1 Introduction and motivation

GW170817 was the first gravitational wave event to have a confirmed electromagnetic counterpart, GRB 170817A, marking a milestone in the era of multi-messenger astronomy [1, 2]. Gamma-ray bursts (GRBs) are among the most promising sources for detecting electromagnetic counterparts to gravitational wave events because of their extreme luminosity and broad energy range, spanning from radio waves to the PeV regime [3]. In recent years, the Gamma-Ray Integrated Detectors (GRID) has emerged as a pioneering space mission concept, designed to monitor the transient gamma-ray sky at energies from 10 keV to 2 MeV using nanosatellites deployed in Low Earth Orbit (LEO) [4]. The mission’s primary scientific objectives include the detection of GRBs associated with gravitational waves and fast radio bursts, along with other high-energy astrophysical transient sources. Extensive ground calibration efforts [5] and detector response matrix (DRM) calculations [6] have been independently implemented for each detector. However, for nanosatellite constellations dedicated to GRB detection, including GRID, cross-calibration with existing large-scale space missions using actual astronomical observations is not only complementary but also essential for validating the DRMs of GRID and verifying the overall performance of the constellation.

Cross-calibration with predecessor missions has been a well-established practice in space-based gamma-ray astronomy, ensuring continuity and reliability in observational data. Notable examples include the Fermi’s Gamma-ray Burst Monitor (GBM), which underwent rigorous cross-calibration with the Swift’s Burst Alert Telescope (BAT) [7]. In this work, the comparative analysis quantified the reciprocal detection rates between GBM and BAT for their observed GRB populations, and validated instrumental consistency through joint spectral fitting of two events, GRB 080804 and GRB 080810 [8]. Furthermore, Fermi-GBM has been cross-calibrated with INTEGRAL Soft Gamma-Ray Imager (ISGRI) [9,10,11]. Similarly, the gravitational wave high-energy electromagnetic counterpart all-sky monitor (GECAM)-C’s gamma-ray detectors implemented comprehensive cross-calibration efforts with Fermi-GBM and Swift-BAT using a sample of GRBs and soft gamma-ray repeaters [12].

Through the successful deployment of 12 detector payloads, GRID has preliminarily established a nanosatellite constellation capable of continuous, all-sky monitoring for GRBs, as shown in Table 1. For nanosatellite constellations, including GRID, cross-calibration addresses two key objectives: ensuring detector uniformity essential for reliable multi-detector data fusion while validating constellation-level scientific performance beyond traditional validation methods. Moreover, in source localization methods such as flux modulation [13, 14] and arrival time signal triangulation [15,16,17], the accuracy of cross-calibration is particularly relevant. Any discrepancies in detector responses could adversely affect the precision of position reconstruction. Additionally, in contrast to traditional single-satellite missions, constellation missions encounter distinct challenges even when utilizing identical detector designs. A practical but significant challenge arises from the phased development and deployment of detector payloads over an extended six-year period (2018-2024, see Table 1). Performance variations can emerge due to differences in design timing, personnel involved in assembly and calibration, materials used, and electronic system, necessitating comprehensive verification of consistency among the constellation’s distributed detectors. Additionally, the variations in the satellite platforms to which the detectors are mounted can also influence the DRMs. These practical considerations underscore the vital significance of cross-calibration in the context of constellation missions.

Table 1 List of the grid detectors already launched [18]
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Fig. 1
figure 1

Structure of the GRID detector [4]

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2 Instruments and DRMs

The 12 launched GRID detector payloads share the same detector design, as shown in Fig. 1, with the only variation in their readout electronic systems. The gamma-ray detection unit is based on a piece of Ce-doped \(\mathrm {Gd_3{(Al,Ga)}_5O_{12}}\) (GAGG) scintillation crystal coupled with \(4 \times 4\) arrays of silicon photomultipliers (SiPMs). Four such detection units are arranged in a \(2 \times 2\) configuration and mounted on top of the preamplifier board and data acquisition board, forming a compact nanosatellite gamma-ray detector [19]. The unit GAGG crystal has a surface area of \(3.8 \times 3.8\) \(\textrm{cm}^2\) and a thickness of 1 \(\textrm{cm}\), constituting a total detection area of approximately 58 \(\textrm{cm}^2\) for a single GRID detector. The detector module is enclosed in a structural housing with an enhanced specular reflector window serving as the entrance for gamma-rays.

