Hilbert‐Huang transform，一种时间尺度的寻找方法

主要内容:
前些天在973年会上宋黎明老师介绍了Hilbert-Huang变换法，简称HHT，说是类似于FFT、小波变换等方法，但最重要的是不需要基底。但也可以发现时变上的一些信息。

于是我也搜一下相关的信息。
从百度百科看到它的主要特点：1.HHT可以分析非平稳信号；2.自适应，无需预先基底（还是有本质函数的，不过是自动去找）。

宋老师当时引用的应该是这篇文章（被引771次）：http://adsabs.harvard.edu/abs/1998RSPSA.454..903E
查了一下，ADS也有好些用这个方法的，比如这篇做QPO的：http://adsabs.harvard.edu/abs/2014ApJ...788...31H
还有做太阳flare的：http://adsabs.harvard.edu/abs/2015MNRAS.451.4360K

这个方法也可以用到GRB的光变上来呀，也可以用到SwJ1644+57等类似的源上去。

精彩摘抄:
从百度百科抄下来：1998年，Norden E. Huang（黄锷：中国台湾海洋学家）等人提出了经验模态分解方法，并引入了Hilbert谱的概念和Hilbert谱分析的方法，美国国家航空和宇航局（NASA）将这一方法命名为Hilbert-Huang Transform，简称HHT，即希尔伯特-黄变换。

HHT主要内容包含两部分，第一部分为经验模态分解（Empirical Mode Decomposition，简称EMD），它是由Huang提出的；第二部分为Hilbert谱分析（Hilbert Spectrum Analysis，简称HSA）。简单说来，HHT处理非平稳信号的基本过程是：首先利用EMD方法将给定的信号分解为若干固有模态函数（以Intrinsic Mode Function或IMF表示，也称作本征模态函数），这些IMF是满足一定条件的分量；然后，对每一个IMF进行Hilbert变换，得到相应的Hilbert谱，即将每个IMF表示在联合的时频域中；最后，汇总所有IMF的Hilbert谱就会得到原始信号的Hilbert谱。

维基百科的介绍更详细：https://zh.wikipedia.org/wiki/%E5%B8%8C%E7%88%BE%E4%BC%AF%E7%89%B9-%E9%BB%83%E8%BD%89%E6%8F%9B




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Title:		A review on Hilbert-Huang transform: Method and its applications to geophysical studies
Authors:		Huang, Norden E.; Wu, Zhaohua
Affiliation:		AA(Research Center for Adaptive Data Analysis, National Central University, Chungli, Taiwan), AB(Center for Ocean-Land-Atmosphere Studies, Calverton, Maryland, USA)
Publication:		Reviews of Geophysics, Volume 46, Issue 2, CiteID RG2006 (RvGeo Homepage)
Publication Date:		06/2008
Origin:		AGU; WILEY
Keywords:		Computational Geophysics: Data analysis: algorithms and implementation, Global Change: Climate variability (1635, 3305, 3309, 4215, 4513), Oceanography: Physical: Surface waves and tides (1222), Seismology: Earthquake ground motions and engineering seismology, Geodesy and Gravity: Earth rotation variations, Hilbert-Huang transform, empirical mode decomposition, Hilbert spectrum analysis, ensemble empirical mode decomposition
Abstract Copyright:		Copyright 2008 by the American Geophysical Union.
DOI:		10.1029/2007RG000228
Bibliographic Code:		2008RvGeo..46.2006H

Abstract

Data analysis has been one of the core activities in scientific research, but limited by the availability of analysis methods in the past, data analysis was often relegated to data processing. To accommodate the variety of data generated by nonlinear and nonstationary processes in nature, the analysis method would have to be adaptive. Hilbert-Huang transform, consisting of empirical mode decomposition and Hilbert spectral analysis, is a newly developed adaptive data analysis method, which has been used extensively in geophysical research. In this review, we will briefly introduce the method, list some recent developments, demonstrate the usefulness of the method, summarize some applications in various geophysical research areas, and finally, discuss the outstanding open problems. We hope this review will serve as an introduction of the method for those new to the concepts, as well as a summary of the present frontiers of its applications for experienced research scientists.

