What is the process of the Church approving a Marian apparition? Say that a person claimed to have had Mary appear to them in a way similar to the Marian apparitions of Fatima, Lourdes, Guadalupe, etc. What is the process of the Church approving a Marian apparition as being authetic?

Some apparition sites like Our Lady of Guadalupe have accompanying miracles that would lend to the authenticity of the apparition (in that case, the tilma of Juan Diego being imprinted with the image of Mary). But others like Our Lady of Good Help appear to only have the eyewitness testimony of a single person.

Is the process similar to canonization in which they look for miracles that have occurred at the site? Or is there some other way of the Church approving a Marian apparition?
What is the process of the Church approving a Marian apparition?

The Sacred Congregation for the Doctrine of the Faith published it guidelines for Marian apparitions in the document entitled : Norms regarding the manner of proceeding in the discernment of presumed apparitions or revelations

The Vatican's "Norms regarding the manner of proceedings in the discernment of presumed apparitions or revelations" were approved by Pope Paul VI in 1978. An official English translation was released in 2011.


  I. CRITERIA FOR JUDGING, AT LEAST WITH PROBABILITY, THE CHARACTER OF THE PRESUMED APPARITIONS OR REVELATIONS
  
  A) Positive Criteria:
  
  a) Moral certitude, or at least great probability of the existence of the fact, acquired by means of a serious investigation;
  
  b) Particular circumstances relative to the existence and to the nature of the fact, that is to say:
  
  
  Personal qualities of the subject or of the subjects (in particular, psychological equilibrium, honesty and rectitude of moral life, sincerity and habitual docility towards Ecclesiastical Authority, the capacity to return to a normal regimen of a life of faith, etc.);
  As regards revelation: true theological and spiritual doctrine and immune from error;
  Healthy devotion and abundant and constant spiritual fruit (for example, spirit of prayer, conversion, testimonies of charity, etc.).
  
  
  B) Negative Criteria:
  
  a) Manifest error concerning the fact.
  
  b) Doctrinal errors attributed to God himself, or to the Blessed Virgin Mary, or to some saint in their manifestations, taking into account however the possibility that the subject might have added, even unconsciously, purely human elements or some error of the natural order to an authentic supernatural revelation (cf. Saint Ignatius, Exercises, no. 336).
  
  c) Evidence of a search for profit or gain strictly connected to the fact.
  
  d) Gravely immoral acts committed by the subject or his or her followers when the fact occurred or in connection with it.
  
  e) Psychological disorder or psychopathic tendencies in the subject, that with certainty influenced on the presumed supernatural fact, or psychosis, collective hysteria or other things of this kind.
  
  It is to be noted that these criteria, be they positive or negative, are not peremptory but rather indicative, and they should be applied cumulatively or with some mutual convergence.
  
  II. INTERVENTION OF THE COMPETENT ECCLESIASTICAL AUTHORITY
  
  
  If, on the occasion of a presumed supernatural fact, there arises in a spontaneous way among the faithful a certain cult or some devotion, the competent Ecclesiastical Authority has the serious duty of looking into it without delay and of diligently watching over it.
  If the faithful request it legitimately (that is, in communion with the Pastors, and not prompted by a sectarian spirit), the competent Ecclesiastical Authority can intervene to permit or promote some form of cult or devotion, if, after the application of the above criteria, nothing stands in the way. They must be careful that the faithful not interpret this practice as approval of the supernatural nature of the fact on the part of the Church (cf. Preliminary note c).
  By reason of its doctrinal and pastoral task, the competent Authority can intervene motu proprio and indeed must do so in grave circumstances, for example in order to correct or prevent abuses in the exercise of cult and devotion, to condemn erroneous doctrine, to avoid the dangers of a false or unseemly mysticism, etc.
  In doubtful cases that clearly do not put the good of the Church at risk, the competent Ecclesiastical Authority is to refrain from any judgment and from any direct action (because it can also happen that, after a certain period of time, the presumed supernatural fact falls into oblivion); it must not however cease from being vigilant by intervening if necessary, with promptness and prudence.
  
  
  III. AUTHORITIES COMPETENT TO INTERVENE
  
  
  Above all, the duty of vigilance and intervention falls to the Ordinary of the place.
  The regional or national Conference of Bishops can intervene:
  
  
  a) If the Ordinary of the place, having done his part, turns to it to judge the matter with greater certainty;
  
  b) If the matter pertains to the national or regional level; always, however, with the prior consent of the Ordinary of the place.
  
  
  The Apostolic See can intervene if asked either by the Ordinary himself, by a qualified group of the faithful, or even directly by reason of the universal jurisdiction of the Supreme Pontiff (cf. infra, no. IV).
  
  
  IV. ON THE INTERVENTION OF THE SACRED CONGREGATION
  FOR THE DOCTRINE OF THE FAITH
  
  
  a) The intervention of the Sacred Congregation can be requested either by the Ordinary, after he has done his part, or by a qualified group of the faithful. In this second case, care must be taken that recourse to the Sacred Congregation not be motivated by suspect reasons (for example, in order to compel the Ordinary to modify his own legitimate decisions, to support some sectarian group, etc.).
  
  
  b) It is up to the Sacred Congregation to intervene motu proprio in graver cases, especially if the matter affects the larger part of the Church, always after having consulted the Ordinary and even, if the situation requires, the Conference of Bishops.
  
  
  It is up to the Sacred Congregation to judge and approve the Ordinary’s way of proceeding or, in so far as it be possible and fitting, to initiate a new examination of the matter, distinct from that undertaken by the Ordinary and carried out either by the Sacred Congregation itself or by a special Commission. - Rome February 25, 1978