The DRM is a mathematical representation, describing the relationship between the true incident flux of photons across discrete energy bins and the measured counts in the detector. Specifically, the DRM is represented as a matrix R, where each element \(R_{ij}\) indicates the probability that a true event in energy bin i will be detected and recorded as a count in channel j.

Each GRID payload undergoes a comprehensive ground calibration campaign to characterize its response characteristics. This calibration process encompasses tests with radioactive sources, tests with the X-ray beam, calibration of the detector gain, and calibration of the angular responses [5]. Based on these calibration measurements, the DRMs are then constructed independently for each detector through a standard process [6]. The first step involves Monte Carlo simulations using Geant4 [20] to calculate the energy deposition of gamma-ray photons in the detector, spanning various incidence angles and discrete photon energy bins. These simulation results are then refined using the detector’s measured energy resolution from the ground calibration to enhance the accuracy of energy deposition values. For initial spectral fitting, a response matrix is obtained via database interpolation, providing an efficient, approximate model of the instrument’s response [21]. When more detailed analysis is required, a dedicated response matrix is generated by rerunning simulations at specific incidence angles, ensuring improved spectral accuracy.

For cross-calibration purposes, we will perform joint spectral fitting of GRBs simultaneously detected by GRID and Fermi-GBM. The GBM comprises 12 NaI(TI) (8 \(\textrm{keV}\)-1 \(\textrm{MeV}\)) and 2 BGO scintillation detectors (200 \(\textrm{keV}\)-40 \(\textrm{MeV}\)) [22]. The wide field-of-view and localization capabilities of the GBM make it a vital scientific instrument for observing GRBs. The DRMs of Fermi-GBM are generated using the GBM Response GeneratorFootnote 1. The effective areas as a function of energy for GRID and Fermi-GBM (NaI) detectors are shown in Fig 2. The substantial overlap in their energy detection regions makes Fermi-GBM an ideal reference for cross-calibration of GRID’s performance. In the higher energy range, GRID shows better effective area, enabling it to provide additional spectral information. With increasing orbital time, radiation damage to the SiPMs elevates dark counts [23], which contributes to a higher noise level and gradually raises GRID’s lower energy threshold [24]. In this context, Fermi-GBM’s stable performance in the lower energy range can effectively complement these observations.

Fig. 2
figure 2

Effective areas of GRID and Fermi-GBM. For GRID, the blue curve refers to the effective area of a single detector when it is pointed directly at the source [5]. The vertical black dashed line marks the energy of the gadolinium K-edge. For Fermi-GBM, the red curve represents the combined on-axis effective area of all 12 NaI detectors, averaged over the unocculted sky [7]

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3 Methodology of joint fitting

A joint fitting methodology is employed to characterize the spectral properties of GRBs across the two observational datasets, taking advantage of empirical and physical models. Joint fitting enables us to use data from multiple instruments with different energy ranges and sensitivities to verify correctness against each other, thereby improving the robustness and completeness of the spectral analysis.

