Find in Library

In atmospheric science we are confronted with inverse problems arising in applications associated with retrievals of geophysical parameters. A nonlinear mapping from geophysical quantities (e.g., atmospheric properties) to spectral measurements can be represented by a forward model. An inversion often suffers from the lack of stability and its stabilization introduced by proper approaches, however, can be treated with sufficient generality. In principle, regularization can enforce uniqueness of the solution when additional information is incorporated into the inversion process. In this paper, we analyze different forms of the regularization matrix <inline-formula><math display="inline"><semantics><mi mathvariant="bold">L</mi></semantics></math></inline-formula> in the framework of Tikhonov regularization: the identity matrix <inline-formula><math display="inline"><semantics><msub><mi mathvariant="bold">L</mi><mn>0</mn></msub></semantics></math></inline-formula>, discrete approximations of the first and second order derivative operators <inline-formula><math display="inline"><semantics><msub><mi mathvariant="bold">L</mi><mn>1</mn></msub></semantics></math></inline-formula> and <inline-formula><math display="inline"><semantics><msub><mi mathvariant="bold">L</mi><mn>2</mn></msub></semantics></math></inline-formula>, respectively, and the Cholesky factor of the a priori profile covariance matrix <inline-formula><math display="inline"><semantics><msub><mi mathvariant="bold">L</mi><mi mathvariant="normal">C</mi></msub></semantics></math></inline-formula>. Each form of <inline-formula><math display="inline"><semantics><mi mathvariant="bold">L</mi></semantics></math></inline-formula> has its intrinsic pro/cons and thus may lead to different performance of inverse algorithms. An extensive comparison of different matrices is conducted with two applications using synthetic data from airborne and satellite sensors: retrieving atmospheric temperature profiles from microwave spectral measurements, and deriving aerosol properties from near infrared spectral measurements. The regularized solution obtained with <inline-formula><math display="inline"><semantics><msub><mi mathvariant="bold">L</mi><mn>0</mn></msub></semantics></math></inline-formula> possesses a reasonable magnitude, but its smoothness is not always assured. The retrieval using <inline-formula><math display="inline"><semantics><msub><mi mathvariant="bold">L</mi><mn>1</mn></msub></semantics></math></inline-formula> and <inline-formula><math display="inline"><semantics><msub><mi mathvariant="bold">L</mi><mn>2</mn></msub></semantics></math></inline-formula> produces a solution in favor of the smoothness, and the impact of the a priori knowledge is less critical on the retrieval using <inline-formula><math display="inline"><semantics><msub><mi mathvariant="bold">L</mi><mn>1</mn></msub></semantics></math></inline-formula>. The retrieval performance of <inline-formula><math display="inline"><semantics><msub><mi mathvariant="bold">L</mi><mi mathvariant="normal">C</mi></msub></semantics></math></inline-formula> is affected by the accuracy of the a priori knowledge.