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有个博客讲怎么使用的：
http://blog.sina.com.cn/s/blog_84024a4a01019pfw.html

有一本书：

Hilbert-Huang Transform and Its Applications

Norden Eh Huang, Samuel S. Shen

World Scientific, 2005 - 311页

2 评价

The HilbertOCoHuang Transform (HHT) represents a desperate attempt to break the suffocating hold on the field of data analysis by the twin assumptions of linearity and stationarity. Unlike spectrograms, wavelet analysis, or the WignerOCoVille Distribution, HHT is truly a time-frequency analysis, but it does not require an a priori functional basis and, therefore, the convolution computation of frequency. The method provides a magnifying glass to examine the data, and also offers a different view of data from nonlinear processes, with the results no longer shackled by spurious harmonics OCo the artifacts of imposing a linearity property on a nonlinear system or of limiting by the uncertainty principle, and a consequence of Fourier transform pairs in data analysis. This is the first HHT book containing papers covering a wide variety of interests. The chapters are divided into mathematical aspects and applications, with the applications further grouped into geophysics, structural safety and visualization.

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Todd 2014 自动从image中得到光变曲线的软件HEIDI

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Title:		HEIDI: An Automated Process for the Identification and Extraction of Photometric Light Curves from Astronomical Images
Authors:		Todd, M.; Wallon Pizarro, H. U.; Tanga, P.; Coward, D. M.; Zadnik, M. G.
Publication:		eprint arXiv:1401.2593
Publication Date:		01/2014
Origin:		ARXIV
Keywords:		Astrophysics - Instrumentation and Methods for Astrophysics
Comment:		6 pages, 6 figures, submitted to MNRAS
Bibliographic Code:		2014arXiv1401.2593T

Abstract

The production of photometric light curves from astronomical images is a very time-consuming task. Larger data sets improve the resolution of the light curve, however, the time requirement scales with data volume. The data analysis is often made more difficult by factors such as a lack of suitable calibration sources and the need to correct for variations in observing conditions from one image to another. Often these variations are unpredictable and corrections are based on experience and intuition. The High Efficiency Image Detection & Identification (HEIDI) pipeline software rapidly processes sets of astronomical images. HEIDI automatically selects multiple sources for calibrating the images using an algorithm that provides a reliable means of correcting for variations between images in a time series. The algorithm takes into account that some sources may intrinsically vary on short time scales and excludes these from being used as calibration sources. HEIDI processes a set of images from an entire night of observation, analyses the variations in brightness of the target objects and produces a light curve all in a matter of minutes. HEIDI has been tested on three different time series of asteroid 939 Isberga and has produced consistent high quality photometric light curves in a fraction of the usual processing time. The software can also be used for other transient sources, e.g. gamma-ray burst optical afterglows. HEIDI is implemented in Python and processes time series astronomical images with minimal user interaction. HEIDI processes up to 1000 images per run in the standard configuration. This limit can be easily increased. HEIDI is not telescope-dependent and will process images even in the case that no telescope specifications are provided. HEIDI has been tested on various Linux . HEIDI is very portable and extremely versatile with minimal hardware requirements.