The primary software tools used throughout the analysis process of the GRID data are GRID Data Tools [25] for data preprocessing and light curve analysis and XSPEC Version 12.13.0c [26]. This process starts with data format unification, which involves unifying file formats, coordinating units, and aligning metadata to ensure compatibility with the GBM data. Next, we employ background subtraction techniques to isolate the burst signal. Background modeling is performed using polynomial fitting and the Bayesian block algorithm [27], according to the characteristics of the data. Polynomial fitting is applied when the background varies smoothly over time, whereas Bayesian blocks are more effective for data with abrupt background changes, enabling adaptive partitioning of the background component. In the next step, we determine the \(T_\mathrm{{90}}\) interval, defined as the duration for which 90% of the GRB’s total observed fluence is accumulated, from 5% to 95% of the cumulative counts. This metric provides a standardized time window, allowing us to capture the core emission phase of the GRB. Subsequently, photon counts and energy information within the \(T_\mathrm{{90}}\) interval are extracted, forming the basis for subsequent spectral and temporal analysis.

The relationship between the true incident flux of photons across discrete energy bins and the measured counts in the detector is shown as follows:

$$\begin{aligned} N_{\text{ det }}\left( E_{\text{ obs }}\right) =\sum _i R\left( E_{\text{ obs }}, E_i\right) \cdot N_{\text{ int }}\left( E_i\right) \end{aligned}$$
(1)

\(N_{\text{ det }}\left( E_{\text{ obs }}\right) \) represents the measured photon counts in the observed energy bin \(E_{\text{ obs }}\), which is the data recorded by the instrument. \(R\left( E_{\text{ obs }}, E_i\right) \) denotes the DRM, which describes the probability of detecting a photon in the observed energy bin \(E_{\text{ obs }}\) given that it was emitted at true energy \(E_{\text{ i }}\). This function accounts for the efficiency and characteristics of the detector, including energy resolution and response at different energies. \(N_{\text{ int }}\left( E_i\right) \) denotes the intrinsic spectrum of the GRB at true energy \(E_{\text{ i }}\), which represents the actual photon flux of the GRB before any instrumental effects are applied.

Although direct unfolding with the DRM is conceptually straightforward, it is often impractical due to high-dimensional response matrices, potential ill-conditioning, and the amplification of errors from noise. Instead, forward modeling fitting is a more reliable approach for spectral analysis [28], which is the approach used in this study.

For each GRB event, three-tier spectral analysis will be performed utilizing chi-square statistics \(\chi ^2\) with three distinct models: individual fitting for GRID and Fermi-GBM detectors, followed by joint fitting with linked spectral parameters (photon index, cutoff energy, etc.) while allowing independent normalization constants (K). This methodology isolates instrument-specific response differences while maintaining physical consistency in spectral shape parameters.

The following models are commonly used for joint fitting of GRB spectra [3] and are also frequently adopted by the Fermi-GBM team in their spectral catalog analysis [29].

(1) The power law is a simple model for spectral analysis. This model is particularly applicable when observational constraints require a simplified analytical framework, specifically in cases involving faint GRBs or those operating near detector threshold limitations [3]. The primary justification for employing this simplified model stems from the inherent limitations in photon statistics, which preclude the reliable determination of parameters required for more sophisticated spectral models. This model describes the relationship between energy E and photon flux N(E) as follows:

$$\begin{aligned} N(E)=K \cdot E^{-\alpha } \end{aligned}$$
(2)

K denotes a normalization constant. \(\alpha \) represents the spectral index, which defines the slope of the spectrum.

(2) The Band Function [30] is a comprehensive model designed for spectral analysis of GRBs. This model captures the characteristic power law observed at lower energies and accommodates the steeper decline at higher energies. The model combines two power laws with an exponential rollover, effectively describing the smooth transition from low-energy to high-energy spectral behavior of the GRBs. The model is calculated as follows:

$$\begin{aligned} N(E)= {\left\{ \begin{array}{ll}K \cdot E^\alpha \cdot e^{-\frac{E}{E_0}}, & E \le E_{\text{ peak } } \\ K \cdot \left( E_{\text{ peak } }\right) ^{\alpha -\beta } \cdot E^\beta \cdot e^{(\beta -\alpha ) \cdot \frac{E_{\text{ peak } }}{E_0}}, & E>E_{\text{ peak } }\end{array}\right. } \end{aligned}$$
(3)

K denotes a normalization constant. \(\alpha \) and \(\beta \) represent the spectral indices for the low- and high-energy regions, respectively. \(E_0\) represents a characteristic energy (often called folding energy). \(E_{\text {peak}}\) denotes the energy at which the spectrum peaks.

