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Non-linear inverse geothermal problems

機(jī)譯：非線性逆地?zé)釂?wèn)題

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摘要

The inverse geothermal problem consist of estimating the temperature distribution below the earth’s surface using temperature and heat-flux measurements on the earth’s surface. The problem is important since temperature governs a variety of the geological processes including formation of magmas, minerals, fosil fuels and also deformation of rocks. Mathematical this problem is formulated as a Cauchy problem for an non-linear elliptic equation and since the thermal properties of the rocks depend strongly on the temperature, the problem is non-linear. This problem is ill-posed in the sense that it does not satisfy atleast one of Hadamard’s definition of well-posedness. We formulated the problem as an ill-posed non-linear operator equation which is defined in terms of solving a well-posed boundary problem. We demonstrate existence of a unique solution to this well-posed problem and give stability estimates in appropriate function spaces. We show that the operator equation is well-defined in appropriate function spaces. Since the problem is ill-posed, regularization is needed to stabilize computations. We demostrate that Tikhonov regularization can be implemented efficiently for solving the operator equation. The algorithm is based on having a code for solving a well- posed problem related to the operator equation. In this study we demostrate that the algorithm works efficiently for 2D calculations but can also be modified to work for 3D calculations.
機(jī)譯：逆地?zé)釂?wèn)題包括使用地球表面的溫度和熱通量測(cè)量來(lái)估算地球表面以下的溫度分布。這個(gè)問(wèn)題很重要，因?yàn)闇囟瓤刂浦鞣N地質(zhì)過(guò)程，包括巖漿，礦物，化石燃料的形成以及巖石的變形。數(shù)學(xué)上將此問(wèn)題公式化為非線性橢圓方程的柯西問(wèn)題，并且由于巖石的熱特性強(qiáng)烈依賴于溫度，因此該問(wèn)題是非線性的。從不滿足哈達(dá)瑪（Hadamard）關(guān)于適定性的定義中至少一個(gè)的意義上講，這個(gè)問(wèn)題是不適當(dāng)?shù)?。我們將該?wèn)題表述為一個(gè)不適定的非線性算子方程，該方程根據(jù)解決一個(gè)適定的邊界問(wèn)題定義。我們證明存在一個(gè)很好的解決此問(wèn)題的方法，并在適當(dāng)?shù)暮瘮?shù)空間中給出穩(wěn)定性的估計(jì)。我們證明了算子方程在適當(dāng)?shù)暮瘮?shù)空間中是定義明確的。由于問(wèn)題不適當(dāng)，因此需要進(jìn)行正則化以穩(wěn)定計(jì)算。我們演示了Tikhonov正則化可以有效地解決算子方程式。該算法基于具有用于解決與算子方程有關(guān)的適當(dāng)問(wèn)題的代碼。在本研究中，我們演示了該算法可有效地用于2D計(jì)算，但也可以進(jìn)行修改以適用于3D計(jì)算。

著錄項(xiàng)

  • 作者

    Wokiyi, Dennis;

  • 作者單位
  • 年度 2017
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  • 原文格式 PDF
  • 正文語(yǔ)種 eng
  • 中圖分類

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