectivity R (f, θ) being 1-E (f, θ);
s13) establishing a rough earth surface microwave radiation model according to the rough earth surface reflectivity R (f, theta), wherein the rough earth surface microwave radiation model is that R (f, theta) is R (f, theta) e-h·cosθCalculating the reflectivity r (f, theta) of the smooth earth surface according to the rough earth surface microwave radiation model, wherein h is a roughness parameter, and e is a natural constant;
s14) establishing a relation between the soil dielectric constant epsilon and the smooth earth surface reflectivity r (f, theta)
Figure BDA0002986709490000022
Figure BDA0002986709490000023
Calculating the dielectric constant of the soil according to the relation between the dielectric constant epsilon of the soil and the reflectivity r (f, theta) of the smooth earth surface
Figure BDA0002986709490000024
S15) establishing a relation between soil moisture and soil dielectric constant by using a soil mixed dielectric constant model
Figure BDA0002986709490000031
vwIndicating the water content of the soil; p represents the porosity of the soil, and the porosity of the soil is the ratio of the volume weight to the density of solid substances in the soil; epsilonaAnd εrAir dielectric constant and soil particle dielectric constant respectively; subscripts a, s, i, w represent air, soil particles, ice and water, respectively, corresponding to constituent substances in soil, the dielectric constant of water
Figure BDA0002986709490000032
εinfDenotes the dielectric constant parameter,. epsilonw0And e represents a first parameter and a second parameter, respectively, related to the temperature, the first parameter ew0=88.045-0.4147·T+6.295·10-4·T2+ 1.075·10-5·T3The second parameter ∈ ═ 1.1109 · 10-10-3.824·10-12·T+6.938· 10-14·T2-5.096·10-16·T3,vtDenotes the transition water content, gamma is an empirical parameter, the transition water content vt0.49 (0.06774-0.00064sand +0.00478clay) +0.165, empirical parameter γ -0.57 (0.06774-0.00064sand +0.00478clay) +0.481, sand and clay representing the soil sand and clay contents, respectively;
s16) solving the soil water content according to the relation between the soil water and the soil dielectric constant if vw≤vtAnd then the water content of the soil
Figure BDA0002986709490000033
a. b and c are respectively a first coefficient, a second coefficient and a third coefficient, wherein the first coefficient
Figure BDA0002986709490000034
Second coefficient b ═ epsiloni-1, third coefficient c ═ (1-P) epsilonr+ P- ε; if v isw>vtAnd then the water content of the soil
Figure BDA0002986709490000035
Further, in step S2), building a penetration depth calculation model of microwave radiation signals with different observation frequencies and different observation angles, and calculating penetration depths under different observation conditions according to the penetration depth calculation model of microwave radiation signals with different observation frequencies and different observation angles, including the following steps:
s21) calculating the dielectric constant of the soil with different observation frequencies by using the soil mixed dielectric constant model in the step S15);
s22) calculating penetration depths of different observation frequencies by using soil dielectric constants of different observation frequencies
Figure BDA0002986709490000036
V represents the speed of light, and epsilon' respectively represent the real part and the imaginary part of the soil dielectric constant calculated in the step S21);
s23) calculating the transmission angle tau of the electromagnetic wave in two media of air and soil under different observation angles, wherein the transmission angle tau is
Figure RE-GDA0003046056030000041
θ represents an observation angle;
s24) the penetration depth d in the vertical direction of the soil layer at different observation angles is calculated, where the penetration depth d is η · cos τ.
Further, in step S3), the method for obtaining the soil moisture profile information includes the following steps:
s31) sequencing the penetration depths under different observation conditions from small to large to obtain N groups of penetration depth and soil moisture data, wherein the N groups of penetration depth and soil moisture data are respectively (d)1,v1)、(d2,v2)、…、 (dN,vN) Where d denotes penetration depth, v denotes soil moisture, and subscripts 1, 2, …, N denote different depth layer numbers, upsilon, respectively1Denotes d1Soil moisture of the thick soil layer;
s32) calculating the soil moisture content of different depth layers on the vertical section of the soil
Figure BDA0002986709490000042
vj-1,jRepresenting a depth dj-1To djJ is 2 to N.
