Spaceborne Synthetic Aperture Radar Systems Research Articles

SAR sensing of the atmosphere: stack-based processing for tropospheric and ionospheric phase retrieval

ABSTRACT This paper is intended to summarize the research conducted during the first 2 years of the Dragon 5 project 59,332 (geophysical and atmospheric retrieval from Synthetic Aperture Radar (SAR) data stacks over natural scenarios). Monitoring atmospheric phenomena, encompassing both tropospheric and ionospheric conditions, holds pivotal significance for various scientific and practical applications. In this paper, we present an exploration of advanced techniques for estimating tropospheric and ionospheric phase screens using stacks of Synthetic Aperture Radar (SAR) images. Our study delves into the current state-of-the-art in atmospheric monitoring with a focus on spaceborne SAR systems, shedding light on their evolving capabilities. For tropospheric phase screen estimation, we propose a novel approach that jointly estimates the tropospheric component from all the images. We discuss the methodology in detail, highlighting its ability to recover accurate tropospheric maps. Through a series of quantitative case studies using real Sentinel-1 satellite data, we demonstrate the effectiveness of our technique in capturing tropospheric variability over different geographical regions. Concurrently, we delve into the estimation of ionospheric phase screens utilizing SAR image stacks. The intricacies of ionospheric disturbances pose unique challenges, necessitating specialized techniques. We dissect our approach, showcasing its capacity to mitigate ionospheric noise and recover precise phase information. Real data from the Sentinel-1 satellite are employed to showcase the efficacy of our method, unraveling ionospheric perturbations with improved accuracy. The integration of our techniques, though presented separately for clarity, collectively contributes to a comprehensive framework for atmospheric monitoring. Our findings emphasize the potential of SAR-based approaches in advancing our knowledge of atmospheric processes, thus fostering advancements in weather prediction, geophysics, and environmental management.

Application of Quasi-Continuous Waveform Coding in Spaceborne Synthetic Aperture Radar

Quasi-continuous wave radar is an attempt to give consideration to the performance of pulse and continuous wave radar signals. However, it also has the shortcomings of both. This paper aims to add a new quasi-continuous-wave coding method to the spaceborne synthetic aperture radar (SAR) system. The technology of improving spaceborne SAR imaging performance by coding quasi-continuous-wave pulses is studied, and some shortcomings of this algorithm are improved. Firstly, the application of quasi-continuous-wave radar in the SAR system is studied, and the coding and reconstruction scheme is provided so that this technology can be successfully applied in spaceborne SAR. Secondly, the effects of different quasi-continuous-wave coding methods on SAR imaging performance are evaluated, including signal-to-noise ratio, resolution, and integration time. Then, several coding schemes are given, and the characteristic changes of the signal after quasi-continuous-wave coding are analyzed. The transmit–receive conversion loss function and azimuth Doppler ambiguity function of the design scheme are analyzed, which proves the advantages of the scheme. Finally, we design the hardware implementation scheme and carry out the practical test.

Analysis and compensation of stop-and-go approximation in high-resolution spaceborne synthetic aperture radar

ABSTRACT With the advancement of spaceborne synthetic aperture radar (SAR) resolution, the conventional stop-and-go approximation-based motion model is no longer suitable for high-resolution imaging algorithms. This paper conducts a comprehensive analysis of the approximation errors resulting from the stop-and-go approximation and proposes corresponding compensation methods. Firstly, an accurate echo model based on continuous platform motion is derived from the SAR geometry configuration. The impact of these errors on range and azimuth imaging is analysed from a spectral perspective, and the applicable boundary conditions are examined. Subsequently, compensation methods are proposed to eliminate the introduced approximation errors, addressing both”fast time” and”slow time” effects, thus enabling high-resolution imaging through improvements in traditional imaging algorithms. Experimental simulations validate the effectiveness of the error analysis and compensation methods, using X-band low Earth orbit SAR as an illustrative example. The proposed methods are not only compatible with existing mature imaging algorithms, but the analytical approach and compensation methods are also applicable to SAR systems operating on different carrier frequencies, orbits, and platforms.

