## 1 Introduction
Under the compound-Gaussian model, it is often necessary to assign a certain prior statistical distribution to the texture component [21], but real clutter environments may not necessarily correspond to the given distribution. In this case, the detector, designed according to the predetermined statistical distribution of the texture component, is bound to result in performance loss. To solve the aforementioned problem, it is assumed in [22] that there also exists unknown power mismatches between the training samples, and the amount of mismatches are unknown non-random and non-negative quantities. This model is an important complement to the compound-Gaussian model, which can also be transformed from the compound-Gaussian model. Based on the GLRT criterion, together with the alternate estimation algorithm, [PERSON] et al. proposed an adaptive detector in [22], which has the CFAR property and offers better detection performance compared with the well-known adaptive normalized matched filter (ANMF). It is important to highlight that the detection problem for only rank-one point targets is considered in [22], and the detector design is solely based on the GLRT criterion. However, in practice, the signal might be categorized as a subspace signal, which is an extension of rank-one signals and refers to the signal located in the known subspace with unknown coordinates. Meanwhile, for target detection problems in real scenarios, there is often no optimal detector due to the presence of too many unknown parameters. In addition to the GLRT criterion, other useful detector design criteria also include the Rao criterion, Wald criterion, Gradient criterion, and Durbin criterion, among others. Therefore, in this paper, we extend the signal model in [22], extending the rank-one model to the subspace signal model, and design detectors based on the GLRT criterion, Rao criterion, Wald criterion, Gradient criterion, and Durbin criterion. In addition, signal mismatch was not considered in [22]. In practical applications, factors such as array element errors, multipath effects, and rapid maneuvers of targets often lead to signal mismatch [23; 24; 25], meaning that the actual steering vector of the target does not match the value assumed by the radar system. There is a type of detector whose detection performance sharply degrades with signal mismatch increases, known as mismatch-sensitive detectors, which are usually applied in fields such as target tracking [26]. Conversely, another type of detector can still detect targets with a high probability when signal mismatch occurs, known as robust detectors, which are usually applied in fields such as searching models [27]. Therefore, in the performance evaluation, we also consider the performance of proposed detectors when signal mismatch occurs. In summary, this paper investigates the detection problem for subspace signals in a power heterogeneous clutter environment. Utilizing the GLRT criterion, [PERSON] criterion, Wald criterion, Gradient criterion, and Durbin criterion, in conjunction with the iterative concept of alternating estimation, we propose a variety of detectors, named the alternating estimation GLRT (AE-GLRT) detector, alternating estimation Rao (AE-Rao) detector, alternating estimation Wald (AE-Wald) detector, alternating estimation Gradient (AE-Gradient) detector, and alternating estimation Durbin (AE-Durbin) detector, respectively. Monte Carlo simulation experiments and IPIX radar data both demonstrate that the AE-Rao detector has the highest probability of detection (PD) in the absence of signal mismatch. When signal mismatch occurs, the AE-GLRT exhibits the best selectivity, while the AE-Durbin offers the most robust performance. Notation: Vectors are represented by bold lowercase letters, whereas matrices are represented by bold uppercase letters.