Version 254 (modified by tbretz, 20 months ago) ( diff )

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# Spectrum Analysis

## Theory

### Spectrum

The differential flux per area, time and energy interval is defined as

Often is also referred to as as observation time and effective collection area is a constant.

For an observation with an effective observation time , this yields in a given Energy interval :

For simplicity, in the following, will be replaced by just but always refers to to a given energy interval. If data is binned in a histogram, the relation between the x-value for the bin and the corresponding interval is not well defined. Resonable definitions are the bin center (usually in logarithmic bins) or the average energy.

### Efficiency

The total area and the corresponding efficiency are of course only available for simulated data. For simulated data, is the production area and the corresponding energy dependent efficiency of the analysis chain. For a given energy bin, the efficiency is then defined as

where is the number of simulated events in this energy bin and the number of *excess* events that are produced by the analysis chain.

Note that the exact calculation of the efficiency depends on prior knowledge of the correct source spectrum . Therefore, it is strictly speaking only correct if the simulated spectrum and the real spectrum are identical. As the real spectrum is unknown, special care has to be taken of the systematic introduced by the assumption of .

### Effective Collection Area

The effective area is then defined as . Note that at large distances the efficiency vanishes, so that the effective area is an (energy dependent) constant while and the efficiency are mutually dependent.

### Excess and Error

The number of excess events, for data and simulations, is defined as

where is the number of events identified as potential gammas from the source direction ('on-source') and the number of gamma-like events measured 'off-source'. Note that for Simulations, is not necessarily zero for wobble-mode observations as an event can survive the analysis for on- and off-events, if this is not prevented by the analysis (cuts).

The average number of background events is the total number of background events from all off-regions times the corresponding weight (often referred to as ). For five off-regions, this yields

Assuming Gaussian errors, the statistical error is thus

For data this immediately resolves to

with the Poisson (counting) error .

### Weights and Error

In the following refers to a number of simulated events and to a number of measured (excess) events.

is the number of produced events in the energy interval and the zenith angle interval . The weighted number of events in that interval is then

Since with the spectral weight to adapt the spectral shape of the simulated spectrum to the real (measured) spectrum of the source and the weight to adapt to oberservation time versus zenith angle.

The weighted number of produced events in the total energy interval and the total zenith angle interval is then

with being the total number of produced events.

For a sum of weights, e.g. the corresponding error is

As the energy is well defined, and thus

The weights are defined as follow:

where is the simulated spectrum and the (unknown) real source spectrum. is a normalization constant. The zenith angle weights in the interval are defined as

where is the number of produced events in the interval and is the total observation time in the same zenith angle interval. is the normalization constant. The error on the weight in each individual -bin with and is then

While is given by the data acquisition and 1s per 5min run, is just the statistical error of the number of events.

As the efficiency for an energy interval is calculated as

and and are both expressed as the sum given above, the constants and cancel.

The differential flux in an energy interval is then given as

Where is total area of production and the total observation time. The number of measured excess events is in that energy interval is .

Using Gaussian error propagation, the error in a given energy interval is then given by

with .

## Conceptual Example

### Get Data File List

A list with file IDs containing the events to be analyed is required a.t.m. The following query retrieves such a list and fills a temporary table (DataFiles) with the IDs.

CREATE TEMPORARY TABLE DataFiles
(
FileId INT UNSIGNED NOT NULL,
PRIMARY KEY (FileId) USING HASH
) ENGINE=Memory
AS
(
SELECT
FileId
FROM
factdata.RunInfo
WHERE
%0:where
ORDER BY
FileId
)


%0:where is a placeholder, for example for

fZenithDistanceMean<30 AND fThresholdMinSet<350 AND
fSourceKEY=5 AND
fRunTypeKEY=1 AND
fNight>20161201 AND fNight<20170201 AND
fR750Cor>0.9e0*fR750Ref


### Get Observation Time

The following query bins the effective observation time of the runs listed above in zenith angle bins and stores the result in a temporary table (ObservationTime). Note that the result contains only those bins which have entries.

CREATE TEMPORARY TABLE ObservationTime
(
.theta SMALLINT UNSIGNED NOT NULL,
OnTime FLOAT NOT NULL,
PRIMARY KEY (.theta) USING HASH
) ENGINE=Memory
AS
(
SELECT
INTERVAL(fZenithDistanceMean, %0:bins) AS .theta,
SUM(TIME_TO_SEC(TIMEDIFF(fRunStop,fRunStart))*fEffectiveOn) AS OnTime
FROM
DataFiles
LEFT JOIN
factdata.RunInfo USING (FileId)
GROUP BY
.theta
ORDER BY
.theta
)


%0:bins is a placeholder for the bin boundaries, e.g. 5, 10, 15, 20, 25, 30 (five bins between 5° and 30° plus underflow and overflow).