The invention has the beneficial effects that: the method comprises the steps of establishing a single-channel microwave radiation data inversion soil moisture algorithm by using a microwave radiation transmission model, performing inversion to obtain soil moisture corresponding to a single channel, obtaining penetration depths under different observation conditions by using microwave radiation signal penetration depth calculation models with different frequencies and different angles, and calculating the soil moisture of different depth layers of a vertical section of the soil by using the soil moisture inverted by different channels and the corresponding penetration depths to obtain soil moisture section information. The method fully utilizes observation information of different frequency and different angles of passive microwave remote sensing, inverts the soil moisture of different depths of the vertical section of the soil according to the characteristic that the penetration capacity of microwave radiation signals of different observation channels to the earth surface is different, further excavates the application potential of passive microwave remote sensing data in soil moisture inversion work, and is beneficial to improving the applicability of passive microwave remote sensing soil moisture products of different depths in a land surface process mode and global change research.
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In order to more clearly illustrate the technical solutions of the embodiments of the present invention, the drawings used in the embodiments are briefly described below, and it is obvious that the drawings in the following description are some embodiments of the present invention, and it is obvious for those skilled in the art to obtain other drawings based on the drawings without creative efforts.
Fig. 1 is a schematic flow chart of a method for inverting a bare soil moisture profile by using multi-channel microwave radiation data according to a first embodiment of the present disclosure.
Fig. 2 is a schematic diagram of a soil moisture profile calculation according to the first embodiment.
Detailed Description
In order to make the objects, technical solutions and advantages of the present invention more apparent, the present invention is further described in detail below with reference to the accompanying drawings. It should be understood that the detailed description and specific examples, while indicating the invention, are intended for purposes of illustration only and are not intended to limit the scope of the invention. It is noted that the terms "comprises" and "comprising," and any variations thereof, in the description and claims of the present invention and the above-described drawings are intended to cover a non-exclusive inclusion, such that a process, method, system, article, or apparatus that comprises a list of elements is not necessarily limited to those elements, but may include other elements not expressly listed or inherent to such process, method, article, or apparatus.
The embodiment I is a method for inverting a bare soil moisture profile by using multi-channel microwave radiation data, and comprises the following steps of:
s1) establishing a single-channel microwave radiation data inversion soil moisture algorithm by using a microwave radiation transmission model, and respectively inverting the soil moisture of different channels according to the single-channel microwave radiation data inversion soil moisture algorithm, wherein the method comprises the following steps:
s11) obtaining the H-polarization microwave radiation bright temperature TB (f, theta) of the bare soil, wherein theta represents an observation angle, and f represents observation frequency; the bands obtained in this example were the L band (1.4GHz), the C band (6.925GHz), the X band (10.65GHz), and the ka band (36.5GHz), and the observation angles were 30 degrees to 60 degrees with 10 degrees intervals.
S12), obtaining the surface temperature T, calculating the rough surface emissivity E (f, theta) according to the H polarization microwave radiation brightness temperature TB (f, theta) of the bare soil and the surface temperature T, and calculating the rough surface emissivity
Figure BDA0002986709490000061
Calculating a rough surface reflectivity R (f, θ) from the rough surface emissivity E (f, θ), the rough surface reflectivity R (f, θ) being 1-E (f, θ); the sum of the rough surface emissivity E (f, θ) and the rough surface reflectivity R (f, θ) is 1, that is, E (f, θ) + R (f, θ) is 1; in this example, the surface temperature is calculated using the ka band (36.5GHz) V-polarized light temperature, and is expressed as: t is 1.11 · TB (36.5) -15.2.
S13) establishing a rough earth surface microwave radiation model according to the rough earth surface reflectivity R (f, theta), wherein the rough earth surface microwave radiation model is that R (f, theta) is R (f, theta) e-h·cosθAnd calculating the reflectivity r (f, theta) of the smooth earth surface according to a rough earth surface microwave radiation model, wherein e is a natural constant and the value of e is about 2.71828. h is a roughness parameter, and values of different wave bands are different in the example, the value of the L wave band is 0.15, the value of the C wave band is 0.11, and the value of the X wave band is 0.1.
