基于坐标-时间函数（CT-PIM）的矿区时序InSAR形变预计—以淮安盐矿为例

张腾飞; 邢学敏; 彭葳; 朱珺; 刘祥彬; 葛家旺; 雷敏超

doi:10.11834/jrs.20222172

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基于坐标-时间函数（CT-PIM）的矿区时序InSAR形变预计—以淮安盐矿为例
Time Series InSAR Deformation Prediction for Salt Mining Area Based on Coordinate-Time Function （CT-PIM） — A Case Study of Huaian Salt Mine
2022年页码：1-16
网络出版日期： 2022-07-07 ，
DOI： 10.11834/jrs.20222172

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张腾飞，邢学敏，彭葳，朱珺，刘祥彬，葛家旺，雷敏超.XXXX.基于坐标-时间函数（CT-PIM）的矿区时序InSAR形变预计—以淮安盐矿为例.遥感学报，XX（XX）： 1-16

ZHANG Tengfei，XING Xuemin，PENG Wei，ZHU Jun，LIU Xiangbin，GE Jiawang，LEI Minchao. XXXX. Time Series InSAR Deformation Prediction for Salt Mining Area Based on Coordinate-Time Function （CT-PIM） — A Case Study of Huaian Salt Mine. National Remote Sensing Bulletin， XX（XX）：1-16
张腾飞，邢学敏，彭葳，朱珺，刘祥彬，葛家旺，雷敏超.XXXX.基于坐标-时间函数（CT-PIM）的矿区时序InSAR形变预计—以淮安盐矿为例.遥感学报，XX（XX）： 1-16 DOI： 10.11834/jrs.20222172.

ZHANG Tengfei，XING Xuemin，PENG Wei，ZHU Jun，LIU Xiangbin，GE Jiawang，LEI Minchao. XXXX. Time Series InSAR Deformation Prediction for Salt Mining Area Based on Coordinate-Time Function （CT-PIM） — A Case Study of Huaian Salt Mine. National Remote Sensing Bulletin， XX（XX）：1-16 DOI： 10.11834/jrs.20222172.

基于坐标-时间函数（CT-PIM）的矿区时序InSAR形变预计—以淮安盐矿为例 Time Series InSAR Deformation Prediction for Salt Mining Area Based on Coordinate-Time Function （CT-PIM） — A Case Study of Huaian Salt Mine 1 2 first-author 张腾飞，研究方向为矿区时序InSAR建模与形变预计。Email： tengfei@stu.csust.edu.cn 张腾飞，研究方向为矿区时序InSAR建模与形变预计。Email： tengfei@stu.csust.edu.cn tengfei@stu.csust.edu.cn 1 2 corresp 邢学敏，研究方向为时序InSAR技术在地表变形监测中的应用。Email： xuemin.xing@csust.edu.cn 邢学敏，研究方向为时序InSAR技术在地表变形监测中的应用。Email： xuemin.xing@csust.edu.cn xuemin.xing@csust.edu.cn 1 2 1 2 1 2 1 2 1 2 长沙理工大学交通运输工程学院,长沙 410114 长沙理工大学交通运输工程学院,长沙 410114 School of Traffic and Transportation Engineering, Changsha University of Science & Technology, Changsha 410114, China School of Traffic and Transportation Engineering, Changsha University of Science & Technology, Changsha 410114, China 长沙理工大学雷达遥感技术应用研究室,长沙 410114 长沙理工大学雷达遥感技术应用研究室,长沙 410114 Laboratory of Radar Remote Sensing Applications, Changsha University of Science & Technology, Changsha 410114, China Laboratory of Radar Remote Sensing Applications, Changsha University of Science & Technology, Changsha 410114, China 对矿区开展长期形变监测和后续形变预计对于预防矿山开采所引发的潜在安全问题和矿山生态环境保护具有十分重要的意义。利用合成孔径雷达干涉测量技术（Interferometric Synthetic Aperture Radar， InSAR）与概率积分法（Probability Integral Method， PIM）结合是较为有效的矿区形变预计方法。然而，目前矿区时序InSAR形变建模环节中大多数采用纯经验数学模型，未顾及开采沉陷机制，严重影响获取的形变观测值的精度；不准确的InSAR形变观测值将误差传递至后续形变预计，进而导致误差累积。本文将坐标-时间函数（Coordinate Time， CT）引入时序InSAR建模环节，构建了坐标-时间函数预计模型（CT-PIM）以取代传统纯经验数学模型；再基于InSAR相位观测值求解未知参数，将构建的CT-PIM直接用于矿区形变预计，避免了误差的传递，以改善形变预计精度。分别开展了模拟实验和真实实验对提出预计方法进行验证。模拟实验显示：模型预计获取的时序形变与模拟真值的均方根误差（RMSE）为±4.6 mm。真实实验利用覆盖淮安市一盐矿区的 35 景 Sentinel-1A SAR卫星影像开展，获取了研究区域2019年3月30日至2019年7月28日的预计结果，结果显示测区最大预计沉降量为152 mm。本文所提方法相比传统多速率模型，建模精度提升了38.2%；相比传统静态概率积分预计方法获取的形变结果，精度提升了39.1%。本文方法为预防盐矿区采矿引起的地质灾害问题提供了有力的工具，为盐矿区生产安全和生态环境保护提供参考。 (Objective) 2 Long-term monitoring and the subsequential prediction of deformation for salt mining area is essential to the safety prevention and environmental protection of mining areas. The combination of Interferometric Synthetic Aperture Radar (InSAR) technique with Probability Integral Method (PIM) has proven to be powerful in predicting the deformation of mining areas. However, the single Multi-Temporal InSAR (MT-InSAR) has a limitation in that it can only obtain the deformation sequences during SAR acquisition dates, and the subsequent future displacement beyond the span of the SAR observations cannot be acquired. Besides, traditional mathematical empirical models are mostly used in time series modelling of mining areas, ignoring the underground mining mechanisms, which seriously affect the accuracy of the observations. Inaccurate InSAR deformation monitoring results will transmit errors to forward predicted subsidence, which may induce more significant errors. (Method) 2 In this study, the Coordinate-Time (CT) Function is introduced into the time series InSAR deformation modelling, and a Coordinate Time Function prediction model (CT-PIM) is constructed to replace the traditional mathematical empirical models, which can better describe the dynamic evolution disciplines of the underground mining subsidence in the InSAR deformation modelling procedure. The unknown CT-PIM parameters can be estimated directly via the InSAR time series phase observations, and the constructed CT-PIM is directly used in the deformation prediction of the mining area, which can avoid the error propagation from the InSAR-generated deformations and improve the deformation prediction accuracy. (Result) 2 The new approach has been tested by both simulation and real data experiments. The simulation results show the Root Mean Square Error (RMSE) between the time series deformation prediction of the model and the simulated true value is estimated to be ±4.6 mm, which implies that the proposed method is of promising accuracy. The real experiment was carried out using a total of 35 Sentinel-1A SAR images covering the salt mining area in Huaian city, and the deformation prediction of the study area from March 30, 2019 to July 28, 2019 were obtained. The results show that the maximum settlement of deformation prediction in the study area is 152 mm. The modelling accuracy with an improvement of 38.2% compared to traditional SBAS-InSAR, and the deformation prediction accuracy with an improvement of 39.1% compared to traditional static PIM prediction method. (Conclusion) 2 The CT-PIM was used as a substitute for traditional MT-InSAR pure empirical models and was applied for predicting the dynamic deformation over salt mining area, which provides a more robust tool for the forecasting of mining-induced hazards. The above results show that the CT-PIM can describe the temporal dynamical characteristics of the mining-induced subsidence more realistically, which can avoid the secondary error propagation, as well as a reference for safety management and ensuring the environment protection. InSAR 矿区坐标-时间函数地表沉陷形变预计 InSAR mine Coordinate-Time function land subsidence deformation prediction 1　引言 1 我国是世界上拥有极为丰富盐类矿产资源的少数国家之一，已探明的储量约为44500亿吨。由于盐矿区开采方式的全方位性和不确定性，采空区易出现塌陷和卤水外泄，严重破坏矿区生态环境（ Zhang等，2019 Zhang等，2019 ）。对盐矿区开展长期动态的形变监测与形变预计对于预防矿山开采所引发的潜在安全问题和矿山生态环境保护具有十分重要的意义。多时相合成孔径雷达差分干涉测量技术（Multi-Temporal Interferometric Synthetic Aperture Radar， MT-InSAR）相比传统InSAR技术有着诸多优势，已在矿区地表形变监测（ Gama等，2017 Gama等，2017 ；肖亮等，2019 肖亮等，2019 ； Chen 等，2020 Chen 等，2020 ）中体现出巨大潜力。然而，单一MT-InSAR技术只能实现SAR影像覆盖时间周期内形变的监测，无法实现SAR影像后期形变的预计。为解决这一局限，大量学者将概率积分法与InSAR技术结合用于矿区形变预计（ Yang 等，2016 Yang 等，2016 ； Fan等，2018 Fan等，2018 ；朱建军等，2013 朱建军等，2013 ； Yang 等，2017a Yang 等，2017a ； Yang 等，2017c Yang 等，2017c ； Xing 等，2021 Xing 等，2021 ）。其基本思想是利用MT-InSAR技术获取的地表形变序列作为概率积分预计模型（Probability Integral Method， PIM）的观测值以反演概率积分预计参数，再用概率积分预计模型实现矿区形变预计。然而，这一方法存在一些局限：首先，MT-InSAR形变建模环节大多采用传统纯经验数学模型（如：线性模型、季节性模型、多项式模型等）来拟合形变随时间演化的规律，没有顾及开采沉陷机制，不能真实准确地反应矿区开采形变规律，造成形变观测值不准确；其次，用不准确的InSAR形变观测值来估计PIM参数，将会导致误差累积传递至后续形变预计；最后，形变监测中采用的数学模型与后续预计采用的PIM在理论意义上并不一致，这不利于形变结果的物理解译。本文采用SBAS-InSAR技术作为盐矿区形变监测的主要手段，利用矿区动态地表形变的坐标-时间函数（Coordinate Time， CT）（朱广轶等，2011 朱广轶等，2011 ），构建基于InSAR差分干涉相位与概率积分参数之间的时序函数模型（CT-PIM），以替换传统InSAR形变模型，新模型融入了概率积分参数，顾及了矿区开采沉陷机制，更具物理意义；这一模型可利用InSAR相位观测值直接求解概率积分预计参数，且可直接用于SAR影像覆盖后期形变的预计，避免了误差的二次传递，可极大提高形变预计精度，为盐矿区生产安全和生态环境保护提供数据支持。 2　方法原理 1 2.1　动态坐标-时间函数概率积分模型（CT-PIM） 2 文献（朱广轶等，2011 朱广轶等，2011 ）将时间参数融入传统静态概率积分预计模型，提出用于矿山地表动态变形预计的坐标-时间函数。