### Get Monte Carlo File List

The next query obtains all Monte Carlo runs which have their ThetaMin or ThetaMax within one of the bins obtained in the previous query (so all MC runs that correspond to bins in which data is available). Strictly speaking, this step is not necessary, but it accelerats further processing. In addition (here as an example) only runs with even FileIDs are obtained as test-runs (assuming that odd runs were used for training). The resulting FileIDs are stored in a temporary table (MonteCarloFiles).

CREATE TEMPORARY TABLE MonteCarloFiles
(
FileId INT UNSIGNED NOT NULL,
PRIMARY KEY (FileId) USING HASH
) ENGINE=Memory
AS
(
SELECT
FileId
FROM
ObservationTime
LEFT JOIN
BinningTheta ON .theta=bin
LEFT JOIN
factmc.RunInfoMC
ON
(ThetaMin>=lo AND ThetaMin<hi) OR (ThetaMax>lo AND ThetaMax<=hi)
WHERE
PartId=1 AND
FileId%%2=0
ORDER BY
FileId
)


### Get Zenith Angle Histogram

The following table creates a temporaray table (EventCount) internally which bins the MonetCarlo files from the file list in MonteCarloFiles in zenith angle bins. This temporary table ois then joined with table containing the binning for the data files (EventCount) and for each bin, the ratio (ZdWeight) and the corresponding error (ErrZdWeight) is calculated (assuming an error on the on-time of 1s per 5min). To have also the in edges in the same table, the binning is joined as well.

CREATE TEMPORARY TABLE ThetaHist
(
.theta    SMALLINT UNSIGNED NOT NULL,
lo          DOUBLE            NOT NULL COMMENT 'Lower edge of zenith distance bin in degree',
hi          DOUBLE            NOT NULL COMMENT 'Upper edge of zenith distance bin in degree',
CountN      INT UNSIGNED      NOT NULL,
OnTime      FLOAT             NOT NULL,
ZdWeight    DOUBLE            NOT NULL COMMENT 'tau(delta theta)',
ErrZdWeight DOUBLE            NOT NULL COMMENT 'sigma(tau)',
PRIMARY KEY (.theta) USING HASH
) ENGINE=Memory
AS
(
WITH EventCount AS
(
SELECT
INTERVAL(DEGREES(Theta), %0:bins) AS .theta,
COUNT(*) AS CountN
FROM
MonteCarloFiles
LEFT JOIN
factmc.OriginalMC USING(FileId)
GROUP BY
.theta
)
SELECT
.theta, lo, hi,
CountN,
OnTime,
OnTime/CountN AS ZdWeight,
(OnTime/CountN)*SQRT(POW(1/300, 2) + 1/CountN) AS ErrZdWeight
FROM
ObservationTime
LEFT JOIN
EventCount USING(.theta)
LEFT JOIN
BinningTheta ON .theta=bin
ORDER BY
.theta
)


%0:bins is a placeholder for the bin boundaries, e.g. 5, 10, 15, 20, 25, 30 (five bins between 5° and 30° plus underflow and overflow). It should be identical to the binning used for data files in DataFiles.

### Summarize Corsika Production

CREATE TEMPORARY TABLE SimulatedSpectrum
(
.energy SMALLINT UNSIGNED NOT NULL COMMENT 'Bin Index [MC Energy]',
CountN    DOUBLE            NOT NULL,
CountW    DOUBLE            NOT NULL,
CountW2   DOUBLE            NOT NULL,
PRIMARY KEY (.energy) USING HASH
) ENGINE=Memory
AS
(
SELECT
INTERVAL(LOG10(Energy), %0:energyest) AS .energy,
COUNT(*) AS CountN,
SUM(    (%2:spectrum)/pow(Energy, SpectralIndex)   ) AS CountW,
SUM(POW((%2:spectrum)/pow(Energy, SpectralIndex),2)) AS CountW2
FROM
MonteCarloFiles
LEFT JOIN
factmc.RunInfoMC USING (FileId)
LEFT JOIN
factmc.OriginalMC USING (FileId)
GROUP BY
.energy
ORDER BY
.energy
)


### Result (Migration)

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