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Hakkila 2014 对BATSE暴的pulse拟合后的残余的统计

主要内容:
用Norris 2005的pusle的函数拟合，然后关心所剩下的，发现残余的形状都一样！
还给出了很多暴的参数数据。

精彩摘抄:
用Norris的函数

剩下的做平均：

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Title:		Gamma-Ray Burst Pulse Shapes: Evidence for Embedded Shock Signatures?
Authors:		Hakkila, Jon; Preece, Robert D.
Publication:		eprint arXiv:1401.4047
Publication Date:		01/2014
Origin:		ARXIV
Keywords:		Astrophysics - High Energy Astrophysical Phenomena
Comment:		52 pages, 20 figures, accepted for publication in The Astrophysical Journal
Bibliographic Code:		2014arXiv1401.4047H

Abstract

A study of a set of well-isolated pulses in Long and Intermediate gamma ray burst light curves indicates that simple pulse models having smooth and monotonic pulse rise and decay regions are inadequate. Examining the residuals of fits of pulses to such models suggests the following patterns of departure from smooth pulses: three separate wavelike peaks found in the residuals of each pulse (the precursor peak, the central peak, and the decay peak) combine with the underlying Norris et al. (2005) pulse model to produce five distinct regions in the temporal evolution of each pulse. The Precursor Shelf occurs prior to or concurrent with the exponential Rapid Rise. The pulse reaches maximum intensity at the Peak Plateau, then undergoes a Rapid Decay. The decay gradually slows into an Extended Tail. Despite these distinct temporal segments, the pulses studied are almost universally characterized by hard to soft spectral evolution, arguing that the new pulse features reflect a single evolution, rather than being artifacts of pulse overlap. The fluctuations can give a single pulse the appearance of having up to three distinct localized peaks, leading to ambiguities in pulse-fitting if an incorrect pulse model is used. The approach demonstrates that complex GRBs may be composed of fewer pulses than indicated by the number of peaks. The large degree of similar spectro-temporal behavior within gamma-ray burst pulses indicates that a single process is responsible for producing pulses spanning a tremendous range of durations, luminosities, and spectral hardnesses, and the correlated characteristics of the wavelike peaks are related to the pulse asymmetry, suggesting kinematic origins that seem supportive of relativistic shocks.

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Lazzati 2013 长暴的中心引擎时间可能就20秒

主要内容:
而看起来的T90的多样性来自1.穿出包层的时间；2.不同观测角；3.不同红移所导致的修正。

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Title:		On the observed duration distribution of gamma-ray bursts from collapsars
Authors:		Lazzati, D.; Villeneuve, M.; López-Cámara, D.; Morsony, B. J.; Perna, R.
Affiliation:		AA(Department of Physics, NC State University, 2401 Stinson Drive, Raleigh, NC 27695-8202, USA; davide_lazzati@ncsu.edu), AB(Department of Astrophysical and Planetary Sciences, University of Colorado, Boulder, CO 80309-0391, USA), AC(Department of Physics, NC State University, 2401 Stinson Drive, Raleigh, NC 27695-8202, USA), AD(Department of Astronomy, University of Wisconsin-Madison, 3321 Sterling Hall, 475 N. Charter Street, Madison, WI 53706-1582, USA), AE(Department of Astrophysical and Planetary Sciences, University of Colorado, Boulder, CO 80309-0391, USA; JILA, University of Colorado and National Institute of Standards and Technology, Boulder, CO 80309-0440, USA)
Publication:		Monthly Notices of the Royal Astronomical Society, Volume 436, Issue 2, p.1867-1872 (MNRAS Homepage)
Publication Date:		12/2013
Origin:		OUP
Astronomy Keywords:		hydrodynamics, methods: numerical, gamma-ray burst: general
Abstract Copyright:		2013 The Authors Published by Oxford University Press on behalf of the Royal Astronomical Society
DOI:		10.1093/mnras/stt1705
Bibliographic Code:		2013MNRAS.436.1867L

Abstract

The duration of the prompt emission of long gamma-ray bursts is generally considered to be fairly similar to the duration of the activity of the engine in the centre of the progenitor star. Here, we investigate the relation between the duration of the engine activity and that of the observed light curve, using inputs from both numerical simulations and observations. We find that the observed burst duration is a good proxy for the engine duration after the time necessary for the jet to break out the star's surface is subtracted. However, the observed duration is a function of the viewing angle and can be significantly shorter than the duration of the engine activity. We also show that the observed, redshift-corrected burst duration evolves only moderately with redshift for both observations and synthetic light curves. We conclude that the broad distribution of the observed duration of long Burst And Transient Sources Experiment (BATSE) gamma-ray bursts is mostly accounted for by an engine lasting ˜20 s, the dispersion being due to viewing and redshift effects. Our results do not rule out the existence of engines with very long duration. However, we find that they are constrained to be a small minority of the BATSE detected bursts.