因此，可构建地表下沉与概率积分参数间的时间函数（CT-PIM）模型如下 <math display="block" id="M1"><mi>W</mi><mfenced separators="|"><mrow><mi>x</mi><mo>,</mo><mi>y</mi><mo>,</mo><mi>t</mi></mrow></mfenced><mo>=</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><msub><mrow><mi>w</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></mrow></mfrac><mi>W</mi><mfenced separators="|"><mrow><mi>x</mi><mo>,</mo><mi>t</mi></mrow></mfenced><mi>W</mi><mfenced separators="|"><mrow><mi>y</mi><mo>,</mo><mi>t</mi></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=30725342&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=30725341&type= 43.60333252 8.04333401 式中： <math id="M2"><mi>W</mi><mfenced separators="|"><mrow><mi>x</mi><mo>,</mo><mi>y</mi><mo>,</mo><mi>t</mi></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=30725344&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=30725343&type= 18.62666702 4.57200003 为工作面推进时刻为 <math id="M3"><mi>t</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=30725346&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=30725345&type= 1.10066664 2.62466669 时矿区地表任一点 <math id="M4"><mfenced separators="|"><mrow><mi>x</mi><mo>,</mo><mi>y</mi></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=30725532&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=30725531&type= 10.15999985 4.57200003 的沉降量。 <math id="M5"><msub><mrow><mi>w</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=30725350&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=30725349&type= 3.47133350 3.72533321 为该地质采矿条件下的最大沉降值 <math id="M6"><msub><mrow><mi>w</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>=</mo><mi>m</mi><mi>q</mi><mo>⋅</mo><mi mathvariant="normal">c</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">s</mi><mi>α</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=30725352&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=30725351&type= 23.36800003 4.57200003 ；其中： <math id="M7"><mi>m</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=30725448&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=30725447&type= 2.70933342 2.62466669 表示开采厚度， <math id="M8"><mi>q</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=30725654&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=30725653&type= 1.69333339 3.47133350 表示下沉系数， <math id="M9"><mi>α</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=30725524&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=30725523&type= 2.20133328 2.62466669 表示矿层倾角。式中： <math id="M10"><mi>W</mi><mfenced separators="|"><mrow><mi>x</mi><mo>,</mo><mi>t</mi></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=30725428&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=30725412&type= 13.29266548 3.89466691 和 <math id="M11"><mi>W</mi><mfenced separators="|"><mrow><mi>y</mi><mo>,</mo><mi>t</mi></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=30725440&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=30725439&type= 13.29266548 4.57200003 可表示如下：（1）　当矿区开采工作面倾向充分采动时，走向未充分采动时 3 <math display="block" id="M12"><mtable columnalign="left"><mtr><mtd><mtext> </mtext><mi>W</mi><mfenced separators="|"><mrow><mi>x</mi><mo>,</mo><mi>t</mi></mrow></mfenced></mtd></mtr><mtr><mtd><mtext> </mtext><mo>=</mo><mfrac><mrow><mi>w</mi><mfenced separators="|"><mrow><mi>x</mi></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac><mfenced open="[" close="]" separators="|"><mrow><mi>e</mi><mi>r</mi><mi>f</mi><mfenced separators="|"><mrow><mroot><mrow><mi>π</mi></mrow><mrow/></mroot></mrow></mfenced><mo>+</mo><mi>e</mi><mi>r</mi><mi>f</mi><mfenced separators="|"><mrow><mroot><mrow><mi>π</mi></mrow><mrow/></mroot><mfrac><mrow><msub><mrow><mi>v</mi></mrow><mrow><mi>m</mi></mrow></msub><mi>t</mi><mo>-</mo><mi>n</mi><mi>H</mi><mo>-</mo><mi>r</mi></mrow><mrow><mi>r</mi></mrow></mfrac></mrow></mfenced></mrow></mfenced></mtd></mtr><mtr><mtd><mtext> </mtext><mi>x</mi><mo>≤</mo><mi>n</mi><mi>H</mi></mtd></mtr><mtr><mtd><mo>=</mo><mfrac><mrow><mi>w</mi><mfenced separators="|"><mrow><mi>x</mi></mrow></mfenced></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac><mfenced open="[" close="]" separators="|"><mrow><mtable columnalign="left"><mtr><mtd><mi>e</mi><mi>r</mi><mi>f</mi><mfenced separators="|"><mrow><mroot><mrow><mi>π</mi></mrow><mrow/></mroot></mrow></mfenced><mo>+</mo></mtd></mtr><mtr><mtd><mi>e</mi><mi>r</mi><mi>f</mi><mfenced separators="|"><mrow><mroot><mrow><mi>π</mi></mrow><mrow/></mroot><mfenced open="[" close="]" separators="|"><mrow><mfrac><mrow><msub><mrow><mi>v</mi></mrow><mrow><mi>m</mi></mrow></msub><mi>t</mi><mo>-</mo><mi>r</mi><mo>-</mo><mi>x</mi><mo>+</mo><mfrac><mrow><mi>H</mi></mrow><mrow><mi>a</mi><mo>+</mo><mi>b</mi><mi>x</mi></mrow></mfrac><mo>-</mo><mfrac><mrow><mi>H</mi><mfenced separators="|"><mrow><mi>a</mi><mo>+</mo><mi>b</mi><mi>x</mi></mrow></mfenced></mrow><mrow><mn mathvariant="normal">3</mn></mrow></mfrac></mrow><mrow><mi>r</mi></mrow></mfrac></mrow></mfenced></mrow></mfenced></mtd></mtr></mtable></mrow></mfenced></mtd></mtr><mtr><mtd><mtext> </mtext><mi>n</mi><mi>H</mi><mo>≤</mo><mi>x</mi><mo>≤</mo><mn mathvariant="normal">1.3</mn><mi>r</mi></mtd></mtr><mtr><mtd><mtext> </mtext><mo>=</mo><mfrac><mrow><msub><mrow><mi>w</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac><mfenced open="[" close="]" separators="|"><mrow><mi>e</mi><mi>r</mi><mi>f</mi><mfenced separators="|"><mrow><mfrac><mrow><mroot><mrow><mi>π</mi></mrow><mrow/></mroot></mrow><mrow><mi>r</mi></mrow></mfrac></mrow></mfenced><mo>+</mo><mi>e</mi><mi>r</mi><mi>f</mi><mfenced separators="|"><mrow><mroot><mrow><mi>π</mi></mrow><mrow/></mroot><mfrac><mrow><msub><mrow><mi>v</mi></mrow><mrow><mi>m</mi></mrow></msub><mi>t</mi><mo>+</mo><mi>H</mi><mo>/</mo><mi mathvariant="normal">t</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">n</mi><mi>ω</mi><mo>-</mo><mn mathvariant="normal">2</mn><mi>x</mi></mrow><mrow><mi>r</mi></mrow></mfrac></mrow></mfenced></mrow></mfenced></mtd></mtr><mtr><mtd><mtext> </mtext><mn mathvariant="normal">1.3</mn><mi>r</mi><mo>≤</mo><mi>x</mi></mtd></mtr></mtable></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=30725320&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=30725319&type= 86.44467163 69.76532745 式中： <math id="M13"><mi>x</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=30725362&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=30725361&type= 1.69333339 2.62466669 表示矿区任意一高相干点的横坐标； <math id="M14"><mi>w</mi><mfenced separators="|"><mrow><mi>x</mi></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=30725364&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=30725363&type= 7.53533268 3.72533321 为静态下沉量可由 <math id="M15"><mi>w</mi><mfenced separators="|"><mrow><mi>x</mi></mrow></mfenced><mo>=</mo><mfrac><mrow><msub><mrow><mi>w</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac><mfenced open="[" close="]" separators="|"><mrow><mi>e</mi><mi>r</mi><mi>f</mi><mfenced separators="|"><mrow><mroot><mrow><mi>π</mi></mrow><mrow/></mroot><mfrac><mrow><mi>x</mi></mrow><mrow><mi>r</mi></mrow></mfrac></mrow></mfenced><mo>+</mo><mn mathvariant="normal">1</mn></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=30725366&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=30725365&type= 42.84133148 9.14400005 计算； <math id="M16"><mi>e</mi><mi>r</mi><mi>f</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=30725368&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=30725367&type= 3.72533321 3.47133350 为概率积分函数，可表示为 <math id="M17"><mi>e</mi><mi>r</mi><mi>f</mi><mfenced separators="|"><mrow><mfrac><mrow><mroot><mrow><mi>π</mi></mrow><mrow/></mroot></mrow><mrow><mi>r</mi></mrow></mfrac><mi>x</mi></mrow></mfenced><mo>=</mo><mfrac><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mroot><mrow><mi>π</mi></mrow><mrow/></mroot></mrow></mfrac><msubsup><mo>∫</mo><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mfrac><mrow><mroot><mrow><mi>π</mi></mrow><mrow/></mroot></mrow><mrow><mi>r</mi></mrow></mfrac><mi>x</mi></mrow></msubsup><mrow><msup><mrow><mi>e</mi></mrow><mrow><mo>-</mo><msup><mrow><mi>u</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow></msup></mrow><mi>d</mi><mi>u</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=30725322&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=30725321&type= 44.70399857 10.83733368 （其中， <math id="M18"><mi>u</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=30725384&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=30725383&type= 1.86266661 2.62466669 为积分参数， <math id="M19"><mi>r</mi><mo>=</mo><mfrac><mrow><mi>H</mi></mrow><mrow><mi mathvariant="normal">t</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">n</mi><mi>β</mi></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=30725386&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=30725385&type= 13.54666710 8.04333401 为主要影响半径， <math