The observation error covariance (R) matrix is a key component in the data assimilation (DA) process for retrieval of atmospheric state parameters (ASPs), also impacting the subsequent numerical weather forecast. However, one commonly used type of R matrix depends on instrument noise, which contravenes reality because the retrieved ASPs would depend on the instrument used. Other types of R matrix rely on the observation operator (H), analyzed state ( $x_a$ ), background error covariance (B) matrix or the background state ( $x_b$ ), and the selected forecast ensemble. All these dependences reduce the representativeness of the R matrix, since the correctness of H needs verification and no true values exist for $x_a$ or $x_b$ . As such, a better method to correctly specify the R matrix is needed. Through the physical mechanism occurring between incident radiation and particles in the atmosphere, which complies with the phenomena of energy absorption and emission, correlations among bands or channels in a detected atmospheric radiance spectrum occur. This paper thus proposes a data-derived R matrix based on a large number (N) of detected atmospheric radiance spectra constructed from N real-time measurements, where N real-time measurements can be acquired by staring at some observation location of interest during a short amount of time. This data-derived R matrix for satellite radiance observations does not rely on any assumed quantities and is unambiguous. Technically, recording N real-time measurements is achievable by modifying the trigger configuration of data recording from ground.

The flavor composition of ultra-high energy cosmic neutrinos (UHECN) carries precious information about the physical properties of their sources, the nature of neutrino oscillations and possible exotic physics involved during the propagation. Since UHECN with different incoming directions would propagate through different amounts of matter in Earth and since different flavors of charged leptons produced in the neutrino-nucleon charged-current (CC) interaction would have different energy-loss behaviors in the medium, measurement of the angular distribution of incoming events by a neutrino observatory can in principle be employed to help determine the UHECN flavor ratio. In this paper we report on our investigation of the feasibility of such an attempt. Simulations were performed, where the detector configuration was based on the proposed Askaryan Radio Array (ARA) Observatory at the South Pole, to investigate the expected event-direction distribution for each flavor. Assuming $\nu_{\mu}$-$\nu_{\tau}$ symmetry and invoking the standard oscillation and the neutrino decay scenarios, the probability distribution functions (PDF) of the event directions are utilized to extract the flavor ratio of cosmogenic neutrinos on Earth. The simulation results are summarized in terms of the probability of flavor ratio extraction and resolution as functions of the number of observed events and the angular resolution of neutrino directions. We show that it is feasible to constrain the UHECN flavor ratio using the proposed ARA Observatory.

Askaryan Radio Array (ARA), a large-scale radio Cherenkov observatory which scientists propose to develop in Antarctica, aims at discovering the origin and evolution of the cosmic accelerators that produce the highest energy cosmic rays by means of observing the ultra high energy (UHE) cosmogenic neutrinos. To optimize ARA's angular resolution of the incoming UHE neutrinos, which is essential for pointing pack to its source, the relation between the reconstruction capabilities of ARA and its design is studied. It is found that with the noise effect taken into account, in order to make this neutrino angular resolution as good as possible and detection efficiency as high as possible, the optimal choice for ARA geometry would be the station spacing of 1.6 km and the antenna spacing of 40 m.