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Nemiroff, Robert J. 2011 可以用一个简单的唯象公式来拟合pulse的光变

主要内容:
就是这个公式

做到了参数少又能很好的拟合，其中应该有什么原因。

精彩摘抄:
拟合的参数：

拟合的光变：

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Title:		Extension of an exponential light-curve gamma-ray burst pulse model across energy bands
Authors:		Nemiroff, Robert J.
Affiliation:		AA(Department of Physics, Michigan Technological University, Houghton, MI 49931, USA)
Publication:		Monthly Notices of the Royal Astronomical Society, Volume 419, Issue 2, pp. 1650-1659. (MNRAS Homepage)
Publication Date:		01/2012
Origin:		WILEY
Astronomy Keywords:		gamma-ray burst: general
Abstract Copyright:		© 2011 The Author Monthly Notices of the Royal Astronomical Society © 2011 RAS
DOI:		10.1111/j.1365-2966.2011.19838.x
Bibliographic Code:		2012MNRAS.419.1650N

Abstract

A simple mathematical model of gamma-ray burst(GRB) pulses in time, suggested by Norris et al., is extended across energy. For a class of isolated pulses, two fit parameters appear to be effectively independent of energy. Specifically, statistical fits indicate that pulse amplitude A and pulse width τ are energy dependent, while pulse start time and pulse shape are effectively energy independent. These results bolster the pulse start and pulse scale conjectures of Nemiroff and add a new pulse shape conjecture which states that a class of pulses all have the same shape. The simple resulting pulse counts model is P(t, E) =A(E) exp[ -t/τ(E) -τ(E)/t], where t is the time since the start of the pulse. This pulse model is found to be an acceptable statistical fit to many of the fluent separable Burst and Transient Source Experiment (BATSE) pulses listed by Norris et al. Even without theoretical interpretation, this cross-energy extension may be immediately useful for fitting prompt emission from GRB pulses across energy channels with a minimal number of free parameters.

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Bhat, P. N. 2011 暴的pulse间隔时间的统计

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Title:		Temporal Deconvolution Study of Long and Short Gamma-Ray Burst Light Curves
Authors:		Bhat, P. N.; Briggs, Michael S.; Hodge, Colleen
Affiliation:		AA(Center for Space Plasma and Aeronomic Research (CSPAR),
Publication:		The Astrophysical Journal, Volume 744, Issue 2, article id. 141 (2012). (ApJ Homepage)
Publication Date:		01/2012
Origin:		IOP
Astronomy Keywords:		gamma-ray burst: general, methods: data analysis, relativistic processes
DOI:		10.1088/0004-637X/744/2/141
Bibliographic Code:		2012ApJ...744..141B

Abstract

The light curves of gamma-ray bursts (GRBs) are believed to result from internal shocks reflecting the activity of the GRB central engine. Their temporal deconvolution can reveal potential differences in the properties of the central engines in the two populations of GRBs which are believed to originate from the deaths of massive stars (long) and from mergers of compact objects (short). We present here the results of the temporal analysis of 42 GRBs detected with the Gamma-ray Burst Monitor onboard the Fermi Gamma-ray Space Telescope. We deconvolved the profiles into pulses, which we fit with lognormal functions. The distributions of the pulse shape parameters and intervals between neighboring pulses are distinct for both burst types and also fit with lognormal functions. We have studied the evolution of these parameters in different energy bands and found that they differ between long and short bursts. We discuss the implications of the differences in the temporal properties of long and short bursts within the framework of the internal shock model for GRB prompt emission.