id="M20"><mi>H</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=30725388&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=30725387&type= 2.53999996 2.62466669 为开采深度， <math id="M21"><mi mathvariant="normal">t</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">n</mi><mi>β</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=30725656&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=30725655&type= 7.02733374 3.38666677 为主要影响角正切值）； <math id="M22"><mi>n</mi><mi>H</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=30725392&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=30725391&type= 4.31799984 2.62466669 为启动距离，即地表开始移动时工作面的推进距离； <math id="M23"><mi>n</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=30725528&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=30725527&type= 1.86266661 2.62466669 为启动系数、 <math id="M24"><mi>ω</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=30725396&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=30725395&type= 2.45533323 2.62466669 为静态超前影响角； <math id="M25"><msub><mrow><mi>v</mi></mrow><mrow><mi>m</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=30725398&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=30725397&type= 3.04800010 3.72533321 是工作面在匀速开采条件下的推进速度；系数 <math id="M26"><mi>a</mi><mo>=</mo><mfrac><mrow><mn mathvariant="normal">117</mn><mi>r</mi><mo>-</mo><mi>ω</mi><mi>n</mi><mi>H</mi></mrow><mrow><mn mathvariant="normal">1.3</mn><mi>r</mi><mo>+</mo><mi>n</mi><mi>H</mi></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=30725400&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=30725399&type= 25.14599991 7.36600018 ， <math id="M27"><mi>b</mi><mo>=</mo><mfrac><mrow><mi>ω</mi><mo>-</mo><mn mathvariant="normal">90</mn><mo>°</mo></mrow><mrow><mn mathvariant="normal">1.3</mn><mi>r</mi><mo>+</mo><mi>n</mi><mi>H</mi></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=30725402&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=30725401&type= 21.59000015 7.78933382 。（2）当矿区开采工作面走向充分采动，倾向未充分采动时，地表下沉 <math id="M28"><mi>W</mi><mfenced separators="|"><mrow><mi>y</mi><mo>,</mo><mi>t</mi></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=30725440&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=30725439&type= 13.29266548 4.57200003 可根据式（3）式（3）计算： <math display="block" id="M29"><mi>W</mi><mfenced separators="|"><mrow><mi>y</mi><mo>,</mo><mi>t</mi></mrow></mfenced><mo>=</mo><mfrac><mrow><msub><mrow><mi>w</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac><mfenced open="[" close="]" separators="|"><mrow><mi>e</mi><mi>r</mi><mi>f</mi><mfenced separators="|"><mrow><mroot><mrow><mi>π</mi></mrow><mrow/></mroot><mfrac><mrow><mi>y</mi></mrow><mrow><msub><mrow><mi>r</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow></mfrac></mrow></mfenced><mo>-</mo><mi>e</mi><mi>r</mi><mi>f</mi><mfenced separators="|"><mrow><mroot><mrow><mi>π</mi></mrow><mrow/></mroot><mfrac><mrow><mi>y</mi><mo>-</mo><mi>L</mi></mrow><mrow><msub><mrow><mi>r</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></mrow></mfrac></mrow></mfenced></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=30725371&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=30725370&type= 68.49533844 9.73666668 式中： <math id="M30"><mi>y</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=30725373&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=30725372&type= 1.69333339 3.47133350 表示矿区任意一高相干点的纵坐标； <math id="M31"><msub><mrow><mi>r</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>=</mo><mfrac><mrow><msub><mrow><mi>H</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow><mrow><mi mathvariant="normal">t</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">n</mi><msub><mrow><mi>β</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=30725375&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=30725374&type= 15.83266735 8.04333401 、 <math id="M32"><msub><mrow><mi>r</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>=</mo><mfrac><mrow><msub><mrow><mi>H</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></mrow><mrow><mi mathvariant="normal">t</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">n</mi><msub><mrow><mi>β</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=30725377&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=30725376&type= 15.83266735 8.04333401 分别为下、上山影响半径； <math id="M33"><msub><mrow><mi>H</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=30725379&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=30725378&type= 3.64066648 3.72533321 、 <math id="M34"><msub><mrow><mi>H</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=30725381&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=30725380&type= 3.64066648 3.72533321 分别为下、上山的采深，可根据式 <math id="M35"><msub><mrow><mi>H</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>=</mo><mi>H</mi><mo>-</mo><mfrac><mrow><msub><mrow><mi>D</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac><mi mathvariant="normal">s</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi><mi>α</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=30725413&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=30725382&type= 27.17799950 7.28133297 、 <math id="M36"><msub><mrow><mi>H</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>=</mo><mi>H</mi><mo>+</mo><mfrac><mrow><msub><mrow><mi>D</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac><mi mathvariant="normal">s</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi><mi>α</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=30725415&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=30725414&type= 27.17799950 7.28133297 计算； <math id="M37"><mi>L</mi><mo>=</mo><mfenced separators="|"><mrow><msub><mrow><mi>D</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>-</mo><msub><mrow><mi>s</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>-</mo><msub><mrow><mi>s</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></mrow></mfenced><mfrac><mrow><mi mathvariant="normal">s</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi><mfenced separators="|"><mrow><msub><mrow><mi>θ</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>-</mo><mi>α</mi></mrow></mfenced></mrow><mrow><mi mathvariant="normal">s</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi><msub><mrow><mi>θ</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=30725417&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=30725416&type= 45.63533020 8.97466660 为倾向工作面计算长度； <math id="M38"><msub><mrow><mi>D</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=30725419&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=30725418&type= 3.64066648 3.72533321 为工作面倾向斜长； <math id="M39"><msub><mrow><mi>s</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=30725623&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=30725622&type= 2.45533323 3.72533321 、 <math id="M40"><msub><mrow><mi>s</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=30725625&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=30725624&type= 2.45533323 3.72533321 分别为下、上山拐点偏移距； <math id="M41"><msub><mrow><mi>θ</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>=</mo><mn mathvariant="normal">90</mn><mo>°</mo><mo>-</mo><mi>k</mi><mi>α</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=30725522&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=30725521&type= 20.91266632 4.23333359 为开采影响传播角； <math id="M42"><mi>k</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=30725526&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=30725525&type= 1.69333339 2.62466669 为小于1的常数。