S14) establishing a relation between the soil dielectric constant epsilon and the smooth earth surface reflectivity r (f, theta)
Figure BDA0002986709490000062
Figure BDA0002986709490000063
Calculating the dielectric constant of the soil according to the relation between the dielectric constant epsilon of the soil and the reflectivity r (f, theta) of the smooth earth surface
Figure BDA0002986709490000064
S15) establishing a relation between soil moisture and soil dielectric constant by using a soil mixed dielectric constant model
Figure BDA0002986709490000065
vwIndicating the water content of the soil; p represents the porosity of the soil, and the porosity of the soil is the ratio of the volume weight to the density of solid substances in the soil; epsilonaAnd εrAir dielectric constant and soil particle dielectric constant respectively; subscripts a, s, i, w represent voids corresponding to constituent substances in the soil, respectivelyIn the embodiment, the real part of the dielectric constant of the air is 1, and the imaginary part is 0; the real part of the dielectric constant of the soil particles is 4.9, and the imaginary part is 0; the real part of the dielectric constant of ice is about 3.2, and the imaginary part is 0.1; the dielectric constant of water is temperature and frequency dependent, the dielectric constant of water
Figure BDA0002986709490000066
εinfDenotes the dielectric constant parameter,. epsiloninfThe value is 4.9, epsilonw0And e represents a first parameter and a second parameter, respectively, related to the temperature, the first parameter ew0=88.045-0.4147·T+6.295·10-4·T2+1.075·10-5·T3The second parameter ∈ ═ 1.1109 · 10-10-3.824· 10-12·T+6.938·10-14·T2-5.096·10-16·T3,vtRepresenting the transition water content, gamma being an empirical parameter, both parameters being determined by the soil texture, the transition water content vt0.49 (0.06774-0.00064sand +0.00478clay) +0.165, empirical parameter γ -0.57 (0.06774-0.00064sand +0.00478clay) +0.481, sand and clay representing the soil sand and clay contents, respectively; in this example, sand is 25% and clay is 18%.
S16) solving the soil water content according to the relation between the soil water and the soil dielectric constant if vw≤vtAnd then the water content of the soil
Figure BDA0002986709490000071
a. b and c are respectively a first coefficient, a second coefficient and a third coefficient, wherein the first coefficient
Figure BDA0002986709490000075
Second coefficient b ═ epsiloni-1, third coefficient c ═ (1-P) epsilonr+ P- ε; if v isw>vtAnd then the water content of the soil
Figure BDA0002986709490000072
S2) establishing microwave radiation signal penetration depth calculation models with different observation frequencies and different observation angles, and calculating penetration depths under different observation conditions according to the microwave radiation signal penetration depth calculation models with different observation frequencies and different observation angles, wherein the method comprises the following steps:
s21) calculating the dielectric constant of the soil with different observation frequencies by using the soil mixed dielectric constant model in the step S15);
s22) calculating penetration depths of different observation frequencies by using soil dielectric constants of different observation frequencies
Figure BDA0002986709490000073
V represents the speed of light, and epsilon' respectively represent the real part and the imaginary part of the soil dielectric constant calculated in the step S21);
s23) calculating the transmission angle tau of the electromagnetic wave in two media of air and soil under different observation angles, wherein the transmission angle tau is
Figure RE-GDA0003046056030000075
θ represents an observation angle;
s24) the penetration depth d in the vertical direction of the soil horizon at different observation angles is calculated, where the penetration depth d is η · cos τ.
S3) calculating the soil moisture content of different depth layers on the vertical profile of the soil by using the soil moisture of different channels and the penetration depth under different observation conditions, and obtaining the soil moisture profile information, as shown in fig. 2, including the following steps:
s31) under the same soil condition, the penetration capacity of different observation frequencies to the soil is different, and generally, the lower the frequency, the stronger the penetration capacity. Sequencing the penetration depths under different observation conditions from small to large to obtain N groups of penetration depth and soil moisture data, wherein the N groups of penetration depth and soil moisture data are respectively (d)1,v1)、 (d2,v2)、…、(dN,vN) Where d denotes penetration depth, v denotes soil moisture, and subscripts 1, 2, …, N denote different depth layer numbers, upsilon, respectively1Denotes d1The soil moisture of the thick soil layer, i.e., the soil moisture content of the shallowest layer.In this example, 3 bands and 4 angles are calculated, and 12 groups of data are calculated, namely N-12.
