Solar extreme ultraviolet (EUV) imaging spectrographs utilize the Doppler effect of spectral lines to derive plasma velocity during solar explosive activities. The multi-order slitless imaging spectrograph has overcome the spatial and spectral desynchronization issues present in traditional slit spectrographs and has become an important tool for observing solar activities. The spectral information obtained by the multi-order diffraction imaging system is mixed within the multi-order image data, and an inversion algorithm is needed to extract and reconstruct it. Due to the limited number of observation orders in multi-order Solar EUV spectrographs (MOSES), the retrieval error of plasma velocity can reach up to 40%. To improve the inversion accuracy of the multi-order slitless imaging system, we design an inversion algorithm based on a five-order slitless imaging spectrograph to obtain solar plasma velocity. By adding observation data, we aim to improve the inversion accuracy of plasma velocity and enhance the practicality of the slitless imaging spectrograph.

To reconstruct a three-dimensional data cube containing spatial dimensions (x, y) and spectral dimensions (λ) from six two-dimensional projection images, an initial data cube is first generated using images I₀ and I_∞. This initial data cube is then used to reproduce six new projection images. The radiation intensity differences between the original observation images and the reproduced images are used to optimize the initial data cube. The spectral dimension of the optimized data cube is fitted to a Gaussian function, and a new input data cube for the next iteration is generated by sampling the Gaussian profile. These steps are repeated until the difference percentage between two consecutive data cubes is less than a predefined threshold. Finally, plasma velocity is derived from the spectral shifts at each spatial position.

To facilitate comparison with the SMART algorithm used in MOSES, the projection angle α of the imaging system is set to 45°, and the dispersion angles θ of images I₁, I₂, I₃, and I₄ are set to 0°, 90°, 180°, and 270°, respectively. Five two-dimensional images projected onto the x-y plane and one prior image on the y-λ plane are used to simulate the five actual observation images and introduce prior knowledge. The convergence threshold t is set to 0.1%, and the algorithm performs 986 iterations. The evaluation parameters for the inversion results of spectral line center offsets are shown in Table 1, including the correlation coefficient R, linear fitting slope s, and root mean square error E_RMS between the retrieved and true center offsets at all spatial positions. For comparison, the SMART algorithm results on MOSES data are also listed. The proposed algorithm achieves better inversion accuracy than the SMART algorithm, with a 4.65% increase in R, an 18.75% increase in slope, and a 17% decrease in E_RMS. The fitting results of the spectral line offsets from the inversion algorithm are shown in Fig. 5. The data points represent reconstructed and simulated spectral line center offsets at various spatial positions. The dashed line represents the fitting result, with a slope of 0.76, indicating that the inversion results are systematically underestimated by 24%. Compared with the SMART algorithm, the systematic error is significantly reduced. According to Eq. (5), the plasma Doppler velocity is derived from the spectral line center offset. The simulated and reconstructed center offsets on the x-y plane are shown in Figs. 6(a) and 6(b), respectively. Through comparison, it is found that the overall retrieved Doppler velocity is lower than the original velocity. The spatial distribution of velocity error, obtained by subtracting the simulated result from the reconstructed one, is shown in Fig. 6(c), with most errors falling within ±5 km/s. The variation of Doppler velocity along the x-axis at y=64 is presented in Fig. 7. The reconstructed curve (solid line) closely matches the original velocity curve (dotted line), indicating high inversion accuracy.

We propose and design an inversion algorithm for extracting spectral information from multi-order diffraction images captured by a five-order slitless spectral imaging spectrograph. This algorithm reconstructs a three-dimensional data cube containing spatial and spectral information and then derives solar plasma velocity from the spectral deviations. Numerical simulation validates the proposed algorithm, showing that it achieves better performance than the SMART algorithm by reducing the plasma velocity retrieval error to 24%. The correlation coefficient increases by 4.65%, the linear fitting slope by 18.75%, and the root mean square error decreases by about 17%. A high-precision inversion algorithm can enhance the practicality of multi-order diffraction imaging systems and serves as an important foundation for optimizing instrument design.