(3) The cut-off power law model [3] includes an exponential cut-off at higher energies, making it more appropriate for spectra that exhibit a sharp decline at high energies. The implementation of the cut-off power law model is primarily necessitated by instrumental constraints, specifically the limited energy bandwidth of detectors and insufficient high-energy photon statistics. The model is represented as follows:

$$\begin{aligned} N(E)=K \cdot E^{-\alpha } \cdot e^{-\frac{E}{E_{\textrm{cut}}}} \end{aligned}$$
(4)

K denotes the normalization constant. \(\alpha \) represents the spectral index. \(E_{\text {cut}}\) denotes the energy at which the exponential cut-off occurs.

4 GRB case studies

For GRID’s GAGG detectors, an energy range of 30-1200 \(\textrm{keV}\) is selected for this study. Due to the increasing leakage current [23, 24] in GRID’s SiPMs caused by radiation damage over time in orbit, the lower energy threshold needs to be adjusted accordingly. For Fermi-GBM’s NaI detectors, an energy range of 8-980 \(\textrm{keV}\) is selected based on Fermi-GBM data analysis documentationFootnote 2. We retrieved the time-tagged event (TTE) dataset of GRBs from the Fermi-GBM public data archiveFootnote 3.

Fig. 3
figure 3

GRID-02 and Fermi-GBM light curves for GRB 210121A, with \(T_0\) = 18:41:49.0 UT. The vertical red dashed lines indicate the joint fit interval. The four channels correspond to the four detection units of the GRID detector

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Table 2 Fitting parameters for GRB 210121A (\(1 \sigma \) uncertainty)
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As an student-driven mission, not all data have undergone the full processing pipeline due to the ongoing development of the time-energy reconstruction and burst search pipelines. Currently, the detectors that have been fully processed include GRID-02, GRID-03B, GRID-04, GRID-07, and GRID-08B. In these datasets, there are 16 events during about 5000 hours of effective observation time. Among these events, we prioritized GRB events with high flux and small incident angles relative to the normal direction of the GRID detector. These selection criteria ensure an optimal signal-to-noise ratio and minimize systematic uncertainties in the initial validation phase. For Fermi-GBM, we exclusively utilized NaI detectors (rather than BGO) due to their overlapping energy band with GRID’s sensitivity range. Then, we specifically selected the two NaI detectors with minimal angular offset from the GRB incident direction, unobstructed viewing geometry confirmed via detector attitude analysis and maximum GRB photon counts. Based on these criteria, GRB 210121A, GRB 230827A, GRB 231215A and GRB 240229A were selected for time-integrated spectral analysis performed over the GRID detector’s \(T_\mathrm{{90}}\) durations, which are calculated in an energy range of 30-1200 \(\textrm{keV}\).

Fig. 4
figure 4

Energy spectrum fitting with residuals for GRB 210121A with GRID and Fermi-GBM detectors

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Fig. 5
figure 5

GRID-04 and Fermi-GBM light curves for GRB 230827A, with \(T_0\) = 18:17:53.0 UT. The vertical red dashed lines lines indicate the joint fit interval. The four channels correspond to the four detection units of the GRID detector

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GRB 210121A

triggered the Hard X-ray Modulation Telescope (HXMT) (GCN 29346 [31]) and GECAM-B (GCN 29347 [32]) on 21 Jan 2021. To further investigate the nature of this event, we employed a burst selection method [33] to analyze both the Fermi-GBM and GRID data. The GRID trigger time was 2021-01-21T18:41:49.000 UTC. The \(T_\mathrm{{90}}\) of GRB 210121A, calculated in the 30-1200 \(\textrm{keV}\) energy range using data from GRID, is \(15.08^{+0.22}_{-0.16} ~ \textrm{s}\). Our analysis indicated that both instruments captured the same event. Based on the consistency of the detection across different instruments and the calculated \(T_\mathrm{{90}}\) duration, we concluded that GRB 210121A is indeed a long GRB [34].

