Abstract. Global high-resolution observations of atmospheric composition from satellites can greatly improve our understanding of surface emissions through inverse analyses. Variational inverse methods can optimize surface emissions at any resolution but do not readily quantify the error and information content of the posterior solution. In fact, the information content of satellite data may be orders of magnitude lower than its coverage suggests because of failed retrievals, instrument noise, and error correlations that propagate through the inversion. Analytical solution to the inverse problem provides closed-form characterization of posterior error statistics and information content but requires the construction of the Jacobian matrix that relates emissions to atmospheric concentrations. Building the Jacobian matrix is computationally expensive at high resolution because it involves perturbing each emission element, typically individual grid cells, in the atmospheric transport model used as forward model for the inversion. We propose and analyze two methods, reduced-dimension and reduced-rank, to construct the Jacobian matrix at greatly decreased computational cost while retaining information content. Both methods begin from an initial native-resolution estimate of the Jacobian matrix constructed at no computational cost by assuming that atmospheric concentrations are most sensitive to local emissions. The reduced-dimension method uses this estimate to construct a Jacobian matrix on a multiscale grid that maintains high resolution in areas with high information content and aggregates grid cells elsewhere. The reduced-rank method constructs the Jacobian matrix at native resolution by perturbing the leading patterns of information content given by the initial estimate. We demonstrate both methods in an analytical Bayesian inversion of GOSAT methane satellite data with augmented information content over North America in July 2009. We show that both methods reproduce the results of the native-resolution inversion while achieving a factor of 4 improvement in computational performance. The reduced-dimension method produces an exact solution at lower spatial resolution while the reduced-rank method solves the inversion at native resolution in areas of high information content and defaults to the prior estimate elsewhere.

中文翻译：

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高分辨率大气反演卫星观测的雅可比矩阵的低成本构造

摘要。通过对卫星大气成分的全球高分辨率观测，可以通过反分析极大地增进我们对地表发射的理解。变分逆方法可以在任何分辨率下优化表面发射，但不能轻易量化后解的误差和信息含量。实际上，由于检索失败，仪器噪声以及通过反演传播的误差相关性，卫星数据的信息内容可能比其覆盖范围低几个数量级。反问题的解析解提供了后验误差统计和信息内容的封闭形式特征，但需要构造将排放与大气浓度相关的雅可比矩阵。建立高分辨率的雅可比矩阵在计算上是昂贵的，因为它涉及在大气传输模型（用作反演的正向模型）中干扰每个发射元素，通常是单个网格单元。我们提出并分析了降维和降秩两种方法，以在保留信息内容的同时以大大降低的计算成本构造雅可比矩阵。两种方法均始于假设大气浓度对局部排放最敏感的情况下对雅可比矩阵的初始原始分辨率估计，该估计无需任何计算即可构建。降维方法使用此估计值在多尺度网格上构造雅可比矩阵，该矩阵在具有高信息含量的区域中保持高分辨率，并聚集其他位置的网格单元。降低秩的方法通过扰动由初始估计值给​​出的信息内容的前导模式，以自然分辨率构造雅可比矩阵。我们在2009年7月对北美地区GOSAT甲烷卫星数据的贝叶斯反演进行了分析，并增加了信息含量。我们展示了这两种方法均能再现原始分辨率反演的结果，同时使计算性能提高4倍。降维方法可在较低的空间分辨率下生成精确的解决方案，而降秩方法可在信息量高的区域中以原始分辨率解决反演问题，并且默认为其他位置的先前估计。