arXiv subjects
Publications and source records attributed to Rehan Siddiqui.
In this work, we prove the irrationality of $\pi$ motivated by the nested radicals with roots of $2$ of kind $c_k = \sqrt{2 + c_{k - 1}}$ and $c_0 = 0$. Sample computations showing how the rational approximation tends to $\pi$ with increasing the integer $k$ are presented.
In this work, we obtain an iterative formula that can be used for computing digits of $π$ and nested radicals of kind $c_n/\sqrt{2 - c_{n - 1}}$, where $c_0 = 0$ and $c_n = \sqrt{2 + c_{n - 1}}$. We also show how with the help of this iterative formula, the two-term Machin-like formulas for $π$ can be generated and approximated. Some examples with Mathematica codes are presented.
In this work, we develop a method for rational approximation of the Fourier transform (FT) based on the real and imaginary parts of the complex error function \[ w(z) = e^{-z^2}(1 - {\rm{erf}}(-iz)) = K(x,y) + iL(x,y), \qquad z = x + iy, \] where $K(x,y)$ and $L(x,y)$ are known as the Voigt and imaginary Voigt functions, respectively. In contrast to our previous rational approximation of the FT, the expansion coefficients in this method are not dependent on the values of a sampled function. As the values of the Voigt functions remain the same, this approach can be used for rapid computation with help of look-up tables. Mathematica codes with some examples are presented.
In this work, we consider the properties of the two-term Machin-like formula and develop an algorithm for computing digits of $π$ by using its rational approximation. In this approximation, both terms are constructed by using a representation of $1/π$ in the binary form. This approach provides the squared convergence in computing digits of $π$ without any trigonometric functions and surd numbers. The Mathematica codes showing some examples are presented.
In this work, we develop a new iterative method for computing the digits of $π$ by argument reduction of the tangent function. This method combines a modified version of the iterative formula for $π$ with squared convergence that we proposed in a previous work and a leading arctangent term from the Machin-like formula. The computational test we performed shows that algorithmic implementation can provide more than $17$ digits of $π$ per increment. Mathematica codes, showing the convergence rate for computing the digits of $π$, are presented.
In this work we derive a generalized series expansion of the acrtangent function by using the enhanced midpoint integration (EMI). Algorithmic implementation of the generalized series expansion utilizes two-step iteration without surd and complex numbers. The computational test we performed reveals that such a generalization improves accuracy in computation of the arctangent function by many orders of the magnitude with increasing integer $M$, associated with subintervals in the EMI formula. The generalized series expansion may be promising for practical applications. It may be particularly useful in practical tasks, where extensive computations with arbitrary precision floating points are needed. The algorithmic implementation of the generalized series expansion of the arctangent function shows a rapid convergence rate in the computation of digits of $π$ in the Machin-like formulas.
Previously we have proposed a new method of transforming quotients into integer reciprocals in the Machin-like formulas for $π$. As a further development, here we show how to generate a multi-term Machin-like formula for $π$ with a reduced Lehmer measure. The Mathematica codes validating these results are presented.
Recently we have reported a new method of rational approximation of the sinc function obtained by sampling and the Fourier transforms. However, this method requires a trigonometric multiplier that originates from shifting property of the Fourier transform. In this work we show how to represent the Fourier transform of a function $f(t)$ in form of a ratio of two polynomials without any trigonometric multiplier. A MATLAB code showing algorithmic implementation of the proposed method for rational approximation of the Fourier transform is presented.
The Radiance Enhancement (RE) method was introduced for efficient detection of clouds from the space. Recently, we have also reported that due to high reflectance of combustion-originated smokes, this approach can also be generalized for detection of the forest fires by retrieving and analyzing datasets collected from a space orbiting micro-spectrometer operating in the near infrared spectral range. In our previous publication, we have performed a comparison of observed and synthetic radiance spectra by developing a method for computation of surface reflectance consisting of different canopies by weighted sum based on their areal coverage. However, this approach should be justified by a method based on corresponding proportions of the upwelling radiance. The results of computations we performed in this study reveal a good match between areal coverage of canopies and the corresponding proportions of the upwelling radiance due to effect of the instrument slit function.
The existing methods for detection of the cloud scenes are applied at relatively small spectral range within shortwave upwelling radiative wavelength flux. We have reported a new method for detection of the cloud scenes based on the Radiance Enhancement (RE). This method can be used to cover a significantly wider spectral range from 1100 nm to 1700 nm by using datasets from the space-orbiting micro-spectrometer Argus 1000. Due to high sunlight reflection of the smoke originated from the forest or field fires the proposed RE method can also be implemented for detection of combustion aerosols. This approach can be a promising technique for efficient detection and continuous monitor of the seasonal forest and field fires. To the best of our knowledge this is the first report showing how a cloud method can be generalized for efficient detection of the forest fires due to combustion-originated reflectance.
In this work we develop a new algorithm for the efficient computation of the Voigt/complex error function. In particular, in this approach we propose a two-domain scheme where the number of the interpolation grid-points is dependent on the input parameter $y$. The error analysis we performed shows that the MATLAB implementation meets the requirements for radiative transfer applications involving the HITRAN molecular spectroscopic database. The run-time test shows that this MATLAB implementation provides rapid computation, especially at smaller ranges of the parameter $x$.
We present a new form of the Machin-like formula for $π$ that can be generated by using iteration. This form of the Machin-like formula may be promising for computation of the constant $π$ due to rapidly increasing integers at each step of the iteration. The computational test we performed shows that, with an integer $k \ge 17$, the Lehmer measure remains small and practically does not increase after $18$ steps of iteration.