Using the Displaced Phase Center Azimuth Multiple Beams Technique with Spaceborne Synthetic Aperture Radar Systems for Multichannel Reconstruction of Accelerated Moving Targets

The displaced phase center multiple azimuth beams (DPCMAB) technique can help spaceborne synthetic aperture radar (SAR) systems obtain the high-resolution wide-swath (HRWS) imaging capacity, and azimuth multichannel reconstruction is usually required due to azimuth non-uniform sampling. Compared with stationary and moving targets, the range history and azimuth signal model of the moving target with an acceleration are obviously different. The azimuth multichannel signal model of an accelerated moving target is established, and the relationship between acceleration and Doppler parameters is analyzed. Furthermore, the impact of the acceleration on azimuth multichannel reconstruction and imaging results is simulated and analyzed. According to the azimuth multichannel signal model, an azimuth multichannel reconstruction approach for accelerated moving targets is proposed. The key point of the proposed reconstruction approach is the modified azimuth multichannel matrix, which is related not only to azimuth and slant velocities but also accelerations. The target’s velocities and accelerations are obtained using multiple Doppler parameter estimations. Compared with the conventional method of processing the raw data of accelerated moving targets, this proposed method could distinctly suppress image defocusing and pairs of false targets. Simulation results on point targets validate the proposed azimuth multichannel reconstruction approach.

Intelligent Detection and Segmentation of Space-Borne SAR Radio Frequency Interference

Space-borne synthetic aperture radar (SAR), as an all-weather observation sensor, is an important means in modern information electronic warfare. Since SAR is a broadband active radar system, radio frequency interference (RFI) in the same frequency band will affect the normal observation of the SAR system. Untangling the above problem, this research explores a quick and accurate method for detecting and segmenting RFI-contaminated images. The purpose of the current method is to quickly detect the existence of RFI and to locate it in massive SAR data. Based on deep learning, the method shown in this paper realizes the existence of RFI by determining the presence or absence of interference in the image domain and then performs pixel-level image segmentation on Sentinel-1 RFI-affected quick-look images to locate RFI. Considering the need to quickly detect RFI in massive SAR data, an improved network based on MobileNet is proposed, which replaces some inverted residual blocks in the network with ghost blocks, reducing the number of network parameters and the inference time to 6.1 ms per image. Further, this paper also proposes an improved network called the Smart Interference Segmentation Network (SISNet), which is based on U2Net and replaces the convolution of the VGG blocks in U2Net with a residual convolution and introduces attention mechanisms and a modified RFB module to improve the segmentation mIoU to 87.46% on average. Experiment results and statistical analysis based on the MID dataset and PAIS dataset show that the proposed methods can achieve quicker detection than other CNNs while ensuring a certain accuracy and can significantly improve segmentation performance under the same conditions compared to the original U2Net and other semantic segmentation networks.

Performance Analysis of Channel Imbalance Control and Azimuth Ambiguity Suppression in Azimuth Dual Receiving Antenna Mode of LT-1 Spaceborne SAR System

The LuTan-1(LT-1), known as the L-band differential interferometric synthetic aperture radar (SAR) satellite system, is an essential piece of civil infrastructure in China, providing extensive applications such as surface deformation monitoring and topographic mapping. To achieve high-resolution and wide-swath (HRWS) observation abilities, the LT-1 takes the dual receiving antenna (DRA) imaging mode as its working mode. However, amplitude and phase errors between channels lead to a mismatch between the reconstruction filter and the multichannel echo signal, worsen the reconstructed azimuth spectrum, and introduce ambiguity targets in the final imaging results, seriously affecting the final imaging quality. In order to better evaluate the channel error and azimuth ambiguity performance of the LT-1 system, this paper proposed an advanced channel consistency correction method and conducted many measured data experiments. The experimental results show that the proposed method is effective, and the LT-1 system has excellent channel error control and azimuth ambiguity performance.