The three-tier spectral analysis of GRB 210121A, detected by GRID-02 during its early in-orbit phase (2.5 months post-launch), utilized joint observations from GRID-02 and Fermi-GBM detectors n0/n3. Triggered GRID-02 at 2021-01-21T18:41:49.0 UTC, the event was analyzed over a 15.08 \(\textrm{s}\) time interval spanning \([\mathrm {T_0}-0.23~\textrm{s},~\mathrm {T_0}+14.85~\textrm{s}]\), as shown in Fig. 3. The cut-off power law model provided the optimal fit among the three models, with the parameters for all three models listed in Table 2. The energy spectrum fitting with residuals is shown in Fig. 4. In particular, the GBM data for GRB 210121A were retrieved from its daily data archive, with the corresponding response files generated using the GBM Response Generator based on the real-time attitude of the detectors.

Table 3 Fitting parameters for GRB 230827A (\(1 \sigma \) uncertainty)
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Fig. 6
figure 6

Energy spectrum fitting with residuals for GRB 230827A with GRID and Fermi-GBM detectors

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Fig. 7
figure 7

GRID-04 and Fermi-GBM light curves for GRB 231215A, with \(T_0\) = 9:47:19.0 UT. The vertical red dashed lines indicate the joint fit interval. The four channels correspond to the four detection units of the GRID detector

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GRB 230827A

triggered Fermi-GBM (GCN 34594 [35]) and GRID (GCN 34642 [36]). The three-tier spectral analysis of GRB 230827A, detected by GRID-04 approximately 18 months after its launch, utilized joint observations from GRID-04 and Fermi-GBM detectors n8/nb. Triggered GRID-04 at 2023-08-27T18:17:53.0 UTC, the event was analyzed over a 51.18 s time interval spanning \([\mathrm {T_0}+0.33~\textrm{s},~\mathrm {T_0}+51.51~\textrm{s}]\), as shown in Fig. 5. The cut-off power law model provided the optimal fit among the three models, with the parameters for all three models listed in Table 3. The energy spectrum fitting with residuals is shown in Fig. 6.

GRB 231215A

triggered Fermi-GBM (GCN 35369 [37]) and GRID (GCN 35413 [38]).The three-tier spectral analysis of GRB 231215A, detected by GRID-04 approximately 21 months after its launch, utilized joint observations from GRID-04 and Fermi-GBM detectors n8/nb. Triggered GRID-04 at 2023-12-15T9:47:19.0 UTC, the event was analyzed over a 16.01 s time interval spanning \([\mathrm {T_0}-0.21~\textrm{s},~\mathrm {T_0}+15.80~\textrm{s}]\), as shown in Fig. 7. The cut-off power law model provided the optimal fit among the three models, with the parameters for all three models listed in Table 4. The energy spectrum fitting with residuals is shown in Fig. 8.

Table 4 Fitting parameters for GRB 231215A (\(1 \sigma \) uncertainty)
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GRB 240229A

triggered Fermi-GBM (GCN 35833 [39]) and GRID (GCN 35904 [40]). The three-tier spectral analysis of GRB 240229A, detected by GRID-04 approximately 24 months after its launch, utilized joint observations from GRID-04 and Fermi-GBM detectors n9/na. Triggered GRID-04 at 2024-02-29T14:07:08.0 UTC, the event was analyzed over a 27.04 s time interval spanning \([\mathrm {T_0}-0.52~\textrm{s},~\mathrm {T_0}+26.52~\textrm{s}]\), as shown in Fig. 9. The cut-off power law model provided the optimal fit among the three models, with the parameters for all three models listed in Table 5. The energy spectrum fitting with residuals is shown in Fig. 10.