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Basak, Rupal 2011 暴本身光变的Timing分析

主要内容:
应用于Fermi时代最亮的暴 090618
说还可以作为标准烛光。

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Title:		Measuring the pulse of GRB 090618: A Simultaneous Spectral and Timing Analysis of the Prompt Emission
Authors:		Basak, Rupal; Rao, A. R.
Publication:		eprint arXiv:1110.1994
Publication Date:		10/2011
Origin:		ARXIV
Keywords:		Astrophysics - High Energy Astrophysical Phenomena
Comment:		18 pages, 4 tables, 11 figures, accepted for publication in ApJ
Bibliographic Code:		2011arXiv1110.1994R

Abstract

We develop a new method for simultaneous timing and spectral studies of Gamma Ray Burst (GRB) prompt emission and apply it to make a pulse-wise description of the prompt emission of GRB 090618, the brightest GRB detected in the Fermi era. We exploit the large area (and sensitivity) of Swift/BAT and the wide band width of Fermi/GBM to derive the parameters for a complete spectral and timing description of the individual pulses of this GRB, based on the various empirical relations suggested in the literature. We demonstrate that this empirical model correctly describes the other observed properties of the burst like the variation of the lag with energy and the pulse width with energy. The measurements also show an indication of an increase in pulse width as a function of energy at low energies for some of the pulses, which is naturally explained as an off-shoot of some particular combination of the model parameters. We argue that these model parameters, particularly the peak energy at the beginning of the pulse, are the natural choices to be used for correlation with luminosity. The implications of these results for the use of GRBs as standard candles are briefly described.

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Tashiro, Makoto S. 2011 7个暴的下降阶段的谱

主要内容:
用的日本Suzaku WAM的数据。
TH0, TH1, TH2, or TH3, which nominally cover the 50–110 keV, 110–220 keV, 220–540 keV, and 540–5000 keV bands

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Title:		Spectral Evolutions in Gamma-Ray Burst Exponential Decays Observed with Suzaku WAM
Authors:		Tashiro, Makoto S.; Onda, Kaori; Yamaoka, Kazutaka; Ohno, Masahiro; Sugita, Satoshi; Uehara, Takeshi; Seta, Hiromi
Publication:		eprint arXiv:1110.4246
Publication Date:		10/2011
Origin:		ARXIV
Keywords:		Astrophysics - High Energy Astrophysical Phenomena
Comment:		11 pages, 7 figures, 5 tables. PASJ accepted
Bibliographic Code:		2011arXiv1110.4246T

Abstract

This paper presents a study on the spectral evolution of gamma-ray burst (GRB) prompt emissions observed with the Suzaku Wide-band All-sky Monitor (WAM). By making use of the WAM data archive, 6 bright GRBs exhibiting 7 well-separated fast-rise-exponential-decay (FRED) shaped light curves are presented and the evaluated exponential decay time constants of the energy-resolved light curves from these FRED peak light curves are shown to indicate significant spectral evolution. The energy dependence of the time constants is well described with a power-law function tau(E) ~ E^gamma, where gamma ~ -(0.34 +/- 0.12) in average, although 5 FRED peaks show consistent value of gamma = -1/2 which is expected in synchrotron or inverse-Compton cooling models. In particular, 2 of the GRBs were located with accuracy sufficient to evaluate the time-resolved spectra with precise energy response matrices. Their behavior in spectral evolution suggests two different origins of emissions. In the case of GRB081224, the derived 1-s time-resolved spectra are well described by a blackbody radiation model with a power-law component. The derived behavior of cooling is consistent with that expected from radiative cooling or expansion of the emission region. On the other hand, the other 1-s time-resolved spectra from GRB100707A is well described by a Band GRB model as well as with the thermal model. Although relative poor statistics prevent us to conclude, the energy dependence in decaying light curve is consistent with that expected in the former emission mechanism model.

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