需要说明的是，当矿区开采工作面走向未达到充分采动时，则要在公式（2）公式（2）获取的地表下沉 <math id="M43"><mi>W</mi><mfenced separators="|"><mrow><mi>x</mi><mo>,</mo><mi>t</mi></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=30725422&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=30725410&type= 13.29266548 3.89466691 中，融入走向采动程度系数 <math id="M44"><msub><mrow><mi>C</mi></mrow><mrow><mi>x</mi><mi>m</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=30725424&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=30725423&type= 4.74133301 3.72533321 ，可根据式 <math id="M45"><msub><mrow><mi>C</mi></mrow><mrow><mi>x</mi><mi>m</mi></mrow></msub><mo>=</mo><mfrac><mrow><msub><mrow><mi>W</mi></mrow><mrow><mi>x</mi><mi>m</mi></mrow></msub></mrow><mrow><msub><mrow><mi>W</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></mrow></mfrac><mo><</mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=30725426&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=30725425&type= 21.75933456 7.95866632 计算；式中 <math id="M46"><msub><mrow><mi>W</mi></mrow><mrow><mi>x</mi><mi>m</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=30725430&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=30725429&type= 5.41866684 3.72533321 是假设倾向已经达到充分采动，利用公式（2）公式（2）中第一式求出 <math id="M47"><mi>W</mi><mfenced separators="|"><mrow><mi>x</mi><mo>,</mo><mi>t</mi></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=30725428&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=30725412&type= 13.29266548 3.89466691 的最大值 <math id="M48"><msub><mrow><mi>W</mi></mrow><mrow><mi>x</mi><mi>m</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=30725430&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=30725429&type= 5.41866684 3.72533321 。而当矿区开采工作面走向未达到充分采动时，需要在公式（3）公式（3）获取的地表下沉 <math id="M49"><mi>W</mi><mfenced separators="|"><mrow><mi>y</mi><mo>,</mo><mi>t</mi></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=30725440&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=30725439&type= 13.29266548 4.57200003 基础上，乘以一个小于1的走向采动程度系数 <math id="M50"><msub><mrow><mi>C</mi></mrow><mrow><mi>y</mi><mi>m</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=30725434&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=30725433&type= 4.74133301 3.72533321 ，可根据式 <math id="M51"><msub><mrow><mi>C</mi></mrow><mrow><mi>y</mi><mi>m</mi></mrow></msub><mo>=</mo><mfrac><mrow><msub><mrow><mi>W</mi></mrow><mrow><mi>y</mi><mi>m</mi></mrow></msub></mrow><mrow><msub><mrow><mi>W</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></mrow></mfrac><mo><</mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=30725436&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=30725435&type= 21.75933456 8.29733276 计算；式中 <math id="M52"><msub><mrow><mi>W</mi></mrow><mrow><mi>y</mi><mi>m</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=30725442&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=30725441&type= 5.41866684 3.72533321 是假设走向已经达到充分采动时利用公式（3）公式（3）求出的 <math id="M53"><mi>W</mi><mfenced separators="|"><mrow><mi>y</mi><mo>,</mo><mi>t</mi></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=30725440&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=30725439&type= 13.29266548 4.57200003 的最大值 <math id="M54"><msub><mrow><mi>W</mi></mrow><mrow><mi>y</mi><mi>m</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=30725442&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=30725441&type= 5.41866684 3.72533321 （朱建军等，2013 朱建军等，2013 ）。 2.2　基于CT-PIM的时序InSAR建模 2 假设获取同一区域 <math id="M55"><mi>N</mi><mo>+</mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=30725534&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=30725533&type= 8.89000034 2.70933342 景SAR影像，共生成 <math id="M56"><mi>M</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=30725494&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=30725493&type= 2.87866688 2.62466669 幅多视差分干涉图，则第 <math id="M57"><mi>m</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=30725448&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=30725447&type= 2.70933342 2.62466669 幅（ <math id="M58"><mn mathvariant="normal">1</mn><mo>≤</mo><mi>m</mi><mo>≤</mo><mi>M</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=30725450&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=30725449&type= 16.51000023 2.62466669 ）差分干涉图上第 <math id="M59"><mi>i</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=30725481&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=30725480&type= 1.10066664 2.62466669 个像素 <math id="M60"><mfenced separators="|"><mrow><mi>x</mi><mo>,</mo><mi>r</mi></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=30725454&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=30725453&type= 9.73666668 3.89466691 处的解缠相位可表示为（ Berardino 等，2002 Berardino 等，2002 ）： <math display="block" id="M61"><mtable columnalign="left"><mtr><mtd><mi>δ</mi><msup><mrow><mi>φ</mi></mrow><mrow><mi>i</mi></mrow></msup><mo>=</mo><mi>δ</mi><msubsup><mrow><mi>φ</mi></mrow><mrow><mi>d</mi><mi>e</mi><mi>f</mi></mrow><mrow><mi>i</mi></mrow></msubsup><mo>+</mo><mi>δ</mi><msubsup><mrow><mi>φ</mi></mrow><mrow><mi>t</mi><mi>o</mi><mi>p</mi><mi>o</mi></mrow><mrow><mi>i</mi></mrow></msubsup><mo>+</mo><mi>δ</mi><msubsup><mrow><mi>φ</mi></mrow><mrow><mi>o</mi><mi>r</mi><mi>b</mi><mi>i</mi><mi>t</mi></mrow><mrow><mi>i</mi></mrow></msubsup><mo>+</mo><mi>δ</mi><msubsup><mrow><mi>φ</mi></mrow><mrow><mi>a</mi><mi>t</mi><mi>m</mi></mrow><mrow><mi>i</mi></mrow></msubsup><mo>+</mo><mi>δ</mi><msubsup><mrow><mi>φ</mi></mrow><mrow><mi>n</mi><mi>o</mi><mi>i</mi><mi>s</mi><mi>e</mi></mrow><mrow><mi>i</mi></mrow></msubsup><mo>+</mo><mi>δ</mi><msubsup><mrow><mi>φ</mi></mrow><mrow><mi>n</mi><mi>o</mi><mi>n</mi></mrow><mrow><mi>i</mi></mrow></msubsup></mtd></mtr><mtr><mtd><mtext> </mtext><mo>≈</mo><mfrac><mrow><mn mathvariant="normal">4</mn><mi>π</mi></mrow><mrow><mi>λ</mi></mrow></mfrac><mi mathvariant="normal">Δ</mi><msup><mrow><mi>d</mi></mrow><mrow><mi>i</mi></mrow></msup><mo>+</mo><mfrac><mrow><mn mathvariant="normal">4</mn><mi>π</mi><msub><mrow><mi>B</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow><mrow><mi>λ</mi><mi>R</mi><mi mathvariant="normal">s</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi><mi>θ</mi></mrow></mfrac><mi mathvariant="normal">Δ</mi><msup><mrow><mi>H</mi></mrow><mrow><mi>i</mi></mrow></msup><mo>+</mo><mi mathvariant="normal">Δ</mi><msubsup><mrow><mi>φ</mi></mrow><mrow><mi>r</mi><mi>e</mi><mi>s</mi></mrow><mrow><mi>i</mi></mrow></msubsup></mtd></mtr></mtable></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=30725456&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=30725455&type= 73.74466705 12.78466606 式中： <math id="M62"><mi>λ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=30725458&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=30725457&type= 2.28600001 2.62466669 表示雷达波长； <math id="M63"><mi mathvariant="normal">Δ</mi><msup><mrow><mi>d</mi></mrow><mrow><mi>i</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=30725460&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=30725459&type= 5.07999992 3.04800010 表示第 <math id="M64"><mi>i</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=30725481&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=30725480&type= 1.10066664 2.62466669 个差分干涉图视线（Line of Sight， LOS）向上的形变分量，称为低通（Low-Pass， LP）形变分量； <math id="M65"><mi>δ</mi><msubsup><mrow><mi>φ</mi></mrow><mrow><mi>t</mi><mi>o</mi><mi>p</mi><mi>o</mi></mrow><mrow><mi>i</mi></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=30725473&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=30725463&type= 7.19666624 4.65666676 表示地形相位 <math id="M66"><mi>δ</mi><msubsup><mrow><mi>φ</mi></mrow><mrow><mi>t</mi><mi>o</mi><mi>p</mi><mi>o</mi></mrow><mrow><mi>i</mi></mrow></msubsup><mo>=</mo><mfrac><mrow><mn mathvariant="normal">4</mn><mi>π</mi><msub><mrow><mi>B</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow><mrow><mi>λ</mi><mi>R</mi><mi mathvariant="normal">s</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi><mi>θ</mi></mrow></mfrac><mi mathvariant="normal">Δ</mi><msup><mrow><mi>H</mi></mrow><mrow><mi>i</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=30725475&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=30725474&type= 30.56466866 7.36600018 ； <math id="M67"><mi>θ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=30725477&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=30725476&type= 1.77800000 2.70933342 和 <math id="M68"><msub><mrow><mi>B</mi></mrow><mrow><mi>i</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=30725479&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=30725478&type= 3.04800010 3.72533321 分别表示雷达入射角和第 <math id="M69"><mi>i</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=30725481&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=30725480&type= 1.10066664 2.62466669 个干涉对垂直基线的长度； <math id="M70"><mi>R</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=30725483&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=30725482&type= 2.28600001 2.62466669 表示相干目标与雷达卫星所在位置之间的距离； <math id="M71"><mi mathvariant="normal">Δ</mi><mi>H</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=30725484&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=30725464&type= 4.99533367 2.79399991 是高程误差； <math id="M72"><mi>δ</mi><msubsup><mrow><mi>φ</mi></mrow><mrow><mi>d</mi><mi>e</mi><mi>f</mi></mrow><mrow><mi>i</mi></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=30725466&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=30725465&type= 6.26533365 4.65666676 表示形变相位； <math id="M73"><mi>δ</mi><msubsup><mrow><mi>φ</mi></mrow><mrow><mi>o</mi><mi>r</mi><mi>b</mi><mi>i</mi><mi>t</mi></mrow><mrow><mi>i</mi></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=30725468&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=30725467&type= 7.45066643 4.65666676 表示轨道误差； <math id="M74"><mi mathvariant="normal">Δ</mi><msubsup><mrow><mi>φ</mi></mrow><mrow><mi>r</mi><mi>e</mi><mi>s</mi></mrow><mrow><mi>i</mi></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=30725470&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=30725469&type= 6.94266701 4.82600021 表示残余相位，主要包括大气延迟相位，噪声相位和高通（High-Pass，HP）形变分量。