A Priori Knowledge Based Ground Moving Target Indication Technique Applied to Distributed Spaceborne SAR System

Through formation flying, the distributed spaceborne SAR(synthetic aperture radar) system can increase the number of spatial degree of freedoms (DOFs) and provide flexible multi-baselines for SAR-GMTI (ground moving target indication), which improves the system performance. This paper proposes an a priori knowledge-based adaptive clutter cancellation and moving target detection technique applied to the distributed spaceborne SAR-GMTI systems. Firstly, the adaptive clutter cancellation technique is exploited to suppress the ground clutter. A priori knowledge, such as road network information, is integrated to the adaptive clutter cancellation processor to reduce any moving target steering vector mismatch. Secondly, adaptive matched filter (AMF) and adaptive beamformer orthogonal rejection test (ABORT) are exploited as adaptive detection techniques for moving target detection. Due to the dense road network, the moving target steering vector estimation may be ambiguous for the different position and orientation of the roads. The multiple hypothesis testing (MHT) technique is proposed to detect the moving targets and resolve the potential ambiguities. A scheme is exploited to detect, classify, and relocate the moving targets. Finally, simulation experiments and performance analysis have demonstrated the effectiveness and robustness of the proposed technique.

Analysis of Geometric Characteristics and Coverage for Moon-Based/Spaceborne Bistatic SAR Earth Observation

With the rapid development of Earth system science, a new understanding of the complete Earth system has highlighted the crucial importance of integrated observations, especially in research involving large-scale geoscience phenomena. As an active sensor with all-time and all-weather capabilities, synthetic aperture radar (SAR) has been widely used in recent decades for Earth observation. However, the existing spaceborne, airborne, and ground-based SAR systems have difficulty providing temporally consistent and spatially continuous Earth observation data on a global scale. As Earth’s only natural satellite, the Moon is a very promising Earth observation platform. By deploying a transmitter on the Moon and a receiver on the high-orbit satellite, a Moon-based/spaceborne bistatic synthetic aperture radar (MS-BiSAR) can be formed. In this paper, the MS-BiSAR geometric model of Earth observation was established using ephemeris and orbit propagators with reference system transformations, and three different MS-BiSAR configurations were used to calculate and analyze their geometric characteristics and Earth observation coverage. The results show that with the advantage of wide swaths, continuous observation capabilities, and large coverage, such an MS-BiSAR could significantly contribute to monitoring and understanding large-scale geoscience phenomena.

Precise Ambiguity Performance Evaluation for Spaceborne SAR with Diverse Waveforms

The ambiguity suppression is a technical challenge for the present generation of spaceborne synthetic aperture radar (SAR) systems since this kind of suppression does not take the high spatial resolution and wide coverage into account simultaneously. The transmitting scheme based on the waveform diversity technique is a promising candidate for the conventional (one transmit, one receive channel) SAR systems and has been widely discussed, because it has almost no extra system costs and the ambiguity suppression performance is not closely related to pulse repetition frequency (PRF). However, the accurate method to evaluate the ratio of the intensities of the ambiguities to that of the signal is still a gap. To this end, starting from the precise signal model formulated in this paper, the ambiguity evaluation for spaceborne SAR with waveform diversity has been analyzed in detail. Particularly, the modified azimuth ambiguity-to-signal ratio (AASR) and range ambiguity-to-signal ratio (RASR) formulas are given for the single polarization SARs and quadrature-polarimetric (quad-pol) SARs, which contributes a lot, for the system designer, to precisely evaluating the ambiguity performance. Finally, detailed simulation experiments exploiting the system parameters of the LuTan (LT-1) system are carried out to corroborate the theoretical developments.

A Novel Method for Staggered SAR Imaging in an Elevation Multichannel System

Synthetic aperture radar (SAR) is an advanced remote sensing technique, capable of observing Earth’s surface independent of weather conditions and sunlight illumination. Restricted by the minimum antenna area, however, conventional spaceborne SAR systems cannot achieve high azimuth resolution in a wide swath. In addition, blind ranges are present as the constant pulse repetition interval (PRI) is used. To solve these problems, a PRI-staggered elevation multichannel SAR (EMC-SAR) system is employed in this article. By transmitting the continuously PRI-varied sequence, the blind ranges are located at different regions in different receive instants, effectively avoiding the loss of coverage in elevation. In this system, three issues are required to be addressed: 1) recovering the missed data located at blind ranges; 2) suppressing range ambiguous components; and 3) restoring the PRI-varied signal into a regular grid. To deal with these problems, we propose a novel SAR imaging method for a PRI-staggered EMC-SAR system. To be applied on-ground, assume downlinking of the individual elevation channels. First, the modified <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\varepsilon $ </tex-math></inline-formula> -insensitive loss tube regression with the L2 regularization method is applied to recover the missed data. Then, the range ambiguous components are suppressed by performing digital beamforming (DBF) based on the elevation multichannel technique, where the covariance matrix is constructed by using an iterative adaptive algorithm. After that, a generalized scaling transform is employed to restore the PRI-varied signal into a uniform sampled grid. Finally, a well-focused SAR image can be obtained by performing the conventional SAR imaging techniques. The effectiveness of the proposed method is validated by both simulated and real SAR data processing results.