The normalized fitting parameters for the four GRBs are shown in Fig. 11. Results demonstrate consistency within the 1 \(\sigma \) error range, suggesting a strong agreement between the observations. This consistency underscores the compatibility of GRID and Fermi-GBM in characterizing GRBs.

5 Conclusion and outlook

Fig. 8
figure 8

Energy spectrum fitting with residuals for GRB 231215A with GRID and Fermi-GBM detectors

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Fig. 9
figure 9

GRID-04 and Fermi-GBM light curves for GRB 240229A, with \(T_0\) = 14:07:08.0 UT. The vertical red dashed lines indicate the joint fit interval. The four channels correspond to the four detection units of the GRID detector

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Herein, we present our validation effort, focusing on the cross-calibration between two GRID detectors and Fermi-GBM using GRB events. By extracting and fitting the energy spectra within the T90 time interval of the two GRBs, excellent consistency between the instruments was achieved. The successful fitting of the GRB energy spectrum shows the accuracy of the GRID DRM and the reliability of its scientific data. The observation and analysis of cases such as GRB 210121A [34] and GRB 230812B [25] illustrate the capacity of nanosatellite missions to effectively support large scientific projects like Fermi-GBM. These cases also underscore the potential benefits of future nanosatellite constellations in collaboration with larger satellites, enhancing coverage, facilitating continuous observation, and achieving meaningful scientific outcomes.

Table 5 Fitting parameters for GRB 240229A (\(1 \sigma \) uncertainty)
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Fig. 10
figure 10

Energy spectrum fitting with residuals for GRB 240229A with GRID and Fermi-GBM detectors

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Fig. 11
figure 11

Normalized parameters for GRBs

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Based on studies, discussed in previous parts of this paper, the new generation of nanosatellite constellations for GRB detection represents an attempt to explore distributed implementations of traditional single instruments. More precisely, the GRID constellation effectively distributes detector units similar to individual Fermi-GBM detectors across different orbits. Discussion and evaluation of the implementation, performance differences, and methodological implications of this distributed approach over the next few years are quite interesting. This distributed detector approach offers both advantages and challenges. Nanosatellite constellations provide a progressive pathway to enhance overall mission sensitivity and performance. The operational lifespan of individual detector payloads presents fundamental limitations that must be considered during the constellation design process. The primary limitation stems from the characteristics of satellite platforms, particularly orbital decay governed by initial injection altitude (typically 300-600 km for Nanosatellites) and solar activity modulation of atmospheric drag. Secondary constraints are the gradual degradation of detector performance, notably radiation damage to SiPMs, as mentioned previously. This ultimately influences the thresholds for science-operational lifetime. The constellation architecture introduces new requirements for data analysis. It is essential to ensure consistent performance across these detectors for reliable data integration, upon integrating data from multiple detectors to enhance sensitivity. Furthermore, accurate cross-calibration between detectors is particularly relevant in source localization methods. Consequently, maintaining detector uniformity is more critical than ever in the field of gamma-ray astronomy.

With several nanosatellite constellations proposed and ongoing, including GRID, our cross-calibration methodology and preliminary analysis results provided a framework for future validation studies. The success of this validation effort, combined with the potential for internal cross-validation within satellite constellations, establishes a foundation for comprehensive validation across distributed detector networks. As one of the first attempts to validate a distributed network of gamma-ray detectors in space, this study offers valuable insights for similar constellation missions in their design and operation phases.

Based on the work presented in this study, the GRID collaboration has already published tens of GRB circulars on the General Coordinates NetworkFootnote 4, accompanied by preliminary analysis. These GRB events are undergoing further analysis. We are also preparing to introduce the GRID Data Tools and a GRB catalog, with more scientific results anticipated. For researchers interested in analyzing GRID data, the data can be accessed via Tsinghua Open Source MirrorFootnote 5.