当忽略地表水平移动时，InSAR 视线向形变值可表示为： <math display="block" id="M75"><mtable columnalign="left"><mtr><mtd><mtext> </mtext><mi mathvariant="normal">Δ</mi><msup><mrow><mi>d</mi></mrow><mrow><mi>i</mi></mrow></msup><mo>=</mo><msub><mrow><mi>d</mi></mrow><mrow><mi>L</mi><mi>O</mi><mi>S</mi></mrow></msub><mfenced separators="|"><mrow><msubsup><mrow><mi>t</mi></mrow><mrow><mi>B</mi></mrow><mrow><mi>i</mi></mrow></msubsup></mrow></mfenced><mo>-</mo><msub><mrow><mi>d</mi></mrow><mrow><mi>L</mi><mi>O</mi><mi>S</mi></mrow></msub><mfenced separators="|"><mrow><msubsup><mrow><mi>t</mi></mrow><mrow><mi>A</mi></mrow><mrow><mi>i</mi></mrow></msubsup></mrow></mfenced></mtd></mtr><mtr><mtd><mo>=</mo><mfenced open="[" close="]" separators="|"><mrow><mi>W</mi><mfenced separators="|"><mrow><mi>x</mi><mo>,</mo><mi>y</mi><mo>,</mo><msub><mrow><mi>t</mi></mrow><mrow><mi>B</mi></mrow></msub></mrow></mfenced><mo>-</mo><mi>W</mi><mfenced separators="|"><mrow><mi>x</mi><mo>,</mo><mi>y</mi><mo>,</mo><msub><mrow><mi>t</mi></mrow><mrow><mi>A</mi></mrow></msub></mrow></mfenced></mrow></mfenced><mo>⋅</mo><mi mathvariant="normal">c</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">s</mi><mi>θ</mi></mtd></mtr></mtable></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=30725486&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=30725485&type= 49.86866760 11.51466751 式中： <math id="M76"><mi>W</mi><mfenced separators="|"><mrow><mi>x</mi><mo>,</mo><mi>y</mi><mo>,</mo><msub><mrow><mi>t</mi></mrow><mrow><mi>B</mi></mrow></msub></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=30725488&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=30725487&type= 20.06599998 4.57200003 和 <math id="M77"><mi>W</mi><mfenced separators="|"><mrow><mi>x</mi><mo>,</mo><mi>y</mi><mo>,</mo><msub><mrow><mi>t</mi></mrow><mrow><mi>A</mi></mrow></msub></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=30725489&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=30725471&type= 19.98133469 4.57200003 分别为在 <math id="M78"><msub><mrow><mi>t</mi></mrow><mrow><mi>B</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=30725491&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=30725490&type= 2.45533323 3.72533321 和 <math id="M79"><msub><mrow><mi>t</mi></mrow><mrow><mi>A</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=30725472&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=30725492&type= 2.28600001 3.72533321 时刻的动态沉降量。将式（5）式（5）代入式（4）式（4），即可得到InSAR差分干涉相位与概率积分预计参数之间的函数模型： <math display="block" id="M80"><mtable columnalign="left"><mtr><mtd><mi>δ</mi><msup><mrow><mi>φ</mi></mrow><mrow><mi>i</mi></mrow></msup><mo>=</mo><mfrac><mrow><mn mathvariant="normal">4</mn><mi>π</mi></mrow><mrow><mi>λ</mi></mrow></mfrac><mfenced open="[" close="]" separators="|"><mrow><mi>W</mi><mfenced separators="|"><mrow><mi>x</mi><mo>,</mo><mi>y</mi><mo>,</mo><msub><mrow><mi>t</mi></mrow><mrow><mi>B</mi></mrow></msub></mrow></mfenced><mo>-</mo><mi>W</mi><mfenced separators="|"><mrow><mi>x</mi><mo>,</mo><mi>y</mi><mo>,</mo><msub><mrow><mi>t</mi></mrow><mrow><mi>A</mi></mrow></msub></mrow></mfenced></mrow></mfenced><mo>⋅</mo><mi mathvariant="normal">c</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">s</mi><mi>θ</mi></mtd></mtr><mtr><mtd><mtext> </mtext><mo>+</mo><mfrac><mrow><mn mathvariant="normal">4</mn><mi>π</mi><msub><mrow><mi>B</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow><mrow><mi>λ</mi><mi>R</mi><mi mathvariant="normal">s</mi><mi mathvariant="normal">i</mi><mi mathvariant="normal">n</mi><mi>θ</mi></mrow></mfrac><mi mathvariant="normal">Δ</mi><msup><mrow><mi>H</mi></mrow><mrow><mi>i</mi></mrow></msup><mo>+</mo><mi mathvariant="normal">Δ</mi><msubsup><mrow><mi>φ</mi></mrow><mrow><mi>r</mi><mi>e</mi><mi>s</mi></mrow><mrow><mi>i</mi></mrow></msubsup></mtd></mtr></mtable></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=30725504&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=30725503&type= 61.55266953 16.51000023 上式为基于CT-PIM的时序InSAR相位模型，其中矿区地质参数 <math id="M81"><mi>G</mi><mi>P</mi><mo>=</mo><mfenced open="[" close="]" separators="|"><mrow><mi>m</mi><mi mathvariant="normal">、</mi><mi>α</mi><mi mathvariant="normal">、</mi><mi>H</mi><mi mathvariant="normal">、</mi><msub><mrow><mi>D</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mi mathvariant="normal">、</mi><mi>ω</mi><mi mathvariant="normal">、</mi><msub><mrow><mi>v</mi></mrow><mrow><mi>m</mi></mrow></msub></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=30725506&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=30725505&type= 46.82066727 4.31799984 可根据矿区地质条件资料确定，视为已知数据。概率积分预计参数 <math id="M82"><mi>U</mi><mi>P</mi><mo>=</mo><mfenced open="[" close="]" separators="|"><mrow><mi>q</mi><mi mathvariant="normal">、</mi><mi mathvariant="normal">t</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">n</mi><mi>β</mi><mi mathvariant="normal">、</mi><msub><mrow><mi>s</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mi mathvariant="normal">、</mi><msub><mrow><mi>s</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>,</mo><mi mathvariant="normal">、</mi><mi>k</mi><mi mathvariant="normal">、</mi><mi>n</mi></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=30725508&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=30725507&type= 51.64666748 4.57200003 为待求参数。其中 <math id="M83"><mi>q</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=30725654&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=30725653&type= 1.69333339 3.47133350 的取值范围为0.01~1； <math id="M84"><mi mathvariant="normal">t</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">n</mi><mi>β</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=30725656&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=30725655&type= 7.02733374 3.38666677 、 <math id="M85"><mi mathvariant="normal">t</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">n</mi><msub><mrow><mi>β</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=30725514&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=30725513&type= 8.21266651 4.48733330 、 <math id="M86"><mi mathvariant="normal">t</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">n</mi><msub><mrow><mi>β</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=30725516&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=30725515&type= 8.21266651 4.48733330 的取值范围均为1~3.8； <math id="M87"><msub><mrow><mi>s</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=30725623&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=30725622&type= 2.45533323 3.72533321 、 <math id="M88"><msub><mrow><mi>s</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=30725625&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=30725624&type= 2.45533323 3.72533321 的取值范围均为0.05 H ~0.3 H ； <math id="M89"><msub><mrow><mi>θ</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>=</mo><mn mathvariant="normal">90</mn><mo>°</mo><mo>-</mo><mi>k</mi><mi>α</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=30725522&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=30725521&type= 20.91266632 4.23333359 ，其中 <math id="M90"><mi>α</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=30725524&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=30725523&type= 2.20133328 2.62466669 为矿层倾角， <math id="M91"><mi>k</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=30725526&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=30725525&type= 1.69333339 2.62466669 的取值范围为0.5~0.8；启动距系数 <math id="M92"><mi>n</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=30725528&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=30725527&type= 1.86266661 2.62466669 的取值范围为1/7~1/2（朱建军等，2013 朱建军等，2013 ）。