Dynamic Predictive Quantization for Staggered SAR: Experiments With Real Data

Present and future spaceborne synthetic aperture radar (SAR) missions are designed to acquire an increasingly large amount of onboard data. This is a consequence of the use of large bandwidths, multiple polarizations, and the acquisition of large swath widths at fine spatial resolutions, which result in challenging requirements in terms of onboard memory and downlink capacity. In this scenario, SAR raw data quantization represents an essential aspect, as it affects the volume of data to be stored and transmitted to the ground as well as the quality of the resulting SAR products. Dynamic predictive block-adaptive quantization (DP-BAQ) is a novel technique, recently proposed by the authors, consisting of a low-complexity data compression method, and its application is particularly suitable for staggered SAR systems. DP-BAQ exploits the existing correlation among the azimuth raw data samples by applying linear predictive coding (LPC). This results in a data rate reduction of up to 25% with respect to state-of-the-art SAR quantization methods. In this letter, we test and validate the potential of DP-BAQ on airborne SAR data which emulates the system scenario of Tandem-L, a German Aerospace Center (DLR) mission proposal for a bistatic L-band system. For this purpose, an experimental SAR image has been acquired at the L-band by the airborne DLR flugzeug-SAR (F-SAR) sensor over the Kaufbeuren area, in Southern Germany. In order to simulate the staggered SAR acquisition mode, we implemented a dedicated resampling and filtering of the data. Our analyses confirm the effectiveness of DP-BAQ for efficient data volume reduction, exhibiting a consistent and promising performance when tested on areas characterized by different land cover types and backscatter statistics.

Mitigating Range Ambiguity Method Based on DDMA for SAR Systems

Range ambiguity can lead to deterioration of imagery quality for space-borne synthetic aperture radar (SAR). To solve this problem, we propose a mitigating range ambiguity method based on Doppler division multiple access (DDMA) in this paper. With the orthogonality of the DDMA waveform, transmit channels are separated at the receiver. Afterwards, a pre-processed operation for definition domains of different transmit channels is put forward to transform all definition domains into the same one. Thereafter, by multiplying measured data in a transmit channel by the complex conjugation of those in the other channel, a coupling phase term between the ambiguous range and the Doppler can be generated. So, utilizing the coupling phase, the echoes of different ambiguous regions can be distinguished in the slow-time domain, and they are extracted by designing time domain passband filters. Moreover, the images of different extracted ambiguous regions can be reconstructed to compose the whole unambiguous image. Finally, simulation results exhibit the effectiveness of the proposed method.

A High-Resolution, Wide-Swath SAR Imaging System Based on Tandem SAR Satellites.

For the spaceborne synthetic aperture radar (SAR), it is difficult to obtain high resolution and wide width at the same time. This paper proposes a novel imaging system based on tandem SAR satellites, where one obtains coarse resolution and wide swath by the scanning mode, and the other obtains the undersampled echo from the same swath. The high resolution is achieved by associating the tandem SARs’ echo and using the minimum-energy-based algorithm. Finally, a high-resolution wide-swath SAR system is designed, and its imaging performance is verified by simulated data and real airborne SAR data.

Smallsat with Offset Reflector Antenna

The employment of smallsats have been increasingly in the last years. Features like low cost, more accessible technologies, and the miniaturization of engineering components are driving the trends to smaller platforms. Following this trend, new spaceborne SAR (Synthetic Aperture Radar) systems have been adapted to suit the new architectures. This paper presents a new design based on the employment of a mesh deployable offset reflector antenna in association with a small spacecraft. In the analysis are taken into account some considerations as geometry, orbit and attachment to the satellite which reinforce the advantages of using mesh offset reflector antennas.