S32) calculating the soil moisture content of different depth layers on the vertical section of the soil
Figure BDA0002986709490000081
vj-1,jRepresenting a depth dj-1To djJ is 2 to N.
With the development of passive microwave remote sensing sensor technology, the observation frequency covers the L wave band (about 1.4GHz) to the Ka wave band (36.5GHz), and the observation angle covers 0-60 degrees, so that multi-frequency and multi-angle observation information is provided. Due to the fact that the penetration capacities of the electromagnetic waves with different frequencies to the soil are different, observation information of different frequencies and different angles of passive microwave remote sensing is fully utilized, soil moisture of different depths of a vertical section of the soil is obtained according to the characteristic that microwave radiation signals of different observation channels have different penetration capacities to the ground surface, and therefore the applicability of passive microwave remote sensing soil moisture products with different depths in a land surface process mode and global change research is improved.
By adopting the technical scheme disclosed by the invention, the following beneficial effects are obtained:
the method comprises the steps of establishing a single-channel microwave radiation data inversion soil moisture calculation method by using a microwave radiation transmission model, performing inversion to obtain soil moisture corresponding to a single channel, obtaining penetration depths under different observation conditions by using microwave radiation signal penetration depth calculation models with different frequencies and different angles, and calculating the soil moisture of different depth layers of a vertical section of the soil by using the soil moisture inverted by different channels and the corresponding penetration depths to obtain soil moisture section information. The method fully utilizes observation information of different frequencies and different angles of passive microwave remote sensing, inverts the soil moisture of different depths of the vertical section of the soil according to the characteristic that the microwave radiation signals of different observation channels have different penetrating abilities to the earth surface, further excavates the application potential of passive microwave remote sensing data in soil moisture inversion work, and is beneficial to improving the applicability of passive microwave remote sensing soil moisture products of different depths in a land surface process mode and global change research.
The foregoing is only a preferred embodiment of the present invention, and it should be noted that it will be apparent to those skilled in the art that various modifications and improvements can be made without departing from the principle of the present invention, and such modifications and improvements should also be considered within the scope of the present invention.

Claims (4)

1. A method for inverting a bare soil moisture profile by using multichannel microwave radiation data is characterized by comprising the following steps:
s1) establishing a single-channel microwave radiation data inversion soil moisture algorithm by using a microwave radiation transmission model, and respectively inverting the soil moisture of different channels according to the single-channel microwave radiation data inversion soil moisture algorithm;
s2) establishing microwave radiation signal penetration depth calculation models with different observation frequencies and different observation angles, and calculating penetration depths under different observation conditions according to the microwave radiation signal penetration depth calculation models with different observation frequencies and different observation angles;
s3) calculating the soil moisture content of different depth layers on the vertical profile of the soil by using the soil moisture of the different channels and the penetration depth under different observation conditions, and obtaining the profile information of the soil moisture.