式（6）式（6）中未知参数的求解是一个非线性参数估计的问题。对于生成的 <math id="M93"><mi>M</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=30725494&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=30725493&type= 2.87866688 2.62466669 个差分干涉图上任意一高相干点 <math id="M94"><mfenced separators="|"><mrow><mi>x</mi><mo>,</mo><mi>y</mi></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=30725532&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=30725531&type= 10.15999985 4.57200003 ，可列出 <math id="M95"><mi>M</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=30725494&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=30725493&type= 2.87866688 2.62466669 个方程。如果获取超过6景的InSAR差分干涉图，即可求解出未知参数。本文采用遗传算法（Genetic Algorithm， GA）与单纯形法（Simplex Method， SM）相结合估计模型的未知参数（肖宏峰等，2008 肖宏峰等，2008 ； Yang 等，2017b Yang 等，2017b ）。 2.3　基于CT-PIM模型的时序InSAR矿区形变预计 2 图1 图1 为基于InSAR相位观测值的CT-PIM模型数据处理流程，具体步骤如下：（1）前期差分干涉处理：对获取覆盖研究区域的 <math id="M96"><mi>N</mi><mo>+</mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=30725534&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=30725533&type= 8.89000034 2.70933342 景SAR影像进行GAMMA前期差分干涉处理生成差分干涉图、相干系数图和强度图。利用差分干涉图进行相位解缠生成相位矩阵，基于双重阈值法（振幅离差指数和时域相干系数）进行高相干点的选取。（2）基于CT-PIM模型的时序InSAR建模。（3）基于遗传算法和单纯形法搜索串行（Genetic Algorithm and Simplex Method，GASM）的CT-PIM模型参数估计。（4） SAR影像覆盖后期形变预计结果的生成：利用模型参数结果和SAR影像覆盖后期任意一时间序列，生成SAR影像覆盖后期矿区垂向时序形变预计结果。 3　实验研究 1 3.1　模拟实验 2 为验证本文方法的可行性和可靠性，开展了模拟实验。模拟实验中采用的卫星参数依据真实实验中Sentinel-1 A SAR影像参数设定（具体见3.2.1节）。矿区地质参数 <math id="M97"><mi>G</mi><mi>P</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=30725539&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=30725538&type= 4.57200003 2.62466669 根据真实矿区的实地调查资料进行设定： <math id="M98"><mi>G</mi><mi>P</mi><mo>=</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=30725567&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=30725566&type= 7.70466709 2.70933342 ［ 3.0 m、7.5°、300 m、200 m、80°、0.24 m/d］。设置待求的预计参数真值为 <math id="M99"><mi>U</mi><mi>P</mi><mo>=</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=30725543&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=30725542&type= 7.87400007 2.70933342 ［0.604、3.120、30.310 m、28.080 m、0.524、0.169］，代入式（1）式（1）生成真实的沉降序列。差分干涉图根据式（6）式（6）模拟生成，并分别加入0~0.65 rad的随机噪声相位。利用前述方法估计的预计参数如表1 表1 所示，并获取了34景形变序列。提取了形变序列中500个像素点进行定量的对比分析（如图2 图2 所示）。预计参数真实值计算值误差相对误差 <math id="M100"><mi>q</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=30725598&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=30725597&type= 1.26999998 2.62466669 0.604 0.626 0.022 3.6% <math id="M101"><mi mathvariant="normal">t</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">n</mi><mi>β</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=30725592&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=30725599&type= 5.33400011 2.53999996 3.120 3.158 0.038 1.2% <math id="M102"><msub><mrow><mi>s</mi></mrow><mrow><mn>1</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=30725609&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=30725608&type= 1.94733346 2.87866688 （m） 30.310 32.320 2.010 6.4% <math id="M103"><msub><mrow><mi>s</mi></mrow><mrow><mn>2</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=30725611&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=30725610&type= 1.94733346 2.87866688 （m） 28.080 27.437 0.643 2.3% <math id="M104"><mi>k</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=30725613&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=30725612&type= 1.26999998 2.03200006 0.524 0.505 0.019 3.6% <math id="M105"><mi>n</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=30725615&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=30725614&type= 1.43933344 2.03200006 0.169 0.175 0.006 3.4% 图2 图2 为500个像素点在0.65 rad高噪声水平时，形变求解值与模拟真实沉降值对比图。很显然，即使在较大噪声的干扰下，本文方法生成的沉降结果与真实值仍保持了较好的一致性。从数据统计来看，图2 图2 （a）到图2 图2 （h）中，误差分布在［-3，3］ mm之间的高相干点分别占100%、99.2%、98.4%、96%、94%、88%、86%、83%，其中的最大误差分别为1.2 mm、2.5 mm、3.2 mm、3.9 mm、4.6 mm、8.2 mm、9.4 mm和10.2 mm。图2 图2 （h）中红色折线为852天时误差情况，最大误差仅占最大形变量的7.7%。为定量评定CT-PIM模型求解参数值精度，分别计算出参数真实与计算值的相对误差（如表1 表1 所示），结果表明：6个未知参数的相对误差均小于6.5%，相对误差越小，求解参数值越接近于模拟真值。CT-PIM模型生成的形变量与模拟沉降量的均方根误差为±4.6 mm。 3.2　真实数据实验 2 3.2.1　测区数据及处理 3 真实实验选取淮安市西顺河镇某盐矿作为研究区域。图3 图3 为研究区域在中国地图及SAR卫星影像上的位置。图中红色矩形表示覆盖测区的Sentinel-1 A（升轨）数据的空间覆盖范围，紫色矩形为本文实验裁剪区域。研究区域岩盐矿产资源极为丰富，不仅储量大，而且含盐品味高，分布范围相对集中。矿床属陆地湖面相蒸发岩型沉积，分布范围较广，沉积韵律频繁，持续的矿产资源开发造成了地下岩层结构的改变，对周围环境及地质造成了巨大的破坏；部分沉降显著区域甚至出现地下卤水涌出，造成土地盐碱化，对周边环境造成了严重的破坏。通过对研究区域工作面上覆岩层岩性特征的调研和搜集到的矿区地质资料可知（朱培等，2020 朱培等，2020 ；李小明等，2018 李小明等，2018 ），研究区域主要分布在淮安盐盆、洪泽盐盆两地，倾角5°~9°，矿层顶板埋深约600~800m，分别设定测区地质参数 <math id="M106"><mi>G</mi><mi>P</mi><mo>=</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=30725567&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=30725566&type= 7.70466709 2.70933342 ［7.5°、600m、5m、416.44m、80°、0.056 m/d］。共收集2017年3月28日至2019年7月28日的35景覆盖研究区域的升轨Sentinel-1A SAR影像用于前期差分干涉处理，影像的参数如表2 表2 所示，时空基线分布如图4 图4 所示。实验处理中，采用多项式拟合方法去除轨道误差，利用最小费用流（Minimum Cost Flow， MCF）方法进行解缠处理，共生成56个解缠后的差分干涉图（ Li 等，2008 Li 等，2008 ），提取了钻井水溶矿区3269个高相干点。编号获取日期时间基线/天空间基线/米 0 2017-03-28 -360 -2.25 1 2017-04-21 -336 -108.15 2 2017-05-15 -312 -97.49 3 2017-06-08 -288 -43.62 4 2017-07-02 -264 -42.06 5 2017-07-26 -240 -25.26 6 2017-08-19 -216 -43.43 7 2017-09-12 -192 -97.56 8 2017-10-06 -168 15.89 9 2017-10-30 -144 -80.69 10 2017-11-23 -120 -64.14 11 2017-12-17 -96 70.59 12 2018-01-10 -72 -18.53 13 2018-02-27 -24 -15.03 14 2018-03-23 0 0 15 2018-04-16 24 -16.53 16 2018-05-10 48 -114.97 17 2018-06-03 72 -19.87 18 2018-06-27 96 8.92 19 2018-07-21 120 -86.34 20 2018-08-14 144 -13.71 21 2018-09-07 168 -62.58 22 2018-10-01 192 -68.64 23 2018-10-25 216 53.06 24 2018-11-18 240 -36.96 25 2018-12-12 264 23.73 26 2019-01-05 288 75.43 27 2019-02-10 324 -16.69 28 2019-03-06 348 -53.46 29 2019-03-30 372 -91.37 30 2019-04-23 396 -147.34 31 2019-05-17 420 -15.52 32 2019-06-10 444 -0.45 33 2019-07-04 468 12.01 34 2019-07-28 492 -72.68 3.2.2　实验结果 3 本文先利用SBAS-InSAR传统多速率模型开展了实验，得到2017年3月28日至2019年7月28日期间的34个时序沉降结果，实验发现，截至2019年7月28日，最大沉降为146 mm，由于盐矿区采用两井或多井连通方式开采，在每个钻井周围都会形成沉降漏斗甚至连通的情况（如图5 图5 （a）所示）。如果对整个测区采用一个CT-PIM模型，很难对该研究区中复杂的沉降漏斗实际情况进行准确建模。在此，将矿区划分为6个子沉降漏斗（如图5 图5 （b）所示），分别对每个漏斗开展本文方法实验。实验处理中，利用前29景影像（2017年3月28日至2019年3月6日）开展2.3节中步骤（2-3）的形变监测过程，获取的CT-PIM参数见表3 表3 ，时序形变结果见图6 图6 ；后6景影像（2019年3月30日至2019年7月28日）用于开展步骤（4）的SAR影像覆盖后期形变的预计，预计结果见图7 图7 。参数 A1 A2 A3 A4 A5 A6 <math id="M107"><mi>q</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=30725598&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=30725597&type= 1.26999998 2.62466669 0.792 0.761 0.829 0.813 0.808 0.902 <math id="M108"><mi mathvariant="normal">t</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">n</mi><mi>β</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=30725592&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=30725599&type= 5.33400011 2.53999996 3.385 3.116 3.325 3.575 3.318 3.125 <math id="M109"><msub><mrow><mi>s</mi></mrow><mrow><mn>1</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=30725609&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=30725608&type= 1.94733346 2.87866688 （m） 79.169 40.170 56.132 50.172 60.170 53.166 <math id="M110"><msub><mrow><mi>s</mi></mrow><mrow><mn>2</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=30725611&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=30725610&type= 1.94733346 2.87866688 （m） 46.481 56.480 48.233 56.479 46.398 66.312 <math id="M111"><mi>k</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=30725613&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=30725612&type= 1.26999998 2.03200006 0.568 0.517 0.677 0.667 0.619 0.576 <math id="M112"><mi>n</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=30725615&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=30725614&type= 1.43933344 2.03200006 0.160 0.166 0.233 0.215 0.213 0.251 从图6 图6 可以看出，从2017年4月21日至2018年3月23日期间，整个测区由蓝色到浅绿色，总体呈缓慢下沉趋势，截至2018年3月23日，最大沉降为62 mm。