A Novel Phase Compensation Method for Urban 3D Reconstruction Using SAR Tomography

Synthetic aperture radar (SAR) tomography (TomoSAR) has been widely used in the three-dimensional (3D) reconstruction of urban areas using the multi-baseline (MB) SAR data. For urban scenarios, the MB SAR data are often acquired by repeat-pass using the spaceborne SAR system. Such a data stack generally has long time baselines, which result in different atmospheric disturbances of the data acquired by different tracks. These factors can lead to the presence of phase errors (PEs). PEs are multiplicative noise for observation data, which can cause diffusion and defocus in TomoSAR imaging and seriously affect the extraction of target 3D information. In this paper, we combine the methods of the block-building network (BBN) and phase gradient autofocus (PGA) to propose a novel phase compensation method called BBN-PGA. The BBN-PGA method can effectively and efficiently compensate for PEs of the MB SAR data over a wide area and improve the accuracy of 3D reconstruction of urban areas. The applicability of this proposed BBN-PGA method is proved by using simulated data and the spaceborne MB SAR data acquired by the TerraSAR-X satellite over an area in Barcelona, Spain.

An Advanced Echo Separation Scheme for Space-Time Waveform-Encoding SAR Based on Digital Beamforming and Blind Source Separation

To achieve high-resolution and wide-swath (HRWS) imaging, a space-time waveform-encoding (STWE) spaceborne synthetic aperture radar (SAR) system is adopted. In rugged terrain, the beam-pointing mismatch problem will appear when the traditional digital beamforming (DBF) technique is used to separate the received echoes. This problem leads to decreasing the received echo’s gain, deteriorating the quality of the image product and making the interpretation of SAR image difficult. To address this problem, an advanced echo separation scheme for STWE spaceborne SAR based on the DBF and blind source separation (BSS) is put forward in this paper. In the scheme, the echoes are transmitted within the adjacent pulse repetition intervals to simulate multiple beams, and the scattered echoes are received by the sixteen-channel antennas in elevation simultaneously. In post-processing, a detailed flow is adopted. In the method, the DBF is firstly performed on received echoes. Due to the error caused by terrain undulation, the degree of echo separation is not enough. Then, the BSS is performed on the multiple echoes obtained after the DBF processing. Finally, the highly separated echo signal can be obtained. In this scheme, there is no need to perform the direction of arrival (DOA) estimation before the DBF processing, which saves valuable computing resources. In addition, to verify the proposed scheme, point target and distributed target simulations based on the 16-channel data of an elevation X-band DBF-SAR system are carried out. The results of point targets indicate that the residual echo caused by rough terrain can be reduced by more than 14 dB using the proposed scheme. The proposed scheme can be directly implemented into existing SAR systems; thus, it does not increase the complexity of the system design. The scheme has the potential to be applied to future spaceborne SAR missions.

A Simulation Method of Two-Dimensional Sea-Surface Current Field for Trajectory Crossing Spaceborne SAR

The trajectory-crossing method is an important and effective method for spaceborne SAR (synthetic aperture radar) to detect a two-dimensional current field. However, in practical applications, there are few spaceborne SAR data that meet the requirements of close acquisition time and overlapping trajectories, which makes it difficult to comprehensively analyze the impact of different systems and environmental parameters on current measurement results from real data processing. With the proposal of a SAR constellation plan in the future, the data resources will be constantly enriched, and the trajectory-crossing method will be widely used. It is, thus, necessary to lay some theoretical foundation at present. A two-dimensional sea-surface-current-field simulation method for trajectory crossing spaceborne SAR is proposed in this paper. Based on the principles of the trajectory-crossing method, the proposed method can realize the simulation of two-dimensional sea-surface-current field under given spaceborne SAR system and environmental parameters. In this paper, the simulation process of this method is given, and the simulation experiment is performed. Compared with the current measurement results of SAR data, the simulation experiment shows that the current velocity error is less than 0.03 m/s and the direction error is less than 10 degrees, which proves the reliability of the proposed simulation method. The proposed method lays a foundation for analyzing the influence of various parameters in the application of the trajectory-crossing method.