2. The method for inverting the bare soil moisture profile by using the multi-channel microwave radiation data according to claim 1, wherein in step S1), a single-channel microwave radiation data inversion soil moisture algorithm is established by using a microwave radiation transmission model, and soil moisture of different channels is inverted respectively according to the single-channel microwave radiation data inversion soil moisture algorithm, and the method comprises the following steps:
s11) obtaining the H-polarization microwave radiation bright temperature TB (f, theta) of the bare soil, wherein theta represents an observation angle, and f represents an observation frequency;
s12) obtaining the surface temperature T, and calculating the rough surface emission according to the H polarization microwave radiation bright temperature TB (f, theta) of the bare soil and the surface temperature TEmissivity E (f, θ), said rough surface emissivity
Figure FDA0002986709480000011
Calculating a rough surface reflectivity R (f, θ) from the rough surface emissivity E (f, θ), the rough surface reflectivity R (f, θ) being 1-E (f, θ);
s13) establishing a rough earth surface microwave radiation model according to the rough earth surface reflectivity R (f, theta), wherein the rough earth surface microwave radiation model is that R (f, theta) is R (f, theta) e-h·cosθCalculating the reflectivity r (f, theta) of the smooth earth surface according to the rough earth surface microwave radiation model, wherein h is a roughness parameter, and e is a natural constant;
s14) establishing a relation between the soil dielectric constant epsilon and the smooth earth surface reflectivity r (f, theta)
Figure FDA0002986709480000021
Figure FDA0002986709480000022
Calculating the dielectric constant of the soil according to the relation between the dielectric constant epsilon of the soil and the reflectivity r (f, theta) of the smooth earth surface
Figure FDA0002986709480000023
S15) establishing a relation between soil moisture and soil dielectric constant by using a soil mixed dielectric constant model
Figure FDA0002986709480000024
vwIndicating the water content of the soil; p represents the porosity of the soil, and the porosity of the soil is the ratio of the volume weight to the density of solid substances of the soil; epsilonaAnd εrAir dielectric constant and soil particle dielectric constant respectively; subscripts a, s, i, w represent air, soil particles, ice and water, respectively, corresponding to constituent substances in the soil, the dielectric constant of water
Figure FDA0002986709480000025
εinfDenotes the dielectric constant parameter,. epsilonw0And e represents a first parameter and a second parameter, respectively, related to the temperature, said first parameter ew0=88.045-0.4147·T+6.295·10-4.T2+1.075·10-5·T3The second parameter e is 1.1109-10-10-3.824·10-12·T+6.938·10-14·T2-5.096·10-16·T3,vtDenotes the transition water content, gamma is an empirical parameter, the transition water content vt0.49 (0.06774-0.00064sand +0.00478clay) +0.165, the empirical parameter γ -0.57 (0.06774-0.00064sand +0.00478clay) +0.481, sand and clay representing the soil sand and clay contents, respectively;
s16) solving the soil water content according to the relation between the soil water and the soil dielectric constant if vw≤vtAnd then the water content of the soil
Figure FDA0002986709480000026
a. b and c are respectively a first coefficient, a second coefficient and a third coefficient, wherein the first coefficient
Figure FDA0002986709480000027
The second coefficient b ═ epsiloni-1, said third coefficient c ═ (1-P) epsilonr+ P- ε; if v isw>vtAnd then the water content of the soil
Figure FDA0002986709480000028
3. The method for inverting the bare soil water profile through the multi-channel microwave radiation data according to claim 2, wherein in step S2), a penetration depth calculation model of microwave radiation signals with different observation frequencies and different observation angles is established, and penetration depths under different observation conditions are calculated according to the penetration depth calculation model of microwave radiation signals with different observation frequencies and different observation angles, and the method comprises the following steps:
s21) calculating the dielectric constant of the soil with different observation frequencies by using the soil mixed dielectric constant model in the step S15);
s22) calculating penetration depths of different observation frequencies by using the soil dielectric constants of different observation frequencies
Figure RE-FDA0003046056020000031
V represents the speed of light, and epsilon' respectively represent the real part and the imaginary part of the soil dielectric constant calculated in the step S21);
s23) calculating the transmission angle tau of the electromagnetic wave in two media of air and soil under different observation angles, wherein the transmission angle tau is
Figure RE-FDA0003046056020000032
θ represents an observation angle;
s24) the penetration depth d in the vertical direction of the soil layer at different observation angles is calculated, where the penetration depth d is η · cos.
4. The method for inverting the bare soil moisture profile according to the multi-channel microwave radiation data of claim 3, wherein in step S3), the soil moisture content of different depth layers on the vertical soil profile is calculated by using the soil moisture of different channels and the penetration depths under different observation conditions, so as to obtain the soil moisture profile information, and the method comprises the following steps:
s31) sequencing the penetration depths under different observation conditions from small to large to obtain N groups of penetration depth and soil moisture data, wherein the N groups of penetration depth and soil moisture data are respectively (d)1,v1)、(d2,v2)、…、(dN,vN) Where d denotes penetration depth, v denotes soil moisture, subscripts 1, 2, …, N denote different depth layer numbers, respectively, v1Denotes d1Soil moisture of the thick soil layer;
s32) calculating the soil moisture content of different depth layers on the vertical section of the soil
Figure FDA0002986709480000033
vj-1,jRepresenting a depth dj-1To djJ is 2 to N.
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