在2018年4月16日至2019年3月6日期间，随着矿区开采的进一步推进，整个测区由浅黄色逐渐加剧变红，到2019年3月6日，开始出现大量级的沉降，最大值达到119 mm。由图7 图7 可知，截至2019年7月28日，随着矿区的持续开采而逐渐扩大，累计最大预计沉降量达152 mm。 4　实验结果讨论 1 为了进一步分析本文获取的结果随时间变化的特征，选取三个特征点（F1-F3，位置如图6 图6 所示）进行时序定量分析，结果如图8 图8 所示（虚线为SBAS-InSAR结果，实线为CT-PIM结果）。从图8 图8 中可以看出，2017年3月28日至2019年7月28日期间，三个特征点都表现出相似的时间变化趋势。SBAS-InSAR获取的形变序列呈现出线性变化趋势。CT-PIM模型获取的形变序列整体下沉且伴随着季节性变化。其中F3的沉降速率最大，截至2019年7月28日，其累计下沉量达146 mm。而离钻井口较远的F1点则相对稳定，最大形变量为72 mm。在暖季：第一阶段（2017年3月28日至2017年10月6日）、第二阶段（2018年3月23日至2018年10月1日）和第三阶段（2019年3月6日至2019年7月28日），特征点下沉速率相对较快，累计最大下沉分别为57 mm， 31 mm和27 mm。在冷季，第一阶段（2017年10月6日至2018年3月23日）和第二阶段（2018年10月1日至2019年3月6日），沉降曲线则趋于平缓，累计下沉量为8 mm和12 mm。产生这一季节性波动原因主要是由于芒硝的溶解速率直接受溶剂温度的影响（如表4 表4 所示）（ Foumelis 等，2016 Foumelis 等，2016 ）。实验区域注入井中的淡水溶剂的温度约为40°C，以保证芒硝达到最大溶解度。然而，溶剂温度容易受到外界空气温度的影响。在夏天，外界环境的高温使得溶剂的溶解度显著提升，加速了芒硝的溶解，进而引发地表快速下沉的现象；相反，冬季时由于外界温度的降低，溶剂的温度随之降低，抑制了芒硝的溶解度，进而使得矿区地表的形变较为缓慢。 2 矿物无水芒硝芒硝钙芒硝 11 温度（°C） 0 - 5.0 0.18 10 - 9.0 0.19 20 - 19.4 0.20 30 50.4 40.8 0.21 40 48.8 - 0.21 50 46.7 - - 60 45.3 - 0.21 70 44.1 - 0.20 80 43.7 - 0.20 90 42.9 - - 100 42.5 - 0.16 从图8 图8 中还可以发现，CT-PIM获取的形变序列中，三个特征点均在相同的五个时段出现了微小隆起（黑色箭头所指向位置），这也与图6 图6 和图7 图7 中的时序沉降图颜色时序变化特征保持了一致。隆起现象除与上述冬季低温导致的溶剂溶解度降低有关外，还与降雨量的增加有关（ Foumelis 等，2016 Foumelis 等，2016 ）。根据江苏省淮安市气象局的降水和气温数据显示，2017年7月2日至7月26日、2018年3月23日至4月16日、2018年7月21日至8月14日、2019年3月6日至3月30日和2019年5月17日至6月10日出现了降雨量的显著增加，如图8 图8 所示。外部雨水的回灌加剧了这几个时段矿区地表形变的抬升现象。 CT-PIM模型获取的形变序列与传统SBAS-InSAR对比图显示在图9 图9 中。从图9 图9 可以看出，相对传统SBAS-InSAR结果，CT-PIM模型获取的时序形变结果颜色变化趋势更为明显，整体沉降量偏大并伴有季节性变化。从图8 图8 可以看出，CT-PIM模型获取结果揭示了形变时序变化的抬升现象：2017年7月2日至7月26日、2018年3月23日至4月16日、2018年7月21日至8月14日、2019年3月6日至3月30日和2019年5月17日至6月10日，而在SBAS-InSAR结果中几乎没有体现。从图9 图9 中也可以看出，2019年3月6日至3月30日和2019年5月17日至6月10日两个抬升时段（如图9 图9 中箭头所示），CT-PIM模型结果有变淡趋势，而SBAS-InSAR结果并不明显。通过定量分析可得，CT-PIM模型获取的五个时段抬升最大值分别为9.6 mm、7.4 mm、9.8 mm、6.9 mm和7.4 mm。与传统SBAS-InSAR相比，CT-PIM模型可更好地描述盐矿区形变随时间呈现的非线性演化特征，辅助盐矿区形变物理解译。 5　敏感性分析 1 5.1　模型参数的敏感性 2 如（ Yang等，2018 Yang等，2018 ）所述，PIM参数的估计精度直接影响着 PIM模型预计获取的形变精度，即参数敏感性高时，即使这一参数出现较小的误差，预计形变也会产生较大扰动；对那些敏感性较低的参数而言，即使模型参数估计的误差较大，对最终的形变精度影响也较弱。因此，本文利用模拟数据对CT-PIM模型进行敏感性分析，以分析哪些参数为敏感参数。共利用八组不确定的参数值作为输入数据开展实验，分别获取最终预计形变。表5 表5 列出了6个未知参数的模拟值和每个参数的误差对应的范围，模型的不确定度是根据函数 <math id="M113"><mfrac><mrow><mi>X</mi></mrow><mrow><msub><mrow><mi>X</mi></mrow><mrow><mi>m</mi><mi>a</mi><mi>x</mi></mrow></msub></mrow></mfrac><mo>×</mo><mn mathvariant="normal">100</mn><mi mathvariant="normal">%</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=30725596&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=30725595&type= 19.64266586 7.95866632 来定义。参数参数模拟值误差范围偏差程度 <math id="M114"><mi>q</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=30725598&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=30725597&type= 1.26999998 2.62466669 0.5 0~1 -40%~40% <math id="M115"><mi mathvariant="normal">t</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">n</mi><mi>β</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=30725592&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=30725599&type= 5.33400011 2.53999996 2 1~3 -60%~60% <math id="M116"><msub><mrow><mi>s</mi></mrow><mrow><mn>1</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=30725609&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=30725608&type= 1.94733346 2.87866688 （m） 20 0~40 -100%~100% <math id="M117"><msub><mrow><mi>s</mi></mrow><mrow><mn>2</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=30725611&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=30725610&type= 1.94733346 2.87866688 （m） 20 0~40 -100%~100% <math id="M118"><mi>k</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=30725613&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=30725612&type= 1.26999998 2.03200006 0.65 0.5~0.8 -70%~70% <math id="M119"><mi>n</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=30725615&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=30725614&type= 1.43933344 2.03200006 0.35 0.2~0.5 -60%~60% 图10 图10 显示了由于模型参数不同程度的偏差导致形变估计误差情况。从图中可以看出，下沉系数 <math id="M120"><mi>q</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=30725654&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=30725653&type= 1.69333339 3.47133350 和影响角正切值 <math id="M121"><mi mathvariant="normal">t</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">n</mi><mi>β</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=30725656&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=30725655&type= 7.02733374 3.38666677 的敏感性最高，对形变结果影响占主导作用。左点偏移距 <math id="M122"><msub><mrow><mi>s</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=30725623&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=30725622&type= 2.45533323 3.72533321 和右点偏移距 <math id="M123"><msub><mrow><mi>s</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=30725625&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=30725624&type= 2.45533323 3.72533321 的敏感性相对较低。因此，准确估计下沉系数 <math id="M124"><mi>q</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=30725654&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=30725653&type= 1.69333339 3.47133350 和影响角正切值 <math id="M125"><mi mathvariant="normal">t</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">n</mi><mi>β</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=30725656&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=30725655&type= 7.02733374 3.38666677 对于后续准矿区形变序列的预计至关重要。 5.2　模型误差传递分析 2 为进一步分析本文方法误差传递特性，分别开展两组误差传递模拟实验：（1）利用2.2节中介绍的（6）式，对等号左侧观测相位加入不同程度噪声，再计算估计的CT-PIM参数，分析对应生成预计的形变序列误差变化特征；（2）利用加入不同程度噪声的相位，用传统SBAS方法生成形变序列，再结合形变序列估计静态概率积分（Static-PIM）参数，分析对应生成预计形变序列误差变化特征。由敏感性分析结果可知，下沉系数 <math id="M126"><mi>q</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=30725654&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=30725653&type= 1.69333339 3.47133350 和影响角正切 <math id="M127"><mi mathvariant="normal">t</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">n</mi><mi>β</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=30725656&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=30725655&type= 7.02733374 3.38666677 敏感性较高，因此，分别探究在不同干扰噪声相位下概率积分预计参数 <math id="M128"><mi>q</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=30725654&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=30725653&type= 1.69333339 3.47133350 和 <math id="M129"><mi mathvariant="normal">t</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">n</mi><mi>β</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=30725656&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=30725655&type= 7.02733374 3.38666677 误差对于后期形变预计精度的影响。图11 图11 分别显示了 CT-PIM模型和传统静态概率积分（Static-PIM）模型在不同相位噪声水平下 <math id="M130"><mi>q</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=30725654&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=30725653&type= 1.69333339 3.47133350 和 <math id="M131"><mi mathvariant="normal">t</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">n</mi><mi>β</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=30725656&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=30725655&type= 7.02733374 3.38666677 误差引起的预计形变 <math id="M132"><mi>W</mi><mfenced separators="|"><mrow><mi>q</mi><mo>,</mo><mi>t</mi></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=30725640&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=30725639&type= 13.20800018 4.57200003 和 <math id="M133"><mi>W</mi><mfenced separators="|"><mrow><mi mathvariant="normal">t</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">n</mi><mi>β</mi><mo>,</mo><mi>t</mi></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=30725637&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=30725636&type= 18.54199982 4.48733330 的均方根误差。从图中可以看出，随着噪声水平的增大，参数 <math id="M134"><mi>q</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=30725654&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=30725653&type= 1.69333339 3.47133350 和 <math id="M135"><mi mathvariant="normal">t</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">n</mi><mi>β</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=30725656&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=30725655&type= 7.02733374 3.38666677 的相对误差逐渐增大，所引起预计形变的均方根误差也随之增大。