In our earlier publication we have shown how to compute by iteration a rational number ${u_{2,k}}$ in the two-term Machin-like formula for pi of kind $$\fracπ{4}=2^{k-1}\arctan\left(\frac{1}{u_{1,k}}\right)+\arctan\left(\frac{1}{u_{2,k}}\right),\qquad k\in \mathbb{Z},\quad k\ge 1,$$ where ${u_{1,k}}$ can be chosen as an integer ${u_{1,k}} = \left\lfloor{{a_k}/\sqrt{2-a_{k-1}}}\right\rfloor$ with nested radicals defined as ${a_k}=\sqrt{2+a_{k-1}}$ and $a_0 = 0$. In this work we report an alternative method for determination of the integer $u_{1,k}$. This approach is based on a simple iteration and does not require any irrational (surd) numbers from the set $\left\{a_k\right\}$ in computation of the integer $u_{1,k}$. Mathematica programs validating these results are presented.
Lehmer defined a measure $$ μ=\sum\limits_{j=1}^J\frac{1}{\log_{10}\left(\left|β_j\right|\right)}, $$ where the $β_j$ may be either integers or rational numbers in a Machin-like formula for $π$. When the $β_j$ are integers, Lehmer's measure can be used to determine the computational efficiency of the given Machin-like formula for $π$. However, because the computations are complicated, it is unclear if Lehmer's measure applies when one or more of the $β_j$ are rational. In this article, we develop a new algorithm for a two-term Machin-like formula for $π$ as an example of the unconditional applicability of Lehmer's measure. This approach does not involve any irrational numbers and may allow calculating $π$ rapidly by the Newton$-$Raphson iteration method for the tangent function.
In our earlier publication we introduced the Spectrally Integrated Voigt Function (SIVF) as an alternative to the traditional Voigt function for the HITRAN-based applications [Quine & Abrarov, JQSRT 2013]. It was shown that application of the SIVF enables us to reduce spectral resolution without loss of accuracy in computation of the spectral radiance. As a further development, in this study we present more efficient SIVF approximations derived by using a new sampling method based on incomplete cosine expansion of the sinc function [Abrarov & Quine, Appl. Math. Comput. 2015]. Since the SIVF mathematically represents the mean value integral of the Voigt function, this method accounts for area under the curve of the Voigt function. Consequently, the total band radiance, defined as the integrated spectral radiance within a given spectral region, can also retain its accuracy even at low spectral resolution. Our numerical results demonstrate that application of the SIVF may be promising for more rapid line-by-line computation in atmospheric models utilizing the HITRAN molecular spectroscopic database. Such an approach may be particularly efficient to implement a retrieval algorithm for the greenhouse gases from the NIR space data collected by Earth-orbiting micro-spectrometers like Argus 1000 for their operation in a real-time mode. The real-time mode operation of the micro-spectrometers can be advantageous for instant decision making during flight for more efficient data collection from space.
The description, imagery and interpretation of cloud scenes by remote sensing datasets from Earth-orbiting satellites have become a great debate for several decades. Presently, there are many models for cloud detection and its classifications have been reported. However, none of the existing models can efficiently detect the clouds within the small band of shortwave upwelling radiative wavelength flux (SWupRF) in the spectral range from 1100 to 1700 nm. Therefore, in order to detect the clouds more effectively, a method known as the radiance enhancement (RE) can be implemented (Siddiqui et al., 2015). This article proposes new approaches how with RE and SWupRF methods to distinguish cloud scenes by space orbiting Argus 1000 micro-spectrometer utilizing the GENSPECT line-by-line radiative transfer model (Quine and Drummond, 2002; Siddiqui, 2017). This RE approach can also be used within the selected wavelength band for the detection of combustion-originated aerosols due to seasonal forest fires.
Carbon Dioxide (CO2) is one of the most important greenhouse gases after water vapor (H2O) which plays significant role in the climate process. Accurate space-based measurement of CO2 is of great significance in inferring the location of CO2 sources and sinks. Uncertainties in greenhouse gases (GHG) retrieval process must be minimized to accurately infer the actual amount of the atmospheric species. Clouds pose a large uncertainty in CO2 space-based retrieval process leading, mostly, to an underestimation in the CO2 absorption amount above the cloud layer provided that photons do not perform multiple paths. In this paper, three different cases of data collected over cloudy and clear skies by Argus 1000 micro-spectrometer were analyzed. Findings show that the CO2 absorption in the absence of clouds is approximately 4.5% higher than when clouds are present.
This new study develops an algorithm for Short Wave upwelling Radiative Flux (SWupRF) for the spectral variations within near infrared (NIR) from 1100 to 1700 nm wavelength band based on remote sensing data set of Argus 1000 micro-spectrometer observations. We calculate the SWupRF by investigating the total radiative flux due to O2, H2O, CO2 and CH4 and also by the individual gas within the selected wavelength bands of interest. A GENSPECT synthetic line by line radiative transfer model is applied to perform radiative transfer simulations to calculate the radiative flux by varying surface albedo, mixing ratios of the selected greenhouse gases, surface temperature, solar sun and zenith angles with different latitude and longitude of the instrument. Finally, the SWupRFsyn estimated from GENSPECT was compared with SWupRFobs from Argus 1000 over a period of four years (2009 and 2013) covering all seasons. We calculate and compare both the synthetic and real measured observed data set. The synthetic model gives SWupRFsyn within the range of [0.3950 to 1.650] W/m2 and the selected Argus observed model gives SWupRFobs within the range of [0.10 to 3.15] W/m2. The simulated range of synthetic model also represents 1 to 32 K rise of temperature within the different concentration of water vapor and other gases. The authors determined that the satellite observed data of week 75 pass 43 & week 08 pass 61 showing minimum 0.850 [0.10 to 1.60] W/m2 and maximum 1.65 [0.12 to 3.15] W/m2 respectively.