## Introduction

In design-based stereology, systematic sampling plays a prominent role. It is therefore important to develop error variance predictors for the different kinds of systematic probes involved in a stereological design. Such predictors are indispensable to plan a stereological design in an objective manner; see, for instance, Cruz-Orive *et al*. (2004).

The Cavalieri sections design is a popular systematic design, and it is equivalent to sampling at systematic points on the real axis ℝ; for a recent review, see Cruz-Orive & García-Fiñana (2005). On ℝ^{2}, the grid of test points to estimate planar areas is also classic; for general references, see Weibel (1979), Baddeley & Jensen (2004) and Howard & Reed (2005). The conceptual extension to ℝ^{n} is straightforward; see, for instance, Cruz-Orive (1989). Furthermore, systematic sampling at point locations on the circle �� (Gual-Arnau & Cruz-Orive, 2000; Cruz-Orive & Gual-Arnau, 2002; Hobolth & Jensen, 2002) and on the sphere (Gual-Arnau & Cruz-Orive, 2000, 2002) have also been studied.

Besides Cavalieri planes, the Cavalieri slabs design is also popular because it is a prerequisite for estimating particle number (West *et al*., 1991, 1996; Cruz-Orive & Geiser, 2004). Cavalieri slabs are actually equivalent to systematic segments on ℝ (Gual-Arnau & Cruz-Orive, 1998). The natural extension is systematic sampling by rectangular quadrats on ℝ^{2} (e.g. Baddeley & Jensen, 2005, and references therein), or by rectangular ‘*n*-boxes’ on ℝ^{n}.

Although the properties of Cavalieri sections and slabs have been studied in considerable depth, no satisfactory variance predictors have been hitherto available for systematic quadrats, or for *n*-boxes in general. Related results can be found, for instance, in Matheron (1962, 1965, 1971), or in Journel & Huijbregts (1978), but it is not obvious how to exploit these results for design stereology. This is regrettable, because in most stereological designs the intermediate steps usually involve systematic quadrats, or systematic ‘boxes’ such as optical disectors within slices; see, for instance, West *et al*. (1991, 1996).

The approach adopted in the present paper creates a new scenario that seems to yield variance approximations for systematic boxes in a natural and relatively simple manner (sections 4 and 5). It also yields analogous results for systematic sectors and points on the circle (section 6). We term it the ‘filtering approach’ because it involves the measurement function regularized or ‘filtered’ by the indicator function of the basic probe − this idea is presented in section 2.2. The choice of the term ‘filtering’ is explained further at the end of section 2.

For Cavalieri sections and slices (section 3), the new variance approximations do not quite coincide with the known ones; in fact, the target is different from the extension term of the variance; for a brief description of this concept, see, for instance, Cruz-Orive & García-Fiñana (2005). Furthermore, our models do not require isotropy of the covariogram for *n* ≥ 2, as was usually the case in the previous theory (e.g. Cruz-Orive, 1989; Eq. 8.2), or, equivalently, an isotropic random design (Baddeley & Jensen, 2005; p. 310). This paper, however, may be regarded as a preliminary report, because it only includes variance representations and models involving the covariogram of the measurement function. The further step will be to choose suitable covariogram models that, inserted into the formulae given here, will yield the desired variance predictors from sampled data. This final step is in preparation.