An Innovative Synthetic Aperture Radar Design Method for Lunar Water Ice Exploration

Owing to the Moon’s rough surface, there is a growing controversy over the conclusion that water ice exists in the lunar permanently shadowed regions (PSRs) with a high circular polarization ratio (CPR). To further detect water ice on the Moon, an innovative design method for spaceborne synthetic aperture radar (SAR) system is proposed, to obtain radar data that can be used to distinguish water ice from lunar regolith with a small difference in the dielectric constants. According to Campbell’s dielectric constant model and the requirement that SAR radiometric resolution is smaller than the contrast of targets in images, a newly defined SAR system function involved in the method is presented to evaluate the influence of some system parameters on the water ice detection capability of SAR. In addition, several simulation experiments are performed, and the results demonstrate that the presented SAR design method may be helpful for lunar water ice exploration.

Snow water equivalent change mapping from slope-correlated synthetic aperture radar interferometry (InSAR) phase variations

Abstract. Area-based measurements of snow water equivalent (SWE) are important for understanding earth system processes such as glacier mass balance, winter hydrological storage in drainage basins, and ground thermal regimes. Remote sensing techniques are ideally suited for wide-scale area-based mapping with the most commonly used technique to measure SWE being passive microwave, which is limited to coarse spatial resolutions of 25 km or greater and to areas without significant topographic variation. Passive microwave also has a negative bias for large SWE. Another method is repeat-pass synthetic aperture radar interferometry (InSAR) that allows measurement of SWE change at much higher spatial resolution. However, it has not been widely adopted because (1) the phase unwrapping problem has not been robustly addressed, especially for interferograms with poor coherence, and (2) SWE change maps scaled directly from repeat-pass interferograms are not an absolute measurement but contain unknown offsets for each contiguous coherent area. We develop and test a novel method for repeat-pass InSAR-based dry-snow SWE estimation that exploits the sensitivity of the dry-snow refraction-induced InSAR phase to topographic variations. The method robustly estimates absolute SWE change at spatial resolutions of &lt; 1 km without the need for phase unwrapping. We derive a quantitative signal model for this new SWE change estimator and identify the relevant sources of bias. The method is demonstrated using both simulated SWE distributions and a 9-year RADARSAT-2 (C-band, 5.405 GHz) spotlight-mode dataset near Inuvik, Northwest Territories (NWT), Canada. SWE results are compared to in situ snow survey measurements and estimates from ERA5 reanalysis. Our method performs well in high-relief areas, thus providing complementary coverage to passive-microwave-based SWE estimation. Further, our method has the advantage of requiring only a single wavelength band and thus can utilize existing spaceborne synthetic aperture radar systems.

A Novel Imaging Algorithm for High-Resolution Wide-Swath Space-Borne SAR Based on a Spatial-Variant Equivalent Squint Range Model

High-Resolution Wide-Swath (HRWS) is an important development direction of space-borne Synthetic Aperture Radar (SAR). The two-dimensional spatial variation of the Doppler parameters is the most significant characteristic of the sliding spotlight space-borne SAR system under the requirements of HRWS. Therefore, the compensation of the two-dimensional spatial variation is the most challenging problem faced in the imaging of HRWS situations. The compensatory approach is then proposed to address this problem in this paper. The spatial distribution of the Doppler parameters for the HRWS space-borne SAR data in the sliding spotlight working mode is firstly analyzed, based on which a Spatial-Variant Equivalent Slant Range Model (SV-ESRM) is put forward to accurately formulate the range history for the distributed target. By introducing an azimuth-varying term, the SV-ESRM can precisely describe the range history for not only central targets but also marginal targets, which is more adaptive to the HRWS space-borne SAR requirements. Based on the SV-ESRM, a Modified Hybrid Correlation Algorithm (MHCA) for HRWS space-borne SAR imaging is derived to focus the full-scene data on one single imaging processing. A Doppler phase perturbation incorporated with the sub-aperture method is firstly performed to eliminate the azimuth variation of the Doppler parameters and remove the Doppler spectrum aliasing. Then, an advanced hybrid correlation is employed to achieve the precise differential Range Cell Migration (RCM) correction and Doppler phase compensation. A range phase perturbation method is also utilized to eliminate the range profile defocusing caused by range-azimuth coupling for marginal targets. Finally, a de-rotation processing is performed to remove the azimuth aliasing and the residual azimuth-variance and obtain the precisely focused SAR image. Simulation shows that the SAR echoes for a 20 km × 20 km scene with a 0.25 m resolution in both the range and azimuth directions could be focused precisely via one single imaging processing, which validates the feasibility of the proposed algorithm.