图11 图11 （a）中CT-PIM模型的误差水平整体低于图11 图11 （b）中（Static-PIM）模型。在0.6 rad高噪声水平影响下，CT-PIM模型参数求解值与模拟真值的相对误差均小于6％，所引起的预计形变值与模拟真实形变值的均方根误差均小于3.6 mm；而传统（Static-PIM）模型在相位有较小变化（0.3 rad相位噪声水平）时，就会引起参数相对误差的较大变化，进而导致预计形变结果产生6.2 mm的均方根误差。从而证实了本文方法利用相位直接估计CT-PIM参数，并将模型直接用于形变的预计，可有效减弱误差的传递。 6　精度验证 1 6.1　形变监测结果的精度分析 2 由于研究区域没有可用的外部水准数据，通过查阅已有文献，本实验获取的最大年平均沉降速率51 mm/a与已有对淮安盐矿区形变监测结果保持良好的一致性（夏斯雨等，2018 夏斯雨等，2018 ），证实了本文获取的形变预计结果量级的准确性。为弥补无外部可用水准数据的局限，在此利用文献（ CH/T 6006 CH/T 6006 -2018，2019）精度评定方法开展内部精度评定。根据间接平差理论，未知参数的协因数 <math id="M136"><msubsup><mrow><mover accent="true"><mi>σ</mi><mo>̂</mo></mover></mrow><mrow><msub><mrow><mover accent="true"><mi>X</mi><mo>̂</mo></mover></mrow><mrow><mi>i</mi></mrow></msub></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=30725648&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=30725647&type= 4.23333359 4.57200003 可衡量未知参数的估计精度。由误差传播定律可知，观测值的误差由式（6）式（6）传递至模型的未知参数后，进而引起形变的误差。通过计算形变的中误差 <math id="M137"><mi>σ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=30725644&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=30725643&type= 2.45533323 2.62466669 衡量内部精度，精度评定结果见表6 表6 。根据表6 表6 可知，6个沉降漏斗观测值单位权方差 <math id="M138"><msubsup><mrow><mover accent="true"><mi>σ</mi><mo>̂</mo></mover></mrow><mrow><mn mathvariant="normal">0</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=30725646&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=30725645&type= 3.64066648 4.31799984 均小于9.2 mm 2 ，未知参数的协因数 <math id="M139"><msubsup><mrow><mover accent="true"><mi>σ</mi><mo>̂</mo></mover></mrow><mrow><msub><mrow><mover accent="true"><mi>X</mi><mo>̂</mo></mover></mrow><mrow><mi>i</mi></mrow></msub></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=30725648&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=30725647&type= 4.23333359 4.57200003 均小于2.5。通过计算六个沉降漏斗的累积形变的中误差可知，漏斗A4的形变中误差最大，为 ±3.019 mm，六个漏斗的平均中误差为 ±2.41 mm，占最大形变的1.5%，从而表明模型具有较高的内部精度。指标 A1 A2 A3 A4 A5 A6 <math id="M140"><msubsup><mrow><mover accent="true"><mi>σ</mi><mo>̂</mo></mover></mrow><mrow><mn>0</mn></mrow><mrow><mn>2</mn></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=30725660&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=30725649&type= 2.87866688 3.30200005 2.328 8.925 6.235 9.116 6.607 3.548 <math id="M141"><msubsup><mrow><mover accent="true"><mi>σ</mi><mo>̂</mo></mover></mrow><mrow><mover accent="true"><mi>q</mi><mo>̂</mo></mover></mrow><mrow><mn>2</mn></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=30725662&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=30725661&type= 2.87866688 3.30200005 0.016 0.029 0.022 0.031 0.019 0.025 <math id="M142"><msubsup><mrow><mover accent="true"><mi>σ</mi><mo>̂</mo></mover></mrow><mrow><mi mathvariant="normal">t</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">n</mi><mover accent="true"><mi>β</mi><mo>̂</mo></mover></mrow><mrow><mn>2</mn></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=30725663&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=30725650&type= 5.33400011 3.30200005 0.043 0.032 0.057 0.039 0.047 0.033 <math id="M143"><msubsup><mrow><mover accent="true"><mi>σ</mi><mo>̂</mo></mover></mrow><mrow><msub><mrow><mover accent="true"><mi>s</mi><mo>̂</mo></mover></mrow><mrow><mn>1</mn></mrow></msub></mrow><mrow><mn>2</mn></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=30725665&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=30725664&type= 3.21733332 3.64066648 （m 2 ） 2.033 2.251 1.026 1.156 1.432 1.166 <math id="M144"><msubsup><mrow><mover accent="true"><mi>σ</mi><mo>̂</mo></mover></mrow><mrow><msub><mrow><mover accent="true"><mi>s</mi><mo>̂</mo></mover></mrow><mrow><mn>2</mn></mrow></msub></mrow><mrow><mn>2</mn></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=30725667&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=30725666&type= 3.21733332 3.64066648 （m 2 ） 1.079 2.322 1.213 1.023 1.398 2.312 <math id="M145"><msubsup><mrow><mover accent="true"><mi>σ</mi><mo>̂</mo></mover></mrow><mrow><mover accent="true"><mi>k</mi><mo>̂</mo></mover></mrow><mrow><mn>2</mn></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=30725669&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=30725668&type= 2.87866688 3.30200005 0.018 0.027 0.013 0.017 0.021 0.036 <math id="M146"><msubsup><mrow><mover accent="true"><mi>σ</mi><mo>̂</mo></mover></mrow><mrow><mover accent="true"><mi>n</mi><mo>̂</mo></mover></mrow><mrow><mn>2</mn></mrow></msubsup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=30725671&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=30725670&type= 2.87866688 3.30200005 0.005 0.003 0.006 0.011 0.009 0.013 <math id="M147"><mi>σ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=30725673&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=30725672&type= 1.86266661 2.03200006 （mm） 1.525 2.987 2.496 3.019 2.570 1.883 根据（ Zhao 等，2016 Zhao 等，2016 ）所述，残余相位的大小可以较好地反应InSAR建模监测结果与真实形变的拟合程度，即可衡量模型的建模精度。干涉对中残余相位越小，表示建模精度越高。残余相位可以通过式（6）式（6）中的差分干涉相位与低通形变相位分量进行相减获得。分别对选取的38个差分干涉对中所有高相干点的残余相位进行标准差（Standard Deviation，STD）计算，再与多速率模型结果进行对比，结果如图12 图12 所示。从图中可以看出，两种模型的残余相位均小于0.9 rad， CT-PIM残余相位均小于0.65 rad。SBAS-InSAR多速率模型的STD为0.56 rad，而CT-PIM模型的STD为0.38 rad，相比提升了38.2%。对比分析结果可知，InSAR-CTPIM模型的残余相位明显小于多速率模型，说明CT-PIM在建模盐矿的形变过程中具有更好的精度。 6.2　CT-PIM模型形变预计精度分析 2 如3.2.1节所述，保留了2019年3月30日至2019年7月28日的6景影像用于验证CT-PIM模型对SAR影像覆盖范围后期形变的预计。在此将传统静态概率积分（Static-PIM）的预计方法获取矿区形变预计结果与CT-PIM模型的形变预计结果分别和SBAS-InSAR获取的沉降结果进行对比。提取形变预计的四个时期（分别为2017年3月28日至2019年3月30日、2019年5月17日、2019年6月10日和2019年7月28日）进行分析（如图13 图13 所示）。从图中可以看出（以A1和A3漏斗为例），CT-PIM模型结果与SBAS-InSAR结果保持一致；而Static-PIM与SBAS-InSAR形变监测结果差异较大。为进一步定量分析，分别对图13 图13 四个时期的形变预计结果与SBAS-InSAR监测结果的形变量偏差进行计算，并对其概率分布进行统计（结果如图 14 14 、 15 15 所示）。从图14 图14 可以看出，CT-PIM模型的形变预计结果在四个时期与SBAS-InSAR的监测结果之间的形变量偏差主要分布在［-20，20］ mm范围之内。形变量的最大偏差为19.6 mm；Static-PIM模型的形变量偏差（如图15 图15 所示）主要分布在［-30，30］ mm范围内，形变量的最大偏差为28.9 mm。显然，CT-PIM模型的误差分布比Static-PIM更集中，总STD为±8.4 mm，而Static-PIM结果总STD为±12.6 mm。CT-PIM模型形变预计结果比传统Static-PIM模型提高39.1%。 7　结论 1 本文将坐标-时间函数（Coordinate Time， CT）引入时序InSAR建模环节，提出了基于CT-PIM预计模型的矿区时序InSAR预计方法，并将其应用于淮安盐矿区地表时序形变预计。研究结果表明：（1）测区最大沉降量达152 mm，在时间序列上，盐矿区形变伴随有显著的季节性波动，这一特征与芒硝矿床溶剂随温度的溶解速率有关。（2）本文方法基于InSAR观测相位直接估计CT-PIM模型预计参数，避免了利用SBAS方法生成形变序列观测值再反演预计参数过程中的误差二次传递，提高了形变预计的精度；（3） CT-PIM模型可直接应用于SAR影像覆盖范围后期形变的预计，弥补了传统InSAR形变模型与后期预计模型不一致性的理论缺陷。为验证本文方法可靠性，开展了与传统SBAS-InSAR多速率模型和传统静态概率积分预计方法的对比。结果表明，本文方法构建的CT-PIM模型建模精度提升了38.2%，形变预计精度提升了39.1%。本文方法没有考虑水溶开采中溶液的溶解特性对地表形变的影响，后续研究中将开展顾及溶液浓度和扩散系数等物理参数的时序InSAR建模研究。参考文献（ References ） Berardino P , Fornaro G , Lanari R and Sansosti E . 2002 . 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J 朱建军，杨泽发，李志伟，胡俊 . 2013 . 一种基于InSAR技术的矿区开采监测方法 . 中国专利： ZL 201310011306.2 一种基于InSAR技术的矿区开采监测方法致谢:感谢欧洲航天局（ESA）提供的Sentinel-1A卫星影像。 10.11834/jrs.20222172 P237 A 张腾飞，邢学敏，彭葳，朱珺，刘祥彬，葛家旺，雷敏超.XXXX.基于坐标-时间函数（CT-PIM）的矿区时序InSAR形变预计—以淮安盐矿为例.遥感学报，XX（XX）： 1-16 张腾飞，邢学敏，彭葳，朱珺，刘祥彬，葛家旺，雷敏超.XXXX.基于坐标-时间函数（CT-PIM）的矿区时序InSAR形变预计—以淮安盐矿为例.遥感学报，XX（XX）： 1-16 ZHANG Tengfei，XING Xuemin，PENG Wei，ZHU Jun，LIU Xiangbin，GE Jiawang，LEI Minchao. XXXX. Time Series InSAR Deformation Prediction for Salt Mining Area Based on Coordinate-Time Function （CT-PIM） — A Case Study of Huaian Salt Mine. National Remote Sensing Bulletin， XX（XX）：1-16 ZHANG Tengfei，XING Xuemin，PENG Wei，ZHU Jun，LIU Xiangbin，GE Jiawang，LEI Minchao. XXXX. Time Series InSAR Deformation Prediction for Salt Mining Area Based on Coordinate-Time Function （CT-PIM） — A Case Study of Huaian Salt Mine. National Remote Sensing Bulletin， XX（XX）：1-16 国家自然科学基金面上项目（No.42074033， 51878078）；国家自然科学基金青年基金（No.41701536，No.41904003， No.61701047）；湖南省自然科学基金青年基金（No.2017JJ3322， No.2019JJ50639， No.2020JJ5571）；湖南省教育厅重点项目（No.18A148， 19C0042）；洞庭湖生态环境遥感监测湖南省重点实验室开放项目（No.2021-011） Supported by Supported by the National Natural Science Foundation of China (No. 42074033, 51878078, 41701536, 41904003, 61701047); Hunan Natural Science Foundation Youth Fund (No. 2017JJ3322, 2019JJ50639, 2020JJ5571); Key Project of Education Department of Hunan Province (No. 18A148, 19C0042), the Open Project Program of the Hunan Key Laboratory of remote sensing of ecological environment in Dongting Lake Area (No. 2021-011). 2022-07-07 2022-07-07 遥感学报

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