to cover / to be strewn with
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to cover
To cover a surface with something so densely that it looks as if it had been scattered over it. It is often used figuratively.
Nach dem Fest war der ganze Park mit Müll übersät.
After the festival, the entire park was covered with trash.
'Übersäen' (to cover/to strew over) is an inseparable verb. The prefix 'über-' remains firmly attached to the verb in all sentence forms and is not placed at the end of the sentence.
Der Künstler übersät die Leinwand mit bunten Farbtupfern.
The artist covers the canvas with colorful dots of paint.
The word is made up of 'über' (over) and 'säen' (to sow). The literal idea is to scatter something over a surface, just as one sows seeds in a field.
Ein Meteoritenschauer hatte den Wüstenboden mit kleinen Kratern übersät.
A meteor shower had covered the desert floor with small craters.
This verb is almost always used in the passive or as an adjectival participle. The typical structure is: 'Etwas ist mit etwas übersät' (something is covered/strewn with something).
Der Nachthimmel war mit funkelnden Sternen übersät.
The night sky was strewn with sparkling stars.
In the general meaning of covering, 'übersät' (covered/strewn) is often combined with visual things: to be covered with stars, with flowers, with jewels, or also with wounds.
Die Wiese war mit Gänseblümchen übersät.
The meadow was covered with daisies.
In the figurative sense, the word is often used with negative things. Typical combinations are: to be riddled with errors, with contradictions, or with scars.
Sein erster Entwurf war mit Korrekturen übersät.
His first draft was covered with corrections.
to push through / to assert oneself / to permeate
'durchsetzen' and 'übersäen' can both describe something occurring in many places within an object or across a surface. In this meaning, people often say 'mit etwas durchsetzt sein' (to be permeated with something) when something is penetrated by something else or contains traces of it throughout. 'übersäen' puts more emphasis on the surface: a surface is densely covered with many individual things, for example spots, stones, or mistakes. The words can be confused because both create an image of 'something everywhere'. A practical guide: 'durchsetzt' fits better for something that is inside something or distributed throughout it; 'übersät' fits better for many visible points on a surface or in a text.
Nach dem Sturm war der Gehweg mit Ästen und Blättern übersät.
After the storm, the sidewalk was covered with branches and leaves.
Der alte Käse ist mit kleinen Kräuterstücken durchsetzt.
The old cheese is permeated with small pieces of herbs.