1/f-Noise Estimation From Microwave Imager Data With Periodic Gaps

Abstract

We describe a method to estimate coefficients ${{{\bm{h}}}_{\bm{n}}}$ of a power spectral density of the form ${\bm{\Sigma}}{{{\bm{h}}}_{\bm{n}}}/{{{\bm{f}}}^{\bm{n}}}$ ( ${\bm{f}}\;{\bm{\ }}\text{is}$ the frequency and integer ${\bm{n}} \geq 0$ ) from corresponding measured time series with periodic gaps. This technique is applied to consistently estimate the amount of $1/{\bm{f}}$ noise present in weather system follow-on microwave microwave imager's (MWI) channels from the time-series data collected with different periodic gaps during prelaunch ground tests. The method assumes that the power spectrum of $1/{\bm{f}}$ noise present in MWI can be represented as a second-order frequency polynomial model of the form ${\bm{\Sigma}}{{{\bm{h}}}_{\bm{n}}}/{{{\bm{f}}}^{\bm{n}}}{\bm{\ }}$ and attempts to retrieve the true spectrum by solving for the ${{{\bm{h}}}_{\bm{n}}}$ coefficients using the power spectrum of the time-series data with periodic gaps. The method also assumes that the periodicity and duration of the data gaps are known and consistent for a given time series. The theoretical basis of the new technique is derived and tested using simulation and the new procedure is then applied to real test data to estimate the coefficients of the frequency polynomial. As a quantitative estimate for the $1/{\bm{f}}$ noise, the radiometer gain fluctuation ( ${\bm{\Delta}}{\bm{G}}/{\bm{G}}$ ) at 1 Hz is then solved from the frequency polynomial of the gain fluctuation power spectrum. The 1 Hz $( {{\bm{\Delta}}{\bm{G}}/{\bm{G}}} )$ values were compared between two sets of ground test data for the same MWI channels but with large (92%) and small (4.89%) duty cycles. The similarity of the 1 Hz $( {{\bm{\Delta}}{\bm{G}}/{\bm{G}}} )$ values extracted from these two disparate datasets establishes confidence in the method. The derived noise power spectrum is then used to simulate MWIs radiometric brightness temperature images and predict the level of unwanted striping in the flight data due to $1/{\bm{f}}$ noise content. This method may be applicable to solve for the polynomial coefficients of the power spectrum of any noise process, which can be modeled as a frequency polynomial, given the polynomial form is known a priori.